Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 004, 16 pages Embedding Theorems for the Dunkl Harmonic Oscillator on the Line Jes´us A. ´ ALVAREZ L ´ OPEZ †and Manuel CALAZA ‡ †Departamento de Xeometr´ıa e Topolox´ıa, Facultade de Matem´aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain E-mail: jesus.alvar[email protected] ‡Laboratorio de Investigaci´on 10, Servicio de Reumatolog´ıa, Instituto de Investigaci´on Sanitaria, Hospital Cl´ınico Universitario, 15706 Santiago de Compostela, Spain E-mail: manuel.c[email protected] Received September 09, 2013, in final form January 06, 2014; Published online January 10, 2014 http://dx.doi.org/10.3842/SIGMA.2014.004 Abstract. Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line. Key words: Dunkl harmonic oscillator; Sobolev embedding; generalized Hermite functions; Schwartz space 2010 Mathematics Subject Classification: 46E35; 47B25; 33C45 1 Introduction The subindex ev/odd is added to any space of functions on Rto indicate its subspace of even/odd functions; in particular, C∞=C∞ ev ⊕C∞ odd for C∞:= C∞(R). The Dunkl operator Tσ(σ > −1/2) on C∞is the perturbation of d dxdefined by Tσ=d dxon C∞ ev and Tσ=d dx+ 2σ1 xon C∞ odd. The corresponding Dunkl harmonic oscillator is the perturbation Lσ=−T2 σ+s2x2of the harmonic oscillator H=−d2 dx2+s2x2(s > 0). The conjugation Eσ=|x|σTσ|x|−σon |x|σC∞is equal to d dx−σx−1on |x|σC∞ ev and d dx+σx−1on |x|σC∞ odd; note that |x|σC∞ ev/odd consists of even/odd functions, possibly not smooth or not even defined at 0. Up to the product by a constant, Eσwas introduced by Yang [39]. In the form Tσ, this operator was generalized to Rnby Dunkl [12,13,14], giving rise to what is now called Dunkl theory (see the survey [31]); in particular, the Dunkl harmonic oscillator on Rnwas studied in [15,25,26,29]. See [27] for further generalizations on R. Sometimes the terms Yang–Dunkl operator and Yang–Dunkl harmonic oscillator are used in the case of R[27]. Let pkbe 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 [32, p. 380, Problem 25]; see also [9,10,11,16,29,30]. The corresponding generalized Hermite functions are φk=pke−sx2/2. For each m∈N, let Smbe the Banach space of functions φ∈Cm(R) with supx|xiφ(j)(x)|<∞ for i+j≤m; the corresponding Fr´echet space S=TmSmis the Schwartz space on R. With domain S,Lσis essentially self-adjoint in L2(R,|x|2σdx), and the spectrum of its selfadjoint extension, Lσ, consists of the eigenvalues (2k+ 1 + 2σ)s(k∈N), with corresponding eigenfunctions φk[29]. For each real m≥0, let Wm σbe the Hilbert space completion of Swith respect to the scalar product hφ, ψiWm σ:= h(1 + Lσ)mφ, ψiσ, where h,iσdenotes the scalar product of L2(R,|x|2σdx), obtaining a Fr´echet space W∞ σ=TmWm σ. We show the following embedding theorems; the second one is of Sobolev type. 2 J.A. ´ Alvarez L´opez and M. Calaza Theorem 1.1. For each m∈N,SMm,ev/odd ev/odd ⊂Wm σ,ev/odd continuously, where Mm,ev/odd =   3m+ 3 2+m+ 1 4dσe(dσe+ 3) + dσeif σ≥0and mis odd, 2m+ 3 if σ < 0and mis odd, Mm,ev =   3m+ 2 2+m 4dσe(dσe+ 3) + dσeif σ≥0and mis even, 2m+ 2 if σ < 0and mis even, Mm,odd =   3m+ 4 2+m+ 2 4dσe(dσe+ 3) + dσeif σ≥0and mis even, 2m+ 4 if σ < 0and mis even. Theorem 1.2. For all m∈Nand mσ=m+1+1 2dσe(dσe+1),Wm0 σ,ev/odd ⊂ Sm ev/odd continuously if m0> Nm,ev/odd, where Nm,ev = 2 and Nm,odd = 5 if mσ= 1,Nm,ev = 6 and Nm,odd = 5 if mσ= 2,Nm,ev = 6 and Nm,odd = 7 if mσ= 3, and Nm,ev/odd =mσ+ 3 for mσ≥4. Corollary 1.3. S=W∞ σas Fr´echet spaces. In other words, Corollary 1.3 states that an element φ∈L2(R,|x|2σdx) is in Sif and only if the “Fourier coefficients” hφ, φkiσare rapidly decreasing on k. This also means that S=TmD(Lm σ) (Sis the smooth core of Lσwith the terminology of [6]) because the sequence of eigenvalues of Lσis in O(k) as k→ ∞. We introduce a version Sm σof every Sm, whose definition involves Tσinstead of d dx. They satisfy much simpler embeddings: Sdme+1 σ⊂Wm σ, and Wm0 σ⊂ Sm σif m0−m > 1. Even though S=TmSm σ, the inclusion relations between the spaces Sm σand Sm0are complicated, giving rise to the complexity of Theorems 1.1 and 1.2. Other Sobolev type embedding theorems, for different operators and with different techniques, were recently proved in [35,36,37]. Next, we consider other perturbations of Hon R+. Let Sev,U denote the space of restrictions of even Schwartz functions to some open U⊂R+, and set φk,U =φk|U. Theorem 1.4. Let P=H−2f1d dx+f2, where f1∈C1(U)and f2∈C(U)for some open U⊂R+of full Lebesgue measure. Assume that f2=σ(σ−1)x−2−f2 1−f0 1for some σ > −1/2. Let h=xσe−F1, where F1∈C2(U)is a primitive of f1. Then the following properties hold: (i)P, with domain hSev,U , is essentially self-adjoint in L2(U, e2F1dx); (ii)the spectrum of its self-adjoint extension, P, consists of the eigenvalues (4k+ 1 + 2σ)s (k∈N)with multiplicity one and normalized eigenfunctions √2hφ2k,U ; and (iii)the smooth core of Pis hSev,U . This theorem follows by showing that the stated condition on f1and f2characterizes the cases where Pcan be obtained by the following process: first, restricting Lσto even functions, then restricting to U, and finally conjugating by h. The term of Pwith d dxcan be removed by conjugation with the product of a positive function, obtaining the operator H+σ(σ−1)x−2; in this way, we get all operators of the form H+cx−2with c > −1/4. The conditions of Theorem 1.4 are satisfied by P=H−2c1x−1d dx+c2x−2(c1, c2∈R) on R+ if and only if there is some a∈Rsuch that a2+ (2c1−1)a−c2= 0 and a+c1>−1/2; in this case, h=xaand e2F1=x2c1. For some c1, c2∈R, there are two values of asatisfying these conditions, obtaining two different self-adjoint operators defined by Pin different Hilbert spaces. For instance, Lσmay define a self-adjoint operator when σ≤ −1/2. Embedding Theorems for the Dunkl Harmonic Oscillator on the Line 3 This example is applied in [2] to prove a new type of Morse inequalities on strata of compact stratifications [20,33,34] with adapted metrics [5,23,24], where the Witten’s perturbation [38] is used for the minimal/maximal ideal boundary conditions of de Rham complex [6,7,8]. The version of Morse functions used in [2] is different from the version of Goresky–MacPherson [18]. More precisely, in the local conic model of a stratification around each critical point, by induction on the depth of the stratification, it is assumed that the Laplacian of the minimal/maximal boundary condition of the de Rham complex of the link (section of the cone) has a nice spectral decomposition. Using this, the Witten’s perturbation of the de Rham complex of the cone splits into an infinite direct sum of elliptic complexes of two simple types, with length one and two, which represent the radial direction of Witten’s perturbed complex. It turns out that the Witten’s perturbed Laplacian of these simple complexes is described by the above operator P, and the two possible choices of the constant agive rise to the minimal/maximal ideal boundary conditions. In this way, Theorem 1.4 becomes a key ingredient of [2]. 2 Preliminaries 2.1 Dunkl operator on the line For any φ∈C∞:= C∞(R), there exists some ψ∈C∞so that φ(x)−φ(0) = xψ(x); moreover ψ(m)(x) = Z1 0 tmφ(m+1)(tx)dt(1) for all1m∈N(see e.g. [19, Theorem 1.1.9]). Let us use the notation ψ=x−1φ. The Dunkl operator on Tσ(σ∈R) on C∞is the perturbation of d dxdefined by (Tσφ)(x) = φ0(x)+2σφ(x)−φ(−x) x. Consider matrix expressions of operators on C∞with respect to the decomposition C∞= C∞ ev ⊕C∞ odd, as direct sum of subspaces of even and odd functions. For each function h, the notation his also used for the operator of multiplication by h. Then d dx=   0d dx d dx0  , x =0x x0, Tσ=   0d dx+ 2σx−1 d dx0  =d dx+ 2σ0x−1 0 0  on C∞. With Σ = σ0 0−σ, we get [Tσ, x] = 1 + 2Σ, TσΣ+ΣTσ=xΣ+Σx= 0.(2) 2.2 Dunkl harmonic oscillator on the line On C∞, the harmonic oscillator, and the annihilation and creation operators are H=−d2 dx2+ s2x2,A=sx +d dxand A0=sx −d dx(s > 0). Their perturbations L=−T2 σ+s2x2,B=sx +Tσ 1We adopt the convention 0 ∈N. 4 J.A. ´ Alvarez L´opez and M. Calaza and B0=sx −Tσare called Dunkl harmonic oscillator, and Dunkl annihilation and creation operators. By (2), L=BB0−(1 + 2Σ)s=B0B+ (1 + 2Σ)s=1 2(BB0+B0B),(3) [L, B] = −2sB, [L, B0]=2sB0,(4) [B, B0] = 2s(1 + 2Σ),(5) [L, Σ] = BΣ+ΣB=B0Σ+ΣB0= 0.(6) For each m∈N, let Smbe the space of functions φ∈C∞such that kφkSm=X i+j≤m sup x|xiφ(j)(x)|<∞. This expression defines a norm k kSmon Sm, which becomes a Banach space. We have Sm+1 ⊂ Smcontinuously2, and S=TmSm, with the induced Fr´echet topology, is the Schwartz space on R. Note that kφ0kSm≤ kφkSm+1 for all m. We can restrict the above decomposition of C∞to every Smand S, obtaining Sm=Sm ev⊕Sm odd and S=Sev ⊕Sodd. The matrix expressions of operators on Sare taken with respect to this decomposition. For φ∈C∞ ev ,ψ=x−1φand i, j ∈N, we get from (1) 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. So 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. Hence x:Sev → Sodd is an isomorphism of Fr´echet spaces, and Tσ,B,B0and Ldefine continuous operators on S. Let h,iσand k kσbe the scalar product and norm of L2(R,|x|2σdx). Suppose from now on that σ > −1/2, obtaining that Sis dense in L2(R,|x|2σdx). The following properties hold considering these operators in L2(R,|x|2σdx) with domain S:−Tσis adjoint of Tσ,B0is adjoint of B, and Lis essentially self-adjoint. Let L, or Lσ, denote the self-adjoint extension of L(with domain S). 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,(7) φk=((2ks)−1/2B0φk−1if kis even, (2(k+ 2σ)s)−1/2B0φk−1if kis odd,k≥1.(8) Furthermore Bφ0= 0,(9) Bφk=((2ks)1/2φk−1if kis even, (2(k+ 2σ)s)1/2φk−1if kis odd,k≥1.(10) These properties follow from (3)–(6), like in the case of H. 2For topological vector spaces Xand Y, it is said that X⊂Ycontinuously if Xis a linear subspace of Yand the inclusion map X ,→Yis continuous. Embedding Theorems for the Dunkl Harmonic Oscillator on the Line 5 2.3 Generalized Hermite polynomials By (7), (8) and the definition of B0, we get φk=pke−sx2/2, where pkis the sequence of polynomials inductively given by p0=s(2σ+1)/4Γ(σ+ 1/2)−1/2and pk=((2ks)−1/2(2sxpk−1−Tσpk−1) if kis even, (2(k+ 2σ)s)−1/2(2sxpk−1−Tσpk−1) if kis odd,k≥1.(11) Up to normalization, pkand φkare the generalized Hermite polynomials and functions [32, p. 380, Problem 25]. Each pkis of degree k, even/odd if kis even/odd, and with positive leading coefficient. Moreover Tσp0= 0 and Tσpk=((2ks)1/2pk−1if kis even, (2(k+ 2σ)s)1/2pk−1if kis odd,k≥1.(12) From (11) and (12), we obtain the recursion formula pk=(k−1/2(2s)1/2xpk−1−(k−1+2σ)1/2pk−2if kis even, (k+ 2σ)−1/2(2s)1/2xpk−1−(k−1)1/2pk−2if kis odd.(13) By (13) and induction on k, we easily get the following when kis odd3: 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`.(14) The following theorem contains a simplified version of the asymptotic estimates satisfied by φkand ξk=|x|σφk. They can be obtained by expressing pnin terms of the Laguerre polynomials [30,31], whose asymptotic estimates are studied in [3,17,21,22]. The method of Bonan–Clark [4] can be also applied [1]. Theorem 2.1. There exist C, C0, C00 >0, depending on σand s, such that: (i)if kis odd or σ≥0, then ξ2 k(x)≤C0k−1/6for all x∈R; (ii)if kis even and positive, and σ < 0, then ξ2 k(x)≤C00k−1/6for |x| ≥ 1; and (iii)if σ < 0, then φ2 k(x)≤C00 for all kand |x| ≤ 1. 3 Perturbed Schwartz space We introduce a perturbed version Sm σof each Smthat will be appropriate to show our embedding results. Since Sm σmust contain the functions φk, Theorem 2.1 indicates 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)|.(15) This defines a norm k kSm σon the linear space of functions φ∈C∞with kφkSm σ<∞, and let Sm σ denote the corresponding Banach space completion. There are direct sum decompositions into subspaces of even and odd functions, Sm σ=Sm σ,ev ⊕Sm σ,odd. 3As a convention, the product of an empty set of factors is 1. Thus (k−1)(k−3) · · · (`+ 2) = 1 for `=k−1 in (14). 6 J.A. ´ Alvarez L´opez and M. Calaza When σ < 0, the even and odd functions are considered separately: let kφkSm σ=X i+j≤m, i+jeven sup |x|≤1|xi(Tj σφ)(x)|+ sup |x|≥1|x|σ|xi(Tj σφ)(x)|! +X i+j≤m, i+jodd sup x6=0 |x|σ|xi(Tj σφ)(x)|(16) for φ∈C∞ ev , and 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|≤1|xi(Tj σφ)(x)|+ sup |x|≥1|x|σ|xi(Tj σφ)(x)|!(17) for φ∈C∞ odd. These expressions define a norm kkSm σon the linear spaces of functions φ∈C∞ ev/odd with kφkSm σ<∞. The corresponding Banach space completions will be denoted by Sm σ,ev/odd, and let Sm σ=Sm σ,ev ⊕Sm σ,odd. In any case, there are continuous inclusions Sm+1 σ⊂ Sm σ, and a perturbed Schwartz space is defined as Sσ=TmSm σ, with the corresponding Fr´echet topology, which decomposes as direct sum of the subspaces of even and odd functions, Sσ=Sσ,ev ⊕Sσ,odd; in particular, S0=S. It easily follows that Sσconsists of functions that are C∞on R\ {0}but a priori possibly not even defined at zero, and Sm σ∩C∞is dense in Sm σfor all m; thus Sσ∩C∞is dense in Sσ. Obviously, Σ defines a bounded operator on each Sm σ. It is also easy to see that Tσdefines a bounded operator Sm+1 σ→ Sm σfor any m; notice that, when σ < 0, the role played by the parity of i+jfits well to prove this property. Similarly, xdefines a bounded operator Sm+1 σ→ Sm σfor any mbecause, by (2), [Tj σ, x] = (jTj−1 σif jis even, (j+ 2Σ)Tj−1 σif jis odd. So Band B0define bounded operators Sm+1 σ→ Sm σ, and La bounded operator Sm+2 σ→ Sm σ. Thus Tσ,x, Σ, B,B0and Ldefine continuous operators on Sσ. In order to prove Theorems 1.1 and 1.2, we introduce an intermediate weakly perturbed Schwartz space Sw,σ. Like Sσ, it is a Fr´echet space of the form Sw,σ =TmSm w,σ, where each Sm w,σ is the Banach space defined like Sm σby using d dxinstead of Tσin the right hand sides of (15)–(17); in particular, S0 w,σ =S0 σas Banach spaces. Let k kSm w,σ denote the norm of Sm w,σ. As before, Sw,σ consists of functions which are C∞on R\{0}but a priori possibly not even defined at zero, Sw,σ ∩C∞is dense in Sw,σ, there is a canonical decomposition Sw,σ =Sw,σ,ev ⊕Sw,σ,odd, and d dx and xdefine bounded operators on Sm+1 w,σ → Sm w,σ. Thus d dxand xdefine continuous operators on Sw,σ. Lemma 3.1. If σ≥0, then Sm+dσe⊂ Sm w,σ continuously. Proof. Let φ∈ S. For all iand j, we have |x|σ|xiφ(j)(x)| ≤ |xi+dσeφ(j)(x)|for |x| ≥ 1, and |x|σ|xiφ(j)(x)|≤|xiφ(j)(x)|for |x| ≤ 1. So kφkSm w,σ ≤ kφkSm+dσe. Lemma 3.2. If σ≥0,Smσ w,σ ⊂ Smcontinuously, where mσ=m+1+1 2dσe(dσe+ 1). Proof. Let φ∈ Sw,σ. For all iand j, xiφ(j)(x)≤ |x|σxiφ(j)(x)(18) for |x| ≥ 1. It remains to prove an inequality of this type for |x| ≤ 1, which will be a consequence of the following assertion. Embedding Theorems for the Dunkl Harmonic Oscillator on the Line 7 Claim 1. For each n∈N, there are finite families of real numbers, cn a,b,dn k,` and en u,v, where the indices a,b,k,`,uand vrun in finite subsets of Nwith b, `, v ≤Mn= 1 + n(n+1) 2and k≥n, such that, for all φ∈C∞, φ(x) = X a,b cn a,bxaφ(b)(1) + X k,` dn k,`xkφ(`)(x) + X u,v en u,vxuZ1 x tnφ(v)(t)dt. Assuming that Claim 1is true, the proof can be completed as follows. Let φ∈ Sw,σ and set n=dσe. For |x| ≤ 1, according to Claim 1,|φ(x)|is bounded by X a,b |cn a,b|φ(b)(1)+X k,` |dn k,`|xkφ(`)(x)+X u,v |en u,v|2 max |t|≤1tnφ(v)(t) ≤X i,j |cn a,b|φ(b)(1)+X k,` |dn k,`||x|σφ(`)(x)+X u,v |en u,v|2 max |t|≤1|t|σφ(v)(t). Let m, i, j ∈Nwith i+j≤m. By applying the above inequality to the function xiφ(j), and expressing each derivative (xiφ(j))(r)as a linear combination of functions of the form xpφ(q)with p+q≤i+j+r, it follows that there is some C≥1, depending only on σand m, such that xiφ(j)(x)≤CkφkSi+j+Mn w,σ (19) for |x| ≤ 1. By (18) and (19), kφkSm≤CkφkSmσ w,σ because mσ=m+Mn. Now, let us prove Claim 1. By induction on nand using integration by parts, it is easy to prove that Z1 x tnφ(n+1)(t)dt= n X r=0 (−1)n−rn! r!φ(r)(1) −xrφ(r)(x).(20) This shows directly Claim 1for n∈ {0,1}. Proceeding by induction, let n≥2 and assume that Claim 1holds for n−1. By (20), it is enough to find appropriate expressions of xrφ(r)(x) for 0< r < n. For that purpose, apply Claim 1for n−1 to each function φ(r), and multiply the resulting equality by xrto get xrφ(r)(x) = X a,b cn−1 a,b xr+aφ(r+b)(1) + X k,` dn−1 k,` xr+kφ(r+`)(x) +X u,v en−1 u,v xr+uZ1 x tn−1φ(r+v)(t)dt, where a,b,k,`,uand vrun in finite subsets of Nwith b, `, v ≤Mn−1and k≥n−1; thus r+k≥nand r+b, r +`, r +v≤n−1 + Mn−1=Mn−1. Therefore it only remains to rise the exponent of tby a unit in the integrals of the last sum. Once more, integration by parts makes the job: Z1 x tnφ(r+v+1)(t)dt=φ(r+v)(1) −xnφ(r+v)(x)−nZ1 x tn−1φ(r+v)dt.  Lemma 3.3. If σ < 0, then Sm+1 ⊂ Sm w,σ continuously. Proof. This is proved by induction on m. For φ∈C∞ ev and |x| ≥ 1, we have |x|σ|φ(x)|≤|φ(x)|, obtaining k kS0 w,σ ≤ k kS0on C∞ ev . On the other hand, for φ∈C∞ odd and ψ=x−1φ∈C∞ ev , we get |x|σ|φ(x)| ≤ (|ψ(x)|if 0 <|x| ≤ 1, |φ(x)|if |x| ≥ 1. So kφkS0 w,σ ≤max{kφkS0,kψkS0} ≤ kφkS1by (1). 8 J.A. ´ Alvarez L´opez and M. Calaza Now, assume that m≥1 and the result holds for m−1. Let i, j ∈Nsuch that i+j≤m, and let φ∈ Sev ∪Sodd. Independently of the parity of φand i+j, we have |x|σ|xiφ(j)(x)| ≤ |xiφ(j)(x)| for |x| ≥ 1. Suppose that φ∈ Sev. If i= 0 and jis odd, then φ(j)∈ Sodd. Thus there is some ψ∈ Sev such that φ(j)=xψ, obtaining |x|σ|φ(j)(x)| ≤ |ψ(x)|for 0 <|x| ≤ 1. If i+jis odd and i > 0, then |x|σ|xiφ(j)(x)| ≤ |xi−1φ(j)(x)|for 0 <|x| ≤ 1. Hence, by (1), there is some C > 0, independent of φ, such that kφkSm w,σ ≤Cmax{kφkSm,kψkS0} ≤ CmaxkφkSm,kφ(j)kS1≤CkφkSm+1 . Finally, assume φ∈ Sodd, and let ψ=x−1φ∈ Sev. If iis even and j= 0, then |x|σ|xiφ(x)| ≤ |xiψ(x)|for 0 <|x| ≤ 1. If i+jis even and j > 0, then |x|σxiφ(j)(x)≤xiψ(j)(x)+j|x|σxiψ(j−1)(x) for 0 <|x| ≤ 1 because [ dj dxj, x] = jdj−1 dxj−1. Therefore, by (1) and the induction hypothesis, there are some C0, C00 >0, independent of φ, such that kφkSm w,σ ≤C0maxkφkSm,kψkSm+kψkSm−1 w,σ ≤C00kφkSm+1 . Lemma 3.4. If σ < 0, then Sm+1 w,σ ⊂ Smcontinuously. Proof. Let i, j ∈Nsuch that i+j≤m. Since xiφ(j)(x)≤(|x|σxiφ(j)(x)if 0 <|x| ≤ 1, |x|σxi+1φ(j)(x)if |x| ≥ 1. for any φ∈C∞, we get kφkSm≤ kφkSm+1 w,σ . Corollary 3.5. S=Sw,σ as Fr´echet spaces. Corollary 3.6. x−1defines a bounded operator Sm0 w,σ,odd → Sm w,σ,ev, where m0=(m+2+1 2dσe(dσe+ 3) if σ≥0, m+ 3 if σ < 0. Proof. If σ≥0, the composite Sm+2+1 2dσe(dσe+3) w,σ,odd ,→ Sm+dσe+1 odd x−1 −−−−→ Sm+dσe ev ,→ Sm w,σ,ev is bounded by Lemmas 3.1 and 3.2. If σ < 0, the composite Sm+3 w,σ,odd ,→ Sm+2 odd x−1 −−−−→ Sm+1 ev ,→ Sm w,σ,ev, is bounded by Lemmas 3.3 and 3.4. Corollary 3.7. x−1defines a continuous operator Sw,σ,odd → Sw,σ,ev. Lemma 3.8. SMm,ev/odd w,σ,ev/odd ⊂ Sm σ,ev/odd continuously, where Mm,ev/odd =   3m 2+m 4dσe(dσe+ 3) if σ≥0and mis even, 2mif σ < 0and mis even, Embedding Theorems for the Dunkl Harmonic Oscillator on the Line 9 Mm,ev =   3m−1 2+m−1 4dσe(dσe+ 3) if σ≥0and mis odd, 2m−1if σ < 0and mis odd, Mm,odd =   3m+ 1 2+m+ 1 4dσe(dσe+ 3) if σ≥0and mis odd, 2m+ 1 if σ < 0and mis odd. Proof. This follows by induction on m. It is true for m= 0 because S0 w,σ =S0 σas Banach spaces. Now, let m≥1, and assume that the result holds for m−1. For φ∈C∞ ev ,i+j≤mwith j > 0, and x∈R, we have |xi(Tj σφ)(x)|=|xi(Tj−1 σφ0)(x)| with φ0∈C∞ odd, obtaining kφkSm σ≤ kφ0kSm−1 σ. But, by the induction hypothesis and since Mm,ev =Mm−1,odd + 1, there are some C, C0>0, independent of φ, such that kφ0kSm−1 σ≤Ckφ0kSMm−1,odd w,σ ≤C0kφkSMm,ev w,σ . For φ∈C∞ odd, let ψ=x−1φ, and take i,jand xas above. Then xi(Tj σφ)(x)≤xi(Tj−1 σφ0)(x)+ 2|σ|xiTj−1 σψ(x) with φ0, ψ ∈C∞ ev , obtaining kφkSm σ≤ kφ0kSm−1 σ+2|σ|kψkSm−1 σ. But, by the induction hypothesis, Corollary 3.6, and since Mm,odd =(Mm−1,ev +2+1 2dσe(dσe+ 3) if σ≥0, Mm−1,ev + 3 if σ < 0, there are some C, C0>0, independent of φ, such that kφ0kSm−1 σ+ 2|σ|kψkSm−1 σ≤Ckφ0kSMm−1,ev w,σ +kψkSMm−1,ev w,σ ≤C0kφkSMm,odd w,σ . Corollary 3.9. Sw,σ ⊂ Sσcontinuously. Corollary 3.10. SM0 m,ev/odd ev/odd ⊂ Sm σ,ev/odd continuously, where, with the notation of Lemma 3.8, M0 m,ev/odd =(Mm,ev/odd +dσeif σ≥0, Mm,ev/odd + 1 if σ < 0. Proof. This follows from Lemmas 3.1,3.3 and 3.8. 4 Perturbed Sobolev spaces Observe that Sσ⊂L2(R,|x|2σdx). Like in the case where Sis considered as domain, it is easy to check that, in L2(R,|x|2σdx), with domain Sσ,Bis adjoint of B0and Lis symmetric. Lemma 4.1. Sσis a core4of L. Proof. Let Rdenote the restriction of Lto Sσ. Then L ⊂ R⊂R∗⊂ L∗=Lin L2(R,|x|2σdx) because S ⊂ Sσby Corollaries 3.5 and 3.9. 4Recall that a core of a closed densely defined operator Tbetween Hilbert spaces is any subspace of its domain D(T) which is dense with the graph norm. 16 J.A. ´ Alvarez L´opez and M. Calaza [29] Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkh¨auser, Basel, 1994, 369–396, math.CA/9307224. 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