Witten’s perturbation on strata with general adapted metrics

WITTEN’S PERTURBATION ON STRATA WITH GENERAL
ADAPTED METRICS
JES´
US A. ´
ALVAREZ L ´
OPEZ, MANUEL CALAZA, AND CARLOS FRANCO
Abs ac . Le Mbe a s a um o a compac s a i ied space A. I is equipped
wi h a gene al adap ed me ic g, which is sligh ly mo e gene al han he
adap ed me ics o Nagase and B assele -Hec o -Sa alegi. In pa icula , g
has a gene al ype, which is an ex ension o he ype o an adap ed me ic.
A es ic ion on his gene al ype is assumed, and hen gis called good. We
conside he maximum/minimum ideal bounda y condi ion, dmax/min, o he
compac ly suppo ed de Rham complex on M, in he sense o B ¨uning-Lesch.
Le H∗
max/min(M) and ∆max/min deno e he cohomology and Laplacian o
dmax/min. The i s main heo em s a es ha ∆max/min has a disc e e spec-
um sa is ying a weak o m o he Weyl’s asymp o ic o mula. The second
main heo em is a e sion o Mo se inequali ies using H∗
max/min(M) and wha
we call el-Mo se unc ions. An ing edien o he p oo s o bo h heo ems is
a e sion o dmax/min o he Wi en’s pe u ba ion o he de Rham complex.
Ano he ing edien is ce ain pe u ba ion o he Dunkl ha monic oscilla o
p e iously s udied by he au ho s using classical pe u ba ion heo y.
The condi ion on g o be good is gene al enough in he ollowing sense.
Assume ha Ais a s a i ied pseudomani old, and conside i s in e sec ion
homology I¯pH∗(A) wi h pe e si y ¯p; in pa icula , he lowe and uppe middle
pe e si ies a e deno ed by ¯mand ¯n, espec i ely. Then, o any pe e si y
¯p≤¯m, he e is an associa ed good adap ed me ic on Msa is ying he Nagase
isomo phism H
max(M)∼
=I¯pH (A)∗( ∈N). I Mis o ien ed and ¯p≥¯n, we
also ge H
min(M)∼
=I¯pH (A). Thus ou e sion o he Mo se inequali ies can
be desc ibed in e ms o I¯pH∗(A).
Con en s
1. In oduc ion 2
2. P elimina ies 11
3. A pe u ba ion o he Dunkl ha monic oscilla o 15
4. Two simple ypes o ellip ic complexes 19
5. Wi en’s pe u ba ion on a cone 32
6. Spli ing o he Wi en’s complex on a cone 34
7. Rela i ely local model o he Wi en’s pe u ba ion 40
8. P oo o Theo em 1.1 41
9. Func ional calculus 41
1991 Ma hema ics Subjec Classi ica ion. 58A14, 32S60.
Key wo ds and ph ases. Mo se inequali ies, ideal bounda y condi ion, s a i ica ion, gene al
adap ed me ic, Wi en’s pe u ba ion.
The i s au ho is pa ially suppo ed by MICINN, G an MTM2011-25656, and by MEC,
G an MTM2014-56950-P.
The hi d au ho has ecei ed inancial suppo om he Xun a de Galicia and he Eu opean
Union (Eu opean Social Fund - ESF).
1
2 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
10. The wa e ope a o 42
11. P oo o Theo em 1.2 42
Re e ences 44
1. In oduc ion
1.1. Ideal bounda y condi ions o he de Rham complex. The ollowing
usual no a ion is used o a densely de ined linea ope a o Tin a Hilbe space. I s
domain and ange a e deno ed by D(T) and R(T). I Tis essen ially sel -adjoin ,
i s closu e is deno ed by T. I Tis sel -adjoin , i s smoo h co e is D∞(T) :=
T∞
m=1 D(Tm), and i s spec um is deno ed by σ(T).
AHilbe complex (D,d) is a di e en ial complex o ini e leng h gi en by a
densely de ined closed ope a o din a g aded sepa able Hilbe space H[9]. Then
he ope a o D=d+d∗, wi h D(D) = D(d)∩D(d∗), is sel -adjoin in H, and
he e o e he Laplacian ∆=D2=dd∗+d∗dis also sel -adjoin . Mo eo e D∞(∆)
is a subcomplex o (D,d) wi h he same homology [9, Theo em 2.12]; i may be
also said ha D∞(∆) is he smoo h co e o d.
The abo e no ion is applied he e in he ollowing case. Fo a Riemannian man-
i old M, le Ω0(M) be he space o compac ly suppo ed di e en ial o ms, and
L2Ω(M) he g aded Hilbe space o squa e in eg able di e en ial o ms. Le dand
δbe he de Rham de i a i e and code i a i e ac ing on Ω0(M), and le D=d+δ
and ∆ = D2=dδ +δd ( he Laplacian). E e y Hilbe complex ex ension d
o din L2Ω(M) is called an ideal bounda y condi ion (i.b.c.) [9], gi ing ise o
sel -adjoin ex ensions Dand ∆o Dand ∆ in L2Ω(M). The e exis s a mini-
mum/maximum i.b.c., dmin =dand dmax =δ∗, inducing sel -adjoin ex ensions
Dmax/min and ∆max/min o Dand ∆. I Mis o ien ed, hen ∆max co esponds o
∆min by he Hodge s a ope a o . The co esponding cohomologies, Hmax/min(M),
a e quasi-isome ic in a ian s o M; o ins ance, Hmax(M) is he usual L2co-
homology H(2)(M) [12]. They gi e ise o e sions o Be i numbe s and Eule
cha ac e is ic, β
max/min =β
max/min(M) and χmax/min =χmax/min(M). These con-
cep s can indeed be de ined o a bi a y ellip ic complexes [9]. I is well known ha
dmin =dmax i Mis comple e. Thus conside ing an i.b.c. becomes in e es ing when
Mis no comple e. Fo example, i Mis he in e io o a compac Riemannian
mani old Nwi h wi h ∂N 6=∅, hen dmax/min is de ined by aking absolu e/ ela i e
bounda y condi ions. Wi h mo e gene ali y, we will assume ha Mis a s a um o
a compac s a i ied space A[41, 31, 32, 42], equipped wi h a gene aliza ion o he
adap ed me ics conside ed in [33, 34, 8]. As we will see, we can assume M=Ai
desi ed (i can be said ha Mis he egula s um in his case).
1.2. S a i ied spaces. Roughly speaking, a (Thom-Ma he )s a i ied space (o
s a i ica ion) is a Hausdo , locally compac and second coun able space Aequipped
wi h a pa i ion in o C∞mani olds ( he s a a), sa is ying ce ain condi ions [41,
31]. In pa icula , an o de ela ion on he amily o s a a is de ined by decla -
ing X≤Ywhen X⊂Y. Wi h espec o his o de ing, he maximum leng h
o chains o s a a less o equal han a s a um Xis called he dep h o X. The
sup emum o he s a a dep h is called he dep h o A, deno ed dep h A. The p e-
cise de ini ion and needed p elimina ies we e collec ed in [4, Sec ion 3], whe e we
WITTEN’S PERTURBATION ON STRATA 3
ha e mainly ollowed [42]. Ins ead o ecalling i , le us desc ibe how he s a a o
A i oge he , desc ibing also mo phisms/isomo phisms o s a i ica ions, and, in
pa icula , he g oup o au omo phisms, Au (A). We p oceed by induc ion on i s
dep h. I dep h A= 0, hen Ais jus a C∞mani old, and Au (A) consis s o i s
di eomo phisms.
Now, gi en any k∈Z+, assume ha any s a i ied space Lis desc ibed i
dep h L < k, as well as Au (L). I Lis compac , he cone wi h link Lis c(L) =
(L×[0,∞))/(L×{0}), whose e ex is he poin ∗=L×{0} ∈ c(L). Le L0be
ano he compac s a i ica ion o dep h < k, and φ:L→L0a mo phism. Then
le c(φ) : c(L)→c(L0) be he map induced by φ×id : L×[0,∞)→L0×[0,∞);
in pa icula , we ge he g oup c(Au (L)) = {c(φ)|φ∈Au (L)}. I is also
decla ed ha c(∅) = {∗}, o he emp y s a i ica ion, and c(∅) = id, o he emp y
map. The cone c(L) is used as a model s a i ied space o dep h ki Lis o dep h
k−1, whose s a a a e {∗} and he mani olds Y×R+ o s a a Yo L. The
second ac o p ojec ion L×[0,∞)→[0,∞) de ines a c(Au (L))-in a ian unc ion
ρ:c(L)→[0,∞), called he adial unc ion. The es ic ions o ρ o he s a a
a e C∞. A conic bundle is a ibe bundle To e a mani old Xwi h ypical ibe
c(L) and s uc u al g oup c(Au (L)). Then ρinduces a adial unc ion on T, also
deno ed by ρ, and he e ex o c(L) de ines he e ex sec ion o T, whose image
is iden i ied wi h X. Mo eo e he s a i ied s uc u e de ined on c(L) can be used
o de ine a s a i ied s uc u e on T, whe e Xbecomes he e ex s a um.
Fo any s a i ica ion Ao dep h k, e e y s a um Xhas an open neighbo hood
(a ube ep esen a i e) ha is isomo phic o an open neighbo hood o Xin some
conic bundle TXo e X(wi h he ob ious es ic ions o s a i ied s uc u es o
open subse s). The ypical ibe o TXis o he o m c(LX) o some compac
s a i ica ion LX( he link o X) wi h dep h LX<dep h A. The e ex and adial
unc ion o c(LX) a e deno ed by ∗Xand ρX. Two such neighbo hoods o X
ep esen he same ube i hei s uc u e is equal on some smalle neighbo hood o
X. No e ha Xis open in Ai and only i LX=∅.
Finally, a mo phism be ween wo s a i ica ions is a con inuous map sending
e e y s a um o ano he s a um, whose es ic ions o he s a a a e C∞, and
whose es ic ions o small enough ube ep esen a i es a e es ic ions o conic
bundle mo phisms. Then isomo phisms and au omo phisms o s a i ica ions ha e
he ob ious meaning. This comple es he desc ip ion because he dep h is locally
ini e by he local compac ness.
The ( opological) dimension o a s a i ica ion Aequals he sup emum o he
dimensions o i s s a a. I may be in ini e, bu i is locally ini e. The codimension
o e e y s a um Xis dim A−dim X. Ou main esul s will assume ha he
s a i ica ion is compac , bu non-compac s a i ica ions will be also used in he
p oo s. In any case, we will only conside s a i ica ions o ini e dimension. I he
abo e desc ip ion o Ais modi ied by equi ing ha , a e e y induc i e s ep, only
s a i ica ions wi h no s a a o codimension 1 a e used, hen Ais called a s a i ied
pseudomani old.
A locally closed subse B⊂Ais called a subs a i ica ion o Ai he es ic ions
o he s a a and ubes o A o Bde ine a s a i ied s uc u e on B. Fo ins ance,
Acan be es ic ed o any open subse , o any locally closed union o s a a, and o
he closu e o any s a um. I mo eo e he e a e ube ep esen a i es o Awhose
es ic ions o Bha e he same ibe s o e poin s o B, hen Bis called sa u a ed.
4 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
Le xbe a poin o a s a um Xo dimension mXin a s a i ica ion A. A local
i ializa ion o TXon some open neighbo hood Uo xde ines a cha O≡O0
o A o some open O0⊂RmX×c(LX). We can assume O0=U0×c(LX),
whe e U0is some open neighbo hood o 0 in RmXand c(LX) is he subse o
c(LX) de ined by he condi ion ρX< , o some  > 0. This cha is said o
be cen e ed a xi x≡(0,∗X)∈O0. The co esponding concep o a las has
he ob ious meaning. These concep s can be gene alized as ollows. Any ini e
p oduc o s a i ica ions has a non-canonical s a i ied s uc u e [4, Sec ion 3.1.2];
in pa icula , any ini e p oduc o cones is isomo phic o a cone [4, Lemma 3.8].
Mo eo e Au (P)×Au (Q) is canonically injec ed in Au (P×Q) o s a i ica ions
Pand Q. Thus i makes sense o conside a decomposi ion c(LX)∼
=QaX
i=1 c(LX,i)
(aX∈N), o compac s a i ica ions LX,i. The e ex and adial unc ion o
e e y c(LX,i) a e deno ed by ∗X,i and ρX,i. Then we can also conside gene al
ube ep esen a i es gi en by bundles TXwi h ypical ibe s QaX
i=1 c(LX,i) and
s uc u al g oups QaX
i=1 c(Au (LX,i)). This gi es ise o a gene al cha O≡O0
a ound x o some open O0⊂RmX×QaX
i=1 c(LX,i), which is cen e ed a xi
x≡(0,∗X,1,...,∗X,aX)∈O0. As abo e, we can assume O0=U0×QaX
i=1 c(LX,i)
o some  > 0. Le ρX,0deno e he no m unc ion on RmX. The unc ion
ρ= (ρ2
X,0+··· +ρ2
X,aX)1/2is called he adial unc ion o RmX×QaX
i=1 c(LX,i),
e en hough, when mX= 0, ρis no he adial unc ion o any cone s uc u e on
QaX
i=1 c(LX,i) [4, Example 3.6 and P oo o Lemma 3.8]. A collec ion o gene al
cha s co e ing Ais called a gene al a las.
We can suppose ha he s a a o Aa e connec ed [4, Rema k 1 ( )]. Fix
a s a um Mo dimension nin A. Since he s a i ied s uc u e o Acan be
es ic ed o M[4, Sec ion 3.1.1], we can also assume wi hou loss o gene ali y
ha M=A(any o he s a um is < M); in pa icula , dep h A= dep h Mand
dim A=n. Wi h he abo e no a ion, o a cha O≡O0cen e ed a x, we ge
M∩O≡M0∩O0, whe e M0=RmX×N×R+ o some dense s a um Non LX.
In he case o a gene al cha O≡O0cen e ed a x, we ha e M∩O≡M0∩O0 o
M0=RmX×QaX
i=1(Ni×R+), whe e e e y Niis some dense s a um o LX,i. We
will use he no a ion kX,i = dim Ni+ 1.
1.3. Gene al adap ed me ics. Agene al adap ed me ic gon Mis de ined by
induc ion on he dep h o M. I is any (Riemannian) me ic i dep h M= 0. Now,
assume ha dep h M > 0 and gene al adap ed me ics a e de ined o lowe dep h.
Gi en any gene al cha O≡O0as abo e, ake any gene al adap ed me ic ˜gion
e e y Ni(dep h Ni<dep h M), and le gi=ρ2uX,i
X,i ˜gi+ (dρX,i)2on Ni×R+ o
some uX,i >0. Le also g0be he Euclidean me ic on RmX. Then gis a gene al
adap ed me ic i , ia any such gene al cha , g|Ois quasi-isome ic o (PaX
i=0 gi)|O0.
In his case, he mapping X7→ uX:= (uX,1, . . . , uX,aX)∈RaX
+(X < M) is called
he gene al ype o g. Such a gene al cha is called compa ible wi h g, o wi h i s
gene al ype.
Le us poin ou ha a gene al me ic does no comple ely de e mine i s gene al
ype. Fo ins ance, suppose uX,i =uX,j = 1 o indices i6=j. W i e c(LX,i)×
c(LX,j)≡c(L), wi h adial unc ion ρ, o some s a i ica ion L. Then Ni×R+×
Nj×R+≡N×R+ o some dense s a um No L. Mo eo e he e is a gene al
adap ed me ic ˜gon Nsuch ha gi+gjis quasi-isome ic o ρ2˜g+ (dρ)2 ia he
abo e iden i y. The e o e we can omi uX,i o uX,j in uX, ob aining a di e en
ype o g. This canno be done i uX,i =uX,j 6= 1 (P oposi ion 2.1).
WITTEN’S PERTURBATION ON STRATA 5
I he abo e de ini ion o gene al adap ed me ic is modi ied by equi ing ha ,
a e e y induc i e s ep, he gene al ype sa is ies uX,i ≤1 o all X < M and
i= 1, . . . , aX, hen he gene al adap ed me ic is called good o he scope o
his pape . On he o he hand, i he de ini ion is modi ied by equi ing a e e y
induc i e s ep ha aX= 1 and uXdepends only on k:= kX,1= codim X o all
X < M, hen we ge he adap ed me ics conside ed in [33, 34, 8]. In his case,
he gene al cha s compa ible wi h he gene al ype a e indeed cha s. W i ing
uk=uX≡uX,1∈R+, he condi ion on an adap ed me ic g o be good becomes
uk≤1 o all k, a e e y induc i e s ep o i s de ini ion. In [33, 34, 8], i is assumed
ha Ais a s a i ied pseudomani old, and hen ˆu= (u2, . . . , un) s ands o he ype
o g. This ˆuis de e mined by g. In pa icula , i he de ini ion is modi ied by aking
uk= 1 o all ka e e y induc i e s ep, we ge he adap ed me ics o conic ype
conside ed in [12, 13, 14]. Be ale ed abou he h ee sligh ly di e en e ms used
o he scope o his pape : adap ed me ics o conic ype, adap ed me ics and
gene al adap ed me ics. The class o (good) gene al adap ed me ics is p ese ed
by p oduc s, as well as he class o adap ed me ics o conic ype, bu he class
o adap ed me ics does no ha e his p ope y. The exis ence o gene al adap ed
me ics wi h any possible gene al ype can be shown like in he case o adap ed
me ics [33, Lemma 4.3], [8, Appendix].
Like in [4], he e m “ ela i e(ly)” (o simply “ el-”) usually means ha some
condi ion is equi ed in he in e sec ion o Mwi h small neighbo hoods o he poin s
in M, o ha some concep can be desc ibed using hose in e sec ions.
Le Mbe equipped wi h a gene al adap ed me ic g, wi h a gene al ype X7→ uX
as abo e. The el-local me ic comple ion c
Mo Mconsis s o he poin s in he
me ic comple ion ep esen ed by Cauchy sequences ha con e ge in M(c
Mis he
me ic comple ion o Mi Mis compac ). Figu e 1 illus a es his concep . The
limi s o Cauchy sequences de ine a con inuous map lim : c
M→M. The ollowing
p ope ies can be p o ed like in he case o conic me ics [4, P oposi ion 3.20 (i),(ii)].
c
Mhas a unique s a i ied s uc u e wi h connec ed s a a so ha lim : c
M→Mis
a mo phism whose es ic ions o he s a a a e local di eomo phisms. Mo eo e g
is also a gene al adap ed me ic wi h espec o c
M.
(a) M(b) c
M
Figu e 1. The s a i ied space c
M.
1.4. Rela i ely Mo se unc ions. A smoo h unc ion on Mis called el-
admissible when he unc ions ,|d |and |Hess |a e el-bounded. In his case,

6 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
may no ha e any con inuous ex ension o M, bu i has a con inuous ex en-
sion o c
M. So i makes sense o say ha x∈c
Mis a el-c i ical poin o when
lim in |d (y)|= 0 as y→xin c
Mwi h y∈M. The se o el-c i ical poin s o is
deno ed by C i el( ). I is said ha is a el-Mo se unc ion i i is el-admissible
and has he ollowing desc ip ion a ound e e y x∈C i el( ):
• he e is a gene al cha O≡O0o c
M, cen e ed a xand compa ible wi h
g, such ha M∩O≡M0∩O0 o M0=RmX×QaX
i=1(Ni×R+), whe e X
is he s a um o c
Mcon aining x; and
• |M∩O≡ (x) + 1
2(ρ2
+−ρ2
−)|M0∩O0, whe e ρ±is he adial unc ion o
Rm±×Qi∈I±c(LX,i) o some exp ession mX=m++m−(m±∈N) and
some pa i ion o {1, . . . , aX}in o se s I±.
This local condi ion is used ins ead o equi ing ha Hess is “ el-non-degene a e”
a he el-c i ical poin s because a “ el-Mo se lemma” is missing. Mo eo e , o
e e y ∈ {0, . . . , n}, le
ν
x,max/min =X
( 1,..., aX)
aX
Y
i=1
β i
max/min(Ni),(1)
whe e ( 1, . . . , aX) uns in he subse o NaXde e mined by
=m−+PaX
i=1 i+|I−|,
i<kX,i−1
2+1
2uX,i i i∈I+
i≥kX,i−1
2+1
2uX,i i i∈I−) o ν
x,max ,
i≤kX,i−1
2−1
2uX,i i i∈I+
i>kX,i−1
2−1
2uX,i i i∈I−) o ν
x,min .

















(2)
When aX= 0 in (1), he single on N0consis s o he emp y sequence, ob aining1
ν
x,max/min =δ ,m−wi h he con en ion ha he alue o emp y p oduc s is 1.
Finally, le ν
max/min =Pxν
x,max/min wi h x unning in C i el( ). The no a ion
ν
x,max/min( ) and ν
max/min( ) may be used i necessa y. The exis ence o el-Mo se
unc ions o gene al adap ed me ics holds like in he case o adap ed me ics [4,
P oposi ion 4.9].
1.5. Main heo ems. The ollowing is ou i s main heo em, whe e p ope y (ii)
is a weak e sion o he Weyl’s asymp o ic o mula.
Theo em 1.1. The ollowing p ope ies hold on any s a um o a compac s a i-
ica ion wi h a good gene al adap ed me ic:
(i) ∆max/min has a disc e e spec um, 0≤λmax/min,0≤λmax/min,1≤ ···, whe e
e e y eigen alue is epea ed acco ding o i s mul iplici y.
(ii) lim in kλmax/min,k k−θ>0 o some θ > 0.
Ou second main esul is he ollowing e sion o Mo se inequali ies o el-Mo se
unc ions.
1K onecke ’s del a symbol is used.
WITTEN’S PERTURBATION ON STRATA 7
Theo em 1.2. Fo any el-Mo se unc ion on a s a um o dimension no a com-
pac s a i ica ion, equipped wi h a good gene al adap ed me ic, we ha e
k
X
=0
(−1)k− β
max/min ≤
k
X
=0
(−1)k− ν
max/min (0 ≤k < n),
χmax/min =
n
X
=0
(−1) ν
max/min .
In he case o adap ed me ics o conic ype, Theo em 1.1 (i) is essen ially due
o Cheege [12, 13] (see also [1, 2, 4]), Theo em 1.1–(ii) was p o ed by he au ho s
[4], and Theo em 1.2 was p o ed by he au ho s [4] and Ludwig [30] (wi h mo e
es ic i e condi ions bu s onge consequences). O he de elopmen s o ellip ic
heo y on s a a we e made in [10, 25, 23, 39, 16, 2, 1], all o hem using adap ed
me ics o conic ype. The main no el y o ou pape is he ex ension o he ellip ic
heo y on s a a o he wide class o good gene al adap ed me ics, including good
adap ed me ics.
1.6. Applica ions o in e sec ion homology. Conside now he case whe e A
is a s a i ied pseudomani old, and he e o e Mis i s egula s a um. Le I¯pH∗(A)
deno e i s in e sec ion homology wi h pe e si y ¯p[19, 20], aking eal coe icien s.
Le β¯p
=β¯p
(A) and χ¯p=χ¯p(A) deno e he e sions o Be i numbe s and Eule
cha ac e is ic o I¯pH∗(A). E e y pe e si y can be conside ed as a sequence ¯p=
(p2, p3, . . . ) in Nsa is ying p2= 0 and pk≤pk+1 ≤pk+ 1. Fo example, he ze o
pe e si y is ¯
0 = (0,0, . . . ), he op pe e si y is ¯
= (0,1,2, . . . ) ( k=k−2), he
lowe middle pe e si y is ¯m= (0,0,1,1,2,2,3, . . . ) (mk=bk
2c−1), and he uppe
middle pe e si y is ¯n= (0,1,1,2,2,3,3, . . . ) (nk=dk
2e−1). Recall also ha wo
pe e si ies ¯pand ¯qa e called complemen a y i ¯p+ ¯q=¯
. W i e ¯p≤¯qi pk≤qk o
all k. Le gbe an adap ed me ic on Mo ype ˆu= (u2, . . . , un). I ˆuis associa ed
wi h a pe e si y ¯p≤¯min he sense
1
k−1−2pk≤uk<1
k−3−2pki 2pk≤k−3,
1≤uk<∞i 2pk=k−2,)(3)
hen H
(2)(M)∼
=I¯pH (A)∗[33, 34, 8], and he e o e β¯p
=β
max. In pa icula ,
H
(2)(M)∼
=I¯mH (A)∗i gis an adap ed me ic o conic ype [14]. Thus he
incompa ibili y o adap ed me ics wi h p oduc s is ela ed o he sub le ies o he
e sions o he K¨unne h heo em o in e sec ion homology [15, 17]. Fo ins ance,
he isomo phism I¯pH∗(P×Q)∼
=I¯pH∗(P)⊗I¯pH∗(Q), o a bi a y pseudomani olds
Pand Q, only holds wi h some special pe e si ies ¯p, including ¯p= ¯m. Acco ding
o (3), he e exis good adap ed me ics on Mwhose ype is associa ed wi h any
gi en pe e si y ≤¯m.
In (3), only he choices 2pk=k−2, k −4, . . . a e possible i kis e en, and only
he choices 2pk=k−3, k −5, . . . a e possible i kis odd. Thus, o e e y k, (3)
es ablishes a bijec ion be ween he possibili ies o pkand a pa i ion o [ 1
k−1,∞)
in o semi-open in e als, whe e ukis aken.
Le be a el-Mo se unc ion on M, le x∈C i el( ), le Xbe he s a um
o c
Mcon aining x, and le k= codim X. Wi h he abo e no a ion o a cha
O≡O0o c
Mcen e ed a x, he e is an adap ed me ic ˜gon Nso ha , ia he
cha , g|Ois quasi-isome ic o he es ic ion o g0+ρ2uk
X˜g+ (dρX)2 o M0∩O0.
8 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
Then he ype o ˜gis also associa ed wi h ¯p. Mo eo e he e is some exp ession,
mX=m++m−(m±∈N), and some decomposi ion, c(LX)≡c(L+)×c(L−), so
ha M0≡Rm+×N+×R+×Rm−×N−×R+ o dense s a a N±o L±, and
|O≡ (x) + 1
2(ρ2
+−ρ2
−)|O0, whe e ρ±is he adial unc ion o Rm±×c(L±). Le
k±= dim N±+ 1; hus k=k++k−. He e, some o he s a i ica ions L±may be
emp y; in ac , L+6=∅ 6=L−only can happen i uk= 1 (Sec ion 1.3). F om (1)
and (2), i ollows ha he numbe s ν
x,max a e independen o he choice o ˆu
associa ed wi h ¯p, and he e o e he no a ion ν¯p
x, =ν¯p
x, ( ) will be used. P ecisely,
hey ha e he ollowing exp essions:
•I L+6=∅ 6=L−(only i uk= 1), hen
ν¯p
x, =X
( +, −)
β¯p
+(L+)β¯p
−(L−),
whe e ( +, −) uns in he subse o N2de e mined by he condi ions
=m−+ ++ −+ 1 , +<k+
2, −≥k−
2.
•I LX=L+6=∅(L−=∅), hen
ν¯p
x, =X
+
β¯p
+(LX),
whe e + uns in he subse o Nde e mined by he condi ions
=m−+ +, +<(k−1−pki uk<1
k
2i uk= 1 .
•I LX=L−6=∅(L+=∅), hen
ν¯p
x, =X
−
β¯p
−(LX),
whe e − uns in he subse o Nde e mined by he condi ions
=m−+ −+ 1 , −≥(k−1−pki uk<1
k
2i uk= 1 .
•I LX=∅, hen ν¯p
x, =δ ,m−.
Finally, le ν¯p
=ν¯p
( ) = Pxν¯p
x, (x∈C i el( )), which equals ν
max.
Suppose now ha Ais o ien ed (Mis o ien ed) and compac . We ha e β
min =
βn−
max o all because ∆min co esponds o ∆max by he Hodge s a ope a o . On
he o he hand, o any pe e si y ¯q≥¯n, i ¯p≤¯mis complemen a y o ¯q, hen
I¯qH (A)∼
=I¯pHn− (A)∗[19, 20], and he e o e β¯q
=β¯p
n− , ob aining β¯q
=β
min. As
be o e, i ollows om (1) and (2) ha he numbe s ν
x,min a e independen o he
choice o ˆuassocia ed wi h ¯p. P ecisely, wi h he no a ion ν¯q
x, =ν¯q
x, ( ) = ν
x,min,
hey ha e he ollowing exp essions:
•I L+6=∅ 6=L−(only i uk= 1), hen
ν¯q
x, =X
( +, −)
β¯q
+(L+)β¯q
−(L−),
whe e ( +, −) uns in he subse o N2de e mined by he condi ions
=m−+ ++ −+ 1 , +≤k+
2−1, −>k−
2−1.
WITTEN’S PERTURBATION ON STRATA 9
•I LX=L+6=∅(L−=∅), hen
ν¯q
x, =X
+
β¯q
+(LX),
whe e + uns in he subse o Nde e mined by he condi ions
=m−+ +, +≤(k−2−qki uk<1
k
2−1 i uk= 1 .
•I LX=L−6=∅(L+=∅), hen
ν¯q
x, =X
−
β¯q
−(LX),
whe e − uns in he subse o Nde e mined by he condi ions
=m−+ −+ 1 , −>(k−2−qki uk<1
k
2−1 i uk= 1 .
•I LX=∅, hen ν¯q
x, =δ ,m−.
Like ν¯p
, we also de ine ν¯q
=ν¯q
( ) = Pxν¯q
x, (x∈C i el( )), which equals ν
min.
Theo em 1.2 has he ollowing di ec consequence.
Co olla y 1.3. Le Abe a compac pseudomani old o dimension n, le Mbe i s
egula s a um, and le ¯pbe a pe e si y. I ¯p≤¯m, o i Ais o ien ed and ¯p≥¯n,
hen, o any el-Mo se unc ion on M(wi h espec o any good adap ed me ic),
we ha e
k
X
=0
(−1)k− β¯p
≤
k
X
=0
(−1)k− ν¯p
(0 ≤k < n),
χ¯p=
n
X
=0
(−1) ν¯p
.
S a i ied Mo se heo y was in oduced by Go esky and MacPhe son [21], and has
a g ea weal h o applica ions. In pa icula , Go esky and MacPhe son ha e p o ed
Mo se inequali ies on complex analy ic a ie ies wi h Whi ney s a i ica ions, in-
ol ing he in e sec ion homology wi h pe e si y ¯m[21, Chap e 6, Sec ion 6.12].
Ludwig also ga e an analy ic in e p e a ion o Mo se heo y in he spi i o Go esky
and MacPhe son o con o mally conic mani olds [26, 27, 28, 29]. Ou e sion o
Mo se unc ions, c i ical poin s and associa ed numbe s is di e en om hose used
in [21], e en in he case o pe e si y ¯m. To he au ho s’ knowledge, Co olla y 1.3
is he i s e sion o Mo se inequali ies o in e sec ion homology wi h pe e si y
6= ¯m.
1.7. Ideas o he p oo s. In he p oo s o Theo ems 1.1 and 1.2, se e al s eps
a e like in he case o adap ed me ics o conic ype [4]. Only b ie indica ions o
hose s eps a e gi en in his pape , whe eas he pa s wi h new ideas a e explained
wi h de ail. We adap he well-known analy ic me hod o Wi en [43]; specially,
as desc ibed in [36, Chap e s 9 and 14]. Thus, gi en a el-Mo se unc ion on
M, we conside he Wi en’s pe u ba ion ds=e−s des =d+s d ∧on Ω0(M)
(s > 0). Le ds,max/min deno e i s maximum/minimum i.b.c., wi h co esponding
Laplacian ∆s,max/min. Since ∆s,max/min −∆max/min is bounded, i is enough o
16 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
and k kc,d, espec i ely. The simple no a ion L2
+,h,iand k k is used when c= 0.
Recall ha he ha monic oscilla o on C∞
+is he ope a o H=−d2
dρ2+s2ρ2(s > 0).
Fo c1, c2, d1, d2∈R, le
P0=H−2c1ρ−1d
dρ +c2ρ−2, Q0=H−2d1d
dρ ρ−1+d2ρ−2.(10)
P oposi ion 3.1 ([3, Theo em 1.4]).I a∈Rsa is ies
a2+ (2c1−1)a−c2= 0 ,(11)
σ:= a+c1>−1
2,(12)
hen he ollowing holds:
(i)P0, wi h D(P0) = ρaSe ,+, is essen ially sel -adjoin in L2
c1,+.
(ii)The spec um o P0:= P0consis s o he eigen alues
λk= (2k+ 1 + 2σ)s , (13)
o k∈2N, wi h mul iplici y one and co esponding no malized eigen unc ions
χk=χs,σ,a,k := √2ρaφs,σ,k,+.
(iii)D∞(P0) = ρaSe ,+.
P oposi ion 3.2 (See [3, Sec ion 5]).I b∈Rsa is ies
b2+ (2d1+ 1)b−d2= 0 ,(14)
τ:= b+d1>−3
2,(15)
hen he ollowing holds:
(i)Q0, wi h D(Q0) = ρbSodd,+, is essen ially sel -adjoin in L2
d1,+.
(ii)The spec um o Q0:= Q0consis s o he eigen alues gi en by he exp es-
sion (13), o k∈2N+ 1 and using τins ead o σ, wi h mul iplici y one and
co esponding no malized eigen unc ions χk=χs,τ,b,k := √2ρbφs,τ,k,+.
(iii)D∞(Q0) = ρbSodd,+.
P oposi ion 3.3 ([5, Co olla y 8.1]).Le ξ > 0and
0< u < 1.(16)
I a∈Rsa is ies (11) and
σ:= a+c1> u −1
2,(17)
hen he e is a posi i e sel -adjoin ope a o Pin L2
c1,+sa is ying he ollowing:
(i)ρaSe ,+is a co e o P1/2and, o all φ, ψ ∈ρaSe ,+,
hP1/2φ, P1/2ψic1=hP0φ, ψic1+ξhρ−uφ, ρ−uψic1.(18)
(ii)Phas a disc e e spec um. Le λ0≤λ2≤ ··· be i s eigen alues, epea ed
acco ding o hei mul iplici y. The e is some D=D(σ, u)>0and, o any
 > 0, he e is some C=C(, σ, u)>0so ha , o all k∈2N,
λk≥(2k+ 1 + 2σ)s+ξDsu(k+ 1)−u,(19)
λk≤(2k+ 1 + 2σ)(s+ξsu) + ξCsu.(20)
P oposi ion 3.4 ([5, Co olla y 8.2]).Fo ξand ulike in P oposi ion 3.3, i b∈R
sa is ies (14) and
τ:= b+d1> u −3
2,(21)
hen he e is a posi i e sel -adjoin ope a o Qin L2
d1,+sa is ying he ollowing:

WITTEN’S PERTURBATION ON STRATA 17
(i)ρbSodd,+is a co e o Q1/2and, o all φ, ψ ∈ρbSodd,+,
hQ1/2φ, Q1/2ψid1=hQ0φ, ψid1+ξhρ−uφ, ρ−uψid1.(22)
(ii)Qhas a disc e e spec um. Le λ1≤λ3≤ ··· be i s eigen alues, epea ed
acco ding o hei mul iplici y. The e is some D=D(τ, u)>0and, o any
 > 0, he e is some C=C(, τ, u)>0so ha (19) and (20) a e sa is ied,
o k∈2N+ 1 and wi h τins ead o σ.
P oposi ion 3.5 ([5, Co olla y 8.3]).Conside he no a ion and condi ions o
P oposi ions 3.3 and 3.4. Fix also some η∈R, and le
θ > −1
2.(23)
Mo eo e suppose ha he ollowing p ope ies hold:
(a)I σ=θ6=τand τ−σ6∈ −N, hen
σ−1< τ < σ + 1,2σ+1
2.(24)
(b)I σ6=θ=τand σ−τ6∈ −N, hen
−τ, τ −1< σ < 3τ+ 1,11τ+ 2, τ + 1 .(25)
(c)I σ6=θ=τ+ 1 and σ−τ−16∈ −N, hen
τ+ 1 < σ < τ + 3,2τ+7
2.(26)
(d)I σ6=θ6=τand σ−θ, τ −θ6∈ −N, hen
σ−τ
2−1,τ−σ
2,σ+τ−1
4,σ+3τ−2
14 ,3σ+τ−4
14 ,σ+τ−1
2< θ < σ+τ+1
2,
τ−1< σ < τ + 3 .(27)
Then he e is a posi i e sel -adjoin ope a o Win L2
c1,d1,+sa is ying he ollowing:
(i)ρaSe ,+⊕ρbSodd,+is a co e o W1/2, and, o φ= (φ1, φ2)and ψ= (ψ1, ψ2)
in ρaSe ,+⊕ρbSodd,+,
hW1/2φ, W1/2ψic1,d1=h(P0⊕Q0)φ, ψic1,d1+ξhρ−uφ, ρ−uψic1,d1
+ηhρ−a−b−1φ2, ψ1iθ+hφ1, ρ−a−b−1ψ2iθ.(28)
(ii)Whas a disc e e spec um. I s eigen alues o m wo g oups, λ0≤λ2≤ ···
and λ1≤λ3≤ ···, epea ed acco ding o hei mul iplici y, such ha he e
is some D=D(σ, τ, u)>0and, o e e y  > 0, he e a e some C=
C(, σ, τ, u)>0and E=E(, σ, τ, θ)>0so ha , o all k∈N,
λk≥(2k+ 1 + 2ςk)s−2|η|s +1
2+ξDsu(k+ 1)−u−2|η|Es +1
2,(29)
λk≤(2k+ 1 + 2ςk)s+ξsu+ 2|η|s +1
2+ξCsu+ 2|η|Es +1
2,(30)
whe e =σ+τ−2θ,ςk=σi kis e en, and ςk=τi kis odd.
(iii)Le ˜u∈Rsuch ha
0, , τ −2θ+1
2, σ −2θ−1
2<˜u < 1, + 1, σ +1
2, τ +3
2,(31)
and le ˆu= max{˜u, + 1 −˜u}. The e is some D=D(σ, τ, u)>0and, o
any  > 0, he e is some e
C=e
C(, σ, τ, u)>0so ha , o all k∈N,
λk≥(2k+ 1 + 2ςk)s−|η|sˆu+ξDsu(k+ 1)−u−|η|e
Csˆu.(32)
18 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
(i )I u= +1
2and ξ≥ |η|, hen he e is some e
D=e
D(σ, τ, u)>0so ha , o all
k∈N,
λk≥(2k+ 1 + 2ςk)s+ (ξ−|η|)e
Dsu(k+ 1)−u.(33)
( )I we add he e m ξ0hφ1, ψ1ic1+ξ00hφ2, ψ2id1 o he igh -hand side o (28),
o some ξ0, ξ00 ∈R, hen he esul holds as well wi h he addi ional e m
max{ξ0, ξ00}in he igh -hand side o (30), and he addi ional e m, ξ0 o
k∈2Nand ξ00 o k∈2N+ 1, in he igh -hand sides o (29),(32) and (33).
Rema k 7.(i) I his a bounded measu able unc ion on R+wi h h(ρ)→1 as
ρ→0, hen hhχ0, χ0ic1→1 as s→ ∞ [4, Lemma 7.3].
(ii) The exis ence o a∈Rsa is ying (11) is cha ac e ized by he condi ion (2c1−
1)2+ 4c2≥0, which holds i c2≥min{0,2c1}. I c2= 0, hen (11) means
ha a∈ {0,1−2c1}. I c2= 2c1, hen (11) means ha a∈ {1,−2c1}.
(iii) The exis ence o b∈Rsa is ying (14) is cha ac e ized by he condi ion (2d1+
1)2+ 4d2≥0, which holds i d2≥min{0,−2d1}. I d2= 0, hen (14) means
ha b∈ {0,−1−2d1}. I d2=−2d1, hen (14) means ha b∈ {−1,−2d1}.
(i ) P oposi ions 3.1 and 3.2 a e indeed equi alen , as well as P oposi ions 3.3
and 3.4, because, i c1=d1+ 1 and c2=d2, hen Q0=ρP0ρ−1by (9), and
ρ:L2
c1,+→L2
d1,+is a uni a y isomo phism.
( ) We ha e P=P,Q=Qand W=W, whe e
P=P0+ξρ−2u, Q =Q0+ξρ−2u,(34)
W=P ηρ2(θ−c1)−a−b−1
ηρ2(θ−d1)−a−b−1Q,(35)
wi h D(P) = D∞(P), D(Q) = D∞(Q) and D(W) = D∞(W) [5, Rema k 1.4 (i)
and Sec ion 8].
( i) We ha e
D(P1/2) = D(P1/2
0),D(Q1/2) = D(Q1/2
0),D(W1/2) = D((P0⊕Q0)1/2).
Thus he exp essions (18), (22) and (28) can be ex ended o φand ψin
D(P1/2), D(Q1/2) and D(W1/2), espec i ely, using
hP1/2
0φ, P1/2
0ψic1,hQ1/2
0φ, Q1/2
0ψid1,h(P0⊕Q0)1/2φ, (P0⊕Q0)1/2ψic1,d1
ins ead o
hP0φ, ψic1,hQ0φ, ψid1,h(P0⊕Q0)φ, ψic1,d1,
espec i ely [5, Rema k 3.21 and Sec ion 8].
( ii) In P oposi ion 3.5 (iii), he condi ion (31) means ha (16), (17) and (21) also
hold wi h ˜uand + 1 −˜uins ead o u. The e exis s ˜usa is ying (31) jus
when
0, , τ −2θ+1
2, σ −2θ−1
2<1, + 1, σ +1
2, τ +3
2.(36)
This p ope y is sa is ied in he cases (b) and (d) by (16), (17), (21), (23), (25)
and (27); in pa icula , we can ake ˜u= +1
2. By (16), (17), (21), (23) and (24)
( espec i ely, (26)), in he case (a) ( espec i ely, in he case (c)), we ha e (36)
i and only i τ < 3σ( espec i ely, σ < 3τ+ 4).
Conside he condi ions and no a ion o P oposi ion 3.3, and he no a ion o
P oposi ion 3.1. Take a comple e o hono mal sys em {ˆχk= ˆχP,k |k∈2N}o
L2
c1,+so ha e e y ˆχkis a λk-eigen unc ion o P. Le ˆχ0
k= ˆχ0
P,k and ˆχ00
k= ˆχ00
P,k
WITTEN’S PERTURBATION ON STRATA 19
deno e he o hogonal p ojec ions o e e y ˆχk o he subspaces spanned by χkand
{χi|k > i ∈2N}, espec i ely; in pa icula , ˆχ00
0= 0. Le also ˆχ000
k= ˆχ000
P,k =
ˆχk−ˆχ0
k−ˆχ00
k.
Lemma 3.6. kˆχ0
P,kkc1→1as s→ ∞ o e e y k∈2N.
P oo . We p oceed by induc ion on k. Fo k= 0, ake some  > 0 and C > 0
sa is ying (20). By P oposi ions 3.1 (ii) and 3.3 (ii), and Rema k 7 ( i),
(1 + 2σ)(s+ξsu) + ξCsu≥λ0=hP1/2ˆχ0,P1/2ˆχ0ic1>hP1/2
0ˆχ0,P1/2
0ˆχ0ic1
=hP1/2
0ˆχ0
0,P1/2
0ˆχ0
0ic1+hP1/2
0ˆχ000
0,P1/2
0ˆχ000
0ic1
≥(1 + 2σ)skˆχ0
0k2
c1+ (5 + 2σ)skˆχ000
0k2
c1= (1 + 2σ)s+ 4skˆχ000
0k2
c1,
gi ing
kˆχ000
0k2
c1<((1 + 2σ)+C)ξ
4s1−u→0
as s→ ∞, and he e o e kˆχ0
0k2
c1→1.
Now, ake any e en in ege k > 0 and suppose ha he esul holds o all e en
indices < k. This yields kˆχ00
kkc1→0 as s→ ∞. Thus, gi en any δ > 0, we ha e
kˆχ00
kk2
c1< δ/k o sla ge enough. Take some  > 0 and C > 0 sa is ying (20). By
P oposi ions 3.1 (ii) and 3.3 (ii), and Rema k 7 ( i),
(2k+ 1 + 2σ)(s+ξsu) + ξCsu≥λk=hP1/2ˆχk,P1/2ˆχkic1>hP1/2
0ˆχk,P1/2
0ˆχkic1
=hP1/2
0ˆχ0
k,P1/2
0ˆχ0
kic1+hP1/2
0ˆχ00
k,P1/2
0ˆχ00
kic1+hP1/2
0ˆχ000
k,P1/2
0ˆχ000
kic1
≥(2k+ 1 + 2σ)skˆχ0
kk2
c1+ (1 + 2σ)skˆχ00
kk2
c1+ (2k+ 5 + 2σ)skˆχ000
kk2
c1
= (1 + 2σ)s+ 2ks(kˆχ0
kk2
c1+kˆχ000
kk2
c1)+4skˆχ000
kk2
c1
>(1 + 2σ)s+ 2ks(1 −δ/k)+4skˆχ000
kk2
c1,
gi ing
kˆχ000
kk2
c1<((2k+ 1 + 2σ)+C)ξ
4s1−u+δ
2< δ
o sla ge enough. Thus kˆχ000
kk2
c1→0 as s→ ∞, and he esul ollows. 
Co olla y 3.7. I his a bounded measu able unc ion on R+such ha h(ρ)→1
as ρ→0, hen hhˆχP,0,ˆχP,0ic1→1as s→ ∞.
P oo . This ollows om Lemma 3.6 and Rema k 7 (i). 
Simila esul s hold o Qand W, bu hey a e omi ed because hey a e no
used.
4. Two simple ypes o ellip ic complexes
He e, we s udy wo simple ellip ic complexes on R+, which will show up in a
di ec sum spli ing o he el-local model o Wi en’s pe u ba ion (Sec ion 6).
20 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
4.1. An ellip ic complex o leng h one. Conside he s anda d me ic on R+.
Le Ebe he g aded Riemannian/He mi ian ec o bundle o e R+whose nonze o
e ms a e E0and E1, which a e eal/complex i ial line bundles equipped wi h
he s anda d Riemannian/Hemi ian me ics. Thus
C∞(E0)≡C∞
+≡C∞(E1), L2(E0)≡L2
+≡L2(E1),
whe e eal-/complex- alued unc ions a e conside ed in C∞
+and L2
+. Fo any ixed
s > 0 and κ∈R, le
C∞(E0)C∞(E1)
d
δ
-

be he di e en ial ope a o s de ined by
d=d
dρ −κρ−1±sρ , δ =−d
dρ −κρ−1±sρ .
I is easy o check ha (E, d) is an ellip ic complex, and ha 4δ=d†.
4.1.1. Sel -adjoin ope a o s de ined by he Laplacian. By (9), he homogeneous
componen s o ∆ (o ∆±) a e:
∆0=H+κ(κ−1)ρ−2∓s(1 + 2κ),(37)
∆1=H+κ(κ+ 1)ρ−2±s(1 −2κ),(38)
whe e His he ha monic oscilla o on C∞
+de ined wi h he cons an s. Then ∆0
and ∆1a e like P0and Q0in (10), wi h c1=0=d1, plus a cons an . Then, by
P oposi ions 3.1 and 3.2, ∆0and ∆1de ine he sel -adjoin ope a o s Aiand Bi
in L2
+indica ed in Table 1, whe e he condi ions come om (12) and (15). The
no a ion A±
iand B±
imay be used as well o speci y ha hese ope a o s a e de ined
by ∆±
0and ∆±
1. In hese cases, we ha e c1=d1= 0, and he e o e σ=aand
τ=b, which a e gi en by (11) and (14).
σ τ Condi ion
∆0A1κ κ > −1
2
A21−κ κ < 3
2
∆1B1κ κ > −3
2
B2−1−κ κ < 1
2
Table 1. Sel -adjoin ope a o s de ined by ∆0and ∆1
The e a e he ollowing o e laps in Table 1:
•Bo h A1and A2a e de ined i −1
2< κ < 3
2, and hey a e equal jus when
κ=1
2.
•Bo h B1and B2a e de ined i −3
2<κ<1
2, and hey a e equal jus when
κ=−1
2.
The co es o Aiand Bi, gi en by P oposi ions 3.1 and 3.2, will be deno ed by E0
iand
E1
i, espec i ely. No e ha he g aded subspace Ei=E0
i⊕E1
io C∞(E)∩L2(E),
whene e de ined, is p ese ed by D=d+δ. P oposi ions 3.1 and 3.2 also desc ibe
he spec a o Aiand Bi:
4The supe index †is used o deno e he o mal adjoin .
WITTEN’S PERTURBATION ON STRATA 21
•The spec um o A1consis s o he eigen alues
(2k+ (1 ∓1)(1 + 2κ))s(k∈2N) (39)
o mul iplici y one.
•The spec um o A2consis s o he eigen alues
(2k+ 4 −(1 ±1)(1 + 2κ))s(k∈2N) (40)
o mul iplici y one.
•The spec um o B1consis s o he eigen alues
(2k+ 2 + (1 ∓1)(−1+2κ))s(k∈2N+ 1) (41)
o mul iplici y one.
•The spec um o B2consis s o he eigen alues
(2k−2−(1 ±1)(−1+2κ))s(k∈2N+ 1) (42)
o mul iplici y one.
These eigen alues ha e no malized eigen unc ions χk, de ined o he co esponding
alues o a=σand b=τ. Fo A+
1, (39) becomes 2ks. Fo A−
1, (39) is 2(k+1+2κ)s.
Fo A+
2, (40) becomes 2(k+ 1 −2κ)s. Fo A−
2, (40) is 2(k+ 2)s. Fo B+
1, (41) is
2(k+ 1)s. Fo B−
1, (41) becomes 2(k+ 2κ)s. Fo B+
2, (42) is 2(k−2κ)s. Fo
B−
2, (42) becomes 2(k−1)s. Using his, we ge he in o ma ion abou he sign o
he eigen alues o Aiand Bigi en in Table 2. In he ables, g ey colo is used o
cases ha will be dis ega ded la e ( o ins ance, i he e may exis some nega i e
eigen alue), and a ques ion ma k is used o unknown in o ma ion.
Sign o eigen alues Sign o eigen alues
A+
1
0 i k= 0 B+
1+∀k∈2N+ 1
+ i k≥2 e en
B−
1
κ > −1
2+∀k∈2N+ 1
A−
1+∀k∈2Nκ=−1
2
0 i k= 1
A+
2
κ > 1
2
−i k < 2κ−1+ i k≥3 odd
0 i k= 2κ−1
κ < −1
2
−i k < −2κ
+ i k > 2κ−1 0 i k=−2κ
κ=1
2
0 i k= 0 + i k > −2κ
+ i k≥2 e en B+
2+∀k∈2N+ 1
κ < 1
2+∀k∈2NB−
2
0 i k= 1
A−
2+∀k∈2N+ i k≥3 odd
Table 2. Sign o he eigen alues o Aiand Bi
4.1.2. Laplacians o he maximum/minimum i.b.c.
P oposi ion 4.1 ([4, P oposi ion 8.4]).Table 3 desc ibes ∆max/min.

22 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
∆max,0∆min,0∆max,1∆min,1
κ≥1
2A1B1
|κ|<1
2A1A2B1B2
κ≤ −1
2A2B2
Table 3. Desc ip ion o ∆max/min
Rema k 8.(i) In [4], he p oo o P oposi ion 4.1 uses he ollowing p ope y
[4, Lemma 8.5]. Suppose ha ei he θ > 1
2, o θ=1
2=κ( espec i ely,
θ=1
2=−κ). Then, o e e y ξ∈ρθSe ,+, conside ed as subspace o C∞(E0)
( espec i ely, C∞(E1)), he e is a sequence (ξn) in C∞
0(E0) ( espec i ely,
C∞
0(E1)), independen o κ, such ha limnξn=ξin L2(E0) ( espec i ely,
L2(E1)) and limndξn=dξ in L2(E1) ( espec i ely, limnδξn=δξ in L2(E0)).
In pa icula , ρθSe ,+is con ained in D(dmin) ( espec i ely, D(δmin)). Mo e-
o e , acco ding o he p oo o [4, Lemma 8.5], gi en 0 < a < b, we can ake
ξn=αnξ o some αn∈C∞
+sa is ying χ[b
n,na]≤αn≤χ[a
n,nb], whe e χS
deno es he cha ac e is ic unc ion o e e y subse S⊂R+.
(ii) E0
i( espec i ely, E1
i) is also a co e o dmax/min ( espec i ely, δmin/max) when
∆max/min,0=Ai( espec i ely, ∆max/min,1=Bi).
4.2. An ellip ic complex o leng h wo. Conside again he s anda d me ic
on R+. Le Fbe he g aded Riemannian/He mi ian ec o bundle o e R+whose
nonze o e ms a e F0,F1and F2, which a e i ial eal/complex ec o bundles o
anks 1, 2 and 1, espec i ely, equipped wi h he s anda d Riemannian/He mi ian
me ics. Thus
C∞(F0)≡C∞
+≡C∞(F2), C∞(F1)≡C∞
+⊕C∞
+,
L2(F0)≡L2
+≡L2(F2), L2(F1)≡L2
+⊕L2
+,
whe e eal-/complex- alued unc ions a e conside ed in C∞
+and L2
+. Fix s, µ > 0,
0<u<1 and κ∈R. Le
C∞(F0)C∞(F1)C∞(F2)
d0≡d0,1
d0,2
δ0≡δ0,1δ0,2
d1≡d1,1d1,2
δ1≡δ1,1
δ1,2
-
-

be he di e en ial ope a o s de ined by
d0,1=µρ−u, d0,2=d
dρ −(κ+u)ρ−1±sρ ,
d1,1=d
dρ −κρ−1±sρ , d1,2=−µρ−u,
δ0,1=µρ−u, δ0,2=−d
dρ −(κ+u)ρ−1±sρ ,
δ1,1=−d
dρ −κρ−1±sρ , δ1,2=−µρ−u.
Obse e ha δ0=d†
0and δ1=d†
1. We may also use he mo e explici no a ion
d±
,δ±
,d±
,i and δ±
,i. A di ec compu a ion shows ha d0and d1de ine an ellip ic
complex (F, d) o leng h wo. No e ha , by (9),
d1,1=ρ−ud0,2ρu, δ0,2=ρ−uδ1,1ρu.(43)
WITTEN’S PERTURBATION ON STRATA 23
4.2.1. Sel -adjoin ope a o s de ined by he Laplacian. By (9), he homogeneous
componen s o he co esponding Laplacian ∆ (o ∆±) a e gi en by
∆0=H+ (κ+u)(κ+u−1)ρ−2+µ2ρ−2u∓s(1 + 2(κ+u)) ,
∆2=H+κ(κ+ 1)ρ−2+µ2ρ−2u±s(1 −2κ),
∆1=∆1,1−2µuρ−u−1
−2µuρ−u−1∆1,2,
∆1,1=H+κ(κ−1)ρ−2+µ2ρ−2u∓s(1 + 2κ),
∆1,2=H+ (κ+u)(κ+u+ 1)ρ−2+µ2ρ−2u±s(1 −2(κ+u)) .
(We may also use (37) and (38) o compu e easily some pa s o he abo e compo-
nen s o ∆.) The ope a o s ∆0, ∆2, ∆1,1and ∆1,2a e like Pand Qin (34), wi h
c1= 0 = d1, plus a cons an e m. W i e ∆1=U∓sV , whe e
V=1+2κ0
0−1 + 2(κ+u).(44)
Then, by P oposi ions 3.3, 3.4 and 3.5, and Rema k 7 ( ), ∆0, ∆2and ∆1de ine
he sel -adjoin ope a o s Piand Qjin L2
+, and Wi,j in L2
+⊕L2
+, indica ed in
Table 4, whe e he condi ions come om (17), (21), (23), (24), (25), (26) and (27).
The no a ion P±
i,Q±
jand W±
i,j may be used as well o speci y ha hese ope a o s
a e de ined by ∆±
0, ∆±
2and ∆±
1. No e ha =u o all Wi,j. The co es o P1/2
i,
Q1/2
jand W1/2
i,j , gi en by P oposi ions 3.3, 3.4 and 3.5, will be deno ed by F0
i,F2
j
and F1
i,j =F1,1
i⊕F1,2
j, espec i ely.
Rema k 9.In con as o Eiin Sec ion 4.1.1, no e ha he g aded subspace F0
i⊕
F1
i,j ⊕F2
jo C∞(F)∩L2(F), whene e de ined, is no p ese ed by D=d+δ. Fo
ins ance, i is p ese ed by dbu no by δwhen i=j= 1, and i is p ese ed by δ
bu no by dwhen i=j= 2.
σ τ θ Condi ion
∆0P1κ+u κ > −1
2
P21−κ−u κ < 3
2−2u
∆2Q1κ κ > u −3
2
Q2−1−κ κ < 1
2−u
∆1
W1,1κ κ +u κ κ > u −1
2
W2,21−κ−1−κ−u−κ−u κ < 1
2−2u
6 ∃ W1,2κ−1−κ−u−1
2−uImpossible
W2,11−κ κ +u1
2−1−u
2< κ < 1−u
2
Table 4. Sel -adjoin ope a o s de ined by ∆0, ∆2and ∆1
Le us explain he con en s o Table 4. Since c1=d1= 0, we ha e σ=aand
τ=b, which a e gi en by (11) and (14). Mo eo e σ,τand ude e mine θin Table 4
so ha Uis o he o m (35) because 2θ−σ−τ=−u. Le us check he condi ions
w i en in his able, which a e gi en by he hypo hesis o P oposi ions 3.3–3.5. Fo
24 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
Piand Qj, only (17) and (21) a e equi ed. Fo Wi,j, we also equi e (23), and he
hypo hesis (a)–(d) o P oposi ion 3.5, ob aining he ollowing:
•Fo W1,1, we ha e σ=θ6=τand τ−σ=u6∈ −N. Thus (a) applies in
his case. No e ha (17), (21) and (23) mean κ>u−1
2. Then (24) holds
because 0 < u < 1 and κ > u −1
2. So (a) is sa is ied.
•Fo W2,2, we ha e σ6=θ=τ+ 1 and σ−τ−1 = 1 + u6∈ −N. Thus (c)
applies in his case. Now, (17), (21) and (23) mean κ < 1
2−2u. Then (26)
holds because 0 < u < 1 and κ < 1
2−2u. So (c) is sa is ied.
•The e is no W1,2because θ < −1
2in ha case.
•Fo W2,1, (17), (21) and (23) mean −3
2< κ < 3
2−u, and we ha e he
ollowing possibili ies:
–The case σ=θ=τis no possible because u6= 0.
–The case σ=θ6=τhappens when κ=1
2. Then σ=1
2and τ=1
2+u,
ob aining τ−σ=u6∈ −N. Thus (a) applies in his case. Mo eo e (24)
holds because 0 <u<1. So (a) is sa is ied.
–The case σ6=θ=τhappens when κ=1
2−u. Then σ=1
2+uand
τ=1
2, ob aining σ−τ=u6∈ −N. Thus (b) applies in his case.
Mo eo e (25) holds because 0 < u < 1. Hence (b) is sa is ied.
–The case σ6=θ=τ+ 1 happens when κ=−1
2−u. Then σ=3
2+u
and τ=−1
2, ob aining σ−τ−1 = 1+u6∈ −N. Thus (c) applies in his
case. Mo eo e (26) holds because 0 < u < 1. Hence (c) is sa is ied.
–Finally, assume ha σ6=θ6=τ. The condi ion σ−θ, τ −θ6∈ −N
means ha κ6∈ (1
2+N)∪(1
2−u−N), which in u n means ha
κ6=1
2,1
2−u, −1
2−ubecause −3
2<κ<3
2−u. Bu σ=θi κ=1
2,
τ=θi κ=1
2−u, and θ=τ+ 1 i κ=−1
2−u, as we ha e seen in
he p e ious cases. So σ−θ, τ −θ6∈ −N, and (d) applies in his case.
Mo eo e , since 0 <u<1, (27) holds jus when −1−u
2<κ<1−u
2.
Thus (d) is sa is ied assuming he s a ed condi ions on κ.
The e o e W2,1is de ined in one o he abo e ways i −1−u
2<κ<1−u
2.
The e a e he ollowing o e laps o he condi ions in Table 4:
•Bo h P1and P2a e de ined o −1
2< κ < 3
2−2u, and P1=P2jus when
κ=1
2−u.
•Bo h Q1and Q2a e de ined o u−3
2< κ < 1
2−u, and Q1=Q2jus when
κ=−1
2.
•Bo h W1,1and W2,2a e de ined o u−1
2< κ < 1
2−2u(i u < 1
3), bu
W1,16=W2,2 o all such κ.
•Bo h W1,1and W2,1a e de ined o u−1
2<κ<1−u
2, and W1,1=W2,1
jus when κ=1
2.
•Bo h W2,2and W2,1a e de ined o −1−u
2< κ < 1
2−2u, and W2,2=W2,1
jus when κ=−1
2−u.
P oposi ions 3.3, 3.4 and 3.5 also gi e he ollowing spec al es ima es, o all  > 0:
•The spec um o P1consis s o eigen alues λ0≤λ2≤ ···, aking mul-
iplici y in o accoun , such ha he e a e some D=D(κ, u)>0 and
WITTEN’S PERTURBATION ON STRATA 25
C=C(, κ, u)>0 so ha , o all k∈2N,
λk≥(2k+ (1 ∓1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(45)
λk≤(2k+ (1 ∓1)(1 + 2(κ+u)))s
+ (2k+ 1 + 2(κ+u))µ2su+µ2Csu.(46)
The i s e m o he igh -hand side o (45) and (46) o P+
1and P−
1is 2ks
and 2(k+ 1 + 2(κ+u))s, espec i ely.
•The spec um o P2consis s o eigen alues λ0≤λ2≤ ···, aking mul-
iplici y in o accoun , such ha he e a e some D=D(κ, u)>0 and
C=C(, κ, u)>0 so ha , o all k∈2N,
λk≥(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(47)
λk≤(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s
+ (2k+ 3 −2(κ+u))µ2su+µ2Csu.(48)
The i s e m o he igh -hand side o (47) and (48) o P+
2and P−
2
becomes 2(k+ 1 −2(κ+u))sand 2(k+ 2)s, espec i ely.
•The spec um o Q1consis s o eigen alues λ1≤λ3≤ ···, aking mul-
iplici y in o accoun , such ha he e a e some D=D(κ, u)>0 and
C=C(, κ, u)>0 so ha , o all k∈2N+ 1,
λk≥(2k+ 2 −(1 ∓1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(49)
λk≤(2k+ 2 −(1 ∓1)(1 −2κ))s+ (2k+ 1 + 2κ)µ2su+µ2Csu.(50)
The i s e m o he igh -hand side o (49) and (50) o Q+
1and Q−
1is
2(k+ 1)sand 2(k+ 2κ)s, espec i ely.
•The spec um o Q2consis s o eigen alues λ1≤λ3≤ ···, aking mul-
iplici y in o accoun , such ha he e a e some D=D(κ, u)>0 and
C=C(, κ, u)>0 so ha , o all k∈2N+ 1,
λk≥(2k−2 + (1 ±1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(51)
λk≤(2k−2 + (1 ±1)(1 −2κ))s+ (2k−1−2κ)µ2su+µ2Csu.(52)
The i s e m o he igh -hand side o (51) and (52) o Q+
2and Q−
2is
2(k−2κ)sand 2(k−1)s, espec i ely.
•Fo W2,1, we can ake ˜u=u+1
2sa is ying (31). Mo eo e he maximum
eigen alue o ∓sV is s(1∓(2κ+u)−u). Thus he spec um o W2,1consis s
o wo g oups o eigen alues, λ0≤λ2≤ ··· and λ1≤λ3≤ ···, epea ed
acco ding o mul iplici y, such ha he e a e some D=D(κ, u)>0, C=
C(, κ, u)>0, e
C=e
C(, κ, u)>0 and E=E(, κ)>0 so ha , o all
k∈2N,
λk≥1−2µusu−1
2(2k+ 3 −2κ)s
+µ2Dsu(k+ 1)−u−2µu e
Csu+1
2∓(1 + 2κ)s , (53)
λk≤(2k+ 4 −(1 ±1)(2κ+u))s
+ (2k+ 3 −2κ)(µ2su+ 4µusu+1
2) + µ2Csu+ 4µuEsu+1
2,(54)
32 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
Since 7→ φ de ines a di e en iable map wi h alues in L2(F), i ollows ha he e
is a sequence a<ci↑ ∞ such ha φ (ci)→0, and
d
d Z∞
a+ |φ (ρ)|2dρ = lim
i
d
d Zci
a+ |φ (ρ)|2dρ ≤lim
i|φ (ci)|2= 0 .
So Z∞
a+ |φ (ρ)|2dρ ≤Z∞
a|φ0(ρ)|2dρ =Z∞
a|φ(ρ)|2dρ = 0 .
5. Wi en’s pe u ba ion on a cone
Fo el-Mo se unc ions, he el-local analysis o he Wi en’s pe u bed Lapla-
cian will be educed o he case o he unc ions ±1
2ρ2on a s a um o a cone wi h
a model adap ed me ic, whe e ρdeno es he adial unc ion. This kind o el-local
analysis begins in his sec ion.
5.1. Wi en’s pe u ba ion. To begin wi h, ecall he ollowing gene ali ies abou
he Wi en’s pe u ba ion. Le M≡(M, g) be a Riemannian n-mani old. Fo all
x∈Mand α∈TxM∗, le
αy= (−1)n +n+1 ? α∧?=−ια]on
^TxM∗,
in ol ing he Hodge s a ope a o ?on VTxM∗de ined by any choice o o ien a-
ion o TxM. Fo any ∈C∞(M), E. Wi en [43] has in oduced he ollowing
pe u ba ions o d,δ,Dand ∆, depending on s≥0:
ds=e−s d es =d+s d ∧,(71)
δs=es δ e−s =δ−s d y,(72)
Ds=ds+δs=D+sR ,
∆s=D2
s=dsδs+δsds= ∆ + s(RD +DR) + s2R2,(73)
whe e R=d ∧−d y. No ice ha δs=d†
s; hus Dsand ∆sa e o mally sel -adjoin .
By analyzing he e ms RD +DR and R2, he exp ession (73) becomes
∆s= ∆ + sHess +s2|d |2,(74)
whe e Hess is an endomo phism de ined by Hess [36, Lemma 9.17], sa is ying
|Hess |=|Hess |[4, Sec ion 9].
5.2. De Rham ope a o s on a cone. Le Lbe a non-emp y compac s a i ica-
ion. Conside a s a um No L, and he co esponding s a um M=N×R+o
c(L). We use he no a ion ˜n= dim Nand n= dim M= ˜n+ 1. Le π:M→N
be he i s ac o p ojec ion, and ρ he adial unc ion on c(L). F om VTM∗=
VTN∗VTR∗
+, we ge a canonical iden i y
^TM∗≡π∗
^TN∗⊕dρ ∧π∗
−1
^TN∗≡π∗
^TN∗⊕π∗
−1
^TN∗(75)
o e e y deg ee . So
Ω (M)≡C∞(R+,Ω (N)) ⊕dρ ∧C∞(R+,Ω −1(N)) (76)
≡C∞(R+,Ω (N)) ⊕C∞(R+,Ω −1(N)) .(77)
He e, smoo h unc ions R+→Ω(N) a e de ined by conside ing Ω(N) as F ´eche
space wi h he weak C∞ opology. In his sec ion, all ma ix exp essions o ec o

WITTEN’S PERTURBATION ON STRATA 33
bundle homomo phisms on V TM∗o di e en ial ope a o s on Ω (M) will be
conside ed wi h espec o he decomposi ions (75) and (77).
Le dand ˜
ddeno e he ex e io de i a i es on Ω(M) and Ω(N), espec i ely. We
ha e [4, Lemma 10.1]
d≡ ˜
d0
d
dρ −˜
d!.(78)
Fix a gene al adap ed me ic ˜gon N. Fo u > 0, he me ic g=ρ2u˜g+dρ2is
a gene al adap ed me ic on M. The induced me ics on VTM∗and VTN∗a e
also deno ed by gand ˜g, espec i ely. Fix some deg ee ∈ {0,1, . . . , n}, and, o
simpli y he exp essions, le
κ= (n−2 −1)u
2.(79)
Acco ding o (75),
g≡ρ−2 u ˜g⊕ρ−2( −1)u˜g(80)
on V TM∗. Choose an o ien a ion on an open subse W⊂N, and le ˜ωdeno e
he co esponding ˜g- olume o m on W. Conside he o ien a ion on W×R+⊂M
so ha he co esponding g- olume o m is
ω=ρ(n−1)udρ ∧˜ω . (81)
The co esponding Hodge s a ope a o s on VT(W×R+)∗and VTW∗will be
deno ed by ?and ˜
?, espec i ely. Like in [4, Lemma 10.2], om (80) and (81), i
ollows ha
?≡0ρ2(κ+u)˜
?
(−1) ρ2κ˜
?0(82)
on V T(W×R+)∗. Le L2Ω (M) = L2Ω (M, g) and L2Ω (N) = L2Ω (N, ˜g).
F om (80) and (81), we also ge ha (77) induces he iden i y o Hilbe spaces6
L2Ω (M)≡L2
κ,+b
⊗L2Ω (N)⊕L2
κ+u,+b
⊗L2Ω −1(N).(83)
Le δand ˜
δdeno e he ex e io code i a i es on Ω(M) and Ω(N), espec i ely. Like
in [4, Lemma 10.3], using (78), (82) and (9), we ge
δ≡ ρ−2u˜
δ−d
dρ −2(κ+u)ρ−1
0−ρ−2u˜
δ!(84)
on Ω (M). Le ∆ and e
∆ deno e he Laplacians on Ω(M) and Ω(N), espec i ely.
Like in [4, Co olla y 10.4], om (78), (84) and (9), i ollows ha
∆≡P−2uρ−1˜
d
−2uρ−2u−1˜
δ Q (85)
on Ω (M), whe e
P=ρ−2ue
∆−d2
dρ2−2κρ−1d
dρ ,(86)
Q=ρ−2ue
∆−d2
dρ2−2(κ+u)d
dρ ρ−1.(87)
6Recall ha , o Hilbe spaces H0and H00, wi h scala p oduc s h,i0and h,i00, he no a ion
H0b
⊗H00 is used o he Hilbe space enso p oduc . This is he Hilbe space comple ion o he
algeb aic enso p oduc H0⊗H00 wi h espec o he scala p oduc de ined by hu0⊗u00, 0⊗ 00i=
hu0, 0i0hu00, 00i00.
34 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
5.3. Wi en’s pe u ba ion on a cone. Le ds,δs,Dsand ∆s(s≥0) deno e
he Wi en’s pe u ba ions o d,δ,Dand ∆ induced by he unc ion =±1
2ρ2on
M. The mo e explici no a ion d±
s,δ±
s,D±
sand ∆±
smay be used i needed. In his
case, d =±ρ dρ. Acco ding o (77),
ρ dρ∧ ≡ 0 0
ρ0,−ρ dρy≡0ρ
0 0.
So, by (78), (84), (71) and (72),
ds≡ ˜
d0
d
dρ ±sρ −˜
d!,(88)
δs≡ ρ−2u˜
δ−d
dρ −2(κ+u)ρ−1±sρ
0−ρ−2u˜
δ!,(89)
on Ω (M). Now,
R=±ρ(dρ∧−dρy)≡ ±0ρ
ρ0,
and he e o e
R2≡ρ20
0ρ2≡ρ2.(90)
Like in [4, Lemma 10.6], we ge
RD +DR =∓V(91)
on Ω (M), whe e Vis gi en by (44). As a consequence o (73), (85) and (91), we
ob ain
∆s≡Ps−2uρ−1˜
d
−2uρ−2u−1˜
δ Qs(92)
on Ω (M), whe e
Ps=ρ−2ue
∆ + H−2κρ−1d
dρ ∓s(1 + 2κ),(93)
Qs=ρ−2ue
∆ + H−2(κ+u)d
dρ ρ−1∓s(−1 + 2(κ+u)) .(94)
6. Spli ing o he Wi en’s complex on a cone
6.1. Spec al decomposi ion on he link o he cone. Theo em 1.1 is p o ed
by induc ion on he dep h. Thus, wi h he no a ion o Sec ion 5, suppose ha ˜g
is good, and e
∆max/min sa is ies he s a emen o Theo em 1.1. Mo eo e suppose
ha gis also good; ha is, u≤1.
Le e
Hmax/min = ke e
Dmax/min = ke e
∆max/min, which is a g aded subspace o
Ω(N)∩L2Ω(N). Fo e e y deg ee , le e
Rmax/min, −1,e
R∗
max/min, ⊂L2Ω (N)
be he images o ˜
dmax/min, −1and ˜
δmax/min, , espec i ely, which a e closed sub-
spaces. By es ic ion, e
∆max/min de ines sel -adjoin ope a o s in e
Rmax/min, −1and
e
R∗
max/min, −1, wi h he same eigen alues [4, Sec ion 5.1]. Fo any eigen alue ˜
λo
WITTEN’S PERTURBATION ON STRATA 35
he es ic ion o e
∆max/min o e
Rmax/min, −1, le e
Rmax/min, −1,˜
λand e
R∗
max/min, −1,˜
λ
deno e he co esponding ˜
λ-eigenspaces. We ha e7
L2Ω (N) = e
H
max/min ⊕[
M
˜
λ,˜
λ0e
Rmax/min, −1,˜
λ⊕e
R∗
max/min, ,˜
λ0,(95)
whe e ˜
λand ˜
λ0 un in he spec um o he es ic ions o e
∆max/min o e
Rmax/min, −1
and e
R∗
max/min, , espec i ely.
6.2. Subcomplexes o leng h one. Gi en 0 6=γ∈e
H
max/min, conside he canon-
ical iden i ies
C∞
+≡C∞
+γ⊂Ω (M), C∞
+≡C∞
+dρ ∧γ⊂Ω +1(M).(96)
The ollowing esul ollows om (88) and (89).
Lemma 6.1. Fo s≥0,dsand δsde ine maps
0C∞
+γ C∞
+dρ ∧γ0..
ds, −1
δs, −1
ds,
δs,
ds, +1
δs, +1
-
-
-

Mo eo e , using (96),
ds, =d
dρ ±sρ , δs, =−d
dρ −2κρ−1±sρ .
Le Eγ,0deno e he subcomplex o leng h one o (Ω(M), ds) de ined by
E
γ,0=C∞
+,0γ≡C∞
+,0,E +1
γ,0=C∞
+,0dρ ∧γ≡C∞
+,0.
The closu e o Eγ,0in L2Ω(M) is deno ed by L2Eγ. By (83),
L2E
γ=L2
κ,+γ≡L2
κ,+, L2E +1
γ=L2
κ,+dρ ∧γ≡L2
κ,+.
Assume now ha s > 0. Wi h he no a ion o Sec ion 4.1, conside he eal
e sion o he ellip ic complex (E, d) de e mined by sand κ(gi en by (79)). Using
Lemma 6.1 and (9), like in [4, P oposi ion 12.3], we ge he ollowing.
P oposi ion 6.2. The ope a o ρκ:L2
κ,+→L2
+de ines a uni a y isomo phism
L2Eγ→L2(E), which es ic s o an isomo phism o complexes, (Eγ,0, ds)→
(C∞
0(E), d), up o a shi o deg ee.
By P oposi ion 6.2, (Eγ,0, ds) has a maximum/minimum Hilbe complex ex-
ension in L2Eγ. Le (Dγ,ds,γ) be he maximum/minimum Hilbe complex ex-
ension o (Eγ,0, ds) i γ∈e
H
max/min, and ∆s,γ he co esponding Laplacian. Le
Hs,γ =H
s,γ ⊕ H +1
s,γ = ke ∆s,γ , wi h he induced g ading. The mo e explici
no a ion d±
s,γ,∆±
s,γ and H±
s,γ =H±,
s,γ ⊕H±, +1
s,γ may be also used.
Co olla y 6.3. (i)∆s,γ has a disc e e spec um.
(ii)The dimensions o H±,
s,γ and H±, +1
s,γ a e gi en in Table 11.
(iii)I es∈ Hs,γ wi h no m one o e e y s, and his a bounded measu able unc ion
on R+wi h h(ρ)→1as ρ→0, hen hhes, esi → 1as s→ ∞.
(i )All nonze o eigen alues o ∆s,γ a e posi i e and in O(s)as s→ ∞.
36 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
γ∈e
H
max γ∈e
H
min
H+,
s,γ H+, +1
s,γ H−,
s,γ H−, +1
s,γ H+,
s,γ H+, +1
s,γ H−,
s,γ H−, +1
s,γ
κ≥1
21001
0
0
|κ|<1
20 1
κ≤ −1
20 1
Table 11. Dimensions o H±,
s,γ and H±, +1
s,γ
P oo . This ollows om P oposi ions 6.2 and 4.1, Co olla y 3.7, Sec ion 4.1.1, and
he choice made o de ine ds,γ.
6.3. Subomplexes o leng h wo. Le µ=p˜
λ o an eigen alue ˜
λo he es ic-
ion o e
∆max/min o e
Rmax/min, −1. Acco ding o [4, Sec ion 5.1], he e a e nonze o
di e en ial o ms,
α∈e
Rmax/min, −1,˜
λ⊂Ω (N), β ∈e
R∗
max/min, −1,˜
λ⊂Ω −1(N),
such ha ˜
dβ =µα and ˜
δα =µβ. Conside he canonical iden i ies
C∞
+≡C∞
+β⊂Ω −1(M), C∞
+≡C∞
+dρ ∧α⊂Ω +1(M),(97)
C∞
+⊕C∞
+≡C∞
+α+C∞
+dρ ∧β⊂Ω (M).(98)
The ollowing esul ollows om (88) and (89).
Lemma 6.4. Fo s≥0,dsand δsde ine maps
0C∞
+β C∞
+α+C∞
+dρ ∧β
C∞
+dρ ∧α0.
ds, −2
δs, −2
ds, −1
δs, −1
ds,
δs,
ds, +1
δs, +1
-
-

-
-

Mo eo e , acco ding o (97) and (98),
ds, −1=µ
d
dρ ±sρ,
δs, −1=µρ−2u−d
dρ −2(κ+u)ρ−1±sρ,
ds, =d
dρ ±sρ −µ,
δs, =−d
dρ −2κρ−1±sρ
−µρ−2u.
7Conside a amily o Hilbe spaces, Hawi h scala p oduc h,ia. Recall ha he Hilbe
space di ec sum, c
LaHa, is he Hilbe space comple ion o he algeb aic di ec sum, LaHa, wi h
espec o he scala p oduc h(ua),( a)i=Pahua, aia. Thus c
LaHa=LaHai and only i
he amily is ini e.
WITTEN’S PERTURBATION ON STRATA 37
Le Fα,β,0=F −1
α,β,0⊕ F
α,β,0⊕ F +1
α,β,0deno e he subcomplex o leng h wo o
(Ω(M), ds) de ined by
F −1
α,β,0=C∞
+,0β≡C∞
+,0,F +1
α,β,0=C∞
+,0dρ ∧α≡C∞
+,0,
F
α,β,0=C∞
+,0α+C∞
+,0dρ ∧β≡C∞
+,0⊕C∞
+,0.
The closu e o Fα,β,0in L2Ω(M) is deno ed by L2Fα,β. By (83),
L2F −1
α,β =L2
κ+u,+β≡L2
κ+u,+, L2F +1
α,β =L2
κ,+dρ ∧α≡L2
κ,+,
L2F
α,β =L2
κ,+α+L2
κ+u,+dρ ∧β≡L2
κ,+⊕L2
κ+u,+.
Assume now ha s > 0. Wi h he no a ion o Sec ion 4.2, conside he eal
e sion o he ellip ic complex (F, d) de e mined by sand κ(gi en by (79)). Using
Lemma 6.4 and (9), we ge he ollowing (c . [4, P oposi ion 12.9]).
P oposi ion 6.5. I u < 1, hen ρκ:L2
κ,+→L2
+and ρκ+u:L2
κ+u,+→L2
+
de ine a uni a y isomo phism L2Fα,β →L2(F), which es ic s o an isomo phism
o complexes, (Fα,β,0, ds)→(C∞
0(F), d), up o a shi o deg ee.
By P oposi ion 6.5, (Fα,β,0, ds) has a maximum/minimum Hilbe complex ex-
ension in L2Fα,β. Le (Dα,β,ds,α,β) be he maximum/minimum Hilbe complex
ex ension o (Fα,β,0, ds) i α∈e
Rmax/min, −1,˜
λand β∈e
R∗
max/min, −1,˜
λ. Le ∆s,α,β
deno e he co esponding Laplacian. The mo e explici no a ion d±
s,α,β and ∆±
s,α,β
may be used.
Co olla y 6.6. (i)∆s,α,β has a disc e e spec um.
(ii)The eigen alues o ∆s,α,β a e posi i e and in O(s)as s→ ∞.
P oo . In he case u < 1, his ollows om P oposi ion 6.5 and Co olla y 4.5. In
he case u= 1, his is he con en o [4, P oposi ion 12.11]. 
Rema k 11.Acco ding o (91)–(94), we ha e
∆s≡H−2κρ−1d
dρ ∓s(1 + 2κ) on C∞
+≡C∞
+γ ,
∆s≡H−2κd
dρ ρ−1∓s(−1+2κ) on C∞
+≡C∞
+dρ ∧γ ,
∆s≡H−2(κ+u)ρ−1d
dρ +µ2ρ−2u∓s(1 + 2(κ+u)) on C∞
+≡C∞
+β ,
∆s≡H−2κd
dρ ρ−1+µ2ρ−2u∓s(−1+2κ) on C∞
+≡C∞
+dρ ∧α ,
and
∆s≡Pµ,s −2µuρ−1
−2µuρ−2u−1Qµ,s 
on C∞
+⊕C∞
+≡C∞
+α+C∞
+dρ ∧β, whe e
Pµ,s =H−2κρ−1d
dρ +µ2ρ−2u∓s(1 + 2κ),
Qµ,s =H−2(κ+u)d
dρ ρ−1+µ2ρ−2u∓s(−1 + 2(κ+u)) .
So he esul s o Sec ion 3 could be applied o hese exp essions. We op ed o
analyzing i s he complexes o Sec ion 4 o he sake o simplici y because we ha e
a=b= 0, L2
+is used ins ead o L2
κ,+o L2
κ+u,+, and Rema k 8 is di ec ly applied.

38 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
6.4. Spli ing in o subcomplexes. Le Cmax/min,0deno e an o hono mal ame
o e
Hmax/min consis ing o homogeneous di e en ial o ms. Fo e e y posi i e eigen-
alue µo e
Dmax/min, le Cmax/min,µ be an o hono mal ame o he µ-eigenspace o
e
Dmax/min consis ing o di e en ial o ms α+βlike in Sec ion 6.3. Then le
ds,max/min =M
γ
ds,γ ⊕d
M
µM
α+β
ds,α,β ,
whe e γ uns in Cmax/min,0,µ uns in he posi i e spec um o e
Dmax/min, and α+β
uns in Cmax/min,µ. The no a ion d±
s,max/min may be also used when d±
s,γ and d±
s,α,β
a e conside ed.
P oposi ion 6.7. We ha e ds,max/min =ds,max/min.
P oo . This ollows like [4, P oposi ion 12.12], using [4, Lemma 5.2], [9, Lemma 3.6
and (2.38b)], (76) and (95). 
Le Hs,max/min =L H
s,max/min = ke ∆s,max/min, wi h he induced g ading.
The supe index “±” may be added o his no a ion o indica e ha we a e e e ing
o ∆±
s,max/min.
Co olla y 6.8. (i) ∆s,max/min has a disc e e spec um.
(ii)Table 12 desc ibes he isomo phism class o H±,∗
s,max/min.
(iii)I es∈ Hs,max/min has no m one o e e y s, and his a bounded measu able
unc ion on R+wi h h(ρ)→1as ρ→0, hen hhes, esi → 1as s→ ∞.
(i )Le 0≤λs,max/min,0≤λs,max/min,1≤ ··· be he eigen alues o ∆s,max/min,
epea ed acco ding o hei mul iplici ies. Gi en k∈N, i λs,max/min,k >0 o
some s, hen λs,max/min,k >0 o all s, and λs,max/min,k ∈O(s)as s→ ∞.
( )The e is some θ > 0such ha lim in kλs,max/min,kk−θ>0.
H+,
s,max H−, +1
s,max H+,
s,min H−, +1
s,min
κ≥1
2H
max(N) 0 H
min(N) 0
|κ|<1
20H
min(N)
κ≤ −1
20H
max(N)
Table 12. Spaces isomo phic o H±,∗
s,max/min
P oo . In he case u= 1, his esul was al eady shown in [4, Co olla y 12.13]. So
we conside only he case 0 <u<1. Fo all γ,µand α+βas abo e, ∆s,γ and
∆s,α,β ha e a disc e e spec um by Co olla ies 6.3 (i) and 6.6 (i). Mo eo e he
union o hei spec a has no accumula ion poin s acco ding o Sec ion 4 and since
e
∆max/min is disc e e. Then (i) ollows by P oposi ion 6.7.
Now, p ope ies (ii)–(i ) ollow di ec ly om Co olla ies 6.3 and 6.6, and P opo-
si ion 6.7.
To p o e ( ), le 0 ≤˜
λmax/min,0≤˜
λmax/min,1≤ ··· deno e he eigen alues
o e
∆max/min, epea ed acco ding o hei mul iplici ies. Since Nsa is ies Theo-
em 1.1 (ii) wi h ˜g, he e is some C0, θ0>0 such ha
˜
λmax/min,` ≥C0`θ0(99)
WITTEN’S PERTURBATION ON STRATA 39
o all `la ge enough. Conside he coun ing unc ion
N±
s,max/min(λ)=#nk∈N|λ±
s,max/min,k < λ o(λ > 0) .
F om P oposi ion 4.3, Co olla y 4.5, (39)–(42), (45), (47), (49), (51), (53), (55)
and (99), and he choices made o de ine dγand dα,β (Sec ions 6.2 and 6.3), i
ollows ha he e a e some C1, C2>0 and C3, C0
3∈Rsuch ha
N±
s,max/min(λ)
≤#n(k, `)∈N2|C1k+C2˜
λmax/min,`(k+ 1)−u+C0
3≤λo
≤#{(k, `)∈N2|C1k+C2C0`θ0(k+ 1)−u+C3≤λ}
≤#((k, `)∈N20≤λ−C3
C1
, ` ≤λ−C3−C1k
C2C01
θ0(k+ 1) u
θ0).
Conside he unc ion
:−1, a := λ−C3
C1→[0,∞), (x) = λ−C3−C1x
C2C01
θ0(x+ 1) u
θ0.
Elemen a y calculus shows ha anishes a x=−1, a, i eaches i s maximum a
x=b:= λu −C3u−C1
C1(1 + u),
and i is s ic ly inc easing ( espec i ely, dec easing) on [−1, b] ( espec i ely, [b, a]).
I ollows ha 8
N±
s,max/min(λ)≤Za
0
(x)dx + 2 (b) + a+ 1 .
Bu
(b) = λ−C3+C1
(1 + u)C2C01
θ0u(λ−C3+C1)
(1 + u)C1u
θ0,
and
Za
0
(x)dx ≤ Zλ−C3
C1
0λ−C3−C1x
C2C02
θ0dx!1
2 Zλ−C3
C1
0
(x+ 1)2u
θ0dx!1
2
≤ θ0(λ−C3)2
θ0+1
(2 + θ0)(C2C0)2
θ0C1!1
2
θ0(λ−C3+C1)2u
θ0+1
(2u+θ0)C
2u
θ0+1
1


1
2
=θ0(λ−C3)1
θ0+1
2(λ−C3+C1)u
θ0+1
2
(2 + θ0)1
2(2u+θ0)1
2(C2C0)1
θ0C1+ u
θ0
1
.
8A simila a gumen is made in he p oo o [4, Co olla y 12.13-( iii)]. In ha case, he au ho s
use a s ic ly dec easing unc ion : (−∞, a]→[0,∞). The esul ing es ima e should be
N±
s,max/min(λ)≤Za
0
(x)dx + (0) + a+ 1 ,
bu he e ms (0) + a+ 1 we e missing in ha publica ion. This co ec ion does no a ec he
inal es ima e o N±
s,max/min(λ) ob ained he e.
40 J.A. ´
ALVAREZ L ´
OPEZ, M. CALAZA, AND C. FRANCO
So N±
s,max/min(λ)≤Cλ1+u
θ0+1 o some C > 0 and all la ge enough λ, gi ing ( )
wi h θ=1+u
θ0+ 1. 
Table 13 desc ibes he abo e condi ions on κin e ms o .
κ≥1
2 ≤n−1
2−1
2u
|κ|<1
2| −n−1
2|<1
2u
κ≤ −1
2 ≥n−1
2+1
2u
Table 13. Co espondence be ween condi ions on κand
7. Rela i ely local model o he Wi en’s pe u ba ion
Le m∈N, and le L1, . . . , Labe compac s a i ica ions. Fo each i= 1, . . . , a,
le Nibe a dense s a um o Li, le ki= dim Ni+1, and le ∗iand ρibe he e ex
and adial unc ion o c(Li). Then M:= Rm×Qa
i=1(Ni×R+) is a dense s a um
o A:= Rm×Qa
i=1 c(Li). Fo any ela i ely compac open neighbo hood Oo
x:= (0,∗1,...,∗a), all gene al adap ed me ics on Ma e quasi-isome ic on M∩O
o a me ic o he o m g=g0+Pa
i=1 ρ2ui
i˜gi+ (dρi)2, whe e g0is he Euclidean
me ic on Rm, e e y ˜giis a gene al adap ed me ic on Ni, and ui>0. Suppose
ha gis good; i.e., he me ics ˜gia e good, and ui≤1. We can assume ha e e y
Niis connec ed, which means ha he ibe o lim : c
M→Mo e xconsis s o
a unique poin , which can be iden i ied o x(see [4, P oo o P oposi ion 3.20]).
Acco ding o Sec ion 1.4, he el-local model o a el-Mo se unc ion a ound a el-
c i ical poin is o he o m =1
2(ρ2
+−ρ2
−), whe e ρ±is he adial unc ion o
Rm±×Qi∈I±c(Li), o some decomposi ion m=m++m−(m±∈N), and some
pa i ion o {1, . . . , a}in o se s I±. The el-c i ical se o consis s only o x. Le
ds,δs,Dsand ∆sbe he Wi en’s pe u ba ions o d,δ,Dand ∆ on Ω(M) induced
by . Le Hs,max/min =L H
s,max/min = ke ∆s,max/min, wi h he induced g ading.
The ollowing esul is a di ec consequence o Co olla y 6.8 and [4, Example 9.1
and Lemma 5.1], aking also in o accoun Table 13.
Co olla y 7.1. (i) ∆s,max/min has a disc e e spec um.
(ii)We ha e
H
s,max/min ∼
=M
( 1,..., a)
a
O
i=1
H i
max/min(Ni),
whe e ( 1, . . . , a) uns in he subse o Nade ined by he condi ions
=m−+
a
X
i=1
i+|I−|,
i<ki−1
2+1
2uii i∈I+
i≥ki−1
2+1
2uii i∈I−) o H
s,max ,
i≤ki−1
2−1
2uii i∈I+
i>ki−1
2−1
2uii i∈I−) o H
s,min .
WITTEN’S PERTURBATION ON STRATA 41
(iii)I es∈ Hs,max/min wi h no m one o e e y s, and his a bounded measu able
unc ion on R+wi h h(ρ)→1as ρ→0, hen hhes, esi → 1as s→ ∞.
(i )Le 0≤λs,max/min,0≤λs,max/min,1≤ ··· be he eigen alues o ∆s,max/min,
epea ed acco ding o hei mul iplici ies. Gi en k∈N, i λs,max/min,k >0 o
some s, hen λs,max/min,k >0 o all sand λs,max/min,k ∈O(s)as s→ ∞.
( )The e is some θ > 0such ha lim in kλs,max/min,k k−θ>0.
Fo e e y ρ > 0, le Bρbe he open ball o cen e 0 and adius ρin Rm, and le
Ux,ρ =Bρ×
a
Y
i=1
(Ni×(0, ρ)) ⊂M .
Taking complex coe icien s, by P oposi ions 6.2, 6.5 and 6.7, he ollowing esul
clea ly boils down o he case o P oposi ion 4.6.
P oposi ion 7.2. Fo α∈L2Ω(M), le α = exp(i Ds,max/min)α. I supp α⊂Ux,a
o some a > 0, hen supp α ⊂Ux,a+| | o all ∈R.
8. P oo o Theo em 1.1
This heo em ollows om Co olla y 7.1 (i),( ) wi h he same a gumen s as [4,
Theo em 1.1]. Mo e p ecisely, [4, P oposi ions 14.2 and 14.3] a e used o glob-
alize he p ope ies o he el-local model, he min-max p inciple (see e.g. [35,
Theo em XIII.1]) is used o show ha he p ope ies o he s a emen a e in a i-
an by aking Wi en’s pe u ba ion de ined by el-admissible unc ions, and Re-
ma k 6 (iii),(i ) is used o p oduce el-admissible cu o unc ions and pa i ions
o uni y wi h bounded di e en ial. These unc ions a e needed o he Wi en’s
pe u ba ion and o apply [4, P oposi ions 14.2 and 14.3].
9. Func ional calculus
Le Mbe a s a um o a compac s a i ica ion, equipped wi h a good gene al
adap ed me ic g. Le be any el-admissible unc ion on M, and le ds,δs,Dsand
∆sbe he co esponding Wi en’s pe u ba ions o d,δ,Dand ∆. Since is el-
admissible, o e e y s, ∆s−∆ is a homomo phism wi h uni o mly bounded no m
by (74). F om (74) and he min-max p inciple (see e.g. [35, Theo em XIII.1]), i
also ollows ha D(∆s,max/min) = D(∆max/min), D∞(∆s,max/min) = D∞(∆max/min),
and ha he p ope ies s a ed in Theo em 1.1 can be ex ended o he pe u ba ion
∆s,max/min.
Fo any apidly decaying unc ion φon R,φ(∆s,max/min) is a Hilbe -Schmid
ope a o on L2Ω(M) by he e sion o Theo em 1.1 (ii) o ∆s,max/min. In ac ,
φ(∆s,max/min) is a ace class ope a o because φcan be gi en as he p oduc
o wo apidly decaying unc ions, |φ|1/2and sign(φ)|φ|1/2, whe e sign(φ)(x) =
sign(φ(x)) ∈ {±1}i φ(x)6= 0.
Like in he case o closed mani olds (see e.g. [36, Chap e s 5 and 8]), φ(∆s,max/min)
is gi en by a Schwa z ke nel Ks, and T φ(∆s,max/min) equals he in eg al o he
poin wise ace o Kson he diagonal. Bu we do no know whe he Ksis uni o mly
bounded because a “ el-Sobole embedding heo em” is missing [4, Sec ion 19].
Theo em 1.1 (ii) becomes impo an in ou a gumen s o make up o his lack.