Dunkl harmonic oscillator and Witten's perturbation on strata
Abstract
The main goal of this work is to use Witten's perturbation method to prove a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata, endowed with adapted metrics, where compact Thom-Mather strati cations are considered. For that purpose, we study rst eigenfunction estimates and embedding results for the Dunkl harmonic oscillator on the line, which are generalized to other related operators on R+. The study of these operators is the key ingredient in our local analysis of the Witten's perturbation. Thus this thesis has two main parts. The rst part is devoted to the study of eigenfunction estimates and embedding results for the Dunkl harmonic oscillator and related operators. The second part deals with the Witten's perturbation on strata, where the rst part is used. This work is published in the preprints [1, 2]. Let us introduce those chapters separately and state their main results.
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Dunkl harmonic oscillator and Witten’s perturbation on strata (Oscilador arm´onico de Dunkl y perturbaci´on de Witten en estratos) Manuel Calaza Cabanas
3 Dunkl harmonic oscillator and Witten’s perturbation on strata (Oscilador arm´onico de Dunkl y perturbaci´on de Witten en estratos) Manuel Calaza Cabanas Tesis realizada en el Departamento de Geometr´ıa y Topolog´ıa de la Facultad de Matem´aticas, bajo la direcci´on del profesor Jes´us Antonio ´ Alvarez L´opez, para obtener el grado de Doctor en Ciencias Matemticas por la Universidad de Santiago de Compostela. Autorizaci´on del director: Fdo.: Jes´us A. ´ Alvarez L´opez El autor: Fdo.: Manuel Calaza Cabanas Santiago de Compostela a de de 2012.
Contents Abstract 1 Introduction 3 Eigenfunction estimates and embedding theorems 3 Witten’s perturbation on strata 7 Part 1. Eigenfunction estimates and embedding theorems 11 Chapter 1. Preliminaries on the Dunkl harmonic oscillator 13 1. Dunkl operator 13 2. Dunkl harmonic oscillator 14 3. Generalized Hermite polynomials 15 4. Proofs of the properties of the Dunkl harmonic oscillator 17 Chapter 2. Estimates of the generalized Hermite functions 21 1. Second perturbation of H21 2. Description of qk22 3. Location of the zeros of ξkand ξ0 k24 4. Estimates of ξk28 Chapter 3. Perturbed Schwartz space 35 Chapter 4. Perturbed Sobolev spaces 41 Chapter 5. Perturbation of Hon R+49 Chapter 6. Examples 51 1. Case where f1is a multiple of x−151 2. Case where f1is a multiple of other potential functions 52 3. Case where f1is a multiple of g0/g for some function g53 4. Transformation of Pby changes of variables 54 Part 2. Witten’s perturbation on strata 55 Chapter 7. Preliminaries on Thom-Mather stratifications 57 1. Thom-Mather stratifications 57 2. Adapted metrics on strata 64 Chapter 8. Relatively Morse functions 69 Chapter 9. Preliminaries on Hilbert complexes 77 1. Hilbert complexes 77 5
6 CONTENTS 2. Elliptic complexes 80 Chapter 10. Sobolev spaces defined by an i.b.c. 83 Chapter 11. Two simple types of elliptic complexes 93 1. Some more results on general elliptic complexes 93 2. An elliptic complex of length two 94 3. An elliptic complex of length three 97 4. Finite propagation speed of the wave equation 101 Chapter 12. Preliminaries on Witten’s perturbation of the de Rham complex 103 Chapter 13. Witten’s perturbation on a cone 105 1. Laplacian on a cone 105 2. Witten’s perturbation on a cone 107 Chapter 14. Domains of the Witten’s Laplacian on a cone 109 1. Domains of first type 109 2. Domains of second type 110 3. Domains of third type 111 4. Domains of fourth type 112 5. Domains of fifth type 113 Chapter 15. Splitting of the Witten complex on a cone 119 1. Subcomplexes defined by domains of first and second types 119 2. Subomplexes defined by domains of third, fourth and fifth types 122 3. Splitting into subcomplexes 126 Chapter 16. Local model of the Witten’s perturbation 131 Chapter 17. Spectral properties of ∆min/max 133 Chapter 18. Functions of the perturbed Laplacian on strata 135 Chapter 19. Finite propagation speed of the wave equation on strata 137 Chapter 20. Morse inequalities on strata 139 1. Analytic inequalities 139 2. Null contribution away from the critical points 140 3. Contribution from the rel-critical points 142 Chapter 21. Remark on the Sobolev spaces on strata 143 Conclusion 145 Resumen 147 Estimaciones de autofunciones y teoremas de embebimiento 147 Perturbaci´on de Witten en estratos 150 Bibliography 155
Abstract In the first part, eigenfunction estimates and embedding results are proved for the Dunkl harmonic oscillator on the line. These kind of results are generalized to operators on R+of the form P=−d2 dx2+sx2−2f1d dx +f2, where s > 0, and f1 and f2are functions satisfying f2=σ(σ−1)x−2−f2 1−f0 1for some σ > −1/2. The second part contains the main result, which is a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata endowed with adapted metrics, where compact Thom-Mather stratifications are considered. An adaptation of the analytic method of Witten is used in the proof. The local analysis is reduced to the study of the operator Pof the first part. 1
Introduction The main goal of this work is to use Witten’s perturbation method to prove a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata, endowed with adapted metrics, where compact Thom-Mather stratifications are considered. For that purpose, we study first eigenfunction estimates and embedding results for the Dunkl harmonic oscillator on the line, which are generalized to other related operators on R+. The study of these operators is the key ingredient in our local analysis of the Witten’s perturbation. Thus this thesis has two main parts, Parts 1 and 2. Part 1 is devoted to the study of eigenfunction estimates and embedding results for the Dunkl harmonic oscillator and related operators. Part 2 deals with the Witten’s perturbation on strata, where the first part is used. This work is published in the preprints [1, 2]. Let us introduce those chapters separately and state their main results. Eigenfunction estimates and embedding theorems The Dunkl operator Tσon C∞(R), depending on some σ > −1/2, is the perturbation of the usual derivative that can be defined by setting Tσ=d dx on even functions and Tσ=d dx + 2σ1 xon odd functions. This kind of operator, more generally on Rn, was introduced by C.F. Dunkl [21, 22, 23, 24, 25]. It gave rise to what is now called Dunkl theory (see the survey article [57]). This area had a big development in the last years, mainly due to its applications in Quantum Calogero- Moser-Sutherland models (see e.g. [10, 52, 37, 38, 61, 3, 4]). In particular, the Dunkl harmonic oscillator [55, 26, 50, 49] is Lσ=−T2 σ+sx2, depending on s > 0; i.e., it is given by using Tσinstead of d/dx in the expression of the usual harmonic oscillator H=−d2 dx2+sx2. On the other hand, let pk(k∈N, including zero1) is the sequence of orthogonal polynomials for the measure e−sx2|x|2σdx, taken with norm one and positive leading coefficient. Up to normalization, these are the generalized Hermite polynomials [59, p. 380, Problem 25]; see also [16, 20, 27, 17, 55, 56]. Let xk,k < xk,k−1<··· < xk,1denote the roots of each pk; in particular, xk,k/2is the smallest positive root if kis even. The corresponding generalized Hermite functions are φk=pke−sx2/2. It is known that Lσ, with domain the Schwartz space S=S(R), is essentially self-adjoint in L2(R,|x|2σdx). Moreover the spectrum of its self-adjoint extension, denoted by Lσ, consists of the eigenvalues (2k+ 1 + 2σ)s, with corresponding eigenfunctions φk. 1We adopt the convention 0 ∈N. 3
10 INTRODUCTION where they consider intersection homology with lower middle perversity of complex analytic varieties with Whitney stratifications. Another analytic proof of Morse inequalities was made by U. Luwig in [41, 42, 43] for the special case of conformally conic manifolds, but her admissible and Morse functions are different from ours: the norm of their differential is bounded away from zero around the frontier of the stratum, and the norm of their Hessian may be unbounded. In the future, we hope to extend this work to the case of other types of adapted metrics (those considered in [47, 48, 8], or even more general ones); in the case of dmin with the adapted metrics of [47, 48, 8], it would give Morse inequalities for the intersection homology with arbitrary perversity. This will require the study of a perturbation of the harmonic oscillator on R+more general than in Part 1. It is also natural to try to extend this work to the case of “rel-Morse-Bott functions”, where the rel-critical point set consists of “rel-non-degenerate rel-critical Thom-Mather substratifications”. Acknowledgment. We thank F. Alcalde for pointing out a mistake in a different previous version of the thesis, dealing with Morse inequalities for orbit spaces, which led us to study the version of this work. We thank Y.A. Kordyukov and M. Saralegui for helpful conversations on topics of this work. We also thank MathOverflow user R. Israel for answering a question concerning a part of this work. Finally, we thank R. Sjamaar for indirectly helping us (via M. Saralegui).
Part 1 Eigenfunction estimates and embedding theorems
CHAPTER 1 Preliminaries on the Dunkl harmonic oscillator Most of the contents of this section are taken or adapted from [55]. 1. Dunkl operator Recall that, for any φ∈C∞=C∞(R), there is some ψ∈C∞such that φ(x)−φ(0) = xψ(x), which also satisfies ψ(m)(x) = Z1 0 tmφ(m+1)(tx)dt (9) for all m∈N(see e.g. [36, Theorem 1.1.9]). The notation ψ=x−1φis used. The Dunkl operator, in the case of dimension one, is the differential-difference operator Tσon C∞, depending on a parameter σ∈R, defined by (Tσφ)(x) = φ0(x)+2σφ(x)−φ(−x) x. It can be considered as a perturbation of the derivative operator d dx . Consider the decomposition C∞=C∞ ev ⊕C∞ odd, as direct sum of subspaces of even and odd functions. The matrix expressions of operators on C∞will be considered with respect to this decomposition. The operator of multiplication by a function hwill be denoted also by h. We can write d dx =0d dx d dx 0, x =0x x0, Tσ=0d dx + 2σx−1 d dx 0=d dx + 2σ0x−1 0 0 on C∞. With Σ = σ0 0−σ, we have [Tσ, x] = 1 + 2Σ ,(10) TσΣ+ΣTσ=xΣ+Σx= 0 .(11) Consider the perturbed factorial m!σof each m∈N, which is inductively defined by setting 0!σ= 1, and m!σ=((m−1)!σmif mis even (m−1)!σ(m+ 2σ) if mis odd for m > 0. Observe that m!σ>0 if σ > −1/2, which will be the case of our interest; otherwise, m!σmay be ≤0. For k≤m, even when k!σ= 0, the quotient m!σ/k!σ 13
14 1. PRELIMINARIES ON THE DUNKL HARMONIC OSCILLATOR can be understood as the product of the factors from the definition of m!σwhich are not included in the definition of k!σ. For any φ∈C∞and m∈N, we have (Tm σφ)(0) = m!σ m!φ(m)(0) .(12) This equality follows by (9) and induction on m. 2. Dunkl harmonic oscillator Recall that, for dimension one, the harmonic oscillator, and the annihilation and creation operators are H=−d2 dx2+s2x2, A =sx +d dx , A0=sx −d dx on C∞. By using Tσinstead of d/dx, we get a perturbations of H,Aand A0called Dunkl harmonic oscillator, and Dunkl annihilation and creation operators: L=−T2 σ+s2x2=H−2σx−1d dx 0 0d dx x−1, B=sx +Tσ=A+ 2σ0x−1 0 0 , B0=sx −Tσ=A0−2σ0x−1 0 0 . By (10) and (11), L=BB0−(1 + 2Σ)s=B0B+ (1 + 2Σ)s=1 2(BB0+B0B),(13) [L, B] = −2sB , [L, B0] = 2sB0,(14) [B, B0] = 2s(1 + 2Σ) ,(15) [L, Σ] = BΣ+ΣB=B0Σ+ΣB0= 0 .(16) Recall also that the Schwartz space S=S(R) is the space of functions φ∈C∞ such that kφkSm=X i+j≤m sup x|xiφ(j)(x)| is finite for all m∈N. This defines a sequence of norms k kSmon S, which is endowed with the corresponding Fr´echet topology. The Banach space completion of Swith respect to each norm k kSmwill be denoted by Sm. We have Sm+1 ⊂ Sm continuously, and S=TmSm. Let us remark that kφ0kSm≤ kφkSm+1 for all m. The above decomposition of C∞can be restricted to each Smand S, giving Sm=Sm ev ⊕Sm odd and S=Sev ⊕Sodd. The matrix expressions of operators on S will be considered with respect to this decomposition. For φ∈C∞ ev ,ψ=x−1ψand i, j ∈N, it follows from (9) that |xiψ(j)(x)| ≤ Z1 0 tj−i|(tx)iφ(j+1)(tx)|dt ≤sup y∈R|yiφ(j+1)(y)| for all x∈R. Thus kψkSm≤ kφkSm+1 for all m∈N, obtaining that Sodd =xSev and x−1:C∞ odd →C∞ ev restricts to a continuous operator x−1:Sodd → Sev. Therefore x:Sev → Sodd is an isomorphism of Fr´echet spaces, and Tσ,B,B0and Ldefine continuous operators on S.
3. GENERALIZED HERMITE POLYNOMIALS 15 Let h,iσand k kσdenote the scalar product and norm of L2(R,|x|2σdx). Assume from now on that σ > −1/2, and therefore Sis a dense subset of L2(R,|x|2σdx). In L2(R,|x|2σdx), with domain S,−Tσis adjoint of Tσ,B0is adjoint of B, and Lis essentially self-adjoint. The self-adjoint extension of L, with domain S, will be denoted by L, or Lσ. Its spectrum consists of the eigenvalues (2k+1+2σ)s (k∈N). The corresponding normalized eigenfunctions φkare inductively defined by φ0=s(2σ+1)/4Γ(σ+ 1/2)−1/2e−sx2/2,(17) φk=((2ks)−1/2B0φk−1if kis even (2(k+ 2σ)s)−1/2B0φk−1if kis odd (18) for k≥1. We also have Bφ0= 0 ,(19) Bφk=((2ks)1/2φk−1if kis even (2(k+ 2σ)s)1/2φk−1if kis odd (20) for k≥1. These assertions follow from (13)–(16) like in the case of H. 3. Generalized Hermite polynomials From (17), (18) and the definition of B0, it follows that the functions φkare the generalized Hermite functions φk=pke−sx2/2, where pkis the sequence of polynomials inductively defined by p0=s(2σ+1)/4Γ(σ+ 1/2)−1/2,(21) pk=((2ks)−1/2(2sxpk−1−Tσpk−1) if kis even (2(k+ 2σ)s)−1/2(2sxpk−1−Tσpk−1) if kis odd ,(22) for k≥1. Up to normalization, these are the generalized Hermite polynomials; i.e., the orthogonal polynomials associated with the measure |x|2σe−sx2dx [59, p. 380, Problem 25]. Each pkis of precise degree k, even/odd if kis even/odd, and with positive leading coefficient, denoted by γk. By (22), γk=(k−1/2(2s)1/2γk−1if kis even (k+ 2σ)−1/2(2s)1/2γk−1if kis odd .(23) We also have Tσp0= 0 ,(24) Tσpk=((2ks)1/2pk−1if kis even (2(k+ 2σ)s)1/2pk−1if kis odd .(25) The following recursion formula follows directly from (22) and (25): pk=(k−1/2(2s)1/2xpk−1−(k−1+2σ)1/2pk−2if kis even (k+ 2σ)−1/2(2s)1/2xpk−1−(k−1)1/2pk−2if kis odd .(26)
16 1. PRELIMINARIES ON THE DUNKL HARMONIC OSCILLATOR We have pk(0) = 0 if and only if kis odd, and p0 k(0) = 0 if and only if kis even. By (26) and induction on k, pk(0) = (−1)k/2s(k−1+2σ)(k−3+2σ)···(1 + 2σ) k(k−2) ···2p0(27) if kis even. When kis odd, by (25) and (27), (Tσpk)(0) = (−1)(k−1)/2s(k+ 2σ)(k−2+2σ)···(1 + 2σ)2s (k−1)(k−3) ···2p0, obtaining p0 k(0) = (−1)(k−1)/2 1+2σs(k+ 2σ)(k−2+2σ)···(1 + 2σ)2s (k−1)(k−3) ···2p0(28) by (12). From (26) and by induction on k, we also get x−1pk=X `∈{0,2,...,k−1} (−1)k−`−1 2s(k−1)(k−3) ···(`+ 2)2s (k+ 2σ)(k−2+2σ)···(`+ 1 + 2σ)p`(29) if kis odd1. The following assertions come from the general theory of orthogonal polynomials [59, Chapter III]. All zeros of each polynomial pkare real and of multiplicity one. Each open interval between consecutive zeros of pkcontains exactly one zero of pk+1, and at least one zero of every p`with `>k. Moreover pkhas exactly bk/2cpositive zeros and bk/2cnegative zeros. The zeros of each pkwill be denoted xk,1> xk,2>··· > xk,k. On each interval (xk,i+1, xk,i), the function pk+1/pkis strictly increasing, and satisfies lim x→x± k,i pk+1(x) pk(x)=∓∞ . For every polynomial pof degree ≤k−1, we have p2(x)≤Z∞ −∞ p2(t)|t|2σe−st2dt · k X `=0 p2 `(x) (30) for all x∈R. The Gauss-Jacobi formula states that there are λk,1, λk,2, . . . , λk,k ∈R such that, for any polynomial pof degree ≤2k−1, Z∞ −∞ p(x)|x|2σe−sx2dx = k X i=1 p(xk,i)λk,i .(31) Lemma 1.1.We have p0 k 2(xk,i)λk,i =(2sif kis even 2s/(1 + 2σ)if kis odd . 1As a convention, the product of an empty set of factors is 1. Thus (k−1)(k−3) ···(`+2) = 1 for `=k−1 in (29). Similarly, (27) and (28) also hold for k= 0 and k= 1, respectively.
4. PROOFS OF THE PROPERTIES OF THE DUNKL HARMONIC OSCILLATOR 17 Proof. This is a direct adaptation of the proof of [6, Corollary 3]. With p=pkpk−1 x−xk,i , the formula (31) becomes γk γk−1 =p0 k(xk,i)pk−1(xk,i)λk,i , and the result follows from (23)–(25). 4. Proofs of the properties of the Dunkl harmonic oscillator For the reader’s convenience, we include in this part the formal statements and proofs of the spectral properties of Lindicated in Section 2. The polynomials pkof Section 3 are also used. The reader familiar with this type of arguments can skip this part; there will not be any further reference to it. Lemma 1.2.With Sas domain, −Tσis adjoint of Tσin L2(R,|x|2σdx). Proof. For φ∈ Seven and ψ∈ Sodd, d dxφ, ψσ =Z∞ −∞ φ0ψ|x|2σdx = 2 Z∞ 0 φ0ψx2σdx =−2Z∞ 0 φ(ψ0x2σ+ψ2σx2σ−1)dx =−2Z∞ 0 φ(ψ0+ 2σx−1ψ)x2σdx =−Z∞ −∞ φ(ψ0+ 2σx−1ψ)|x|2σdx =−φ, d dx + 2σx−1ψσ . Corollary 1.3.With Sas domain, B0is adjoint of Bin L2(R,|x|2σdx), and Lis symmetric in L2(R,|x|2σdx). Proof of (19) and (20).By (17), Bφ0=s(2σ+1)/4Γ((2σ+ 1)/2)−1/2sx +d dxe−sx2/2= 0 . Next, we proceed by induction on k≥1. By (13) and (18), Bφ1= (2(1 + 2σ)s)−1/2BB0φ0 = (2(1 + 2σ)s)−1/2(B0B+ 2(1 + 2Σ)s)φ0 = (2(1 + 2σ)s)−1/22(1 + 2σ)sφ0) = (2(1 + 2σ)s)1/2φ0. Now, let k≥2 and suppose that the statement holds for φk−1. To simplify the notation, let νk= 1 −(−1)k. Observe that νk=νk−1+ 2 (−1)k−1. Then, by (13)
18 1. PRELIMINARIES ON THE DUNKL HARMONIC OSCILLATOR and (18) again, Bφk= (2(k+νkσ)s)−1/2BB0φk−1 = (2(k+νkσ)s)−1/2(B0B+ 2(1 + 2Σ)s)φk−1 = (2(k+νkσ)s)−1/2((2(k−1 + νk−12σ)s)1/2B0φk−2 + 2(1 + (−1)k−12σ)sφk−1) = (2(k+νkσ)s)−1/2(2(k−1 + νk−12σ)s+ 2(1 + (−1)k−12σ)s)φk−1 = (2(k+νkσ)s)1/2φk−1. Proposition 1.4.For each k∈N,φkis an eigenfunction of L, normalized in L2(R,|x|2σdx), with corresponding eigenvalue (2k+ 1 + 2σ)s. Proof. This follows by induction on k. For k= 0, Lφ0=kψ0k−1 σLψ0=kψ0k−1 σ(Hψ0−2σx−1ψ0 0) =kψ0k−1 σ(1 + 2σ)s ψ0= (1 + 2σ)s φ0, and kφ0kσ= 1 because Z∞ −∞ e−sx2|x|2σdx = 2 Z∞ 0 e−sx2x2σdx =s−(2σ+1)/2Γ((2σ+ 1)/2) . Now suppose that k≥1 and the result holds for φk−1. Let νk= 1 −(−1)k, like in the proof of (19) and (20). By (13), (14), (18) and Corollary 1.3, Lφk= (2(k+νkσ)s)−1/2LB0φk−1 = (2(k+νkσ)s)−1/2(B0L+ 2sB0)φk−1 = (2(k+νkσ)s)−1/2((2(k−1) + 1 + 2σ)s+ 2s)B0φk−1 = (2k+ 1 + 2σ)s φk, kφkk2 σ= (2(k+νkσ)s)−1hBB0φk−1, φk−1iσ = (2(k+νkσ)s)−1h(L+ (1 + 2Σ)s)φk−1, φk−1iσ = (2(k+νkσ)s)−12(k+σ+ (−1)k−1σ)skφk−1k2 σ = 1 . The functions φkform a base of the linear subspace P=p e−sx2/2|pis a polynomial ⊂ S . The density of Pin L2(R,|x|2σdx) does not follow from the general theory of orthogonal polynomials [59, Section 3.1], and therefore a particular proof must be given like in the case of the Hermite polynomials [59, Theorem 5.7.1]. Proposition 1.5.Pis dense in L2(R,|x|2σdx).
4. PROOFS OF THE PROPERTIES OF THE DUNKL HARMONIC OSCILLATOR 19 Proof. For each integer j≥0, let fj(x) = xje−sx2/2. We have kfjk2 σ=Z∞ ∞ x2je−sx2|x|2σdx = 2 Z∞ 0 x2(j+σ)e−sx2dx =s−1/2Z∞ 0 yj+2σ−1 2e−ydy =s−1/2Γj+2σ+ 1 2 ≤s−1/2(j+bσc)! , where we have used the substitution y=sx2. Hence k(iλ)j(j!)−1/2fjkσ≤s−1/4(bσc! 2bσc)1/2(21/2|λ|)j(j!)−1/2 for each λ∈Rbecause (j+bσc)! j!=bσc!j+bσc j≤ bσc! 2j+bσc. It follows that the series eiλx−sx2/2=∞ X j=0 (iλ)j j!fj is convergent in L2(R,|x|2σdx); indeed, it belongs to Pbecause fj∈ P. Therefore any forthogonal to Pin L2(R,|x|2σdx) satisfies Z∞ −∞ f(x)eiλx−sx2/2|x|2σdx = 0 for all λ∈R, obtaining f(x)e−sx2/2|x|2σ= 0 almost everywhere with respect to dx by Plancherel’s theorem. So f= 0 almost everywhere with respect to |x|2σdx. The following result is a direct consequence of Propositions 1.4 and 1.5, and Corollary 1.3. Corollary 1.6.With domain S, the operator Lis essentially self-adjoint in L2(R,|x|2σdx), and its spectrum consists of the eigenvalues and eigenfunctions stated in Proposition 1.4.
26 2. ESTIMATES OF THE GENERALIZED HERMITE FUNCTIONS and we get Ic= [ak,c, bk,c]. Since s(bk,c −ak,c)2=cmax −c , (43) we have length(Ic)≥2π/√cs if and only if c(cmax −c)≥4π2, which means that cmax ≥4πand c−≤c≤c+for c±=cmax ±pc2 max −16π2 2. Since cmax ∈O(k) as k→ ∞, there is some C0>0, depending on σ, such that cmax ≥4πfor all k≥C0. Assuming k≥C0, let ak,±=ak,c±and bk,±=bk,c±, which satisfy ak< ak,−< ak,+< bk,+< bk,−< bk. Fix any x∈Iand let qk(x) = cs. First, x∈[ak,−, ak,+]∪[bk,+, bk,−] if and only if length(Ic)≥2π/√cs, and in this case xsatisfies the condition of (i) with xk,i ∈Icand C1= 2πby Claim 1. Second, if x∈(ak, ak,−)∪(bk,−, bk), then length(Ic)<2π/√cs,Ic⊃Ic−, and we already know that Ic−contains some zero of ξk. Hence xalso satisfies the condition of (i) with C1= 2π. And third, if x∈(ak,+, bk,+), then s(bk,+−ak,+)2=cmax −c+=c−=16π2 c+≤32π2 cmax ≤32π2 c by (43), obtaining length(Ic+)≤4√2π √cs . Since Ic⊂Ic+and it is already proved that Ic+contains some zero of ξk, it follows that xalso satisfies the condition of (i) with C1= 4√2π. Summarizing, (i) holds in this case with J=Iand C1= 4√2πif cmax ≥4π. In this case, (ii) is obvious because J=I. The second type of oscillation interval is of the form I= (0, bk), which corresponds to the condition ¯σk<0. Now, cs ∈qk(I) for any c > 0, the set q−1 k(cs) consists of the points ±bk,c, defined like in (42), and we have Ic= (0, bk,c]. The equality cs =qk(2π/√cs) holds when (2k+ 1 + 2σ)2−4¯σk−16π2>0 (44) and cis c±= 2π22k+ 1 + 2σ±p(2k+ 1 + 2σ)2−4¯σk−16π2 ¯σk−4π2. Assuming (44), we have length(Ic)≥2π/√cs if and only if c−≤c≤c+. Let bk,±=bk,c±, satisfying 0 < bk,+< bk,−< bk. Fix any x∈Iand let qk(x) = cs. First, x∈[bk,+, bk,−] if and only if length(Ic)≥2π/√cs; in this case, xsatisfies the condition of (i) with xk,i ∈Ic and C1= 2πby Claim 1. And second, if x∈(bk,−, bk), then length(Ic)<2π/√cs, Ic⊃Ic−, and we already know that Ic−contains some zero of ξk. Hence xalso satisfies the condition of (i) with C1= 2π. So, when (44) is true, (i) holds with J= [bk,+, bk) and C1= 2π. Notice that c+∈O(k) as k→ ∞. Then there are some C0, C2>0, depending on σ, such that, if k≥C0, then (44) holds and sb2 k,+= 4π2/c+≤C2k−1, showing (ii) in this case.
3. LOCATION OF THE ZEROS OF ξkAND ξ0 k27 The third and final type of oscillation interval is I= (−bk, bk), which corresponds to the condition ¯σk= 0. We have cs ∈qk(I) when 0 < c ≤cmax. Then q−1 k(cs) consists of the points ±bk,c, defined like in (42), and we get Ic= [−bk,c, bk,c]. Since sb2 k,c =cmax −c , (45) we have length(Ic)≥2π/√cs if and only if c(cmax −c)≥π2, which means that cmax ≥πand c−≤c≤c+for c±=cmax ±pc2 max −4π2 2. Since cmax ∈O(k) as k→ ∞, there is some C0>0, depending on σ, such that cmax ≥4πfor all k≥C0. Assuming k≥C0, let bk,±=bk,c±, which satisfy 0< bk,+< bk,−< bk. Fix any x∈Iand let qk(x) = cs. First, bk,+≤ |x| ≤ bk,−if and only if length(Ic)≥2π/√cs; in this case, xsatisfies the condition of (i) with xk,i ∈Icand C1= 2πby Claim 1. Second, if |x|> bk,−, then length(Ic)<2π/√cs,Ic⊃Ic−, and we already know that Ic−contains some zero of ξk. Hence xalso satisfies the condition of (i) with C1= 2π. And third, if |x|< bk,+, then sb2 k,+=cmax −c+=c−=4π2 c+≤8π2 cmax ≤8π2 c by (45), obtaining length(Ic+)≤√2π √cs . Since Ic⊂Ic+and it is already proved that Ic+contains some zero of ξk, it follows that xalso satisfies the condition of (i) with C1=√2π. Summarizing, (i) holds in this case with J=Iand C1= 2π. In this case, (ii) is also obvious because J=I. Lemma 2.3.There exist C0 0, C0 1, C0 2>0, depending on σand s, such that, if k≥C0 0and Iis any oscillation interval of ξk, then there is some subinterval J0⊂I so that: (i) qk≥C0 1k1/3on J0; and (ii) each connected component of I\J0is of length ≤C0 2k−1/6. Proof. We use the notation of the proof of Lemma 2.2. The same type of argument can be used for all types of oscillation intervals. Thus, e.g., suppose that Iis of the type (0, bk). Since bk∈O(k1/2) as k→ ∞, we have b0 k=bk−k−1/6∈I for klarge enough, and qk(b0 k) = −s2(k−1/3−2bkk−1/6)−4¯σk((bk−k−1/6)−2−b−2 k)∈O(k1/3) as k→ ∞. So there are C0 0, C0 1>0, depending on σand s, such that b0 k∈Iand c0=qk(b0 k)≥C0 1k1/3for k≥C0 0. Then (i) and (ii) hold with J0=Ic0= (0, b0 k]. Corollary 2.4.There exist C00 0, C00 1>0, depending on σand s, such that, if k≥C00 0and Iis any oscillation interval of ξk, then, for each x∈I, there exists some zero xk,i of ξkin Iso that |x−xk,i| ≤ C00 1k−1/6.
28 2. ESTIMATES OF THE GENERALIZED HERMITE FUNCTIONS Proof. With the notation of Lemmas 2.2 and 2.3, let C00 0= max{C0, C0 0}and C00 2= max{C2, C0 2}. Assume k≥C00 0and consider the subinterval J00 =J∩J0⊂I. By Lemmas 2.2-(ii) and 2.3-(ii), each connected component of I\J00 is of length ≤C00 2k−1/6. Then, for each x∈I, there is some x00 ∈J00 such that |x−x00| ≤ C00 2k−1/6. By Lemmas 2.2-(i) and 2.3-(i), there is some zero xk,i of ξkin Isuch that |x00 −xk,i|=C1 pqk(x00)≤C1 pC0 1 k−1/6. Hence |x−xk,i| ≤ C00 2+C1/pC0 1k−1/6. 4. Estimates of ξk Lemma 2.5.Let Ibe an oscillation interval of ξk, let x∈Iand let xk,i be a zero of ξkin I. Then ξ2 k(x)≤(8s 3|x−xk,i|if kis even 8s 3(1+2σ)|x−xk,i|if kis odd . Proof. We can assume that there are no zeros of ξkbetween xand xk,i. For the sake of simplicity, suppose also that xk,i < x and ξk>0 on (xk,i, x); the other cases are analogous. The key observation of [6] is that then the graph of ξkon [xk,i, x] is concave down, and therefore 1 2ξk(x)(x−xk,i)≤Zx xk,i ξk(t)dt . By Schwartz’s inequality and (31), it follows that 1 2ξk(x)(x−xk,i)2 ≤ Z∞ −∞ p2 k(t)|t|2σe−st2 (t−xk,i)2dt! Zx xk,i (t−xk,i)2dt! =p0 k 2(xk,i)λk,i (x−xk,i)3 3, and the result follows by Lemma 1.1. With the notation of Lemma 2.2, for each k≥C0, let b Ikdenote the union of the oscillation intervals of ξk, and let b Jk⊂b Ikdenote the union of the corresponding subintervals Jdefined in the proof of Lemma 2.2. More precisely: •if ¯σk>0 and cmax >0, then b Jk=b Ik= (−ak,−bk)∪(ak, bk); •if ¯σk<0, then b Ik= (−bk,0) ∪(0, bk) and b Jk= (−bk,−bk,+]∪[bk,+, bk); and •if ¯σk= 0, then b Jk=b Ik= (−bk, bk). If k < C0, we also use the notation b Jk=b Ikfor the union of the oscillation intervals, which may be empty if there are no oscillation intervals. Proof of Theorem A. Part (i) follows from Lemmas 2.2 and 2.5. In any case, ξk(x)→0 as x→ ∞. If moreover kis odd or σ≥0, then ξk is continuous on R. Thus ξ2 kis bounded and reaches its maximum at some point ¯x∈R. Since ξk(0) = 0 (if ¯σk6= 0) or 0 ∈b Ik(if ¯σk= 0), it follows from Lemma 2.1 that ¯x∈b Ik. Then (ii) follows by Corollary 2.4 and Lemma 2.5.
4. ESTIMATES OF ξk29 If kis even and σ < 0, then ξkis not defined at 0 and ξ2 k(x)→ ∞ as x→0. So we can only conclude as above that the restriction of ξ2 kto the set defined by |x| ≥ xk,k/2is bounded, and reaches its maximum at some point ¯xof this set. Then ¯x∈b Ikby Lemma 2.1, and therefore (iii) holds by Corollary 2.4 and Lemma 2.5. Consider the case σ < 0 and keven, when Theorem A does not provide any estimate of ξ2 karound zero. According to Section 3 of Chapter 1, the function p2 k(x) on the region |x| ≤ xk,k/2reaches its maximum at x= 0, and moreover p2 k(0) < p2 0 by (27). Hence φ2 k(x)< p2 0for |x| ≤ xk,k/2, which complements Theorem A-(iii). On the other hand, φ2 k(x)≤ξ2 k(x) for |x| ≤ 1. Moreover xk,k/2≤1 for klarge enough by Corollary 2.4 since ak→0 as k→ ∞. So Theorem B follows from Theorem A-(iii). The following lemmas will be used in the proof of Theorem C. Lemma 2.6.There is some F > 0such that, for k≥1and x≥bk+1, ξk(x)≤Fk−5/12 (x−bk)2. Proof. Let x0∈(xk,1, bk) such that ξ0 k(x0) = 0. Since ξ0 k(x) = Zx x0 ξ00 k(t)dt and ξ0 k(x)<0 for x>bk, we get Zx x0 qk(t)ξk(t)dt > 0 for x > bk. Because ξk(x)>0 for x>x0,qk(x)>0 for x0< x < bkand qk(x)<0 for x>bk, it follows that Zbk x0 qk(t)ξk(t)dt > −Zx bk qk(t)ξk(t)dt . (46)
30 2. ESTIMATES OF THE GENERALIZED HERMITE FUNCTIONS According to Corollary 2.4 and Theorem A-(ii),(iii), for k≥C00 0and with ¯ C= max{C0, C00}, we get Zbk x0 qk(t)ξk(t)dt ≤¯ C1/2k−1/12 Zbk x0 qk(t)dt =¯ C1/2k−1/12(2k+ 1 + 2σ)s(bk−x0) −s2 3(b3 k−x3 0) + ¯σk(b−1 k−x−1 0) ≤¯ C1/2k−1/12(2k+ 1 + 2σ)sC00 1k−1/6 −s2 3(b3 k−(bk−C00 1k−1/6)3) + |¯σk|C00 1k−1/6 bk(bk−C00 1k−1/6) ≤¯ C1/2k−1/12(2k+ 1 + 2σ)sC00 1k−1/6 −s2C00 1b2 kk−1/6−C00 1 2bkk−1/3−C00 1 3k−1/2 3 +|¯σk|C00 1k−1/6 bk(bk−C00 1k−1/6). Since 2k+ 1 + 2σ−sb2 k=¯σk sb2 k , there is some F0>0 such that Zbk x0 qk(t)ξk(t)dt ≤F0k1/12 (47) for all k∈N. On the other hand, −Zx bk qk(t)ξk(t)dt ≥ −ξk(x)Zx bk qk(t)dt . With the substitution u=t−bk, we get qk(t) = −s2u(u+ 2bk) + ¯σk b2 k−¯σk(u+bk)−2, giving −ξk(x)Zx bk qk(t)dt =ξk(x)s21 3(x−bk)3+bk(x−bk)2 −¯σk b2 k (x−bk)−¯σk(x−1−b−1 k) ≥ξk(x)s2bk(x−bk)2−|¯σk| b2 k (x−bk)−|¯σk|b−1 k ≥ξk(x)s2bk−|¯σk| b2 k(bk+1 −bk)(x−bk)2−|¯σk|b−1 k
4. ESTIMATES OF ξk31 for x≥bk+1. By (40), it follows that there is some F1>0 such that −Zx bk qk(t)ξk(t)dt ≥F1ξk(x)k1/2(x−bk)2(48) for all kand x≥bk+1. Now the result follows from (46)–(48). Lemma 2.7.For each > 0, there is some G > 0such that, for all k∈N, max |x−xk,1|≤k−1/6 k−1 X `=0 ξ2 `(x)≤Gk1/6. Proof. Take any x∈Rsuch that |x−xk,1| ≤ k−1/6. By Corollary 2.4, |x−bk|≤|x−xk,1|+|xk,1−bk| ≤ (+C00 1)k−1/6(49) for k≥C00 0. In particular, bk< x if kis large enough. With this assumption, let `0, `1, `2∈Nsatisfying 0 < `0< `1< `2−1, where `0and `1will be determined later, and `2is the maximum of the naturals `<kwith b`0≤xfor all `0≤`. Let f±(t) = √2t+ 1 + 2σ±1 for t≥1. We have f±(`)−√sb`= 2 ±(2`+ 1 + 2σ) + 1 + ¯σ` 2`+ 1 + 2σ±2−p(2`+ 1 + 2σ)2−4¯σ`f±(`) + √sb` for `∈Z+. So, assuming that kis large enough, we can fix `0, independently of k and x, so that f−(`)<√sb`< f+(`) for all `≥`0. We have f+(`1)< f−(`2) because `1< `2−1. Moreover observe that f0 +(t) = (2(t+1+σ))−1/2>0, f00 +(t) = −(2(t+1+σ))−3/2<0 for all t≥1. Then, by Lemma 2.6, `1−1 X `=`0 ξ2 `(x)≤ `1−1 X `=`0 F2`−5/6 (x−b`)4≤F2 `1−1 X `=`0 `−5/6 (b`2−b`)4 ≤F2√s `1−1 X `=`0 `−5/6 (f−(`2)−f+(`))4≤F2√sZ`1 `0 t−5/6dt (f−(`2)−f+(t))4. After integrating by parts four times, we get Z`1 `0 t−5/6dt (f−(`2)−f+(t))4≤`−5/6 1f0 +−1(`1) 3(f−(`2)−f+(`1))3+5`−11/6 1f0 +−2(`1) 36 (f−(`2)−f+(`1))2 +55 `−17/6 1f0 +−3(`1) 216 (f−(`2)−f+(`1)) +935 1296`−23/6 1f0 +−4(`1) ln(f−(`2)) +21505 7776 ln(f−(`2)) Z`1 `0 t−29/6f0 +−4(t)dt .
32 2. ESTIMATES OF THE GENERALIZED HERMITE FUNCTIONS Therefore, since f0 +(t)∈O(t−1/2) as t→ ∞, there exists some G1>0, independent of kand x, such that `1−1 X `=`0 ξ2 `(x)≤G1`−1/3 1 (f−(`2)−f+(`1))3+`−5/6 1 (f−(`2)−f+(`1))2 +`−4/3 1 f−(`2)−f+(`1)+`−11/6 1ln(f−(`2)) + ln(f−(`2)). We have `−11/6 1ln(f−(`2)) + ln(f−(`2)) ≤`1/6 2 for klarge enough. Then P`0−1 `=1 ξ2 `(x) has an upper bound of the type of the statement if `1satisfies max`−1/3 1 (f−(`2)−f+(`1))3,`−5/6 1 (f−(`2)−f+(`1))2,`−4/3 1 f−(`2)−f+(`1)≤`1/6 2.(50) On the other hand, according to Theorem A-(ii),(iii), `2 X `1 ξ2 `(x)≤¯ C `2 X `1 `−1/6≤¯ CZ`2 `1 y−1/6dy =6¯ C 5(`5/6 2−`5/6 1), where ¯ C= max{C0, C00}. Then P`2 `=`1ξ2 `(x) has an upper bound of the type of the statement if `5/6 2−`5/6 1≤G2`1/6 2 for some G2>0, independent of kand x, which is equivalent to `1≥`21−G2`−2/3 26/5.(51) Thus we must check the compatibility of (50) with (51) for some `1and G2. By (51) and since, for each G2, δ > 0, we have G2`−2/3 2≤`−2 3+δ 2for klarge enough, we can replace (50) with max`−1/3 21−`−2 3+δ 2−2/5 (f−(`2)−f+(`1))3,`−5/6 21−`−2 3+δ 2−1 (f−(`2)−f+(`1))2, `−4/3 21−`−2 3+δ 2−8/5 f−(`2)−f+(`1)≤`1/6 2 for some δ > 0, which is equivalent to `1≤1 2p2(`2+σ)−`a 21−`−2 3+δ 2b2 −1−σ for (a, b)∈ {(−1/6,−2/15),(−1/2,−1/2),(−3/2,−8/5)}. Thus the compatibility of (50) with (51) holds if there is some G2, δ > 0 such that `21−G2`−2/3 26/5≤1 2p2(`2+σ)−`a 21−`−2 3+δ 2b2 −2−σ , which is equivalent to G2≥`2/3 21−1 2q2(1 + σ`−1 2)−`a−1 2 21−`−2 3+δ 2b2 −(2 + σ)`−1 25/6.
4. ESTIMATES OF ξk33 There is some G2>0 satisfying this condition because the l’Hˆospital rule shows that, for δsmall enough, each function t2/31−1 2p2(1 + σt−1)−ta−1 21−t−2 3+δb2 −(2 + σ)t−15/6 is convergent in Ras t→ ∞. Now, if `2< k −1, let `3denote the minimum integer ` < k such that b`0> x for all `0≥`. Also, let ¯σmin/max denote the minimum/maximum values of ¯σ`for `∈N. Then s2(`3−1) + 1 + 2σ+p(2(`3−1) + 1 + 2σ)2+ 4¯σmin 2s≤x <s2(`2+1)+1+2σ+p(2(`2+1)+1+2σ)2+ 4¯σmax 2s, obtaining 2(`3−`2)−4 <p(2(`2+1)+1+2σ)2+ 4¯σmax −p(2(`3−1) + 1 + 2σ)2+ 4¯σmin . If `3> `2+ 1, it follows that (2(`2+1)+1+2σ)2+ 4¯σmax >(2(`3−1) + 1 + 2σ)2+ 4¯σmin , giving 2√¯σmax −¯σmin >p(2(`2+1)+1+2σ)2−(2(`3−1) + 1 + 2σ)2 ≥2(`3−`2)−4. Therefore P`3 `=`2+1 ξ2(x) has an upper bound of the type of the statement by Theorem A-(ii),(iii). Let h(t) = (2t+ 1 + 2σ)s−s2x2−¯σmaxx−2 for t≥0. According to Theorem A-(i), if `3< k −1, then k−1 X `=`3+1 ξ2 `(x)≤C k−1 X `=`3+1 1 pq`(x)≤C k−1 X `=`3+1 1 ph(`)≤CZk−1 `3 dt ph(t) =C 2sph(k−1) −ph(`3)≤C 2sp2(k−1−`3). Hence Pk−1 `=`3+1 ξ2 `(x) also has an upper bound like in the statement because, by (41), (40) and (49), there is some G3, G4>0 such that G3(k−1−`3)k−1/2≤bk−1−b`3≤bk−1−x≤G4k−1/6. Proof of Theorem C. By (31), 1 = Z∞ −∞ pk(x) x−xk,12|x|2σe−sx2 p0 k 2(xk,1)λk,1 dx .
34 2. ESTIMATES OF THE GENERALIZED HERMITE FUNCTIONS Thus, by (30) and Lemma 2.7, Z|x−xk,1|≤k−1/6pk(x) x−xk,12|x|2σe−sx2 p0 k 2(xk,1)λk,1 dx ≤Z|x−xk,1|≤k−1/6 k−1 X `=0 ξ2 `(x)dx ≤2k−1/6max |x−xk,1|≤k−1/6 k−1 X `=0 ξ2 `(x)≤2G for any > 0. It follows that Z|x−xk,1|≥k−1/6pk(x) x−xk,12|x|2σe−sx2 p0 k 2(xk,1)λk,1 dx ≥1 2(52) when ≤1 4G, which implies part (i) by Lemma 1.1. When kis even and σ < 0, either 0 < xk,k/2< akor |xk,k/2−ak| ≤ C00 1k−1/6 for klarge enough according to Corollary 2.4. Moreover |xk,1−bk| ≤ C00 1k−1/6 for klarge enough by Corollary 2.4 as well. So, by (39) and (38), there are some C0, C1>0, independent of k, such that xk,k/2≤ak+C00 1k−1/6≤C0k−1/2, xk,1−xk,k/2≥bk−ak−2C00 1k−1/6=rcmax s−2C00 1k−1/6≥C1k1/2 On the other hand, by (27), there is some C2>0, independent of k, such that ξ2 k(x)≤C2|x|2σfor |x| ≤ xk,k/2. Therefore Z|x|≤xk,k/2 ξ2 k(x)dx (x−xk,1)2≤C2 (xk,1−xk,k/2)2Z|x|≤xk,k/2|x|2σdx =2C2x2σ+1 k,k/2 (2σ+ 1)(xk,1−xk,k/2)2≤2C2C2σ+1 0 (2σ+ 1)C2 1 k−2σ+3 2<2C2C2σ+1 0 (2σ+ 1)C2 1 k−1. This inequality and (52) imply part (ii).
CHAPTER 3 Perturbed Schwartz space We introduce a perturbed version Sσof S. It will be shown that Sσ=Safter all, but the relevance of this new definition to study Lwill become clear in the next section; in particular, the norms used to define Sσwill be appropriate to show embedding results, like a version of the Sobolev embedding theorem. Since Sσmust contain the functions φk, Theorems A and B indicate that different definitions must be given for σ≥0 and σ < 0. When σ≥0, for any φ∈C∞and m∈N, let kφkSm σ=X i+j≤m sup x|x|σ|xiTj σφ(x)|.(53) This defines a norm kkSm σon the linear space of functions φ∈C∞with kφkSm σ<∞, and let Sm σdenote the corresponding Banach space completion. There is a canonical inclusion Sm+1 σ⊂ Sm σ, and the perturbed Schwartz space is defined as Sσ=TmSm σ, endowed with the corresponding Fr´echet topology. In particular, S0is the usual Schwartz space S. Like in the case of S, there are direct sum decompositions into subspaces of even and odd functions, Sm σ=Sm σ,ev ⊕ Sm σ,odd for each m∈N, and Sσ=Sσ,ev ⊕Sσ,odd. When σ < 0, the spaces of even and odd functions are considered separately. Let kφkSm σ=X i+j≤m, i+jeven sup x|xi(Tj σφ)(x)| +X i+j≤m, i+jodd sup x6=0 |x|σ|xi(Tj σφ)(x)|(54) for φ∈C∞ ev , and let kφkSm σ=X i+j≤m, i+jeven sup x6=0 |x|σ|xi(Tj σφ)(x)| +X i+j≤m, i+jodd sup x|xi(Tj σφ)(x)|(55) for φ∈C∞ odd. These expressions define a norm k kSm σon the linear spaces of functions φin C∞ odd and C∞ ev with kφkSm σ<∞. The corresponding Banach space completions will be denoted by Sm σ,odd and Sm σ,ev. Let Sm σ=Sm σ,ev ⊕Sm σ,odd, which is also a Banach space by considering e.g. the norm, also denoted by k kSm σ, defined by the maximum of the norms on both components. There are canonical inclusions Sm+1 σ⊂ Sm σ, and let Sσ=TmSm σ, endowed with the corresponding Fr´echet topology. We have Sσ=Sσ,ev ⊕Sσ,odd for Sσ,ev =TmSm σ,ev and Sσ,odd =TmSm σ,odd. 35
42 4. PERTURBED SOBOLEV SPACES for all φ∈ S. Then, by (14), kBφk2 Wm σ=h(1 + L)Bφ, BφiWm−1 σ =kBφk2 Wm−1 σ+hLBφ, BφiWm−1 σ = (1 −2s)kBφk2 Wm−1 σ+hBLφ, BφiWm−1 σ ≤(1 −2s)kBφk2 Wm−1 σ+kBLφkWm−1 σkBφkWm−1 σ ≤Cm−1((1 −2s)kφk2 Wm−1 σ+kLφkWm σkφkWm σ) ≤Cmkφk2 Wm+1 σ for some Cm>0 independent of φ. Similarly, kB0φk2 Wm σ≤C0 mkφk2 Wm+1 σ for some C0 m>0 independent of φ. Remark 1.B0is not adjoint of Bin Wm σfor m > 0. Land Σ preserve Wm σ,ev and Wm σ,odd for each m, whilst Band B0interchange these subspaces. The motivation of our tour through perturbed Schwartz spaces is the following embedding results; the second one is a version of the Sobolev embedding theorem. Proposition 4.3.Sm0 σ⊂Wm σcontinuously if m0−m > 1/2. Proposition 4.4.Wm0 σ⊂ Sm σcontinuously if m0−m > 1. Corollary 4.5.Sσ=W∞ σas Fr´echet spaces. For each non-commutative polynomial p(of two variables, Xand Y), let p0 denote the non-commutative polynomial obtained by reversing the order of the variables in p; e.g., if p(X, Y ) = X2Y3X, then p0(X, Y ) = XY 3X2. It will be said that pis symmetric if p(X, Y ) = p0(Y, X). Notice that any non-commutative polynomial of the form p0(Y, X)p(X, Y ) is symmetric. Given any non-commutative polynomial p, the continuous operators p(B, B0) and p0(B0, B) on Sσare adjoint from each other in L2(R,|x|2σdx); thus p(B, B0) is a symmetric operator if pis symmetric. The following lemma will be used in the proof of Proposition 4.3 Lemma 4.6.For each non-negative integer m, we have (1 + L)m=X a q0 a(B0, B)qa(B, B0) for some finite family of homogeneous non-commutative polynomials qaof degree ≤m. Proof. The result follows easily from the following assertions. Claim 3.If mis even, then Lm=gm(B, B0)2for some symmetric homogeneous non-commutative polynomial gmof degree m. Claim 4.If mis odd, then Lm=g0 m,1(B0, B)gm,1(B, B0) + g0 m,2(B0, B)gm,2(B, B0) for some homogeneous non-commutative polynomials gm,1and gm,2of degree m.
4. PERTURBED SOBOLEV SPACES 43 If mis even, then Lm/2=gm(B, B0) for some symmetric homogeneous noncommutative polynomial gmof degree ≤mby (13). So Lm=gm(B, B0)2, showing Claim 3. If mis odd, then write Lbm/2c=fm(B, B0) as above for some symmetric homogeneous non-commutative polynomial fmof degree ≤m−1. Then, by (13), Lm=1 2fm(B, B0)(BB0+B0B)fm(B, B0). Thus Claim 4 follows with gm,1(B, B0) = 1 √2B0fm(B, B0), gm,2(B, B0) = 1 √2Bfm(B, B0). Proof of Proposition 4.3 when σ≥0.By the definitions of Band B0, for each non-commutative polynomial pof degree ≤m0(of three variables), there exists some Cp>0 such that |x|σ|p(x, B, B0)φ|is uniformly bounded by CpkφkSm0 σfor all φ∈ Sσ. Write (1 + L)m=X a q0 a(B0, B)qa(B, B0) according to Lemma 4.6, and let ¯qa(x, B, B0) = xm0−mqa(B, B0). Then, for each φ∈ Sσ, kφk2 Wm σ=X akqa(B, B0)φk2 σ =X aZ∞ −∞ |(qa(B, B0)φ)(x)|2|x|2σdx ≤2X aC2 qa+C2 ¯qaZ∞ 1 x−2(m0−m)dxkφk2 Sm0 σ, where the integral is finite because −2(m0−m)<−1. Proof of Proposition 4.3 when σ < 0.Now, for each homogeneous noncommutative polynomial pof degree d≤m0, there is some Cp>0 such that: • |p(x, B, B0)φ|is uniformly bounded by CpkφkSm0 σ,ev for all φ∈ Sσ,ev if dis even, and by CpkφkSm0 σ,odd for all φ∈ Sσ,odd if dis odd; and • |x|σ|p(x, B, B0)φ|is uniformly bounded by CpkφkSm0 σ,odd for all φ∈ Sσ,odd if dis even, and by CpkφkSm0 σ,ev for all φ∈ Sσ,ev if dis odd. With the notation of Lemma 4.6, let dadenote the degree of each homogenous non-commutative polynomial qa, and let ¯qa(x, B, B0) be defined like in the previous case. Then, as above, kφk2 Wm σ≤2X awith daeven C2 qaZ1 0 x2σdx +C2 ¯qaZ∞ 1 x−2(m0−m)+2σdxkφk2 Sm0 σ,ev + 2 X awith daodd C2 qa+C2 ¯qaZ∞ 1 x−2(m0−m)dxkφk2 Sm0 σ,ev
44 4. PERTURBED SOBOLEV SPACES for φ∈ Sσ,ev, and kφk2 Wm σ≤2X awith daeven C2 qa+C2 ¯qaZ∞ 1 x−2(m0−m)dxkφk2 Sm0 σ,odd + 2 X awith daodd C2 qaZ1 0 x2σdx +C2 ¯qaZ∞ 1 x−2(m0−m)+2σdxkφk2 Sm0 σ,odd for φ∈ Sσ,ev, where the integrals are finite because −1/2< σ < 0 and −2(m0−m)< −1. Let Cdenote the space of rapidly decreasing sequences of real numbers. Recall that a sequence c= (ck)∈RNis rapidly decreasing if kckCm= sup k|ck|(1 + k)m is finite for all m≥0. These expressions define norms k kCmon C. Let Cmdenote the completion of Cwith respect to k kCm, which consists of the sequences c∈RN with kckCm<∞. So C=TmCmwith the induced Fr´echet topology. Also, for each m≥0, let `2 mdenote the Hilbert space completion of Cwith respect to the scalar product h,i`2 mdefined by hc, c0i`2 m=X k ckc0 k(1 + k)m for c= (ck) and c0= (c0 k). The corresponding norm will be denoted by k k`2 m. Thus `2 mis a weighted version of `2; in particular, `2 0=`2. Let `2 ∞=Tm`2 mwith the corresponding Fr´echet topology. A sequence c= (ck) will be called even/odd if ck= 0 for all odd/even k. We get the following direct sum decompositions into subspaces of even and odd sequences: Cm=Cm,ev ⊕Cm,odd ,C=Cev ⊕Codd , `2 m=`2 m,ev ⊕`2 m,odd , `2 ∞=`2 ∞,ev ⊕`2 ∞,odd . Lemma 4.7.`2 2m⊂ Cmand Cm0⊂`2 mcontinuously for all mif 2m0−m > 1. Proof. It is easy to see that kckCm≤ kck`2 2m,kck`2 m≤ kckCm0X k (1 + k)m−2m01/2 for any c∈ C, where the last series is convergent because m−2m0<−1. Corollary 4.8.`2 ∞=Cas Fr´echet spaces. According to Section 2 of Chapter 1, the “Fourier coefficients” mapping φ7→ (hφk, φiσ) defines a quasi-isometry Wm σ→`2 mfor all m, and therefore an isomorphism W∞ σ→ C of Fr´echet espaces. Notice that the “Fourier coefficients” mapping can be restricted to the even and odd subspaces. Corollary 4.9.Any φ∈L2(R,|x|2σdx)is in Sσif and only if its “Fourier coefficients” hφk, φiσare rapidly degreasing on k. Proof. By Corollary 4.5, the “Fourier coefficients” mapping defines an isomorphism Sσ→ C of Fr´echet spaces. There is also a version of the Rellich theorem stated as follows.
4. PERTURBED SOBOLEV SPACES 45 Proposition 4.10.The operator Wm0 σ,→Wm σis compact for m0> m. By using the “Fourier coefficients” mapping, Proposition 4.10 follows from the following lemma (see e.g. [54, Theorem 5.8]). Lemma 4.11.The operator `2 m0,→`2 mis compact for m0> m. Proof of Proposition 4.4. For φ∈ Sσ, its “Fourier coefficients” ck=hφk, φiσ form a sequence c= (ck) in C, and X k|ck|(1 + k)m/2≤ kck`2 m0X k (1 + k)m−m01/2 by Cauchy-Schwartz inequality, where the last series is convergent since m−m0< −1. Therefore X k|ck|(1 + k)m/2≤CkφkWm0 σ(59) for some C > 0 independent of φ. On the other hand, for all i, j ∈Nwith i+j≤m, there is some homogeneus non-commutative polynomial pi,j of degree i+jsuch that xiTj σ=pi,j(B, B0). Then, by (18)–(20), |hφk, xiTj σφiσ| ≤ Ci,j(1 + k)m/2X |`−k|≤m|c`|(60) for some Ci,j >0 independent of φ. Now suppose that σ≥0. By (59), (60) and Theorem A-(ii), there is some C0 i,j >0 independent of φand xso that |x|σ|xiTj σφ(x)|≤|x|σX k|hφk, xiTj σφiσ||φk(x)| =X k|hφk, xiTj σφiσ||ξk(x)| ≤ C0 i,j kφkWm0 σ(61) for all x. Hence kφkSm σ≤C0kφkWm0 σfor some C0>0 independent of φ. Finally assume that σ < 0. By (59), (60) and Theorem B, there is some C0 i,j >0, independent of φand x, so that |xiTj σφ(x)| ≤ X k|hφk, xiTj σφiσ||φk(x)| ≤ C0 i,j kφkWm0 σ for all xif φ∈ Sσ,ev and i+jis even, or φ∈ Sσ,odd and i+jis odd. On the other hand, by (59), (60) and Theorem A-(ii), there is some C00 i,j >0, independent of φ and x, such that, like in (61), |x|σ|xiTj σφ(x)| ≤ C00 i,j kφkWm0 σ for all x6= 0 if φ∈ Sσ,odd and i+jis even, or φ∈ Sσ,ev and i+jis odd. Therefore there is some C0>0 such that kφkSm σ,ev ≤C0kφkWm0 σfor all φ∈ Sσ,ev, and kφkSm σ,odd ≤C0kφkWm0 σfor all φ∈ Sσ,odd. As suggested by (29), consider the mapping c= (ck)7→ Ξ(c)=(d`), where cis odd and Ξ(c) is even with d`=X k∈{`+1,`+3,... } (−1)k−`−1 2s(k−1)(k−3) ···(`+ 2)2s (k+ 2σ)(k−2+2σ)···(`+ 1 + 2σ)ck
46 4. PERTURBED SOBOLEV SPACES for `even, assuming that this series is convergent. Lemma 4.12.Ξdefines a bounded map `2 m0,odd → Cm,ev if m0−m > 1. Proof. By the Cauchy-Schwartz inequality, kdkCm= sup `X k∈{`+1,`+3,... }s(k−1)(k−3) ···(`+ 2)2s (k+ 2σ)(k−2+2σ)···(`+ 1 + 2σ)|ck|(1 + `)m ≤√2ssup `X k∈{`+1,`+3,... }|ck|(1 + `)m ≤√2skck`2 m0sup `X k∈{`+1,`+3,... } (1 + k)−m0(1 + `)m1/2 ≤√2skck`2 m0X k (1 + k)m−m01/2 , where the last series is convergent since m−m0<−1. Corollary 4.13.x−1defines a bounded operator Sm0 σ,odd → Sm σ,ev if 2m0> m+ 5. Proof. Since 2m0> m + 5, there are m1, m2, m3≥0 such that m0−m3>1/2, m3−m2>1,2m2−m1>1, m1−m > 1. Then, by Propositions 4.3 and 4.4, Lemmas 4.7 and 4.12, and using the “Fourier coefficients” mapping, we get the following composition of bounded maps: Sm0 σ,odd ,→Wm3 σ,odd →`2 m3,odd Ξ −→ Cm2,ev ,→`2 m1,ev →Wm1 σ,ev ,→ Sm σ,ev . By (29), this composite is an extension of the map x−1:Sodd → Sev. Question 4.14.The proof of Corollary 4.13 is very indirect. Is it possible to prove it without using (29) and the perturbed Sobolev spaces? Corollary 4.15.x−1defines a continuous operator Sσ,odd → Sσ,ev. Lemma 4.16.S1 σ,ev ⊂ S1 w,σ,ev and Sm+2 σ⊂ Sm w,σ continuously for m≥1. Proof. Let us construct a sequence of naturals Mm,ev/odd such that SMm,ev/odd σ,ev/odd ⊂ Sm w,σ,ev/odd continuously for all m. Like in the proof of Lemma 3.8, we proceed by induction on m, with M0,ev/odd = 0. For m > 0, assume that the terms Mm−1,ev/odd are constructed. For φ∈C∞ ev ,i+j≤mwith j > 0 and x∈R, we have xiφ(j)(x)=xi(Tσφ)(j−1)(x), obtaining kφkSm w,σ ≤ kTσφkSm−1 w,σ +kφkSm σ. But there are some C, C0>0, independent of φ, such that kTσφkSm−1 w,σ ≤CkTσφkSMm−1,odd σ≤C0kφkSMm,ev σ with Mm,ev =Mm−1,odd + 1 .(62)
4. PERTURBED SOBOLEV SPACES 47 For φ∈C∞ odd, and i,jand xas above, we have xiφ(j)(x)≤xi(Tσφ)(j−1)(x)+ 2σxi(x−1φ)(j−1)(x), obtaining kφkSm w,σ ≤ kTσφkSm−1 w,σ + 2 |σ|kx−1φkSm−1 w,σ +kφkSm σ, But, by Corollary 4.13, there are some C, C0>0, independent of φ, such that kTσφkSm−1 w,σ + 2 |σ|kx−1φkSm−1 w,σ ≤Ckφ0kSMm−1,ev σ+kx−1φkSMm−1,ev σ ≤C0kφkSMm,odd σ if Mm,odd ≥Mm−1,ev + 1 ,2Mm,odd > Mm−1,ev + 5 .(63) The conditions (62) and (63) are satisfied with M1,ev = 1, M1,odd = 3 and Mm,ev/odd =m+ 2 for m≥2. Corollary 4.17.SMm,ev/odd ev/odd ⊂ Sm σ,ev/odd continuously for all m, where Mm,ev/odd =(3m 2+m 4dσe(dσe+ 3) + dσeif σ≥0and mis even 5m 2+ 2 if σ < 0and mis even , Mm,ev =(3m−1 2+m−1 4dσe(dσe+ 3) + dσeif σ≥0and mis odd 5m+1 2if σ < 0and mis odd , Mm,odd =(3m+1 2+m+1 4dσe(dσe+ 3) + dσeif σ≥0and mis odd 5m+7 2if σ < 0and mis odd . Proof. This follows from Lemmas 3.1, 3.3 and 3.8. Corollary 4.18.Sm0 σ⊂ Smcontinuously for all m, where m0=m+3+dσe(dσe+ 1) 2. Moreover S1 σ,ev ⊂ S0 ev continuously. Proof. This follows from Lemmas 3.2, 3.4 and 4.16. Corollary 4.19.Sσ=Sas Fr´echet spaces. Proof. This is a consequence of Corollaries 4.17 and 4.18 Now, Theorems D and E follow from Corollaries 4.17 and 4.18 and Propositions 4.3 and 4.4.
CHAPTER 5 Perturbation of Hon R+ More general perturbations of Hcan be obtained with conjugation of Lby the operator of multiplication by functions which are defined and positive almost everywhere (with respect to the Lebesgue measure), like we did in Section 1 of Chapter 2 with the function |x|σ. We will only consider conjugations of the even and odd components of Lseparately, and acting on spaces of functions on R+. This will be also enough for the application in Part 2. Let Lev/odd, or Lσ,ev/odd, denote the restriction of Lto Sev/odd. Since the function |x|2σis even, there is an orthogonal decomposition L2(R,|x|2σdx) = L2 ev(R,|x|2σdx)⊕L2 odd(R,|x|2σdx) as direct sum of subspaces of even and odd functions. Then Lev/odd is essentially self-adjoint in L2 ev/odd(R,|x|2σdx), and its self-adjoint extension Lev/odd, or Lσ,ev/odd, is obtained by restriction of L. We also get an obvious version of Corollary F for Lev/odd. Fix open subset U⊂R+of full Lebesgue measure. Let Sev/odd,U denote the linear subspace of C∞(R+) consisting of the restrictions to Uof the functions in Sev/odd. The restriction to Udefines a linear isomorphism Sev/odd ∼ =Sev/odd,U ,(64) and a unitary isomorphism L2 ev/odd(R,|x|2σdx)∼ =L2(R+,2x2σdx).(65) Let Lev/odd,U , or Lσ,ev/odd,U , denote the operator defined by Lev/odd on Sev/odd,U via (64). Let also φk,U =φk|U, whose norm in L2(R+, x2σdx) is 1/√2 since (65) is unitary. When U=R+, the notation Sev/odd,+,Lev/odd,+, or Lσ,ev/odd,+, and φk,+ will be used. Moreover let Lev/odd,+, or Lσ,ev/odd,+, be the self-adjoint operator in L2(R+, x2σdx) that corresponds to Lev/odd via (65). Going one step further, for any positive function h∈C2(U), the operator (of multiplication by) hdefines a unitary isomorphism h:L2(R+, x2σdx)∼ = −→ L2(R+, x2σh−2dx).(66) We get that hLev/odd,U h−1, with domain hSev/odd,U , is essentially self-adjoint in L2(R+, x2σh−2dx), and its self-adjoint extension is hLev/odd,+h−1. Via (65) and (66), we obtain an obvious version of Corollary F for hLev/odd,+h−1. By using d dx, h=h0,d2 dx2, h= 2h0d dx +h00 ,(67) it easily follows that hLev/odd,U h−1is of the form (1) with f1∈C1(U) and f2∈ C(U). Then Theorem G is a consequence of the following. Lemma 5.1.For σ > −1/2, a positive function h∈C2(U), and an operator Pof the form (1) with f1∈C1(U)and f2∈C(U), we have P=hLσ,ev,U h−1on 49
50 5. PERTURBATION OF HON R+ hSev,U if and only if (2) and (3) are satisfied with some primitive F1∈C2(U)of f1. Proof. By (67), h−1Ph =−h−1d2 dx2h+sx2−2h−1f1 d dx h+f2 =−d2 dx2−h−12h0d dx +h00+sx2 −2f1 d dx −2h−1f1h0+f2 =H−2(h−1h0+f1)d dx −h−1h00 −2h−1f1h0+f2. So P=hLσ,ev,U h−1if and only if h−1h0=σx−1−f1,(68) f2=h−1h00 + 2h−1h0f1.(69) The equality (68) is equivalent to (3), and gives h−1h00 = (σx−1−f1)2−σx−2+f0 1. So, by (69), f2= (σx−1−f1)2−σx−2+f0 1+ 2(σx−1−f1)f1 =σ(σ−1)x−2−f2 1−f0 1. It follows that (68) and (69) are equivalent to (3) and (2). Remark 2.By (67), we get an operator of the same type if hand d dx is interchanged in (1). Remark 3.By using (67) with h=x−1on R+, it is easy to check that Lσ,odd,+=xL1+σ,ev,+x−1on Sodd,+=xSev,+for all σ > −1/2. So no new operators are obtained with the conjugation Lσ,odd,U by h. Remark 4.If f1is a rational function, then the function f2, given by (2), is also rational. Remark 5.The term of Pwith d dx can be removed by conjugation, obtaining the operator H+σ(σ−1)x−2, given by restricting Kσ, first to even functions and second to R+. In this way, we get all operators of the form H+cx−2with c > −1/4.
CHAPTER 6 Examples 1. Case where f1is a multiple of x−1 A particular class of (1) is given by the operators of the form P=H−2c1x−1d dx +c2x−2(70) for c1, c2∈R. In this case, we can take F1=c1log x. Then eF1=xc1, (3) gives h=xawith a=σ−c1, and (2) becomes c2x−2= (a2+a(2c1−1))x−2. Therefore Corollary H follows from Theorem G. Remark 6.According to Remark 2, we get an operator of the same type if x−1and d dx is interchanged in (70). We may also use that, with the function xa (a∈R), (67) becomes d dx, xa=axa−1,d2 dx2, xa= 2axa−1d dx +a(a−1)xa−2.(71) Remark 7.By Corollary H-(iii), we have hD∞(P)⊂ D∞(P) for all h∈ C∞(R+) such that h0∈C∞ 0(R+). The existence of a∈Rsatisfying (4) is characterized by the condition (2c1−1)2+ 4c2≥0.(72) Observe that (72) is satisfied if c2≥min{0,2c1}. In particular, we have the following special cases. Example 6.1.Suppose that c2= 0; i.e., P=H−2c1x−1d dx . Thus P=Lc1,ev,+ if c1>−1/2; however, this inequality is not required a priori. Then (4) means that a∈ {0,1−2c1}, and (5) gives σ=(c1if a= 0 1−c1if a= 1 −2c1. In the case a= 0 and σ=c1, the condition c1>−1/2 is needed to apply Corollary H. In this case, Corollary H holds for P=Lc1,ev,+on Sev,+, which is a direct consequence of the known properties of Lc1(Section 2 of Chapter 1 and Corollary F). Nevertheless, Corollary H gives new information in the case a= 1 −2c1and σ= 1 −c1: we have σ > −1/2 just when c1<3/2 (c1≤ −1/2 is allowed!). When this inequality is satisfied, Corollary H states that P, with domain x1−2c1Sev,+, is also essentially self-adjoint in L2(R+, x2c1dx); the spectrum of its self-adjoint extension Pconsists of the eigenvalues (4k+3−2c1)s(k∈N) with multiplicity one; the corresponding normalized eigenfunctions are √2x1+2c1φ2k,+; and D∞(P) = x1−2c1Sev,+. 51
58 7. PRELIMINARIES ON THOM-MATHER STRATIFICATIONS (ii) By the normality of A, we can also assume that, if X, Y ∈ S and TX∩TY6= ∅, then X≤Yor Y≤X. (iii) The frontier of a stratum Xequals the union of the strata Y < X. (iv) The connected components of each stratum may have different dimensions. (v) The connected components of the strata, with the corresponding restrictions of the tubes, define an induced Thom-Mather stratification Acon ≡ (A, Scon, τcon); in this way, we can assume that the strata are connected if desirable. Remark 9.The following are some variants of the concept “stratification” and related notions: (i) A weak Thom-Mather stratification is defined by removing the condition ρXπY=ρXfrom Definition 7.1-(vi). (ii) A stratification is a pair (A, S) satisfying Definition 7.1-(i),(ii),(iv); it is also said that Sis a stratification of A. Definition 7.1-(iv) is called the frontier condition. If moreover τsatisfies the other conditions of Definition 7.1, then it is called Thom-Mather structure on (A, S). (iii) If Ais a subspace of a smooth manifold M, then a stratification Sof Ais usually required to consist of regular submanifolds of M; the term stratified subspace of Mis used in this case. In [29], a weaker version of this notion is defined by requiring local finiteness of Sinstead of the frontier condition. (iv) For a stratified subspace (A, S) of a smooth manifold M, the condition (B), introduced by H. Whitney [66, 67], is defined as follows2. In the case M=Rm, it requires that, for all X6=Yin S, if (xi) and (yi) are sequences in Xand Y, respectively, both of them converging in Ato some x∈X, if the sequence of tangent spaces TyiYconverges3to a linear subspace T⊂Rn, and if the sequence of lines R(xi−yi) converges to a line L⊂Rm, then L⊂T. This property is preserved by local diffeomorphisms of Rm, and therefore generalizes to arbitrary smooth manifolds. This condition gives rise to the concept of Whitney stratification of a subspace (or Whitney stratified subspace) of M. Example 7.2.(i) Any smooth manifold is a Thom-Mather stratification with one stratum and the trivial tube. (ii) Any smooth manifold with boundary is a stratification with two strata, the interior and the boundary. It can be endowed with a Thom-Mather structure by using a collar of the boundary. (iii) Any subanalytic subset of Rmhas a primary and secondary stratifications; the secondary one satisfies condition (B) [40, 45, 33, 32, 34]. (iv) J. Mather [44] has proved that any Whitney stratified subspace of a smooth manifold admits a Thom-Mather structure (see also [29, Proposition 2.6 and Corollary 2.7]). For a stratification A≡(A, S), the depth of any X∈ S, denoted by depth X, is the supremum of the naturals nsuch that there exist strata X0, . . . , Xnwith X0< X1<··· < Xn=X. Notice that depth X≤dim X. Moreover depth X= 0 2Certain condition (A) was also introduced by H. Whitney in [66, 67], but J. Mather [44] has observed that it follows from condition (B). 3The convergence of linear subspaces of Rmis considered in the appropriate Grassmannians.
1. THOM-MATHER STRATIFICATIONS 59 (Xis minimal in S) if and only if Xis closed in A. The depth and dimension of Aare the supremum of the depths and dimensions of its strata, respectively. The dimension of Aequals its topological dimension, which may be infinite. The depth of Ais zero if and only if all strata are open and closed. Let A≡(A, S, τ) be a Thom-Mather stratification. Let B⊂Abe a locally closed subset. Suppose that, for all X∈ S,X∩Bis a smooth submanifold of X, and B∩π−1 X(X∩B), endowed with the restrictions of πXand ρX, defines a tube τX∩Bof X∩Bin B. Then let S|B={X∩B|X∈ S}, and let τ|Bbe defined by the assignment of τX∩Bto each X∩B∈ S|B. If (B, S|B, τ|B) satisfies the conditions of a stratification, it is said that the stratification A(or (S, τ)) can be restricted to B, and B≡(B, S|B, τ|B) is called a restriction of A(or (S|B, τ|B) is called the restriction of (S, τ)); it may be also said that Bis a Thom-Mather substratification of A. For instance, Acan be restricted to any open subset and to any locally closed union of strata. A restriction of a restriction of Ais a restriction of A. For a stratum Xof A, we can consider the restriction of Ato X. In this way, to study X, we can assume that Xis dense in Aand dim X= dim Aif desirable. A locally closed subset B⊂Ais said to be saturated if the stratification Acan be restricted to Band, for every X∈ S, there is a representative (TX, πX, ρX) of τXsuch that π−1 X(X∩B) = TX∩B. Let A0≡(A0,S0, τ0) be another Thom-Mather stratification. A continuous map f:A→A0is called a morphism if, for any X∈ S, there is some X0∈ S0such that f(X)⊂X0, the restriction f:X→X0is smooth, and there are (TX, πX, ρX)∈τXand (T0 X0, π0 X0, ρ0 X0)∈τ0 X0such that f(TX)⊂T0 X0,fπX=π0 X0f and fρX=ρ0 X0. Notice that the continuity of a morphism follows from the other conditions. Morphisms between stratifications form a category with the operation of composition; in particular, we have the corresponding concepts of isomorphism and automorphism. The set of morphisms A→A0is denoted by Mor(A, A0), and the group of automorphisms of Ais denoted by Aut(A). The other variants of the concept “stratification” given in Remark 9 also have obvious corresponding versions of morphisms, isomorphisms and automorphisms; in particular, we get the concept of weak morphism between weak Thom-Mather stratifications. A (weak) morphism is called submersive when it restricts to smooth submersions between the strata. Example 7.3.Let Gbe a compact Lie group Gacting smoothly on a closed manifold M. Consider the orbit type stratifications of Mand G\M[9]. It is well known that G\Madmits a Thom-Mather structure [64, Introduction], which can be seen as follows. G\Mis locally isomorphic to a semi-algebraic subset of an Euclidean space whose primary and secondary stratifications are equal [5]. By using an invariant smooth partition of unity of M, like in the Whitney’s embedding theorem, it follows that G\Mis isomorphic to a Whitney stratified subspace of some Euclidean space, and therefore it admits a Thom-Mather structure. This can also be seen by observing that the stratification of Msatisfies condition (B), and the proof of [29, Proposition 2.6] can be adapted to produce an invariant4 Thom-Mather structure on M, which induces a Thom-Mather structure on G\M. The following two lemmas are easy to prove. 4Gacts by automorphisms.
60 7. PRELIMINARIES ON THOM-MATHER STRATIFICATIONS Lemma 7.4.Let Abe a Hausdorff, locally compact and second countable space, {Ui}an open covering of A, and (Si, τi)a Thom-Mather stratification of each Ui. (i) If (Si, τi)and (Sj, τj)have the same restrictions to Uij := Ui∩Ujfor all iand j, then there is a unique Thom-Mather stratification (S, τ)on A whose restriction to each Uiis (Si, τi). (ii) If ((Si|Uij )con,(τi|Uij )con) = ((Sj|Uij )con,(τj|Uij )con)for all iand j, then there is a unique Thom-Mather stratification (S, τ)on Awith connected strata such that ((S|Ui)con,(τ|Ui)con)=(Si,con, τi,con). Lemma 7.5.Let (A0,S0, τ0)be another Thom-Mather stratification. (i) With the notation of Lemma 7.4-(i), let fi: (Ui,Si, τi)→(A0,S0, τ0)be a morphism for each i. If fi|Uij =fj|Uij for all iand j, then the combination of the maps fiis a morphism f: (A, S, τ)→(A0,S0, τ0). (ii) With the notation of Lemma 7.4-(ii), let fi: (Ui,Si,con, τi,con)→(A0,S0, τ0) be a morphism for each i. If fi|Uij =fj|Uij for all iand j, then the combination of the maps fiis a morphism f: (A, S, τ)→(A0,S0, τ0). Remark 10.As a particular case of Lemma 7.4, given a countable family of Thom-Mather stratifications, {Ai≡(Ai,Si, τi)}, there is a unique Thom-Mather stratification (S, τ) on the topological sum FiAiwhose restriction to each Aiis (Si, τi); this (S, τ) will be called the sum of the Thom-Mather stratifications (Si, τi). 1.2. Products. The product of two weak Thom-Mather stratifications, Aand A0, has a weak Thom-Mather stratification A×A0≡(A×A0,S00, τ00) with S00 = {X×X0|X∈ S, X0∈ S0}and τ00 X×X0= [T00 X×X0, π00 X×X0, ρ00 X×X0], where T00 X×X0= TX×T0 X0,π00 X×X0=πX×π0 X0and ρ00 X×X0(x, x0) = ρX(x) + ρ0 X0(x0). If Aand A0are Thom-Mather stratifications and the depth of at least one of them is zero, then A×A0is a Thom-Mather stratification, but this is not true when the depths of Aand A0are positive [64, Section 1.2.9, pp. 5–6]. Another choice of ρX×X0is needed to get the second equality of Definition 7.1-(vi). For instance, ρ00 X×X0= max{ρX, ρ0 X0}satisfies that condition, but it is not smooth on the intersection of the strata with T00 X×X0. To solve this problem, pick up a function h: [0,∞)2→[0,∞) that is continuous, homogeneous of degree one, smooth on R2 +, with h−1(0) = {(0,0)}, and such that, for some C > 1, we have h(r, s) = max{r, s} if Cmin{r, s}<max{r, s}. Then A×A0becomes a Thom-Mather stratification by setting ρ00 X×X0(x, x0) = h(ρX(x), ρ0 X0(x0)); it will be called a product of Aand A0. 1.3. Cones. Recall that the cone with link a non-empty topological space Lis the quotient space c(L) = L×[0,∞)/L × {0}. The class ∗=L× {0}is called the vertex or summit of c(L). The element of c(L) represented by each (x, ρ)∈L×[0,∞) will be denoted by [x, ρ]. The function on c(L) induced by the second factor projection L×[0,∞)→[0,∞) will be called its canonical function, and will be usually denoted by ρ. Notice that c(L) is locally compact if and only if Lis compact. It is also declared that c(∅) is the singleton space {∗}, and the above terminology can be obviously adapted to this case. Now, suppose that Lis a compact Thom-Mather stratification. Then c(L) has a canonical Thom-Mather stratification so that {∗} is a stratum, its restriction to c(L)\{∗} =L×R+is the product Thom-Mather stratification, and the tube of {∗} is [c(L), π, ρ], where ρis the canonical function and πis the unique map c(L)→ {∗}.
1. THOM-MATHER STRATIFICATIONS 61 If L6=∅, then depth c(L) = depth L+ 1 and dim c(L) = dim L+ 1. For any > 0, let c(L) = ρ−1([0, )). Let L0be another compact Thom-Mather stratification, and let ∗0denote the vertex of c(L0). If L6=∅, the cone of any morphism f:L→L0is the morphism c(f) : c(L)→c(L0) induced by f×id : L×[0,∞)→L0×[0,∞). If L=∅,c(f) is defined by mapping ∗to ∗0. Reciprocally, it is easy to check that, for any morphism h:c(L)→c(L0), there is some morphism f:L→L0such that h=c(f) near ∗; in particular, h(∗) = ∗0. Let c(Aut(L)) = {c(f)|f∈Aut(L)} ⊂ Aut(c(L)). Example 7.6.For each integer m≥1, there is a canonical homeomorphism can : c(Sm−1)→Rmdefined by can([x, ρ]) = ρx. Of course, this is not an isomorphism of Thom-Mather stratifications, but it restricts to a diffeomorphism of the stratum Sm−1×R+of c(Sm−1) to Rm\{0}. Via can : c(Sm−1)→Rm, the canonical function of c(Sm−1) corresponds to the function ρ0(x) = |x|on Rm, which will be also called the canonical function on Rmfor the scope of this work. If ρ1is the canonical function on c(L) for some compact Thom-Mather stratification L, then the function ρ=pρ2 0+ρ2 1will be called the canonical function on Rm×c(L). The following argument shows that a product of two cones is isomorphic to a cone. With the above notation, let ρ:c(L)→[0,∞) and ρ0:c(L0)→[0,∞) be the canonical functions, and let ρ00 =h(ρ×ρ0) : c(L)×c(L0)→[0,∞) for a function h like in Section 1.2. Since the restrictions ρ:L×R+→R+and ρ0:L0×R+→R+ are submersive weak morphisms, and h:R2 +→R+is non-singular, it follows that ρ00 :c(L)×c(L0)\ {(∗,∗0)} → R+is a submersive weak morphism. Hence L00 =ρ00−1(1) is saturated in c(L)×c(L0) [64, Lemma 2.9, p. 17]. Let ∗00 denote the vertex of c(L00). Since his homogeneous of degree one, the mapping [([x, r],[x0, r0]), s]7→ ([x, rs],[x0, r0s]) defines an isomorphism c(L00)→c(L)×c(L0), whose inverse is given by (∗,∗0)7→ ∗00, and ([x, r],[x0, r0]) 7→ x, r h(r, r0),x0,r0 h(r, r0), h(r, r0) if (r, r0)6= (0,0). 1.4. Conic bundles. Let Xbe a smooth manifold, La compact Thom- Mather stratification, and π:T→Xa fiber bundle whose typical fiber is c(L) and whose structural group can be reduced to c(Aut(L)). Thus there is a family of local trivializations of π,{(Ui, φi)}, such that the corresponding transition functions define a cocycle with values in c(Aut(L)); i.e., for all iand j, there is a map hij :Uij := Ui∩Uj→c(Aut(L)) such that φjφ−1 i(x, y) = (x, hij(x)(y)) for every x∈Uij and y∈c(L). Thus we get another cocycle consisting of maps gij :Uij →Aut(L) so that hij(x) = c(gij(x)) for all x∈Uij. Consider the Thom- Mather stratification on each open subset π−1(Ui)⊂Tthat corresponds by φito the product Thom-Mather stratification on Ui×c(L). For each connected open V⊂Uij and every stratum N0of L, there is an stratum N1of Lsuch that gij(x)(N0) = N1 for all x∈V, and suppose also that, in this case, the map V×N0→N1, (x, y)7→ gij(x)(y), is smooth. Then each mapping (x, y)7→ (x, gij(x)(y)) defines an automorphism of Uij ×L. This means that the induced Thom-Mather stratifications on π−1(Ui) and π−1(Uj) have the same restriction to π−1(Uij). By Lemma 7.4-(i), it follows that there is a unique Thom-Mather stratification on Twhose restriction
62 7. PRELIMINARIES ON THOM-MATHER STRATIFICATIONS to each π−1(Ui) is the above Thom-Mather stratification. Furthermore there is a canonical section of π, called the vertex (or summit)section, which is well defined by x7→ ∗x=φ−1 i(x, ∗) if x∈Ui, where ∗denotes the vertex of c(L); each ∗xcan be called the vertex of the fiber over x. The image of the vertex section is a stratum of T, called the vertex (or summit)stratum, which is diffeomorphic to X. If π:T→Xis endowed with a maximal family Φ of trivializations satisfying the above conditions, it will be called a conic bundle, and the corresponding Thom- Mather stratification on Tis called its conic bundle Thom-Mather stratification. It will be also said that Φ is the conic bundle structure of π. Let ρ:c(L)→[0,∞) be the canonical function. Its lift to each Ui×c(L) is also denoted by ρ. The functions φ∗ρon the sets π−1 X(Ui) can be combined to define a function ρ:T→[0,∞). The tubular neighborhood of Xin Tis [T, π, ρ], and (T, π, ρ) is called its canonical representative. Let π0:T0→X0be another conic bundle, whose structure is given by a family Φ0of trivializations as above. Let F:T→T0be a fiber bundle morphism over a map f:X→X0. Then we can choose {(Ui, φi)}as above and a family {(U0 i, φ0 i)} ⊂ Φ0such that f(Ui)⊂U0 ifor all i, and therefore F(π−1(Ui)) ⊂π0−1(U0 i). Let h0 ij =c(g0 ij) : U0 ij := U0 i∩U0 j→c(Aut(L0)) be the maps defined by the transition maps φ0 jφ0 i−1as above. Suppose that there are maps κi:Ui→Mor(L, L0) such that κj(x)gij(x) = g0 ij(f(x)) κj(x) for all x∈Uij. For each connected open V⊂Ui and every stratum Nof L, there is an stratum N0of L0such that κi(x)(N)⊂ N0for all x∈V, and assume also that, in this case, the map V×N→N0, (x, y)7→ κi(x)(y), is smooth. Then Fis called a morphism of conic bundles. In this case, each mapping (x, y)7→ (f(x), κi(x)(y)) defines a morphism Ui×c(L)→ U0 i×c(L0). So each restriction F:π−1(Ui)→π0−1(U0 i) is a morphism of Thom- Mather stratifications, and therefore F:T→T0is a morphism of Thom-Mather stratifications by Lemma 7.5-(i). According to Section 1.3, any morphism of Thom- Mather stratifications between conic bundles, preserving the vertex stratum, equals a conic bundle morphism near the vertex stratum. The case of conic bundles is specially important because, as pointed out in [7, Chapitre A, Remarque 3], the proof of [64, Theorem 2.6, pp. 16–17] can be easily adapted to get the following. Proposition 7.7.Let A≡(A, S, τ)be a Thom-Mather stratification with connected strata. Then, for any X∈ S, there is some (T, π, ρ)∈τXsuch that π:T→Xadmits a structure Φof conic bundle such that the corresponding conic bundle Thom-Mather stratification is (S|T, τ|T). Remark 11.(i) The notation TX,πX,ρX,LXand ΦXwill be used when a reference to the stratum Xis desired. (ii) The connectedness of the strata is assumed for the sake of simplicity. In the general case, the description of Proposition 7.7 holds around the connected components of the strata. (iii) We can choose ρso that (T, π, ρ) is the canonical representative of the tube around Xin Twith its conic bundle Thom-Mather stratification. Definition 7.8.Achart or distinguished neighborhood of Ais a pair (O, ξ), where Ois open in Aand, for some X∈ S and > 0, with the notation and conditions of Proposition 7.7, ξis an isomorphism O→B×c(L) defined by some (U, φ)∈Φ and some chart (U, ζ) of Xwith ζ(U) = B, where Bis an open subset
1. THOM-MATHER STRATIFICATIONS 63 of Rmfor m= dim X. It is said that (O, ξ) is said to be centered at x∈Xif B is an open ball centered at 0 and ξ(x) = (0,∗), where ∗is the vertex of c(L). A collection of charts that cover Ais called an atlas of A. Remark 12.Definition 7.8 also includes the case where any factor of the product Rm×c(L) is missing by taking m= 0 or L=∅. Remark 13.The following two assertions follow by using charts and induction on the depth of the strata: (i) In any Thom-Mather stratification, there is at most one dense stratum, which is open. (ii) Any stratum with compact closure has a finite number of connected components. 1.5. Uniqueness of Thom-Mather stratifications. Lemma 7.9.Let Abe a Hausdorff, locally compact and second countable space, let (A0,S0, τ0)be a Thom-Mather stratification with connected strata, and let f: A→A0be a continuous map. Then there is at most one Thom-Mather stratification (S, τ)on Awith connected strata so that f: (A, S, τ)→(A0,S0, τ0)is a morphism that restricts to local diffeomorphism between corresponding strata. Proof. Let (S, τ) be a Thom-Mather stratification on Asatisfying the conditions of the statement. Then the elements of Sare the connected components Xof the sets f−1(X0) for X0∈ S, endowed with the differential structure so that f:X→X0is a local diffeomorphism. Thus the elements of Sare determined by fand the elements of S0. Let X∈ S and X0∈ S0with f(X)⊂X0, and let (T, π, ρ)∈τXand (T0, π0, ρ0)∈ τ0 X0with f(T)⊂T0,π0f=f π and ρ0f=ρ; in particular, ρis determined by f and ρ0. Let x∈Tand x0=f(x)∈T0, and let Y∈ S such that x∈Y. Then f π(x) = π0(x0), obtaining that π(x) is the unique point of X∩f−1(π0(x0)) that is contained in the connected component of xin f−1π0−1(π0(x0)). It follows that πis also determined by fand π0, and therefore τXis determined by fand τ0 X0. 1.6. Relatively local properties on strata. The following kind of terminology will be used for a subspace Xof an arbitrary topological space A. Let Pbe a property that may hold on open subsets U⊂X; for the sake of simplicity, let us say that “Uis P” when Pholds on U. It is is said that Xis relatively locally (or simply, rel-locally)Pat some x∈Xif there is a base Uof open neighborhoods of xin Asuch that U∩Xis Pfor all U∈ U; if Xis rel-locally Pat all points of X, then Xis said to be relatively locally (or simply, rel-locally)P. Similarly, Pis said to be a relatively local (or simply, rel-local) property when Xis Pif and only if it is rel-locally P. We will apply this terminology to the case where Ais a Thom-Mather stratification and Xis a stratum of A. For instance, on X, we will consider functions that are rel-locally bounded or rel-locally bounded away from zero, rel-locally finite open coverings, and rel-local connectedness at points of X. Any locally finite covering of Xby open subsets of Arestricts to a rel-locally finite open covering of X; thus there exist rel-locally finite open coverings of Xby the paracompactness of A. Observe that Xis compact if and only if any rel-locally finite open covering of Xis finite.
64 7. PRELIMINARIES ON THOM-MATHER STRATIFICATIONS 2. Adapted metrics on strata The definition of adapted metrics was given for the regular stratum of any Thom-Mather stratification that is a pseudomanifold [13, 14, 47, 48]. But its definition has an obvious version for any stratum of a Thom-Mather stratification. In this work, we will consider only the simplest type of adapted metrics, whose definition is recalled. The corresponding (componentwise) metric completion of strata will be specially studied. 2.1. Adapted metrics on strata and local quasi-isometries between Thom-Mather stratifications. Let Abe a Thom-Mather stratification. The adapted metrics on its strata are combinations of the adapted metrics on their connected components with respect to the Thom-Mather stratification defined by those connected components. Thus we can assume that the strata of Aare connected to define adapted metrics. This definition is given by induction on the depth of the strata. Definition 7.10.Let Mbe a stratum of A. If depth M= 0, then Mis a closed manifold, and any Riemannian metric on Mis called adapted. If depth M > 0 and adapted metrics are defined for strata of lower depth, then an adapted metric on M is a Riemannian metric gsuch that, for any point x∈M\M, there is some chart (O, ξ) of Acentered at x, with ξ(O) = B×c(L) and ξ(O∩M) = B×N×(0, ) for some stratum Nof L, so that gis quasi-isometric to ξ∗(g0+ρ2˜g+ (dρ)2) on O, where g0is the standard Riemannian metric on Rm,ρis the standard coordinate of R+, and ˜gis some adapted metric on N, which is defined because the depth of Nin Lis smaller than the depth of Min A. Remark 14.Since all Riemannian metrics on a smooth manifold are locally quasi-isometric, any metric on Rmcould be used in Definition 7.10 instead of g0. Remark 15.The following properties follow by taking charts and using induction on the depth of the strata: (i) Any pair of adapted metrics on M,gand g0, are rel-locally quasi-isometric; in particular, if Mis compact, then any pair of adapted metrics on Mare quasi-isometric. (ii) Any point in Mhas a countable base {Om|m∈N}of open neighborhoods such that, with respect to any adapted metric, vol(M∩Om)→0 and max{diam P|P∈π0(M∩Om)} → 0 as m→ ∞; in particular, if M is compact, then, with respect to any adapted metric, we have vol M < ∞ and diam P < ∞for all P∈π0(M). (iii) Any morphism of Thom-Mather stratifications restricts to rel-locally uniformly continuous maps between corresponding strata with respect to arbitrary adapted metrics. (iv) If gand g0are adapted metrics on strata Mand M0of Thom-Mather stratifications Aand A0, respectively, then g⊕g0is an adapted metric on the stratum M×M0of any product Thom-Mather stratification on A×A0 (Section 1.2). In [8, Appendix], it was proved that there exist adapted metrics on the regular stratum of any Thom-Mather stratification that is a pseudomanifold. It can be easily checked that the same argument proves the existence of adapted metrics on any stratum Mof every Thom-Mather stratification A.
2. ADAPTED METRICS ON STRATA 65 Example 7.11.The proof in [8, Appendix] also shows the following: (i) With the notation of Definition 7.10, the metric g=g0+ρ2˜g+ (dρ)2is adapted on the stratum M=Rm×N×R+of c(L); it will be called a model adapted metric. (ii) Given a rel-locally finite atlas {(Oa, ξa)}of M, a smooth partition of unity {λa}subordinated to the open covering {M∩Oa}of M, and an adapted metric gaon each M∩Oa, then the metric Paλagaon Mis adapted. Example 7.12.For an integer m≥1, let ˜g0be the restriction to Sm−1of the standard metric g0of Rm. Then, via can : c(Sm−1)→Rm(Example 7.6), the model adapted metric g1=ρ2˜g0+ (dρ)2on the stratum Sm−1×R+of c(Sm−1) corresponds to g0on Rm\{0}. Example 7.13.With the notation of Example 7.3, for any invariant Riemannian metric gon M, consider the Riemannian metric ¯gon the strata of G\Mso that the canonical projection of the strata of Mto the strata of G\Mis a Riemannian submersion. The proof of [29, Proposition 2.6] can be easily adapted to produce an invariant Thom-Mather structure on Mso that the restriction of gto any stratum is adapted. Hence ¯gis adapted for the induced Thom-Mather structure of G\M. A weak isomorphism between Thom-Mather stratifications is called a local quasi-isometry if it restricts to rel-local quasi-isometries between their strata with respect to adapted metrics; this is independent of the choice of adapted metrics by Remark 15-(i). In particular, a local quasi-isometry between compact Thom- Mather stratifications restricts to quasi-isometries between their strata; thus a local quasi-isometry between compact Thom-Mather stratifications will be called a quasiisometry. The condition of being locally quasi-isometric defines an equivalence relation on the family of Thom-Mather stratifications on any Hausdorff, locally compact and second countable space; each equivalence class will be called a quasiisometry type of Thom-Mather stratifications. By Remark 15-(iv), the product of Thom-Mather stratifications is unique up to local quasi-isometries. Definition 7.14.Consider an adapted metric on a connected stratum Mof a Thom-Mather stratification A, and let ddenote the corresponding distance function on M. For each x∈Mand ρ > 0, the relative ball (or rel-ball) of radius ρand center xis the set consisting of the points y∈Msuch that there is a sequence (zk) in Mwith limkzk=xin Mand lim supkd(y, zk)< ρ. The term ρ-relative neighborhood (or ρ-rel-neighborhood) of xwill be also used for this concept. Example 7.15.(i) The rel-balls centered at points of Mare the usual balls. (ii) In the case of a model adapted metric on the stratum M=N×R+of c(L), the ρ-rel-neighborhood of the vertex ∗is N×(0, ρ). 2.2. Relatively local completion. Let Mbe a stratum of a Thom-Mather stratification A, and fix an adapted metric gon M. Definition 7.16.Assume first that Mis connected, and consider the distance function don Minduced by g. The relatively local completion (or simply, rel-local completion) is the subspace c Mof the metric completion of Mwhose points can be represented by Cauchy sequences in Mthat converge in A; the limits in Mof those sequences define a canonical continuous map lim : c M→M. The canonical dense
66 7. PRELIMINARIES ON THOM-MATHER STRATIFICATIONS injection of Minto its metric completion restricts to a canonical dense injection ι:M→c Msatisfying lim ι= idM. The more specific notation limMand ιMmay be also used. If Mis not connected, then c Mis defined as the disjoint union of the rel-local completions of its connected components. Remark 16.(i) If Mis compact, then c Mis independent of the choice of the adapted metric by Remark 15-(i). (ii) For any open O⊂A,\ M∩Ocan be canonically identified to the open subspace lim−1(M∩O)⊂c M. Example 7.17 (Relatively local completion of the strata of cones).Let Lbe a compact Thom-Mather stratification and Ma stratum of c(L). With the notation of Section 1.3, if M={∗}, then c M=M, obviously. Now, suppose that M=N×R+ for some stratum Nof L. Consider the model adapted metric g=ρ2˜g+ (dρ)2 for some adapted metric ˜gon N, and the corresponding rel-local completion c M. π0(N) is finite by Remark 13-(ii). For each P∈π0(N), let b Pdenote the rel-local completion of Pwith respect to Lcon, which is independent of the choice of ˜g. Then it is easy to check that M≡FPP×R+ FPιP×id −−−−−−→ FPb P×R+,→FPc(b P) extends to a homeomorphism c M→FP∈π0(N)c(b P). Remark 17.The following properties follow easily by using charts, induction on the depth of the strata, Example 7.17 and Remark 15-(ii): (i) lim : c M→Mis surjective with finite fibers. (ii) Mis rel-locally connected with respect to c M. (iii) If Mis compact, then c Mis compact, and therefore its connected components are the metric completions of the connected components of M. Proposition 7.18.(i) c Mhas a unique Thom-Mather stratification with connected strata such that lim : c M→Mis a morphism that restricts to local diffeomorphisms between corresponding strata. In particular, the connected components of Mcan be considered as strata of c Mvia ιM. (ii) The restriction of gto the connected components of Mare adapted metrics with respect to c M. (iii) Let M0be a connected stratum of another Thom-Mather stratification A0 endowed with an adapted metric. Then, for any morphism f:A→A0 with f(M)⊂M0, the restriction f:M→M0extends to a morphism ˆ f:c M→c M0. Moreover ˆ fis an isomorphism if fis an isomorphism. Proof. This is proved by induction on depth M. If depth M= 0, then c M≡ M=M, and there is nothing to prove. Suppose that depth M > 0 and the statement holds for strata of lower depth. We can assume that the strata of Mis connected. For each stratum Xof M, let (TX, πX, ρX) be a representative of the tube around Xin Msatisfying the conditions of Section 1.4 with a compact Thom-Mather stratification LXand a family {(Ui, φi)}of local trivializations of πX. The corresponding cocycle with values in c(Aut(LX)) consists of the maps hij :Ui∩Uj→c(Aut(LX)) defined by
2. ADAPTED METRICS ON STRATA 67 hij(x)=(φjφ−1 i)(x, ·). We have hij(x) = c(gij(x)) for a cocycle consisting of maps gij :Ui∩Uj→Aut(Lx). By the density of Min Mand Remark 13-(i), there is a dense stratum Nof LX so that φi(M∩π−1 X(Ui)) = Ui×N×R+for all i. Consider triples (x, i, P ) such that x∈Uiand P∈π0(N). Two triples of this type, (x, i, P) and (y, j, Q), are declared to be equivalent if x=yand gij(x)(P) = Q. The equivalence class of each triple (x, i, P) is denoted by [x, i, P], and let X0denote the corresponding quotient set. There is a canonical map fX:X0→X, defined by fX([x, i, P ]) = x. Consider the topology on X0determined by requiring that the sets U0 i,P ={[x, i, P]|x∈Ui}are open, and the restrictions fX:U0 i,P →Uiare homeomorphisms. Notice that fXis a finite fold covering map; in particular, in the case X=M,fMis a homeomorphism. Consider the differential structure on each X0so that fXis a local diffeomorphism. By the induction hypothesis, for each P∈π0(N), b Psatisfies the statement of the proposition with some Thom-Mather stratification. Consider quadruples (x, i, P, u) such that x∈Ui,P∈π0(N) and u∈c(b P). Two such quadruples, (x, i, P, u) and (y, j, Q, v), are said to be equivalent if x=y,gij(x)(P) = Qand c(\ gij(x))(u) = v. The equivalence class of each quadruple (x, i, P, u) is denoted by [x, i, P, u], and let T0 Xdenote the corresponding quotient set. There are canonical maps, π0 X:T0 X→X0, lim0 X:T0 X→TX,ρ0 X:T0 X→[0,∞) and ι0 X:M∩TX→ T0 Xdefined by π0 X([x, i, P, u]) = [x, i, P], lim0 X([x, i, P, u]) = φ−1 i(x, c(limP)(u)), ρ0 X([x, i, P, u]) = ρ(u), and ι0 X(z) = [x, i, P, (ιP(v), r)] if z∈M∩π−1 X(Ui) and φi(z)=(x, v, r)∈Ui×P×R+. Notice that fXπ0 X=πXlim0 Xand ρXπ0 X=ρ0 X. Let G⊂Aut(LX) be the subgroup generated by the above elements gij(x). Since the canonical action of Gon LXpreserves N, we get an induced action of G on π0(N). Since Xis connected, there is a bijection between G\π0(N) and the set π0(X0) of connected components of X0, where any orbit O ∈ G\π0(N) corresponds to the connected component X0 O∈π0(X0) consisting of the points [x, i, P ]∈X0 with P∈ O. Also, let T0 X,O= (π0 X)−1(X0 O)⊂T0 X. Given any O ∈ G\π0(N), fix some P0∈ O. For any other P∈ O, there is some gP∈Gsuch that gP(P) = P0. Thus the restriction gP:P→P0induces a map cgP:b P→c P0, and let φ0 i,P : (π0 X)−1(U0 i,P )→U0 i,P ×c(c P0) be the bijection defined by φ0 i,P ([x, i, P, u]) = ([x, i, P], c(cgP)(u)). Consider the topology on T0 X,O determined by requiring that the sets (π0 X)−1(U0 i,P ) are open, and the maps φ0 i,P are homeomorphisms. Then the maps φ0 i,P are local trivializations of the restriction π0 X,O:T0 X,O→X0 Oof π0 X, obtaining that π0 X,Ois a fiber bundle with typical fiber c(c P0). The associated cocycle has values in c(Aut(c P0)); in fact, it consists of the functions h0 i,P ;j,Q :U0 i,P ∩U0 j,Q →c(Aut(c P0)) defined by h0 i,P ;j,Q([x, i, P])(u) = c(g0 i,P ;j,Q([x, i, P]))(u), where g0 i,P ;j,Q :U0 i,P ∩U0 j,Q →Aut(c P0) is the cocycle given by g0 i,P ;j,Q([x, i, P]) = cgQ\ gij(x)cgP−1. The conditions of Section 1.4 are satisfied, obtaining that π0 X,Ois a conic bundle, and therefore T0 X,Ocan be endowed with the corresponding conic bundle Thom- Mather stratification. Since NX,O:= SP∈O Pis G-invariant, the set NX,O×R+is invariant by all transformations hij(x) for x∈Uij, and therefore it defines an open subspace
74 8. RELATIVELY MORSE FUNCTIONS Proof. If depth M= 0, then the statement holds by the density of the Morse functions in C∞(M) with the strong C∞topology [35, Theorem 6.1.2]. Thus suppose that depth M > 0. Let the sets Fkbe defined like in the proof of Lemma 8.4. Claim 6.For 0 ≤k≤depth M, there is an open neighborhood Ukof Fkin A and some fk∈C(Uk∩M) such that, for each stratum X≤M, (i) fkrestricts to a rel-Morse function on Uk∩X; and, (ii) if depth X > k, then: (a) the restriction of fkto Uk∩Xhas no critical points, and (b) there is some (TX, πX, ρX)∈τXsuch that fkis constant on the fibers of πX:Uk∩M∩TX→X. This assertion is proved by induction on k. To simplify its proof, observe that it is also satisfied for k=−1 with F−1=U−1=∅and f−1=∅. Now, assume that Claim 6 holds for some k∈ {−1,0,...,depth M−1}. Let Vk be another open neighborhood of Fkin Asuch that Vk⊂Uk. We can assume that the strata of Aare connected by Remark 8-(v). For the strata X⊂Fk+1\Fk, choose representatives (TX, πX, ρX)∈τXsatisfying the properties stated in the proof of Claim 5. We can also suppose that these (TX, πX, ρX) satisfy Claim 6-(ii)-(b) with fk. A fixed adapted metric gon Mwill be used. Let Xbe a stratum contained in Fk+1 \Fk. By the density of the Morse functions in C∞(X) with the strong C∞topology and since the restriction of fk to Uk∩Xhas no critical points by Claim 6-(iii), it is easy to construct a Morse function hXon Xsuch that hX=fkon Vk∩X. Since (TX, πX, ρX) satisfies Claim 6-(ii)-(b) with fk, we get π∗ XhX=fkon Uk∩M∩TX. Let Uk+1 be the open neighborhood of Fk+1 given as the union of Vkand the sets TXfor strata X⊂Fk+1 \Fk. The function fkon Vk∩Mand the functions π∗ XhX+ρ2 Xon the sets TX∩Mcan be combined to define a function fk+1 ∈ C(Uk+1∩M). The function fk+1 satisfies Claim 6-(i) and Claim 6-(ii)-(a). Moreover it satisfies Claim 6-(ii)-(b) by Definition 7.1-(vi). Finally, let us complete the proof of Proposition 8.10. A basic neighborhood Nof any h∈C∞(M) with respect to the weak C∞topology can be determined by a finite family of charts (Ui, φi) of M, compact subsets Ki⊂Ui, some k∈N and some > 0. Precisely, Nconsists of the functions h0∈C∞(M) such that |D`((h0−h)φ−1 i)|< on φi(Ki) for all iand 0 ≤`≤k. By Claim 6, there is some open neighborhood Uof M\Min Aand some f∈C(U∩M) that restricts to rel-Morse functions on U∩Xfor all strata X≤M, and whose restriction to U∩M has no critical points. By shrinking Uif necessary, we can assume that U∩Ki=∅ for all i. Let Vbe another open neighborhood of M\Min Aso that V⊂U. By the density of the Morse functions in C∞(M) with the strong C∞topology, it is easy to check that there is a Morse function h0∈ N such that h0=fon V∩M. Therefore h0∈ F ∩N. For rel-Morse functions, a much better density result should be true as suggested by the following. Problem 8.11.By using the ideas of this section, define and study a “relstrong C∞topology” on the set of rel-admissible functions on M, and show that the rel-Morse functions form a dense subset.
8. RELATIVELY MORSE FUNCTIONS 75 An approach to Problems 8.9 and 8.11 would take us too far from the main goals of the work.
CHAPTER 9 Preliminaries on Hilbert complexes Here, we recall from [11] some basic definitions and needed results about Hilbert and elliptic complexes. Some elementary observations are also made. 1. Hilbert complexes For each r∈N, let Hrbe a separable (real or complex) Hilbert space such that, for some N∈N, we have Hr= 0 for all r > N. They give rise to the graded Hilbert space H=LrHr, where the terms Hrare mutually orthogonal. For each degree r, let drbe a densely defined closed operator of Hrto Hr+1. Let Dr=D(dr) and Rr=dr(Dr) for each r, and let D=LrDrand d=Lrdr. Assume that Rr⊂ Dr+1 and dr+1dr= 0 for all r. Then the complex 0−−−−→ D0 d0 −−−−→ D1 d1 −−−−→ ··· dN−1 −−−−→ DN−−−−→ 0 is called a Hilbert complex; its notation is abbreviated as (D,d), or simply as d. Assuming that D06= 0, the maximum N∈Nsuch that DN6= 0 will be called the length of (D,d). We may also consider Hilbert complexes with spaces of negative degree or with homogeneous operators of degree −1 without any essential change. For the adjoint operator d∗ rof each dr, let D∗ r=D(d∗ r)⊂Hr+1 and R∗ r= d∗ r(D∗ r)⊂Hr, and set D∗=LrD∗ rand d∗=Lrd∗ r. Then we get a Hilbert complex 0←−−−− D∗ −1 d∗ 0 ←−−−− D∗ 0 d∗ 1 ←−−−− ··· d∗ N−1 ←−−−− D∗ N−1←−−−− 0, denoted by (D∗,d∗) (or simply d∗), which is called dual or adjoint of (D,d). If (D0,d0) is another Hilbert complex in the graded Hilbert space H0=LrH0 r, a homomorphism of complexes, ζ=Lrζr: (D,d)→(D0,d0), is called a map of Hilbert complexes if it is the restriction of a bounded map ζ:H→H0. If moreover ζis an isomorphism of complexes and ζ−1is a Hilbert complex map, then ζis called an isomorphism of Hilbert complexes. If ζ: (D,d)→(e D0,d0) is an isomorphism, where e D0 r=D0 r+r0for all rand some fixed r06= 0, then it will be said that ζ: (D,d)→(D0,d0) is an isomorphismm up to a shift of degree. Let Hev =M r H2r,Hodd =M r H2r+1 , Dev =M rD2r,D∗ odd =M rD∗ 2r−1, dev =M r d2r,d∗ odd =M r d∗ 2r−1. 77
78 9. PRELIMINARIES ON HILBERT COMPLEXES Note that D∗ odd ⊂Hev. The operator Dev =dev+d∗ odd, with domain Dev∩D∗ odd, is a densely defined closed operator of Hev to Hodd, whose adjoint is Dodd =dodd +d∗ ev. Thus D=0Dev Dodd 0=d+d∗ is a self-adjoint operator in H=Hev ⊕Hodd with D(D) = D ∩D∗, and ∆=D2=DoddDev ⊕DevDodd =d∗d+dd∗ is a self-adjoint non-negative operator, which can be called the Laplacian of (D,d). Observe that (D,d) and (D∗,d∗) define the same Laplacian. The Hilbert complex (D,d) can be reconstructed from Dev [11, Lemma 2.3]. The restriction of ∆to each space Drwill be denoted by ∆r. Notice that ker ∆r= ker dr∩ker d∗ r−1for all r. Moreover we have a weak Hodge decomposition [11, Lemma 2.1] Hr= ker ∆r⊕Rr−1⊕R∗ r. The smooth core D∞(∆), also denoted by D∞(d) or D∞, is a subcomplex of (D,d), and (D∞,d),→(D,d) induces an isomorphism in homology [11, Theorem 2.12]. It will be also said that D∞(respectively, D∞ r) is the smooth core of d (respectively, dr); notice that it is a core of d(respectively, dr). Let R∞ r=dr(D∞ r) and R∗∞ r=d∗ r(D∞ r), which are dense subspaces of Rrand R∗ r. The following properties are equivalent [11, Theorem 2.4]: •The homology of (D,d) is of finite dimension and Ris closed in H. •The homology of (D,d) is of finite dimension. •Dev is a Fredholm operator. •06∈ specess(∆) (the essential spectrum of ∆). In this case, (D,d) is called a Fredholm complex and satisfies the following properties: • Rand R∗are closed in H[11, Corollary 2.5], obtaining the stronger Hodge decompositions Hr= ker ∆r⊕Rr−1⊕R∗ r,D∞= ker ∆r⊕R∞ r−1⊕R∗∞ r. •dr:R∗∞ r→ R∞ rand d∗ r:R∞ r→ R∗∞ rare isomorphisms. •ker ∆ris isomorphic to the homology of degree rof (D,d). It is said that (D,d) is discrete when ∆has a discrete spectrum (specess(∆) = ∅). The following properties hold when (D,d) is discrete: •For each λ∈spec(∆|R∞ r), we get isomorphisms dr:Eλ(∆|R∗∞ r)→Eλ(∆|R∞ r),d∗ r:Eλ(∆|R∞ r)→Eλ(∆|R∗∞ r) between the corresponding eigenspaces. Thus spec(∆|R∞ r) = spec(∆|R∗∞ r). •We have spec(dr|R∗∞ r⊕d∗ r|R∞ r) = {±√λ|λ∈spec(∆|R∞ r)}, and, for each λ∈spec(∆|R∞ r), E±√λ(dr|R∞ r⊕d∗ r|R∗∞ r) consists of the elements of the form u±vwith u∈Eλ(∆|R∞ r) and v∈Eλ(∆|R∗∞ r) satisfying d∗u=√λ v and dv=√λ u. Moreover the mapping u+v7→ u−v, for uand vas above, defines an isomorphism E√λ(dr|R∗∞ r⊕d∗ r|R∞ r)→E−√λ(dr|R∗∞ r⊕d∗ r|R∞ r).
1. HILBERT COMPLEXES 79 •Any Hilbert complex (D0,d0) isomorphic to (D,d) is also discrete, and, if spec(∆r) and spec(∆0 r) consist of the eigenvalues 0 ≤λ0≤λ1≤ ··· and 0 ≤λ0 0≤λ0 1≤ ···, respectively, then there is some C≥1 such that C−1λk≤λ0 k≤Cλkfor all k∈N[11, Lemma 2.17]. Consider Hilbert complexes, (D0,d0) and (D00,d00), in respective graded Hilbert spaces, H0and H00. The Hilbert space tensor product1,H=H0b ⊗H00, has a canonical grading (Hr=Lp+q=rH0 pb ⊗H00 q), and e D= (D0⊗H00)∩(H0⊗D00)⊂H is a dense graded subspace. Let ˜ d=d0⊗1 + w⊗d00 with domain e D, where w denotes the degree involution on H0, and let d=˜ d, whose domain is denoted by D. Then (D,d) is a Hilbert complex in Hcalled the tensor product of (D0,d0) and (D00,d00). If ∆0,∆00 and ∆denote the Laplacians of (D0,d0), (D00,d00) and (D,d), respectively, then ∆=∆0⊗1+1⊗∆00 on e D. The following result is elementary. Lemma 9.1.If (D0,d0)and (D00,d00)are discrete, then (D,d)is discrete. More precisely, given complete orthonormal systems of H0and H00 consisting of eigenvectors e0 kand e00 k(k∈N)of ∆0and ∆00, with corresponding eigenvalues λ0 kand λ00 k, respectively, we get a complete orthonormal system of Hconsisting of the eigenvectors e0 k⊗e00 `∈e Dof ∆with corresponding eigenvalues λ0 k+λ00 `. Let (E, d) be a densely defined complex in a graded separable Hilbert space H(Eis a dense graded linear subspace of H). Consider the family of Hilbert complexes (D,d) in Hextending (E, d) ((E, d) is a subcomplex of (D,d)) endowed with the order relation defined by “being a subcomplex”. We will be interested in its minimum/maximum elements. Notice that, if (E, d) has some Hilbert complex extension, then dis a Hilbert complex; thus, in this case, dis the minimum Hilbert complex extension of (E, d). Another complex of the form (E, δ), with δr:Er+1 → Er for each degree r, will be called a formal adjoint of (E, d) if hdu, vi=hu, δvifor all u, v ∈ E; there is at most one formal adjoint by the density of Ein H. In this case, if (E, δ) has some Hilbert complex extension, then the adjoint of the minimum Hilbert complex extension of (E, δ) is the maximum Hilbert complex extension of (E, d). Now, consider a countable family of densely defined complexes (Ea, da) in separable graded Hilbert spaces Ha(a∈N), and let (Da,da) be a Hilbert complex extension of each (Ea, da) in Ha. Suppose that the Hilbert complexes (Da,da) are of uniformly finite length (there is some N∈Nsuch that Da r= 0 for all r≥N and all a). Let (E, d) be the complex defined by E=LaEaand d=Lada. The Hilbert space direct sum2,H=c LaHa, has an induced grading (Hr=c LaHa r). Let d=c Lada(the graph of dis the Hilbert space direct sum of the graphs of the maps da). The domain Dof dconsists of the points (ua)∈Hsuch that ua∈ Da for all aand (daua)∈H. Moreover dis defined by (ua)7→ (daua). Clearly, (D,d) 1Recall that this is the Hilbert space completion of the algebraic tensor product H0⊗H00 with respect to the scalar product defined by hu0⊗u00, v0⊗v00i=hu0, v0i0hu00, v00i00, where h,i0and h,i00 are the scalar products of H0and H00, respectively. 2Recall that this is the Hilbert space completion of the algebraic direct sum, LaHa, with respect to the scalar product h(ua),(va)i=Pahua, vaia, where each h,iais the scalar product of Ha. We have H=LaHaif the number of terms Hais finite.
80 9. PRELIMINARIES ON HILBERT COMPLEXES is a Hilbert complex extension of (E, d) in Hwith D∞(d) = d M aD∞(da),(79) d∗=d M a da∗.(80) Lemma 9.2.(i) If each (Da,da)is a minimum Hilbert complex extension of (Ea,da)in Ha, then (D,d)is a minimum Hilbert complex extension of (E,d)in H. (ii) If each (Ea, da)has a formal adjoint (Ea, δa)with some Hilbert complex extension, and each (Da,da)is a maximum Hilbert complex extension of (Ea,da)in Ha, then (D,d)is a maximum Hilbert complex extension of (E,d)in H. Proof. Property (i) follows because dis dense in dif each dais dense in da. Now, assume the conditions of (ii) and let δ=Laδa. Then each da∗is a minimum Hilbert complex extension of (Ea, δa). So, by (80) and (i), (D∗,d∗) is a minimum Hilbert complex extension of (E, δ), and therefore (D,d) is a maximum Hilbert complex extension of (E, d). 2. Elliptic complexes Let Mbe a possibly non-complete Riemannian manifold, and let E=LrErbe a graded Riemannian (or Hermitean) vector bundle over M, with Er= 0 if r < 0 or r > N for some N∈N. The space of smooth sections of each Erwill be denoted by C∞(Er), its subspace of compactly supported smooth sections will be denoted by C∞ 0(Er), and the Hilbert space of square integrable sections of Erwill be denoted by L2(Er); then C∞(E) = LrC∞(Er), C∞ 0(E) = LrC∞ 0(Er) and L2(E) = LrL2(Er). For each r, let dr:C∞(Er)→C∞(Er+1) be a first order differential operator, and set d=Lrdr. Suppose that (C∞(E), d) is an elliptic complex3; however, ellipticity is not needed for several elementary properties stated in this section. The simpler notation (E, d) (or even d) will be preferred. Elliptic complexes with non-zero terms of negative degrees or homogeneous differential operators of degree −1 may be also considered without any essential change. Consider the formal adjoint δr=tdr:C∞(Er+1)→C∞(Er) for each r, and set δ=Lrδr. Then (E, δ) is another elliptic complex that will be called the formal adjoint of (E, d), and its subcomplex (C∞ 0(E), δ) is formal adjoint of (C∞ 0(E), d) in L2(E) in the sense of Section 1. Let D=d+δand ∆ = D2=dδ +δd on C∞(E); ∆ can be called the Laplacian defined by (E, d). The components of ∆ are ∆r=dr−1δr−1+δrdr. Any Hilbert complex extension of (C∞ 0(E), d) in L2(E) is called an ideal boundary condition (shortly, i.b.c.) of (E, d). There always exist a minimum and maximum i.b.c., dmin =dand dmax =δ∗ min [11, Lemma 3.1]. The complex dmin/max defines the operator Dmin/max =dmin/max +δmax/min and the Laplacian ∆min/max = D2 min/max, which extend Dand ∆ on C∞ 0(E). The homogeneous components of ∆min/max are ∆min/max,r =δmax/min,r dmin/max,r +dmin/max,r−1δmax/min,r−1.(81) 3Recall that this means that it is a complex and the sequence of principal symbols of the operators dris exact in the fiber over each non-zero cotangent vector
2. ELLIPTIC COMPLEXES 81 The notation dr,min/max and δr,max/min also makes sense for dmin/max,r and δmax/min,r by considering drand δras differential complexes of length one (ellipticity is not needed here); similarly, any first order differential operator can be considered as a differential complex of length one and denote its minimum/maximum i.b.c. with the the min/max subindex, regardless of ellipticity. For any i.b.c. (D,d) of (E, d), the map of complexes, (D∩C∞(E), d),→(D,d), induces an isomorphism in homology [11, Theorem 3.5]. We have D∞⊂ D∩C∞(E) by elliptic regularity. Let (E0, d0) be another elliptic complex over another Riemannian manifold M0. Consider a vector bundle isomorphism ζ:E→E0over a quasi-isometric diffeomorphism ξ:M→M0such that the restrictions of ζto the fibers are quasi-isometries. It induces a map ζ:C∞(E)→C∞(E0) defined by (ζu)(x0) = ζ(u(ξ−1(x0)) for u∈C∞(E) and x0∈M0. If moreover ζ: (C∞(E0), d0)→(C∞(E), d) is a homomorphism of complexes, then it will be called a quasi-isometric isomorphism of elliptic complexes, and the simpler notation ζ: (E0, d0)→(E, d) will be preferred. In this case, ζinduces a quasi-isometric isomorphism ζ:L2(E0)→L2(E), which restricts to an isomorphism of complexes, ζ: (C∞ 0(E0), d0)→(C∞ 0(E), d). Moreover, for any i.b.c. (D0,d0) of (E0, d0), there is a unique i.b.c. (D,d) of (E, d) so that ζ: L2(E0)→L2(E) restricts to a Hilbert complex isomorphism ζ: (D0,d0)→(D,d). In particular, ζinduces Hilbert complex isomorphisms between the corresponding minimum/maximum i.b.c. If ξis isometric and the restrictions to the fibers of ζare isometries, then ζ: (E0, d0)→(E, d) is called an isometric isomorphism of elliptic complexes. For instance, for any quasi-isometric (respectively, isometric) diffeomorphism ξ:M→M0, the induced isomorphism ξ∗between the corresponding de Rham complexes is quasi-isometric (respectively, isometric). Now, let (E0, d0) and (E00, d00) be elliptic complexes on Riemannian manifolds M0and M00, respectively, and consider the exterior tensor product E=E0E00 on M=M0×M00 with its canonical grading (Er=Lp+q=rE0 pE00 q). With the weak C∞topology, C∞(E0)⊗C∞(E00) can be canonically realized as a dense subspace of C∞(E). Then d=d0⊗1 + w⊗d00 on C∞(E0)⊗C∞(E00) has a unique continuous extension to C∞(E), also denoted by d. It turns out that (E, d) is an elliptic complex. Moreover the minimum/maximum i.b.c. of (E, d) is the tensor product, in the sense of Section 1, of the minimum/maximum i.b.c. of (E0, d0) and (E00, d00) [11, Lemma 3.6]. Example 9.3.A particular case of elliptic complex on Mis its de Rham complex (Ω(M), d). In this case, δis the de Rham coderivative, the subcomplex of compactly supported differential forms is denoted by Ω0(M), and the Hilbert space of L2 differential forms is denoted by L2Ω(M). Let Hmin/max(M) denote the cohomology of the minimum/maximum i.b.c., dmin/max, of (Ω0(M), d), which is a quasi-isometric invariant of M.Hmin(M) is canonically isomorphic to the L2-cohomology H(2)(M) [13]; (a generalization to arbitrary elliptic complexes is given in [11, Theorem 3.5]). The dimensions βr min/max(M) = dim Hr min/max(M) can be called min/max-Betti numbers; if they are finite, then χmin/max(M) = Pr(−1)rβr min/max(M) is defined and can be called min/max-Euler characteristic; the simpler notation βr min/max and χmin/max may be used. Is is known that dmin/max satisfies the following properties for special classes of Riemannian manifolds: •If Mis complete, then dmin =dmax (a particular case of [11, Lemma 3.8]).
82 9. PRELIMINARIES ON HILBERT COMPLEXES •If Mis the interior of a compact manifold with boundary, then dmin/max is given by the relative/absolute boundary conditions [11, Theorem 4.1]. •Suppose that M=f M\Σ, where f Mis a closed Riemannian manifold of dimension >2 and Σ is a closed finite union of submanifolds with codimension ≥2. Then dmin =dmax [11, Theorem 4.4]. •Let Abe a compact Tham-Mather stratification that is a pseudomanifold. If Mis the regular stratum of Aendowed with an adapted metric, then H(2)(M) is isomorphic to the intersection homology of Awith lower middle perversity [15]. There is a more general isomorphism of this type involving more general types of adapted metrics and intersection homologies with other perversities [47, 48, 8].
CHAPTER 10 Sobolev spaces defined by an i.b.c. Let Tbe a self-adjoint operator in a Hilbert space H. For each m∈N, the Sobolev space of order kassociated to Tis the Hilbert space completion Wm= Wm(T) of D∞=D∞(T) with respect to the scalar product h,imon D∞defined by hu, vim=hu, (1 + T)mvi. The notation k kmand Clm(or k kWmand ClWm) will be used for the norm and closure in Wm. There are continuous inclusions Wm+1 ,→Wm, and we have D∞=TmWm. Moreover Tdefines a bounded operator Wm+2 →Wm. Now, let (D,d) be an i.b.c. of an elliptic complex (E, d) on a Riemannian manifold M. Its adjoint (D∗,d∗) is an i.b.c. of the elliptic complex (E, δ), where δ=td. We get the operators D=d+δand D=d+d∗, and the Laplacians ∆ = D2 and ∆=D2. Then Wm=Wm(∆) can be called the Sobolev space of order m associated to (D,d), and may be also denoted by Wm(d); the notation Wm(dr) will be also used when we consider its subspace of homogeneous elements of degree r. Since (D,d) and (D∗,d∗) define the same Laplacian, we have Wm(d) = Wm(d∗) for all m. For u∈ D∞ r, we have kuk2 1=kuk2+kDuk2=kuk2+kdruk2+kδr−1uk2. So W1=D(D) = D∩D∗,(82) kuk2 1=kuk2+kDuk2=kuk2+kdruk2+kd∗ r−1uk2(83) for u∈W1(dr). Lemma 10.1.The following properties are equivalent: (i) (D,d)is discrete. (ii) W1,→W0=L2(E)is compact. (iii) Wm+1 ,→Wmis compact for all m. Proof. The part “(i) ⇒(iii)” follows with the arguments of the proof of the Rellich’s theorem on a torus (see e.g. [54, Theorem 5.8]). The part “(ii) ⇒(i)” follows with the arguments to prove that any Dirac operator on a closed manifold has a discrete spectrum (see e.g. [54, pp. 81–82]). The following refinement of Lemma 10.1 is obtained with a deeper analysis. Lemma 10.2.Suppose that (D,d)is discrete, and let 0≤λ1≤λ2≤ ··· be the eigenvalues of ∆, repeated according to their multiplicities. Let B1be the standard unit ball of W1, and Brthe standard ball of radius r > 0in L2(E). Then the following properties are equivalent for θ > 0: (i) lim infkλkk−θ>0. 83
90 10. SOBOLEV SPACES DEFINED BY AN I.B.C. is dense in L2(E) by (92) for m= 1. It follows that, given any u∈Ya n, to check that Dmin/maxu∈W1, its enough to check that the mapping v7→ hDmin/maxu, Dmin/maxvi is bounded on W1∩Oa⊥,Ya nand W1∩Za00 n. This mapping vanishes on W1∩Oa⊥ because Dmin/max(W1∩Oa)⊂ Oa, Dmin/max(W1∩Oa⊥)⊂ Oa⊥. Moreover it is bounded on Ya nbecause this space is of finite dimension. Finally, for v∈W1∩Za00 n, we have hDmin/maxu, Dmin/maxvi=−hu, vi because u⊥1v. Thus the above mapping is bounded on W1∩Za00 n, which completes the proof of Claim 13. Claim 14.Dmin/max(Ya n)⊂Ya n. For u∈Ya nand v∈W2∩Za00 n, we have hDmin/maxu, vi1=hDmin/maxu, vi+h∆min/maxu, Dmin/maxvi =hu, Dmin/maxvi+hDmin/maxu, ∆min/maxvi=hu, Dmin/maxvi1= 0 by Claim 13 and because Dmin/max is self-adjoint. Then Claim 14 follows by Claim 12. Claim 15.Ya n=Oa∩(Za00 n)⊥. Let u∈Ya nand v∈W1∩Za00 n. By Claim 14, ∆min/max is a self-adjoint operator on Ya n. Then u= (1 + ∆min/max)u0for u0= (1 + ∆min/max)−1u∈Ya n, obtaining hu, vi=h(1 + ∆min/max)u0, vi=hu0, vi1= 0 . This shows Claim 15 by Claim 12 and (91). Let Πa n:Oa→Za00 ndenote the orthogonal projection. The following claim follows from (92) for m= 1, and Claims 11-(b), 14 and 15. Claim 16.Πa n(W1∩Oa)⊂W1∩Oa, and [Dmin/max,Πa n] = 0 on W1∩Oa. Consider each function faas the corresponding bounded multiplication operator on L2(E). Assuming that aruns in {1, . . . , A}for some A∈Z+, we get the bounded operator T= (f1, . . . , fA) : L2(E)→LAL2(E). Also, let Σ : LAL2(E)→L2(E) be the bounded operator defined by Σ(u1, . . . , uA) = Paua. We have ΣT= 1 because {fa}is a partition of unity. Claim 17.The image of Tis closed. Let (ui) be a sequence in L2(E) such that (T ui) converges to some vin LAL2(E). Then ui= ΣTui→Σvas i→ ∞, obtaining Tui→TΣvas i→ ∞. Hence v=TΣv∈T(L2(E)), showing Claim 17. By Claim 17 and the open mapping theorem (see e.g. [18, Chapter III, 12.1] or [58, Chapter III, 2.1]), we get that Tis a topological homomorphism3. So T: L2(E)→T(L2(E)) is a quasi-isometric isomorphism; its inverse is Σ : T(L2(E)) → 3Recall that a bounded operator between topological vector spaces, T:H→G, is called a topological homomorphism if the map T:H→T(H) is open, where T(H) is endowed with the restriction of the topology of G.
10. SOBOLEV SPACES DEFINED BY AN I.B.C. 91 L2(E). Since Πn:= LaΠa nis an orthogonal projection of LAL2(E), it follows that Rn:= ΠnTsatisfies Lemma 10.2-(iii)-(b),(c). Moreover, by Claim 11-(a), dim ker Rn≤dim ker Πn=X a dim ker Πa n≤X a C0,a n1/θa≤C0n1/θ with C0=PaC0,a and θ= minaθa, which shows that Rnsatisfies Lemma 10.2- (iii)-(a). We have Rn= (R1 n, . . . , RA n) with Ra n= Πa nfa. Since each function |[d, fa]|is uniformly bounded, it follows that faW1⊂W1and [Dmin/max, fa] : W1→L2(E) extends to a bounded operator on L2(E). Therefore each Ra nsatisfies Lemma 10.2- (iii)-(d) by Claim 16. Finally, Ra nsatisfies Lemma 10.2-(iii)-(e) by Claim 11-(c). Now, the result follows from Lemma 10.2.
CHAPTER 11 Two simple types of elliptic complexes Here, we study the two types of simple elliptic complexes. They will show up in the direct sum splitting of the local model of Witten’s perturbation (Chapter 15). We could describe better the spectra of the Laplacians associated to the minimum/maximum i.b.c. of these simple elliptic complexes, but this will be done with the local model of the Witten’s perturbation (Chapter 14). 1. Some more results on general elliptic complexes Consider the notation of the beginning of Section 2 in Chapter 9. Lemma 11.1.Let G ⊂ C∞(E)∩L2(E)be a graded linear subspace containing C∞ 0(E), preserved by dand δ, and such that hdu, vi=hu, δvifor all u, v ∈ G. Let dG,δGand ∆Gdenote the restrictions of d,δand ∆to G. Assume that ∆G is essentially self-adjoint in L2(E), and Gis the smooth core of ∆G. Then the following properties hold: (i) If Gr⊂ D(dmin,r)and Gr−1⊂ D(dmin,r−1)for some degree r, then Gris the smooth core of dmin,r. (ii) If Gr⊂ D(δmin,r−1)and Gr+1 ⊂ D(δmin,r)for some degree r, then Gris the smooth core of dmax,r. Proof. For each degree r, the restrictions dr:Gr→ Gr+1,δr:Gr+1 → Gr and ∆r:Gr→ Grwill be denoted by dG,r,δG, r and ∆G,r, respectively. Suppose that Gr⊂ D(dmin,r) and Gr−1⊂ D(dmin,r−1), and therefore dG,r ⊂dmin,r and dG,r−1⊂dmin,r−1. Since C∞ 0(E)⊂ G and hdu, vi=hu, δvifor all u, v ∈ G, it follows that Gr+1 ⊂ D(δmax,r) and Gr⊂ D(δmax,r−1), and therefore δG,r ⊂δmax,r and δG,r−1⊂δmax,r−1. By (81), we get ∆G,r ⊂∆min,r. So ∆G,r ⊂∆min,r, and therefore ∆G,r = ∆min,r because these operators are self-adjoint in L2(Er). Then Gris the smooth core of dmin,r, completing the proof of (i). Now, assume that Gr⊂ D(δmin,r−1) and Gr+1 ⊂ D(δmin,r), and therefore δG,r−1⊂δmin,r−1and δG,r ⊂δmin,r. As above, it follows that dG,r−1⊂dmax,r−1 and dG,r ⊂dmax,r. By (81), we get ∆G,r ⊂∆max,r. So ∆G,r ⊂∆max,r, obtaining ∆G,r = ∆max,r as before. Thus Gris the smooth core of dmax,r, completing the proof of (ii). Now, suppose that there is an orthogonal decomposition Er+1 =Er+1,1⊕Er+1,2 for some degree r+ 1. Thus C∞(Er+1)≡C∞(Er+1,1)⊕C∞(Er+1,2), C∞ 0(Er+1)≡C∞ 0(Er+1,1)⊕C∞ 0(Er+1,2), L2(Er+1)≡L2(Er+1,1)⊕L2(Er+1,2), 93
94 11. TWO SIMPLE TYPES OF ELLIPTIC COMPLEXES giving dr=dr,1 dr,2, δr=δr,1δr,2. Lemma 11.2.We have: D(dmax,r) = D(dr,1,max)∩D(dr,2,max), dmax,r =dr,1,max|D(dmax,r) dr,2,max|D(dmax,r). Proof. Let u∈L2(Er). We have u∈ D(dmax,r) if and only if there is some w∈L2(Er+1) such that hu, δvi=hw, vifor all v∈C∞ 0(Er+1), and moreover dmax,ru=win this case. Writing w=w1⊕w2and v=v1⊕v2, this condition on umeans that hu, δ0,ivii=hwi, viifor all vi∈C∞ 0(Ei r+1) and i∈ {1,2}. In turn, this is equivalent to u∈ D(dr,1,max)∩D(dr,2,max) with dr,i,maxu=wi. For i∈ {1,2}, let ∆r,i =δr,idr,i +dr−1δr−1on C∞(Er). Corollary 11.3.If a∆r=b∆r,i +cfor some a, b, c ∈Rwith a, b 6= 0,dmin,r and dr,i,min have the same smooth core, and dr,i,min =dr,i,max for some i∈ {0,1}, then dmin,r =dmax,r. Proof. By Lemma 11.2 and since dr,i,min =dr,i,max, we get D(dmax,r)⊂ D(dr,i,min). Because a∆r=b∆r,i +cfor some a, b, c ∈Rwith a, b 6= 0, it follows that {u∈ D(dmax,r)∩C∞(Er)|∆k ru∈L2(Er)∀k∈N} ⊂ {u∈ D(dr,i,min)∩C∞(Er)|∆k r,iu∈L2(Er)∀k∈N}. This means that the smooth core of dmax,r is contained in the smooth core of dr,i,min, which equals the smooth core of dmin,r. Then dmax,r =dmin,r. 2. An elliptic complex of length two Consider the standard metric on R+. Let Ebe the graded Riemannian/Hermitian vector bundle over R+whose non-zero terms are E0and E1, which are real/complex trivial line bundles endowed with the standard Riemannian/Hemitian metrics. Thus C∞(E0)≡C∞(R+)≡C∞(E1), L2(E0)≡L2(R+, dρ)≡L2(E1), where real/complex valued functions are considered in C∞(R+) and L2(R+, dρ). For any fixed s > 0 and κ∈R, let C∞(E0)C∞(E1) d δ - be the differential operators defined by d=d dρ −κρ−1±sρ , δ =−d dρ −κρ−1±sρ .
2. AN ELLIPTIC COMPLEX OF LENGTH TWO 95 It is easy to check that (E, d) is an elliptic complex, whose formal adjoint is (E, δ). By (71), the homogeneous components of the corresponding Laplacian ∆ are: ∆0=δd ≡H+κd dρ, ρ−1∓sd dρ, ρ+κ2ρ−2∓2sκ =H+κ(κ−1)ρ−2∓s(1 + 2κ),(94) ∆1=dδ =H−κd dρ, ρ−1±sd dρ, ρ+κ2ρ−2∓2sκ =H+κ(κ+ 1)ρ−2±s(1 −2κ),(95) where His the harmonic oscillator on C∞(R+) defined with the constant s. Then ∆0and ∆1are of the form of Pin (1) (with c1= 0) plus a constant; in particular, for κ= 0, they are equal to Hplus a constant. For ∆0, the condition (4) means that a∈ {κ, 1−κ}, and (5) gives σ=κif a=κ, and σ= 1 −κif a= 1 −κ. By Corollary H, the following holds: •If κ > −1/2, then ∆0, with domain ρκSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρκSev,+. •If κ < 3/2, then ∆0, with domain ρ1−κSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρ1−κSev,+. For ∆1, the condition (4) means that a∈ {1+κ, −κ}, and (5) becomes σ= 1+κ if a= 1 + κ, and σ=−κif a=−κ. Now Corollary H states the following: •If κ > −3/2, then ∆1, with domain ρ1+κSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρ1+κSev,+. •If κ < 1/2, then ∆1, with domain ρ−κSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρ−κSev,+. When κ > −1/2, let E1⊂C∞(E)∩L2(E) be the dense graded linear subspace with E0 1≡ρκSev,+,E1 1≡ρ1+κSev,+. When κ < 1/2, let E2⊂C∞(E)∩L2(E) be the dense graded linear subspace with E0 2≡ρ1−κSev,+,E1 2≡ρ−κSev,+. Observe that, by restricting dand δ, we get complexes (E1, d) and (E1, δ) when κ > −1/2, and complexes (E2, d) and (E2, δ) when κ < 1/2. Thus ∆ preserves E1 when κ > −1/2, and preserves E2when κ < 1/2. Proposition 11.4.(i) If |κ|<1/2, then E1and E2are the smooth cores of dmax and dmin, respectively. (ii) If |κ| ≥ 1/2, then (E, d)has a unique i.b.c., whose smooth core is E1when κ≥1/2, and E2when κ≤ −1/2. The following lemma will be used in the proof of Proposition 11.4. Lemma 11.5.Suppose that θ≥1/2. Then, for each ξ∈ρθSev,+, considered as subspace of C∞(E0) (respectively, C∞(E1)), there is a sequence (ξn)in C∞ 0(E0) (respectively, C∞ 0(E1)), independent of κ, such that limnξn=ξand limndξn=
96 11. TWO SIMPLE TYPES OF ELLIPTIC COMPLEXES dξ in L2(E0) (respectively, limnδξn=δξ in L2(E1)). In particular, ρθSev,+is contained in D(dmin) (respectively, D(δmin)). Remark 20.In Lemma 11.5, the independence of κmeans that (ξn) depends only on θand ξ, whilst the convergences limndξn=dξ and limnδξn=δξ hold with dand δdefined by any κ. Proof of Lemma 11.5. The proof is made for D(dmin); the case of D(δmin) is analogous. Let 0 < a < b and f∈C∞ 0(R+) such that 0 ≤f≤1, f(ρ) = 1 for ρ≤a, and f(ρ) = 0 for ρ≥b. For each n∈N, let gn, hn∈C∞(R+) be defined by gn(ρ) = f(nρ) and hn(ρ) = f(ρ/n). It is clear that χ[b n,na]≤(1 −gn)hn≤χ[a n,nb],(96) where χSdenotes the characteristic function of each subset S⊂R+. Let φ∈ Sev,+. From (96), we get (1−gn)hnρθφ∈C∞ 0(E0) and (1−gn)hnρθφ→ ρθφin L2(E0) as n→ ∞. Observe that d((1 −gn)hnρθφ) = −g0 nhnρθφ+ (1 −gn)h0 nρθφ+ (1 −gn)hnd(ρθφ). In right hand side of this equality, the last term converges to d(ρθφ) in L2(E1) as n→ ∞ by (96). Moreover k(1 −gn)h0 nρθφk2=Z∞ 0 (1 −gn)2h0 n 2(ρ)ρ2θφ2(ρ)dρ ≤(max ρ2θφ2)n−2Z∞ 0 f02(ρ/n)dρ = (max ρ2θφ2)n−1Z∞ 0 f02(x)dx = (max ρ2θφ2)n−1kf0k2, which converges to zero as n→ ∞, and kg0 nhnρθφk2=Z∞ 0 g0 n 2(ρ)h2 n(ρ)ρ2θφ2(ρ)dρ ≤(max φ2)n2Z∞ 0 f02(nρ)ρ2θdρ = (max φ2)n1−2θZ∞ 0 f02(x)x2θdx = (max φ2)n1−2θkf0ρθk2, which converges to zero as n→ ∞ if θ > 1/2. In the case θ= 1/2, it is enough to prove that fcan be chosen so that kf0ρ1/2k is as small as desired. For m > 1 and 0 < < 1, observe that there is some fas above such that: •the support of f0is contained in [e−, em], • − 1 mρ ≤f0≤0, and •f0(ρ) = −1 mρ if 1 ≤ρ≤em−. Then kf0ρ1/2k2=Zem e− f02(ρ)ρ dρ ≤1 m2Zem e− dρ ρ=m+ m2, which converges to zero as m→ ∞. Proof of Proposition 11.4. Suppose that |κ|<1/2. Since 1 ±κ > 1/2, by Lemma 11.5, E0 2⊂ D(dmin) and E1 1⊂ D(δmin). The other conditions of Lemma 11.1 are satisfied by dwith G=E2, and by δwith G=E1by the discussion previous
3. AN ELLIPTIC COMPLEX OF LENGTH THREE 97 to Proposition 11.4. So E2is the smooth core of dmin and E1is the smooth core of dmax by Lemma 11.1. Now, assume that κ≥1/2 (respectively, κ≤ −1/2), giving also 1 + κ > 1/2 (respectively, 1−κ > 1/2). Then, by Lemma 11.5, E0 1⊂ D(dmin) and E1 1⊂ D(δmin) (respectively, E0 2⊂ D(dmin) and E1 2⊂ D(δmin)). By the discussion previous to Proposition 11.4, the other conditions of Lemma 11.1 are satisfied by dand δwith G=E1(respectively, G=E2). So, by Lemma 11.1, E1(respectively, E2) is the smooth core of dmin and dmax. Remark 21.In the proof of Lemma 11.5 and Proposition 11.4, we have borrowed ideas from the proof of [11, Theorem 4.1]; in fact, in the case with κ= 0, Proposition 11.4 could be proved exactly like [11, Theorem 4.1]. 3. An elliptic complex of length three Consider again the standard metric on R+. Let Fbe the graded Riemannian/Hermitian vector bundle over R+whose non-zero terms are F0,F1and F2, which are trivial real/complex vector bundles of ranks 1, 2 and 1, respectively, endowed with the standard Riemannian/Hermitian metrics. Thus C∞(F0)≡C∞(R+)≡C∞(F2), C∞(F1)≡C∞(R+)⊕C∞(R+), L2(F0)≡L2(R+, dρ)≡L2(F2), L2(F1)≡L2(R+, dρ)⊕L2(R+, dρ), where real/complex valued functions are considered in C∞(R+) and L2(R+, dρ). Fix s, c > 0 and κ∈R, and let C∞(F0)C∞(F1)C∞(F2) d0 δ0 d1 δ1 - - be the differential operators defined by d0=d0,1 d0,2, δ0=δ0,1δ0,2, d1=d1,1d1,2, δ1=δ1,1 δ1,2,
98 11. TWO SIMPLE TYPES OF ELLIPTIC COMPLEXES where d0,1=c √1 + c2d dρ +κρ−1±sρ, d0,2=1 √1 + c2d dρ −(κ+ 1)ρ−1±sρ, δ0,1=c √1 + c2−d dρ +κρ−1±sρ, δ0,2=1 √1 + c2−d dρ −(κ+ 1)ρ−1±sρ, d1,1=1 √1 + c2d dρ −κρ−1±sρ, d1,2=c √1 + c2−d dρ −(κ+ 1)ρ−1∓sρ, δ1,1=1 √1 + c2−d dρ −κρ−1±sρ, δ1,2=c √1 + c2d dρ −(κ+ 1)ρ−1∓sρ. A direct computation shows that d0and d1define an elliptic complex (F, d) of length three. Its formal adjoint is the complex (F, δ) given by δ0and δ1. The homogeneous components ∆0and ∆2of the corresponding Laplacian ∆ can be computed as follows, where the notation of Section 1 is used. By (94) and (95), ∆0,1=δ0,1d0,1=c2 1 + c2−d dρ +κρ−1±sρd dρ +κρ−1±sρ =c2 1 + c2H+κ(κ+ 1)ρ−2∓s(1 −2κ), ∆0,2=δ0,2d0,2=1 1 + c2−d dρ −(κ+ 1)ρ−1±sρd dρ −(κ+ 1)ρ−1±sρ =1 1 + c2H+ (κ+ 1)κρ−2∓s(1 + 2(κ+ 1)), ∆2,1=d1,1δ1,1=1 1 + c2d dρ −κρ−1±sρ−d dρ −κρ−1±sρ =1 1 + c2H+κ(κ+ 1)ρ−2±s(1 −2κ), ∆2,2=d1,2δ1,2=c2 1 + c2−d dρ −(κ+ 1)ρ−1∓sρd dρ −(κ+ 1)ρ−1∓sρ =c2 1 + c2H+ (κ+ 1)κρ−2±s(1 + 2(κ+ 1)), ∆0=δ0d0= ∆0,1+ ∆0,2=H+κ(κ+ 1)ρ−2∓s2 + 1−c2 1 + c2(1 + 2κ), ∆2=d1δ1= ∆2,1+ ∆2,2=H+κ(κ+ 1)ρ−2±s2 + 1−c2 1 + c2(1 + 2κ).
3. AN ELLIPTIC COMPLEX OF LENGTH THREE 99 Thus ∆0can be identified to ∆2, and they are of the form of Pin (1) (with c1= 0) plus a constant. For ∆0and ∆2, the condition (4) means that a∈ {1 + κ, −κ}, and (5) gives σ= 1 + κif a= 1 + κ, and σ=−κif a=−κ. By Corollary H, the following holds: •If κ > −3/2, then ∆0and ∆2, with domain ρ1+κSev,+, are essentially self-adjoint in L2(R+, dρ), the spectra of their closures are discrete, and the smooth core of their closures is ρ1+κSev,+. •If κ < 3/2, then ∆0and ∆2, with domain ρ1−κSev,+, are essentially selfadjoint in L2(R+, dρ), the spectra of their closures are discrete, and the smooth core of their closures is ρ−κSev,+. Write ∆1=d0δ0+δ1d1 =d0,1δ0,1+δ1,1d1,1d0,1δ0,2+δ1,1d1,2 d0,2δ0,1+δ1,2d1,1d0,2δ0,2+δ1,2d1,2=∆1,1A B∆1,2. By (94) and (95), ∆1,1=1 1 + c2c2d dρ +κρ−1±sρ−d dρ +κρ−1±sρ +−d dρ −κρ−1±sρd dρ −κρ−1±sρ =1 1 + c2c2H+κ(κ−1)ρ−2∓s(1 + 2κ) +H+κ(κ−1)ρ−2∓s(1 + 2κ) =H+κ(κ−1)ρ−2∓s1−c2 1 + c2(1 + 2κ), ∆1,2=1 1 + c2d dρ −(κ+ 1)ρ−1±sρ−d dρ −(κ+ 1)ρ−1±sρ +c2d dρ −(κ+ 1)ρ−1∓sρ−d dρ −(κ+ 1)ρ−1∓sρ =1 1 + c2H+ (κ+ 1)(κ+ 2)ρ−2±s(1 −2(κ+ 1)) +c2H+ (κ+ 1)(κ+ 2)ρ−2∓s(1 −2(κ+ 1)) =H+ (κ+ 1)(κ+ 2)ρ−2∓s1−c2 1 + c2(1 + 2κ). So ∆1,1and ∆1,2also are of the form of Pin (1) (with c1= 0) plus a constant. For ∆1,1, the condition (4) means that a∈ {κ, 1−κ}, and (5) gives σ=κif a=κ, and σ= 1 −κif a= 1 −κ. By Corollary H, the following holds: •If κ > −1/2, then ∆1,1, with domain ρκSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρκSev,+. •If κ < 3/2, then ∆1,1, with domain ρ1−κSev,+, is essentially self-adjoint in L2(R+, dρ), the spectrum of its closure is discrete, and the smooth core of its closure is ρ1−κSev,+.
106 13. WITTEN’S PERTURBATION ON A CONE Lemma 13.2.According to (105), ?≡0ρn−2r+1˜ ? (−1)rρn−2r−1˜ ?0 on VrT(W×R+)∗. Proof. Let α, α0∈π∗VTN∗, at the same point (ρ, x)∈R+×W. If αand α0are of degree r, then α0∧ρn−2r−1dρ ∧˜ ?α = (−1)rρn−2r−1dρ ∧α0∧˜ ?α = (−1)rρn−2r−1˜g(α0, α)dρ ∧˜ω= (−1)rg(α0, α)ω by (108) and (109), giving ?α = (−1)rρn−2r−1dρ ∧˜ ?α. Similarly, if αand α0are of degree r−1, then dρ ∧α0∧ρn−2r+1˜ ?α =ρn−2r+1˜g(α0, α)dρ ∧˜ω=g(dρ ∧α0, dρ ∧α)ω , obtaining ?(dρ ∧α) = ρn−2r+1˜ ?α. Let L2Ωr(M, g) and L2Ωr(N, ˜g) be simply denoted by L2Ωr(M) and L2Ωr(N). From (108) and (109), it follows that (107) induces a unitary isomorphism L2Ωr(M)∼ =(L2(R+, ρn−2r−1dρ)b ⊗L2Ωr(N)) ⊕(L2(R+, ρn−2r+1 dρ)b ⊗L2Ωr−1(N)) ,(110) which will be considered as an identity. Let δand ˜ δdenote the exterior coderivatives on Ω(M) and Ω(N), respectively. Lemma 13.3.According to (107), δ≡ ρ−2˜ δ−d dρ −(n−2r+ 1)ρ−1 0−ρ−2˜ δ! on Ωr(M). Proof. For an oriented open subset W⊂N, consider the orientation on W×R+defined as above, and let ?and ˜ ?denote the corresponding star operators on VT(W×R+)∗and VTW∗. By Lemmas 13.1 and 13.2, on Ωr(W×R+), δ= (−1)nr+n+1 ? d? ≡(−1)nr+n+1 0ρ−n+2r−1˜ ? (−1)n−r+1ρ−n+2r−3˜ ?0 ˜ d0 d dρ −˜ d! ×0ρn−2r+1˜ ? (−1)rρn−2r−1˜ ?0 = (−1)nr+n+1 −(−1)rρ−2˜ ?˜ d˜ ? ρ−n+2r−1d dρ ρn−2r+1˜ ?2 0 (−1)n−r+1ρ−2˜ ?˜ d˜ ?! = ρ−2˜ δ−ρ−n+2r−1d dρ ρn−2r+1 0−ρ−2˜ δ!, which equals the matrix of the statement by (71). Let ∆ and e ∆ denote the Laplacians on Ω(M) and Ω(N), respectively.
2. WITTEN’S PERTURBATION ON A CONE 107 Corollary 13.4.According to (107), ∆≡P−2ρ−1˜ d −2ρ−3˜ δ Q on Ωr(M), where P=ρ−2e ∆−d2 dρ2−(n−2r−1)ρ−1d dρ , Q=ρ−2e ∆−d2 dρ2−(n−2r+ 1) d dρ ρ−1. Proof. By Lemmas 13.1 and 13.3, δd ≡ ρ−2˜ δ−d dρ −(n−2r−1)ρ−1 0−ρ−2˜ δ! ˜ d0 d dρ −˜ d! = ρ−2˜ δ˜ d−d2 dρ2−(n−2r−1)ρ−1d dρ (d dρ + (n−2r−1)ρ−1)˜ d −ρ−2˜ δd dρ ρ−2˜ δ˜ d!, dδ ≡ ˜ d0 d dρ −˜ d! ρ−2˜ δ−d dρ −(n−2r+ 1)ρ−1 0−ρ−2˜ δ! = ρ−2˜ d˜ δ−˜ d(d dρ + (n−2r+ 1)ρ−1) d dρ ρ−2˜ δ−d2 dρ2−(n−2r+ 1) d dρ ρ−1+ρ−2˜ d˜ δ! = ρ−2˜ d˜ δ−˜ d(d dρ + (n−2r+ 1)ρ−1) ρ−2d dρ ˜ δ−2ρ−3˜ δ−d2 dρ2−(n−2r+ 1) d dρ ρ−1+ρ−2˜ d˜ δ!. The sum of these matrices is the matrix of the statement. 2. Witten’s perturbation on a cone Let d± s,δ± s,D± sand ∆± s(s≥0) denote the Witten’s perturbations of d,δ, Dand ∆ induced by the function f=±1 2ρ2on M. In this case, df =±ρ dρ. According to (107), ρ dρ∧ ≡ 0 0 ρ0,−ρ dρy≡0ρ 0 0. So the following is a consequence of Lemmas 13.1 and 13.3, (101) and (102). Corollary 13.5.According to (107), d± s≡ ˜ d0 d dρ ±sρ −˜ d!, δ± s≡ ρ−2˜ δ−d dρ −(n−2r+ 1)ρ−1±sρ 0−ρ−2˜ δ! on Ωr(M). With the notation of Chapter 12, R=±ρ(dρ∧−dρy)≡ ±0ρ ρ0,
108 13. WITTEN’S PERTURBATION ON A CONE and therefore R2≡ρ20 0ρ2≡ρ2.(111) Lemma 13.6.RD +DR =±(2r−n)on Ωr(M). Proof. By Lemmas 13.1 and 13.3, and according to (107), RD ≡ ±0ρ ρ0 ˜ d+ρ−2˜ δ−d dρ −(n−2r+ 1)ρ−1 d dρ −˜ d−ρ−2˜ δ! =± ρd dρ −ρ˜ d−ρ−1˜ δ ρ˜ d+ρ−1˜ δ−ρd dρ −n+ 2r−1!, DR ≡ ± ˜ d+ρ−2˜ δ−d dρ −(n−2r−1)ρ−1 d dρ −˜ d−ρ−2˜ δ!0ρ ρ0 =± −d dρ ρ−n+ 2r+ 1 ρ˜ d+ρ−1˜ δ −ρ˜ d−ρ−1˜ δd dρ ρ!. So RD +DR ≡ ±2r−n0 0 2r−n≡ ±(2r−n) by (71). Remark 23.The expression of RD +DR can be also obtained by computing Hess f(Chapter 12). The following is a consequence of (104), Corollary 13.4 and Lemma 13.6. Corollary 13.7.According to (107), ∆± s≡P± s−2uρ−1˜ d −2ρ−3˜ δ Q± s on Ωr(M), where P± s=ρ−2e ∆ + H−(n−2r−1)ρ−1d dρ ∓s(n−2r), Q± s=ρ−2e ∆ + H−(n−2r+ 1)ρ−1d dρ + (n−2r+ 1)ρ−2∓s(n−2r).
CHAPTER 14 Domains of the Witten’s Laplacian on a cone Theorem I is proved by induction on the dimension. Thus, with the notation of Chapter 13, suppose that ˜ dmin/max satisfies the statement of Theorem I. Let e Hmin/max = ker e Dmin/max = ker e ∆min/max , which is a graded subspace of Ω(N). For each degree r, let e Rmin/max,r−1,e R∗ min/max,r ⊂L2Ωr(N) be the images of ˜ dmin/max,r−1and ˜ δmin/max,r, respectively, whose intersections with D∞(e ∆) are denoted by e R∞ min/max,r−1and e R∗∞ min/max,r. According to Section 1 of Chapter 9, e ∆ preserves e R∞ min/max,r−1and e R∗∞ min/max,r, and its restrictions to these spaces have the same eigenvalues. For any eigenvalue ˜ λof the restriction of e ∆ to e R∞ min/max,r−1, let e Rmin/max,r−1,˜ λ=E˜ λ(e ∆min/max)∩e R∞ min/max,r−1, e R∗ min/max,r,˜ λ=E˜ λ(e ∆min/max)∩e R∗∞ min/max,r . Moreover L2Ωr(N) = e Hr min/max ⊕d M ˜ λe Rmin/max,r−1,˜ λ⊕e R∗ min/max,r,˜ λ,(112) where ˜ λruns in the spectrum of e ∆min/max on e R∞ min/max,r−1; i.e., the positive spectrum of e ∆min/max,r. Now, consider the Witten’s perturbed Laplacian ∆± s. In the following, suppose that s > 0. 1. Domains of first type For some degree r, let 0 6=γ∈e Hr min/max. By Corollary 13.7, ∆± s≡H−(n−2r−1)ρ−1d dρ ∓s(n−2r) on C∞(R+)≡C∞(R+)γ⊂Ωr(M). This operator is of the type of Pin (1) with c2= 0. Thus (72) is satisfied, and (4) means that a∈ {0,−n+ 2r+ 2}. For a= 0, we have 2σ=n−2r−1. When σ > −1/2, which means r≤ n−1 2, Corollary H asserts that ∆± s, with domain Sev,+, is essentially self-adjoint in L2(R+, ρn−2r−1dρ); the spectrum of its closure consists of the eigenvalues (4k+ (1 ∓1)(n−2r))s(113) 109
110 14. DOMAINS OF THE WITTEN’S LAPLACIAN ON A CONE of multiplicity one, with corresponding normalized eigenfunctions χk; and the smooth core of its closure is Sev,+. For ∆+ s, (113) becomes 4ks, which is ≥0 for all kand = 0 just for k= 0. For ∆− s, (113) becomes (4k+ 2(n−2r))s, which is >0 for all k. For a=−n+ 2r+ 2, we have 2σ=−n+ 2r+ 3. When σ > −1/2, which means r≥n−3 2, Corollary H asserts that ∆± s, with domain ρ−n+2r+2 Sev,+, is essentially self-adjoint in L2(R+, ρn−2r−1dρ); the spectrum of its closure consists of the eigenvalues (4k+ 4 −(1 ±1)(n−2r))s(114) of multiplicity one, with normalized eigenfunctions χk; and the smooth core of its closure is ρ−n+2r+2 Sev,+. For ∆+ s, (114) becomes (4k+ 4 −2(n−2r))s, which is: •>0 for all kif r≥n−1 2, • ≥ 0 for all kand = 0 just for k= 0 if r=n 2−1, and •<0 for k= 0 if r=n−3 2. For ∆− s, (114) becomes (4k+ 4)s, which are >0 for all k. When n−3 2≤r≤n−1 2, we have got two essentially self-adjoint operators, with a= 0 and a=−n+ 2r+ 2. These two operators are equal just when r=n 2−1. All of the above operators defined by ∆± s, as well as their domains, will be said to be of first type. 2. Domains of second type With the notation of Section 1, ∆± s≡H−(n−2r−1)ρ−1d dρ + (n−2r−1)ρ−2∓s(n−2r−2) on C∞(R+)≡C∞(R+)dρ ∧γ⊂Ωr+1(M) by Corollary 13.7. This is an operator of the type of Pin (1) with c2=c1. Thus (72) is also satisfied, and (4) becomes a∈ {1,−n+ 2r+ 1}. For a= 1, we have 2σ=n−2r+ 1 according to (5). When σ > −1/2, which means r≤n+1 2, Corollary H asserts that ∆± s, with domain ρSev,+=Sodd,+, is essentially self-adjoint in L2(R+, ρn−2r+1 dρ); the spectrum of its closure consists of the eigenvalues (4k+ 4 + (1 ∓1)(n−2r−2))s(115) of multiplicity one, with normalized eigenfunctions χk; and the smooth core of its closure is ρSev,+. For ∆+ s, (115) is >0 for all k. For ∆− s, (115) is: •>0 for all kif r≤n−1 2, • ≥ 0 for all kand = 0 just for k= 0 if r=n 2, and •<0 for k= 0 if r=n+1 2. For a=−n+2r+1, we have 2σ=−n+2r+1 according to (5). When σ > −1/2, which means r≥n−1 2, Corollary H asserts that ∆± s, with domain ρ−n+2r+1 Sev,+, is essentially self-adjoint in L2(R+, ρn−2r−1dρ); the spectrum of its closure consists of the eigenvalues (4k−(1 ±1)(n−2r−2))s(116) of multiplicity one, with normalized eigenfunctions χk; and the smooth core of its closure is ρ−n+2r+1 Sev,+. For ∆+ s, (116) is >0 for all k. For ∆− s, (116) is ≥0 for all kand = 0 just for k= 0.
3. DOMAINS OF THIRD TYPE 111 For n−1 2≤r≤n+1 2, we have obtained two essentially self-adjoint operators, with a= 1 and a=−n+ 2r+ 1. These operators are equal just when r=n 2. All of the above operators defined by ∆± s, as well as their domains, will be said to be of second type. 3. Domains of third type Let µ=p˜ λfor an eigenvalue ˜ λof the restriction of e ∆min/max to e R∞ min/max,r−1. According to Section 1 of Chapter 9, there are non-zero differential forms, α∈e Rmin/max,r−1,λ ⊂Ωr(N), β ∈e R∗ min/max,r−1,λ ⊂Ωr−1(N), such that ˜ dβ =µα and ˜ δα =µβ. By Corollary 13.7, ∆± s≡ − d2 dρ2−(n−2r+ 1)ρ−1d dρ +µ2ρ−2∓(n−2r+ 2)s such that ˜ dβ =µα and ˜ δα =µβ. By Corollary 13.7, ∆± s≡ − d2 dρ2−(n−2r+ 1)ρ−1d dρ +µ2ρ−2∓(n−2r+ 2)s on C∞(R+)≡C∞(R+)β⊂Ωr−1(M). This operator is of the type of Pin (1) with c2=µ2>0. Thus (72) is satisfied, and (4) becomes a=−n+ 2r±p(n−2r)2+ 4µ2 2.(117) These two possibilities for ahave different sign because µ > 0. For the choice of positive square root in (117), we get σ=1 + p(n−2r)2+ 4µ2 2>1 2(118) according to (5). Then Corollary H asserts that ∆± s, with domain ρaSev,+, is essentially self-adjoint in L2(R+, ρn−2r+1 dρ); the spectrum of its closure consists of the eigenvalues4k+2+p(n−2r)2+ 4µ2∓(n−2r+ 2)s , (119) with multiplicity one and corresponding normalized eigenfunctions χk; and the smooth core of its closure is ρaSev,+. Notice that (119) is >0 for all k. For the choice of negative square root in (117), we get σ=1−p(n−2r)2+ 4µ2 2(120) according to (5). Then σ > −1/2 if and only if µ < 1 and |n−2r|<2p1−µ2,(121) which is equivalent to √3 2≤µ < 1 and r=n 2, or µ < √3 2and n−1 2≤r≤n+1 2. In this case, Corollary H asserts that ∆± s, with domain ρaSev,+, is essentially selfadjoint in L2(R+, ρn−2r+1 dρ); the spectrum of its closure consists of the eigenvalues 4k+ 2 −p(n−2r)2+ 4µ2∓(n−2r+ 2)s , (122)
112 14. DOMAINS OF THE WITTEN’S LAPLACIAN ON A CONE with multiplicity one and corresponding normalized eigenfunctions ρaφ2k,+; and the smooth core of its closure is ρaSev,+. For ∆+ s, (122) is <0 for k= 0. For ∆− s, (122) is >0 for all k. When (121) is satisfied, we have got two different essentially self-adjoint operators defined by the two different choices of ain (117). All of the above operators defined by ∆± s, as well as their domains, will be said to be of third type. 4. Domains of fourth type Let µ,αand βbe like in Section 3. By Corollary 13.7, ∆± s≡ − d2 dρ2+s2ρ2−(n−2r−1)ρ−1d dρ + (µ2+n−2r−1)ρ−2 ∓(n−2r−2)s on C∞(R+)≡C∞(R+)dρ ∧α⊂Ωr+1(M). This is another operator of the type of Pin (1), which satisfies (72) because (1 −(n−2r−1))2+ 4(µ2+n−2r−1) = (n−2r)2+ 4µ2>0. Moreover (4) becomes a=−n+ 2r+ 2 ±p(n−2r)2+ 4µ2 2.(123) These two possibilities for aare different because µ > 0. With the choice of positive square root in (123) and according to (5), σis also given by (118), which is >1/2. Then Corollary H asserts that ∆± s, with domain ρaSev,+, is essentially self-adjoint in L2(R+, ρn−2r−1dρ); the spectrum of its closure consists of the eigenvalues 4k+2+p(n−2r)2+ 4µ2∓(n−2r−2)s , (124) with multiplicity one and corresponding normalized eigenfunctions χk; and the smooth core of its closure is ρaSev,+. Observe that (124) is >0 for all k. With the choice of negative square root in (123) and according to (5), σis also given by (120), which is >−1/2 if and only if (121) is satisfied. In this case, Corollary H asserts that ∆± s, with domain ρaSev,+, is essentially self-adjoint in L2(R+, ρn−2r−1dρ); the spectrum of its closure consists of the eigenvalues 4k+ 2 −p(n−2r)2+ 4µ2∓(n−2r−2)s , (125) with multiplicity one and corresponding normalized eigenfunctions χk; and the smooth core of its closure is ρaSev,+. For ∆+ s, (125) is >0 for all k. For ∆− s, (125) is <0 for k= 0. When (121) is satisfied, we have got two different essentially self-adjoint operators defined by the two different choices of ain (123). All of the above operators defined by ∆± s, as well as their domains, will be said to be of fourth type.
5. DOMAINS OF FIFTH TYPE 113 5. Domains of fifth type Let µ,αand βbe like in Sections 3 and 4. By Corollary 13.7, ∆± s≡P± µ,s −2ρ−1µ −2ρ−3µ Q± µ,s on C∞(R+)⊕C∞(R+)≡C∞(R+)α+C∞(R+)dρ ∧β⊂Ωr(M), where P± µ,s =H−(n−2r−1)ρ−1d dρ +µ2ρ−2∓(n−2r)s , Q± µ,s =H−(n−2r+ 1)ρ−1d dρ + (µ2+n−2r+ 1)ρ−2∓(n−2r)s . We will conjugate this matrix expression of ∆± sby some non-singular matrix Θ, whose entries are functions of ρ, to get a diagonal matrix whose diagonal entries are operators of the type of Pin (1). This matrix will be of the form Θ = BC with B=1 0 0ρ−1, C =c11 c12 c21 c22, where cij are constants to be determined. Let P± µ,s and Q± µ,s be simply denoted by Pand Q. A key observation here is that, by (71), Q−ρ−1P ρ =−d2 dρ2−(n−2r+ 1)ρ−1d dρ + (n−2r+ 1)ρ−2 +ρ−1d2 dρ2ρ+ (n−2r−1)ρ−2d dρ ρ =−d2 dρ2−(n−2r+ 1)ρ−1d dρ + (n−2r+ 1)ρ−2 +d2 dρ2+ 2 d dρ + (n−2r−1)ρ−1d dρ + (n−2r−1)ρ−2 = 2(n−2r)ρ−2, obtaining B−1∆± sB=1 0 0ρ P−2µ ρ−1 −2µ ρ−3Q1 0 0ρ−1 =P−2µρ−2 −2µρ−2ρ Q ρ−1 =P−2µρ−2 −2µρ−2P+ 2(n−2r)ρ−2. On the other hand, Cmust be non-singular and C−1=1 det Cc22 −c12 −c21 c11 .
114 14. DOMAINS OF THE WITTEN’S LAPLACIAN ON A CONE Therefore Θ−1∆± sΘ=(Xij) with X11 =P+2 det C(µ(−c22c21 +c12c11)−(n−2r)c12c21)ρ−2, X12 =2 det Cµ(−c2 22 +c2 12)−(n−2r)c12c22ρ−2, X21 =2 det Cµ(c2 21 −c2 11)+(n−2r)c11c21ρ−2, X22 =P+2 det C(µ(c21c22 −c11c12)+(n−2r)c11c22)ρ−2. We want (Xij) to be diagonal, so we require µ(c2 12 −c2 22)−(n−2r)c12c22 =µ(c2 11 −c2 21)−(n−2r)c11c21 = 0 . Both of these equations are of the form µ(x2−y2)−(n−2r)xy = 0 ,(126) with x=c12 and y=c22 in the first equation, and x=c11 and y=c21 in the second one. There is some c∈R r {0}such that x2−y2−n−2r µxy = (x+cy)x−y c.(127) In fact, since (x+cy)x−y c=x2−y2+c−1 cxy , we need c−1 c=−n−2r µ, giving µc2+ (n−2r)c−µ= 0 ,(128) whose solutions are c±=−n+ 2r±p(n−2r)2+ 4µ2 2µ.(129) Observe that c+c−=−1. Let c=c+>0, and therefore −1/c =c−. By (127), the solutions of (126) are given by x+cy = 0 and cx −y= 0. Then we can take C=1−c c1, with det C= 1 + c2>0. So, for Θ = 1 0 0ρ−11−c c1=1−c cρ−1ρ−1, we get X12 =X21 = 0, and X11 =P+2(−2µc + (n−2r)c2) 1 + c2ρ−2, X22 =P+2(2µc +n−2r) 1 + c2ρ−2.
5. DOMAINS OF FIFTH TYPE 115 The notation X=X11 and Y=X22 will be used; thus Θ−1∆± sΘ = X⊕Y. The above expressions of Xand Ycan be simplified as follows. We have 1 + c2= 2 −n−2r µc=2µ−(n−2r)c µ by (128), obtaining 2(−2µc + (n−2r)c2) 1 + c2=2µc(−2µ+ (n−2r)c) 2µ−(n−2r)c=−2µc , 2(2µc +n−2r) 1 + c2=2µ(2µc +n−2r) 2µ−(n−2r)c. Moreover (2µc +n−2r)2= (n−2r)2+ 4µ2>0 by (129), and (2µ−(n−2r)c)(2µc +n−2r) = 4µ2c+ 2µ(n−2r)−(n−2r)2µc2−(n−2r)2c = 4µ2c+ 2µ(n−2r)−(n−2r)2µ(1 −n−2r µc)−(n−2r)2c =c(4µ2c+ (n−2r)2) by (128). Therefore 2(2µc +n−2r) 1 + c2=2µ(2µc +n−2r)2 (2µ−(n−2r)c)(2µc +n−2r) =2µ((n−2r)2+ 4µ2) c(4µ2c+ (n−2r)2) =2µ c. It follows that X=P−2µcρ−2 =H+s2ρ2−(n−2r−1)ρ−1d dρ + (µ2−2µc)ρ−2∓(n−2r)s , Y=P+2µ cρ−2 =H+s2ρ2−(n−2r−1)ρ−1d dρ + (µ2+2µ c)ρ−2∓(n−2r)s . These operators are of the type of Pin (1), and satisfy (72) because (1 −(n−2r−1))2+ 4(µ2−2µc) = 4 + (n−2r)2+ 4µ2−4p(n−2r)2+ 4µ2 = (2 −p(n−2r)2+ 4µ2)2≥0 and (1 −(n−2r−1))2+ 4(µ2+2µ c) = 4 + (n−2r)2+ 4µ2+ 4p(n−2r)2+ 4µ2) = (2 + p(n−2r)2+ 4µ2)2>0.
122 15. SPLITTING OF THE WITTEN COMPLEX ON A CONE The following result follows from Sections 1 and 2 of Chapter 14, Lemma 6.3 and the choices made to define Eγ. Proposition 15.6.(i) (Dγ,d± s,γ)is discrete. (ii) H+,r+1 s,γ = 0,dim H+,r s,γ = 1 if r≤ n 2−1if nis even n−3 2if nis odd and γ∈e Hr min n−1 2if nis odd and γ∈e Hr max , and H+,r s,γ = 0 otherwise. (iii) H−,r s,γ = 0,dim H−,r+1 s,γ = 1 if r≥ n 2if nis even n−1 2if nis odd and γ∈e Hr min n+1 2if nis odd and γ∈e Hr max , and H−,r+1 s,γ = 0 otherwise. (iv) If e± s∈ H± s,γ with norm one for each s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhe± s, e± si → 1as s→ ∞. (v) All non-zero eigenvalues of ∆± s,γ are in O(s)as s→ ∞. 2. Subomplexes defined by domains of third, fourth and fifth types Consider the notation of Sections 3–5 of Chapter 14. The following result follows from Corollary 13.5. Lemma 15.7.For s≥0,d± sand δ± sdefine maps 0C∞(R+)β C∞(R+)α+C∞(R+)dρ ∧β C∞(R+)dρ ∧α0, d± s,r−2 δ± s,r−2 d± s,r−1 δ± s,r−1 d± s,r δ± s,r d± s,r+1 δ± s,r+1 - - - - which are given by d± s,r−1=µ d dρ ±sρ, δ± s,r−1=µρ−2−d dρ −(n−2r+ 1)ρ−1±sρ, d± s,r =d dρ ±sρ −µ, δ± s,r =−d dρ −(n−2r−1)ρ−1±sρ −µρ−2, according to the canonical identities C∞(R+)β≡C∞(R+)dρ ∧α≡C∞(R+), C∞(R+)α+C∞(R+)dρ ∧β≡C∞(R+)⊕C∞(R+).
2. SUBOMPLEXES DEFINED BY DOMAINS OF THIRD, FOURTH AND FIFTH TYPES 123 Consider only the choices of agiven by the positive square roots in (117) and (123) for domains of third and fourth types, and (140) for domains of fifth type; the other choices of aare rejected because they are very restrictive on µand r, and give rise to some negative eigenvalues. If these values of aare denoted by a3, a4and a5according to the types of domains, then a5=a3=a4−1, and therefore the notation a5=a3=aand a4=a+ 1 will be used. Recall also that we only have the choice (138) for b, which equals a+ 2. So we only consider the following domains of third, fourth and fifth types defined by αand β: Fr−1 α,β =ρaSev,+β≡ρaSev,+, Fr+1 α,β =ρa+1 Sev,+dρ ∧α≡ρa+1 Sev,+, Fr α,β =ρa(φ−cρ2ψ)α+ (cρ−1φ+ρψ)dρ ∧β|φ, ψ ∈ Sev,+ ≡ρa(φ−cρ2ψ, cρ−1φ+ρψ)|φ, ψ ∈ Sev,+. Lemma 15.8.For any s≥0,d± sand δ± sdefine maps 0Fr−1 α,β Fr α,β Fr+1 α,β 0 d± s,r−2 δ± s,r−2 d± s,r−1 δ± s,r−1 d± s,r δ± s,r d± s,r+1 δ± s,r+1 - - - - Proof. Lemma 15.7 gives δ± s(Fr−1 α,β ) = d± s(Fr+1 α,β ) = 0. Observe that a=cµ , (146) obtaining c(a+n−2r) = µ(147) by (128). By Lemma 15.7, (146) and (147), for h∈ Sev,+, d± s(ρah β) = ρaµh α +d dρ +cµρ−1±sρ(h)dρ ∧β,(148) δ± s(ρa+1h dρ ∧α) = ρa−ρd dρ −µ c±sρ2(h)α−µρ−1h dρ ∧β.(149) The inclusion d± s(Fr−1 α,β )⊂ Fr α,β follows from (148) if we can find φ, ψ ∈ Sev,+such that φ−cρ2ψ=µh , (150) cρ−1φ+ρψ =d dρ +cµρ−1±sρ(h).(151) Subtract cρ−2times (150) from ρ−1times (151) to get ψ=1 1 + c2ρ−1d dρ ±s(h), which is well defined in Sev,+. Then φ=µh +cρ2ψ by (150). These functions φand ψsatisfy (150) and (151).
124 15. SPLITTING OF THE WITTEN COMPLEX ON A CONE The inclusion δ± s(Fr+1 α,β )⊂ Fr α,β follows from (149) if we can find φ, ψ ∈ Sev,+ such that φ−cρ2ψ=−ρd dρ −µ c±sρ2(h),(152) cρ−1φ+ρψ =−µρ−1h . (153) The sum (152) and cρ times (153) gives φ=1 1 + c2−ρd dρ −1 + c2 cµ±sρ2(h), which belongs to Sev,+. The even extensions of hand φto R, also denoted by h and φ, satisfy cφ(0) = −µh(0), and therefore µh +cφ ∈ρ2Sev. It follows that ψ=ρ−2(µh +cφ), obtained from (153), is well defined in Sev,+. These functions φand ψsatisfy (152) and (153). For arbitrary φ, ψ ∈ Sev,+, let ζ=ρaφ−cρ2ψα+cρ−1φ+ρψdρ ∧β.(154) By Corollary 13.5, (146) and (147), d± s(ζ) = ρa+1ρ−1d dρ ±s(φ) +c−ρ−1d dρ −c2+ 1 cµ+ 2±sρ2(ψ)dρ ∧α , δ± s(ζ) = ρac−ρ−1d dρ ±s(φ) +−ρd dρ −c2+ 1 cµ+ 2±sρ2(ψ)β , showing d± s(Fr α,β)⊂ Fr+1 α,β and δ± s(Fr α,β)⊂ Fr−1 α,β . By Lemma 15.8, Fα,β =Fr−1 α,β ⊕Fr α,β ⊕Fr+1 α,β is a subcomplex of length three of Ω(M) with d± sand δ± s. Moreover let Fα,β,0 denote the dense subcomplex of Fα,β defined by Fr−1 α,β,0=C∞ 0(R+)β≡C∞ 0(R+), Fr+1 α,β,0=C∞ 0(R+)dρ ∧α≡C∞ 0(R+), Fr α,β,0=C∞ 0(R+)α+C∞ 0(R+)dρ ∧β≡C∞ 0(R+)⊕C∞ 0(R+). The closure of Fα,β (and Fα,β,0) in L2Ω(M) is denoted by L2Fα,β. We have L2Fr−1 α,β =L2(R+, ρn−2r+1 dρ)β≡L2(R+, ρn−2r+1 dρ), L2Fr+1 α,β =L2(R+, ρn−2r−1dρ)dρ ∧α≡L2(R+, ρn−2r−1dρ), L2Fr α,β =L2(R+, ρn−2r+1 dρ)α+L2(R+, ρn−2r−1dρ)dρ ∧β ≡L2(R+, ρn−2r+1 dρ)⊕L2(R+, ρn−2r−1dρ).
2. SUBOMPLEXES DEFINED BY DOMAINS OF THIRD, FOURTH AND FIFTH TYPES 125 Assume now that s > 0. With the notation of Section 3 in Chapter 11, consider the real version of the elliptic complex (F, d), as well as its subcomplex F1, determined by the constants s,cand κ=−1 + p(n−2r)2+ 4µ2 2>−1 2.(155) By (129), κ=cµ +n−2r−1 2=µ c−n−2r+ 1 2.(156) Proposition 15.9.There is a unitary isomorphism L2Fα,β →L2(F), which restricts to isomorphisms of complexes up to a shift of degree, (Fα,β, d± s)→(F1, d) and (Fα,β,0, d± s)→(C∞ 0(F), d). Proof. As an intermediate step, let b Fr−1 α,β =ρFr−1 α,β =ρa+1 Sev,+,b Fr+1 α,β =Fr+1 α,β =ρa+1 Sev,+, b Fr α,β = Θ−1(Fr α,β) = ρaSev,+⊕ρa+2 Sev,+, b Fα,β =b Fr−1 α,β ⊕b Fr α,β ⊕b Fr+1 α,β ,b Fα,β,0=Fα,β,0, L2b Fr−1 α,β =L2b Fr+1 α,β =L2Fr+1 α,β =L2(R+, ρn−2r−1dρ), L2b Fr α,β ≡L2(R+, ρn−2r−1dρ)⊕L2(R+, ρn−2r−1dρ), L2b Fα,β =L2b Fr−1 α,β ⊕L2b Fr α,β ⊕L2b Fr+1 α,β . Moreover let Ξ : L2Fα,β →L2b Fα,β be the unitary isomorphism defined by ρ:L2Fr−1 α,β →L2b Fr−1 α,β ,1 √1 + c2Θ−1:L2Fr α,β →L2b Fr α,β and the identity map L2Fr+1 α,β →L2b Fr+1 α,β . It restricts to isomorphisms Fα,β →b Fα,β and Fα,β,0→b Fα,β,0. Thus, by Lemma 15.8, (Fα,β, d± s) induces via Ξ a complex 0b Fr−1 α,β b Fr α,β b Fr+1 α,β 0. ˆ d± s,r−2ˆ d± s,r−1ˆ d± s,r ˆ d± s,r+2 - - - - By Lemma 15.7 and (71), ˆ d± s,r−1=1 √1 + c2Θ−1d± s,r−1ρ−1 =1 √1 + c21cρ −c ρ µ d dρ ±sρρ−1 =1 √1 + c2 cd dρ + (µ−c)ρ−1±csρ d dρ + (cµ + 1)ρ−1±sρ !,(157) ˆ d± s,r =1 √1 + c2Θd± s,r =1 √1 + c2d dρ ±sρ −µ1−c cρ−1ρ−1 =1 √1 + c2d dρ −cµρ−1±sρ −cd dρ −µρ−1∓csρ.(158)
126 15. SPLITTING OF THE WITTEN COMPLEX ON A CONE Now, the unitary isomorphism ρn−2r−1 2:L2(R+, ρn−2r−1dρ)→L2(R+, dρ) induces a unitary isomorphism L2b Fα,β →L2(F), which restricts to isomorphisms b Fα,β → F1and b Fα,β,0→C∞ 0(F). Moreover, by (157), (158), (71) and (156), ρn−2r−1 2ˆ d± s,r−1ρ−n−2r−1 2 =1 √1 + c2ρn−2r−1 2 cd dρ + (µ−c)ρ−1±csρ d dρ + (cµ + 1)ρ−1±sρ !ρ−n−2r−1 2 =1 √1 + c2 cd dρ +κρ−1±sρ d dρ −(κ+ 1)ρ−1±sρ!, ρn−2r−1 2ˆ d± s,r ρ−n−2r−1 2 =1 √1 + c2ρn−2r−1 2d dρ −cµρ−1±sρ −cd dρ −µρ−1∓csρρ−n−2r−1 2 =1 √1 + c2d dρ −κρ−1±sρ c −d dρ −(κ+ 1)ρ−1∓sρ, which are the operators d0and d1of Section 3 in Chapter 11. Corollary 15.10.(Fα,β,0, d± s)has a unique Hilbert complex extension in L2Fα,β, whose smooth core is Fα,β. Proof. This follows from Propositions 11.6 and 15.9. Let (Dα,β,d± s,α,β) denote the unique Hilbert complex extension of (Fα,β,0, d± s), according to Corollary 15.10, and let ∆± s,α,β denote the corresponding Laplacian. The following result follows from Sections 3–5 of Chapter 14. Proposition 15.11.(i) (Dα,β,d± s,α,β)is discrete. (ii) The eigenvalues of ∆± s,α,β are positive and in O(s)as s→ ∞. 3. Splitting into subcomplexes Let Bmin/max,0denote an orthonormal frame of e Hmin/max consisting of homogeneous differential forms. For each positive eigenvalue µof e Dmin/max, let Bmin/max,µ be an orthonormal frame of Eµ(e Dmin/max) consisting of differential forms α+βlike in Section 2. Then let d± s,min/max =M γ d± s,γ ⊕d M µM α+β d± s,α,β , where γruns in Bmin/max,0,µruns in the positive spectrum of e Dmin/max, and α+β runs in Bmin/max,µ. Observe that the domain of d± s,min/max is independent of s, and therefore it is denoted by Dmin/max. Let also Gmin/max =M γEγ,0⊕M µM α+βFα,β,0. Proposition 15.12.D(d± s,min/max) = Dmin/max and d± s,min/max =d± s,min/max.
3. SPLITTING INTO SUBCOMPLEXES 127 Proof. By Corollaries 15.5 and 15.10, Lemma 9.2 and (112), (Dmin/max,d± s,min/max) is the minimum/maximum Hilbert complex extension of (Gmin/max, d± s). Then the result easily follows from the following assertions. Claim 18.Gmin/max ⊂ D(d± s,min/max). Claim 19.Ω0(M)⊂ Dmin/max. Let ˆ d± s,min/max denote the minimum/maximum Hilbert complex extension of (Ω0(M), d± s) with respect to the product metric ˆg= ˜g+(dρ)2on M=N×R+. With the terminology of [11, p. 110], observe that (Ω(M), d± s) is the product complex of the de Rham complex of N, (Ω(N),˜ d), and the Witten deformation of the de Rham complex of R+, defined by the function 1 2ρ2. Then, by [11, Lemma 3.6 and (2.38b)], D(ˆ d± s,min/max)⊃C∞ 0(R+)D(˜ dmin/max) + C∞ 0(R+)dρ ∧D(˜ dmin/max) ⊃ Gmin/max .(159) On the other hand, for 0 <a<b<∞, let L2 a,bΩ(M, g) and L2 a,bΩ(M, ˆg) denote the Hilbert subspaces of L2Ω(M, g) and L2Ω(M, ˆg), respectively, consisting of L2 differential forms supported in N×[a, b]. Since gand ˆgare quasi-isometric on N×(a0, b0) for 0 < a0< a and b < b0<∞, it follows that D(d± s,min/max)∩L2 a,bΩ(M, g) = D(ˆ d± s,min/max)∩L2 a,bΩ(M, ˆg).(160) Moreover Gmin/max ⊂[ 0<a<b<∞ L2 a,bΩ(M, g).(161) Now Claim 18 follows from (159)–(161). Finally, Claim 19 follows from Ω0(M)⊂M γEγ,0⊕d M µM α+βFα,β,0,(162) where γ,µand α+βvary as above. The inclusion (162) can be proved as follows. According to (106), any ξ∈Ω0(M) can be written as ξ=ξ0+dρ ∧ξ1, where ξ0, ξ1∈C∞ 0(R+,Ω0(N)). Then, by (112), we get functions fk,γ, fk,`,α,β ∈C∞ 0(R+), for k, ` ∈ {0,1}, defined by fk,γ(ρ) = hξk(ρ), γi˜g, fk,0,α,β(ρ) = hξk(ρ), βi˜g, fk,1,α,β(ρ) = hξk(ρ), αi˜g, and moreover α=X γ (f0,γ γ+f1,γ dρ ∧γ) +X µX α+β (f0,0,α,β β+f1,0,α,β α+f1,0,α,β dρ ∧β+f1,1,α,β dρ ∧α) in L2Ω(M, g), where γ,µand α+βvary as above. Thus ξbelongs to the space in the right hand side of (162). Remark 26.From (79), Remark 7, and Propositions 11.4, 11.6 and 15.12, it follows that, with the notation of Example 8.2, h(ρ)D∞(d± s,min/max)⊂ D∞(d± s,min/max) for all h∈C∞(R+) such that h0∈C∞ 0(R+).
128 15. SPLITTING OF THE WITTEN COMPLEX ON A CONE Let H± s,min/max =LrH±,r s,min/max = ker ∆± s,min/max. Corollary 15.13.(i) d± s,min/max is discrete. (ii) H+,r min ∼ =Hr min(N)if r≤(n 2−1if nis even n−3 2if nis odd , and H+,r min = 0 otherwise. (iii) H+,r max ∼ =Hr max(N)if r≤(n 2−1if nis even n−1 2if nis odd , and H+,r max = 0 otherwise. (iv) H−,r+1 min ∼ =Hr min(N)if r≥(n 2if nis even n−1 2if nis odd , and H+,r+1 min = 0 otherwise. (v) H−,r+1 max ∼ =Hr max(N)if r≥(n 2if nis even n+1 2if nis odd , and H+,r+1 max = 0 otherwise. (vi) If e± s∈ H± s,min/max with norm one for each s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhe± s, e± si → 1as s→ ∞. (vii) Let 0≤λ± s,min/max,0≤λ± s,min/max,1≤ ··· be the eigenvalues of ∆s,min/max, repeated according to their multiplicities.Given k∈N, if λ± s,min/max,k >0 for some s, then λ± s,min/max,k ∈O(s)as s→ ∞. (viii) There is some θ > 0such that lim infkλ± s,min/max,kk−θ>0. Proof. For γ,µand α+βas above, the spectra of ∆± son Eγand Fα,β is discrete by Propositions 15.6-(i) and 15.11-(i). Moreover the union of all of these spectra has no accumulation points according to Sections 1–5 of Chapter 14 and since e ∆min/max has a discrete spectrum. Then (i) follows by Proposition 15.12. Now, properties (ii)–(vii) follow directly from Propositions 15.6, 15.11 and 15.12. To prove (viii), let 0 ≤˜ λmin/max,0≤˜ λmin/max,1≤ ··· denote the eigenvalues of e ∆s,min/max, repeated according to their multiplicities, and let µmin/max,` = q˜ λmin/max,` for each `∈N. Since Nsatisfies Theorem I-(ii) with ˜g, there is some C0,˜ θ > 0 such that ˜ λmin/max,` ≥C2 0`˜ θ(163) for all `. Consider the counting function N± s,min/max(λ)=#nk∈N|λ± s,min/max,k < λ o
3. SPLITTING INTO SUBCOMPLEXES 129 for λ > 0. From (113)–(116), (119), (124), (139), (141) and (163), and the choices made in Chapter 15, it follows that there are some C1, C2, C3>0 such that N± s,min/max(λ)≤#(k, `)∈N2|C1k+C2µmin/max,` +C3≤λ ≤#{(k, `)∈N2|C1k+C2C0`˜ θ/2+C3≤λ} ≤#((k, `)∈N2`≤λ−C3 C2C0−C1k C2C02/˜ θ) ≤Zλ−C3 C1 0λ−C3 C2C0−C1x C2C02/˜ θ dx =˜ θ(λ−C3)(2+˜ θ)/˜ θ (2 + ˜ θ)(C2C0)2/˜ θC1 . So N± s,min/max(λ)≤Cλ(2+˜ θ)/˜ θfor some C > 0 and all large enough λ, giving (viii) with θ=˜ θ 2+˜ θ. Example 15.14.Consider the notation of Examples 7.6, 7.12 and 12.1. On the stratum Sm−1×R+of c(Sm−1), the model rel-Morse function ±1 2ρ2and the metric g1define the Witten’s perturbed operators d± s,δ± s,D± sand ∆± s. Since ρ0 and g0respectively correspond to ρand g1by can : Sm−1×R+→Rm\ {0}, it follows that d± s,δ± s,D± sand ∆± srespectively correspond to d± 0,s,δ± 0,s,D± 0,s, ∆± 0,s by can∗: Ω(Rm\{0})→Ω(Sm−1×R+), and moreover L2Ω(Rm, g0)≡L2Ω(Rm\{0}, g0)can∗ −−−−→ L2Ω(Sm−1×R+, g1)(164) is a unitary isomorphism. The extension by zero defines a canonical injection Ω0(Rm\{0})→Ω0(Rm), whose composite with (can∗)−1is an injective homomorphism of complexes, (Ω0(Sm−1×R+), d± s)→(Ω0(Rm), d± 0,s). Thus the unique i.b.c. of (VTRm∗, d± 0,s) in L2Ω(Rm, g0) corresponds to d± s,max via (164). If m≥2, then Hm−1 2(Sm−1) = 0 for odd m. So (VT(Sm−1×R+)∗, d± s) has a unique i.b.c. by Corollaries 15.5 and 15.10, and Proposition 15.12. If m= 1, then Ω(S0)=Ω0(S0)≡R2, and therefore, according to (106), (107) and Corollary 13.5, Ω0(S0×R+)≡C∞(R+,R2), Ω1(S0×R+)≡dρ ∧C∞(R+,R2)≡C∞(R+,R2), d± s≡d dρ ±sρ , δ± s≡ − d dρ ±sρ , giving d± s,min 6=d± s,max by Proposition 11.4-(i).
CHAPTER 16 Local model of the Witten’s perturbation The local model of our version of Morse functions around their critical points will be as follows. Let m±∈N, let L±be a compact Thom-Mather stratification, and let M±be a stratum in c(L±). Thus, either M±=N±×R+for some stratum N±of L±, or M±is the vertex stratum of c(L±). On the stratum M=Rm+× Rm−×M+×M−of the Thom-Mather stratification Rm+×Rm−×c(L+)×c(L−) (for any choice of product Thom-Mather structure on c(L+)×c(L−)), consider an adapted metric given as product of standard metrics on the Euclidean spaces Rm± and model adapted metrics on the strata M±. Let dsdenote the Witten’s perturbed differential map on Ω(M) induced by the model rel-Morse function 1 2(ρ2 +−ρ2 −) (Remark 19-(iii)). Let ∆s,min/max be the Laplacian defined by ds,min/max, and Hs,min/max =LrHr s,min/max = ker ∆s,min/max. The following result is a direct consequence of Example 12.1, Corollary 15.13 and Lemma 9.1. Corollary 16.1.(i) ds,min/max is discrete. (ii) If M+=N+×R+and M−=N−×R+, then Hr s,min/max ∼ =M r+,r− Hr+ min/max(N+)⊗Hr− min/max(N−), where (r+, r−)runs in the subset of Z2defined by (6)–(8). (iii) If M+is the vertex stratum of c(L+)and M−=N−×R+, then Hr s,min/max ∼ =M r− Hr− min/max(N−), where r−runs in the subset of Zdefined by r=m−+r−+ 1 and (8). (iv) If M+=N+×R+and M−is the vertex stratum of c(L+), then Hr s,min/max ∼ =M r+ Hr+ min/max(N+), where r+runs in the subset of Zdefined by r=m−+r+and (7). (v) If M+and M−are the vertex strata of c(L+)and c(L−), then we have dim Hr s,min/max =δr,m−. (vi) If es∈ Hs,min/max with norm one for each s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhe± s, e± si → 1as s→ ∞. (vii) Let 0≤λs,min/max,0≤λs,min/max,1≤ ··· be the eigenvalues of ∆s,min/max, repeated according to their multiplicities.Given k∈N, if λs,min/max,k >0 for some s, then λs,min/max,k ∈O(s)as s→ ∞. (viii) There is some θ > 0such that lim infkλs,min/max,k k−θ>0. 131
138 19. FINITE PROPAGATION SPEED OF THE WAVE EQUATION ON STRATA with αt∈Ω(M0 x) depending smoothly on t∈R. By Propositions 15.3, 15.9 and 15.12, the following result clearly boils down to the case of Proposition 11.7. Proposition 19.1.For 0< a < b, suppose that αt∈ D∞(d0 x,s,min/max), depending smoothly on t∈R, satisfies (166) on Ux,b. The following properties hold: (i) If supp α0⊂M0 x\Ux,a, then supp αt⊂M0 x\Ux,a−|t|for 0<|t| ≤ a. (ii) If supp α0⊂Ux,a, then supp αt⊂Ux,a+|t|for 0<|t| ≤ b−a. There is some ρ0>0 such that each Ux,ρ0is contained in the image of the rel-Morse chart centered at x, and moreover these charts are disjoint one another. We will identify each Ux,ρ0with an open subset of Mvia the rel-Morse chart. According to Example 7.11, we can choose gso that its restriction to each Ux,ρ0is identified to the restriction of gx. Proposition 19.2.Let 0< a < b < ρ0and α∈L2Ω(M). The following properties hold for αt= exp(itDs,min/max)α: (i) If supp α⊂M\Ux,a, then supp αt⊂M\Ux,a−|t|for 0<|t| ≤ a. (ii) If supp α⊂Ux,a, then supp αt⊂Ux,a+|t|for 0<|t| ≤ b−a. Proof. Since exp(itDs,min/max) is bounded, we can take α∈ D∞(ds,min/max), and therefore αt∈ D∞(ds,min/max) for all t. According to Remark 26, there is some h∈C∞(M) such that supp h⊂Ux,ρ0,h= 1 on Ux,b, and hD∞(ds,min/max)⊂ D∞(ds,min/max). Then hαtsatisfies (166) on Ux,b and belongs to D∞(d0 s,min/max). So, by Proposition 19.1, •hαt= 0 on Ux,a−|t|for 0 <|t| ≤ aif supp α⊂M\Ux,a, and •supp hαt⊂Ux,a+|t|for 0 <|t| ≤ b−aif supp α⊂Ux,a. Thus the result follows because h= 1 on Ux,b.
CHAPTER 20 Morse inequalities on strata Here, we prove Theorem J. Consider the notation of Chapter 19. 1. Analytic inequalities By (101), we have the isomorphism of complexes esf : (Ω0(M), ds)→(Ω0(M), d). Since fis bounded, we also have the quasi-isometric isomorphism esf :L2Ω(M)→ L2Ω(M). So we obtain the isomorphism of Hilbert complexes esf : (D(ds,min/max), ds,min/max)→(D(dmin/max), dmin/max), and therefore βr min/max = dim Hr(D(ds,min/max), ds,min/max) (167) for all s≥0. In fact, since |df|is bounded, it also follows from (101) that D(ds,min/max) = D(dmin/max), ds,min/max =dmin/max +s df ∧. Thus esf D(dmin/max) = D(dmin/max). Let φbe a smooth rapidly decreasing function on Rwith φ(0) = 1. Then the operator φ(∆s,min/max) is of trace class (Chapter 18), and set µr s,min/max = Tr(φ(∆s,min/max,r)) . By (167), the following result follows with the obvious adaptation of the proof of [54, Proposition 14.3]. Proposition 20.1.We have the inequalities β0 min/max ≤µ0 min/max , β1 min/max −β0 min/max ≤µ1 s,min/max −µ0 s,min/max , β2 min/max −β1 min/max +β0 min/max ≤µ2 s,min/max −µ1 s,min/max +µ0 s,min/max , etc., and the equality χmin/max =X r (−1)rµr s,min/max . Proof. The proof is reproduced for the reader’s convenience. By Theorem I, ∆min/max,r is discrete with kernel of dimension βr min/max. Then there is some nonnegative rapidly decreasing ˜ φ∈C∞(R) such that ˜ φ(0) = 1 and ˜ φ(λ) = 0 for all non-zero eigenvalue of ∆min/max,r; there is no loss of generality in assuming also that ˜ φ≤φ. Then βr min/max = Tr(˜ φ(∆min/max,r)), so that µr min/max −βr min/max = Tr((φ−˜ φ)(∆min/max,r)) .(168) 139
140 20. MORSE INEQUALITIES ON STRATA We may write (φ−˜ φ)(λ) = λψ2(λ), where ψ∈C∞(R) is non-negative and rapidly decreasing, and vanishes at zero, obtaining (φ−˜ φ)(∆min/max,r)=∆min/max,r ψ2(∆min/max,r).(169) Then Tr dmin/max,r−1δmax/min,r−1ψ2(∆min/max,r) = Tr ψ(∆min/max,r)dmin/max,r−1δmax/min,r−1ψ(∆min/max,r) = Tr δmax/min,r−1ψ2(∆min/max,r)dmin/max,r−1 = Tr δmax/min,r−1dmin/max,r−1ψ2(∆min/max,r−1). By (168), (169) and (81), it follows that (µr min/max −βr min/max)−(µr−1 min/max −βr−1 min/max)+(µr−2 min/max −βr−2 min/max)−··· = Tr dmin/max,r−1δmax/min,r−1ψ2(∆min/max,r) = Tr (dmin/max,r ψ(∆min/max,r))∗dmin/max,r ψ(∆min/max,r) ≥0. For r=n, this is an equality. 2. Null contribution away from the critical points By (104) and because |df|and |Hess f|are bounded on M, we have D(∆s,min/max) = D(∆min/max),(170) ∆s,min/max = ∆min/max +sHessf+s2|df|2(171) for all s≥0. For ρ≤ρ0, let Uρ=SxUx,ρ, with xrunning in Critrel(f). Fix some ρ1>0 such that 3ρ1< ρ0. Let Gand Hbe the Hilbert subspaces of L2Ω(M) consisting of forms essentially supported in M\Uρ1and M\U2ρ1, respectively. It follows from (170) and (171) that there is some C > 0 such that1 ∆s,min/max ≥∆min/max +Cs2on G∩D(∆min/max) (172) if sis large enough. Let hbe a rel-admissible function on Msuch that h≤0, h≡1 on Uρ0and h≡0 on M\U2ρ1(see Example 8.2). Then Ts,min/max = ∆s,min/max +hCs2, with domain D(e ∆min/max), is essentially self-adjoint in L2Ω(M) with a discrete spectrum, and moreover Ts,min/max ≥∆min/max +Cs2(173) for sis large enough by (172). Fix some2φ∈ Sev such that φ≥0, φ(0) = 1 and supp ˆ φ⊂[−ρ1, ρ1], and let ψ∈ S such that φ(x) = ψ(x2). By using Proposition 19.2-(i), the argument of the 1Recall that, for symmetric operators Sand Tin a Hilbert space, with the same domain D, it is said that S≤Tif hSu, ui≤hT u, uifor all u∈ D. 2The Schwartz functions with compactly supported Fourier transform are characterized by the Paley-Wiener-Schwartz theorem (see e.g. [36, Theorem 7.3.1]); they form a dense subalgebra of S, which is invariant by linear changes of variables.
2. NULL CONTRIBUTION AWAY FROM THE CRITICAL POINTS 141 first part of the proof of [54, Lemma 14.6] can be obviously adapted to show the following. Lemma 20.2.ψ(∆s,min/max) = ψ(Ts,min/max)on H. Proof. This proof is also reproduced for the reader’s convenience. For α∈ D∞(∆min/max) supported in M\U2ρ1, consider the time-dependent differential form αt= cos(tDs,min/max)α=1 2eitDs,min/max +e−itDs,min/max α in D∞(∆min/max). It is a solution of the differential equation ∂2αt ∂t2+ ∆sαt= 0 with initial conditions α0=αand ˙α0= 0; in fact, it is the unique solution, as one can easily check by verifying that the “energy” ∂αt ∂t 2 +h∆sαt, αti is preserved. By Proposition 19.2-(i), αtis supported on M\Uρ1if |t|< ρ1, and therefore ∆sαt=Ts,min/maxαt. Thus αtfor |t|< ρ1is also the unique solution to the equation ∂2αt ∂t2+Ts,min/maxαt= 0 with the same initial conditions. So αt= cos(tpTs,min/max)α. Now, ˆ φhas support in [−ρ1, ρ1] and is even (since φis). Therefore ψ(∆s,min/max)α=φ(Ds,min/max)α =1 2πZρ1 −ρ1 eitDs,min/max αˆ φ(t)dt =1 πZρ1 0 ˆ φ(t) cos(tDs,min/max)α dt =1 πZρ1 0 ˆ φ(t)αtdt =1 πZρ1 0 ˆ φ(t) cos tqTs,min/maxα dt =··· =ψ(Ts,min/max)α . Then the result follows because D∞(∆min/max)∩His dense in H. Let Π : L2Ω(M)→Hdenote the orthogonal projection. According to Chapter 18, ψ(∆s,min/max) is of trace class for all s≥0. Then the self-adjoint operator Πψ(∆s,min/max) Π is also of trace class (see e.g. [54, Proposition 8.8]). Lemma 20.3.Tr(Π ψ(∆s,min/max) Π) →0as s→ ∞. Proof. Let 0≤λmin/max,0≤λmin/max,1≤ ··· ,0≤λs,min/max,0≤λs,min/max,1≤ ··· be the eigenvalues of ∆min/max and Ts,min/max, respectively, repeated according to their multiplicities. By (173) and the min-max principle, we have λs,min/max,k ≥λmin/max,k +Cs2
142 20. MORSE INEQUALITIES ON STRATA for slarge enough. So Tr(ψ(Ts,min/max)) = X k ψ(λs,min/max,k)≤X k ψ(λmin/max,k +Cs2) for slarge enough, giving Tr(ψ(Ts,min/max)) →0 as s→ ∞ since ψis rapidly decreasing. Then the result follows because Tr(Π ψ(∆s,min/max) Π) = Tr(Π ψ(Ts,min/max) Π) ≤Tr(ψ(Ts,min/max)) by Lemma 20.2. 3. Contribution from the rel-critical points The following is a direct consequence of Corollary 16.1. Corollary 20.4.If his a bounded measurable function on R+such that h(ρ)→1as ρ→0, then lim s→∞Tr(h(ρ)φ(∆0 x,s,min/max,r)) = lim s→∞Tr φ(∆0 x,s,min/max,r) = νr x,min/max . For each x∈Critrel(f), let e Hx⊂L2Ω(M) be the Hilbert subspace of differential forms supported in Ux,2ρ1; it can be also considered as a Hilbert subspace of L2Ω(M0 x) since gand gxhave identical restrictions to Ux,ρ0. Moreover ∆sand ∆0 x,s can be identified on differential forms supported in Ux,ρ0. By using Proposition 19.2-(ii), the argument of Lemma 20.2 can be obviously adapted to show the following. Lemma 20.5.φ(∆s,min/max)≡φ(∆0 x,s,min/max)on e Hxfor all x∈Critrel(f). For each x∈Critrel(f), let e Πx:L2Ω(M)→e Hxand e Π0 x:L2Ω(M0 x)→e Hx denote the orthogonal projections. Since the subspaces e Hxare orthogonal to each other, Π := Pxe Πx:L2Ω(M)→e H:= Pxe Hxis the orthogonal projection. Lemma 20.6.Tr(e Πφ(∆s,min/max,r)e Π) →νr min/max as s→ ∞. Proof. By Corollary 20.4 and Lemma 20.5, and because Π0 xis the multiplication operator by the characteristic function of Uρ1in M0 x, lim s→∞Tr(e Πφ(∆s,min/max,r)e Π) = lim s→∞ X x∈Critrel(f) Tr(e Πxφ(∆s,min/max,r)e Πx) = lim s→∞ X x∈Critrel(f) Tr(e Π0 xφ(∆0 x,s,min/max,r)e Π0 x) =X x∈Critrel(f) νr x,min/max =νr min/max . Now, lim s→∞Tr(φ(∆s,min/max,r)) = νr min/max by Lemmas 20.3 and 20.6, and because Π + e Π = 1, showing Theorem J by Proposition 20.1.
CHAPTER 21 Remark on the Sobolev spaces on strata Our version of the Sobolev spaces on strata, Wm(dmin/max), may depend on the chosen adapted metric; thus there is no “rel-version” of the elliptic estimate. By taking local charts and arguing like in Chapter 17, it is enough to check this assertion for the perturbed local models d± s,min/max. With the notation of Section 1 in Chapter 14, consider the case where nis odd, r=n−1 2and a= 0; thus σ= 0. We have χ0γ∈W∞(d± s,min/max) with the metric g. Let ˜g0be another adapted metric on Nsuch that e ∆0γ6= 0, and consider the corresponding adapted metric g0=ρ−2˜g0+dρ2on M. Let e ∆0be the laplacian on Ω(N) defined by ˜g0, ∆0the Laplacian on Ω(M) defined by g0, and ∆0± sthe Witten’s perturbation of ∆0induced by the function ±1 2ρ2. Let h,i˜0and h,i0denote the scalar products of L2Ω(N, ˜g0) and L2Ω(M, g0), respectively, and let k k˜0denote the norm defined by h,i˜0. By Corollary 13.7, we have ∆0± s=ρ−2e ∆0+H∓son C∞(R+)γ. Then h∆0± s(χ0γ), χ0γi0=he ∆0γ, γi˜0Z∞ 0 ρ−2χ2 0dρ +kγk˜02(1 ∓1)s=∞ according to (110) and Section 1 of Chapter 14, and because χ0(ρ) = √2p0e−sρ2/2 is bounded away from zero for 0 < ρ ≤1. So χ0γ6∈ W1(d± s,min/max) with the metric g0, obtaining different spaces W1(d± s,min/max) by using gand g0. The above observation is related with the following problem. Problem 21.1.Let Mbe a stratum of an arbitrary compact stratification endowed with an adapted metric, and let L1Ω(M) denote the Banach space of uniformly bounded measurable differential forms on M. Is there a continuous inclusion of Wm(dmin/max) into L1Ω(M) for mlarge enough? For the perturbation Pof harmonic oscillator indicated in Chapter 5, the corresponding version of this problem has an affirmative answer when a≥0 (Corollary H-(iii)). If the spaces Wm(dmin/max) were independent of the adapted metric, we could give an affirmative answer to Problem 21.1 by using the local arguments of this chapter and induction. An affirmative solution of Problem 21.1 would allow to adapt the nice arguments of [54, Lemma 14.6] to show a stronger version of Lemma 20.3: the Schwartz kernel of ψ(∆s,min/max) would converge uniformly to zero on (M\U2ρ1)×(M\U2ρ1). 143
Conclusion The main goal of the thesis was to prove a version of Morse inequalities for the minimum and maximum ideal boundary conditions of the de Rham complex on strata endowed with adapted metrics, taken in compact Thom-Mather stratifications. The analytic method of Witten was used, involving his perturbation of the de Rham complex induced by our version of Morse functions on strata. The cohomology of this minimum ideal boundary condition is isomorphic to the intersection homology with lower middle perversity; there are analogous isomorphisms with other types of adapted metrics and other perversities. Several new features have shown up in this work. First, the local analysis around our version of critical points was reduced to the study of an operator related to the so called Dunkl harmonic oscillator, which was recently very much used in Quantum Mechanics to describe the interaction among several particles. This led us to prove eigenfunction estimates and embedding results for the Dunkl harmonic oscillator on the line, which have their own interest. Second, it turns out that, surprisingly, the Sobolev spaces defined by the perturbed Laplacian depend on the choice of the metric, and therefore the usual way to prove Sobolev inequalities does not work, even though they could be true. Because of this lack of Sobolev inequalities, some of the arguments of Witten’s method cannot be made. Thus new types of arguments were produced to solve that problem, mainly using certain weak version of the Weyl’s assymptotic formula. This formula is proved first by showing that it has a local character and holds for the local models. Our Morse inequalities on strata seem to be new. Their expressions are a priori different from those of the Morse inequalities of Goresky-MacPherson, which involve intersection homology with lower middle perversity on complex analytic varieties with Whitney stratifications. Also, our version of Morse functions is different from those of U. Luwig, who studied Witten’s perturbation for the special case of conformally conic manifolds. We hope there will be future applications, specially when we consider functions canonically associated to geometric or physical situations. In particular, our version of Morse inequalities applies to the case of a smooth action of a compact Lie group Gon a closed manifold M, and functions on the orbit type strata of G\Minduced by invariant Morse-Bott functions on Mwhose critical manifolds are orbits. This provides a rich family of examples. Several open problems emerge from our work: a “rel-Morse lemma”, a “relstrong C∞topology” where the rel-Morse functions should form a dense subset, a “rel-Sobolev lemma”, a version with “rel-Morse-Bott functions” , etc. But the main one is the possible generalization to other types of adapted metrics. We hope that even completely new types of adapted metrics could be tackled, which could correspond to generalizations of intersection homology still to be defined, whose 145
146 CONCLUSION perversities would be a sequence of functions instead of naturals. This would require a generalization of our results on the Dunkl harmonic oscillator to other kind of perturbations of the harmonic oscillator, which seem to be perfectly possible.
Resumen El principal objetivo de la tesis es usar el m´etodo de la perturbaci´on de Witten para probar una versi´on de las desigualdades de Morse para la condici´on ideal de frontera m´ınima y m´axima del complejo de de Rham en estratos, dotados con m´etricas adaptadas, donde se consideran estratificaciones de Thom-Mather compactas. Para lograrlo, se estudian primero estimaciones de autofunciones y resultados de embebimiento para el oscilador arm´onico de Dunkl en la recta, que se generalizan a otros operadores en R+. El estudio de estos operadores es el ingrediente clave en nuestro an´alisis local de la perturbaci´on de Witten. As´ı, esta tesis tiene dos partes principales, Partes 1 y 2. La primera est´a dedicada al estudio de estimaciones de autofunciones y resultados de embebimiento para el oscilador harm´onico de Dunkl y operadores relacionados. La segunda aborda el estudio de la perturbaci´on de Witten en estratos, en donde se usa la primera parte. Este trabajo aparece en los preprints [1, 2]. Pasamos a comentar los cap´ıtulos por separado y enunciar sus resultados principales. Estimaciones de autofunciones y teoremas de embebimiento El operador de Dunkl Tσen C∞(R), dependiendo de un par´ametro σ > −1/2, es la perturbaci´on de la derivada usual que se puede definir como Tσ=d dx en funciones pares y Tσ=d dx + 2σ1 xen funciones impares. Este tipo de operador, m´as generalmente en Rn, fue introducido por C.F. Dunkl [21, 22, 23, 24, 25]. Dio lugar a lo que ahora se denomina teor´ıa de Dunkl (v´ease el panorama presentado en [57]). Este ´area tuvo un gran desarrollo en los ´ultimos a˜nos, principalmente debido a sus aplicaciones en modelos cu´anticos de Calogero-Moser-Sutherland (v´ease por ejemplo [10, 52, 37, 38, 61, 3, 4]). En particular, el oscilador arm´onico de Dunkl [55, 26, 50, 49] es Lσ=−T2 σ+sx2, dependiendo de s > 0; es decir, se define usando Tσen vez de d/dx en la expresi´on del oscilador arm´onico usual, H=−d2 dx2+sx2. Por otra parte, sea pkla sucesi´on de polinomios ortogonales para la medida e−sx2|x|2σdx, considerados con norma uno y coeficiente principal positivo. Salvo normalizaci´on, ´estos son los polinomios de Hermite generalizados [59, p. 380, Problema 25]; v´ease tambi´en [16, 20, 27, 17, 55, 56]. Denotemos por xk,k < xk,k−1< ··· < xk,1las ra´ıces de cada pk; en particular, xk,k/2es la ra´ız positiva m´as peque˜na si kes par. Las correspondientes funciones de Hermite generalizadas son φk=pke−sx2/2. Se sabe que Lσ, con dominio el espacio de Schwartz S=S(R), es esencialmente auto-adjunto en L2(R,|x|2σdx). Adem´as el espectro de su extensi´on autoadjunta, denotada por Lσ, est´a formada por los autovalores (2k+ 1 + 2σ)s(k∈N), cuyas autofunciones correspondientes son las funciones φk. 147