Witten’s perturbation and Lefschetz formula on singular spaces

CENTRO INTERNACIONAL DE ESTUDOS DE DOUTORAMENTO E AVANZADOS DA USC (CIEDUS) TESE DE DOUTORAMENTO WITTEN’S PERTURBATION AND LEFSCHETZ FORMULA ON SINGULAR SPACES Carlos Luis Franco Sanmartín ESCOLA DE DOUTORAMENTO INTERNACIONAL PROGRAMA DE DOUTORAMENTO EN MATEMÁTICAS SANTIAGO DE COMPOSTELA ANO 2019 . DECLARACIÓN DO AUTOR DA TESE Witten’s Perturbation and Lefschetz Formula on Singular Spaces D. Carlos Luis Franco Sanmartín Presento a miña tese, seguindo o procedemento adecuado ao Regulamento, e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) No seu caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. 4) Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. En Santiago de Compostela, 2 de maio de 2019 Asdo. Carlos Luis Franco Sanmartín . AUTORIZACIÓN DO DIRECTOR DA TESE Witten’s Perturbation and Lefschetz Formula on Singular Spaces D. Jesús Antonio Álvarez López INFORMA: Que a presente tese se corresponde co traballo realizado por D. Carlos Luis Franco Sanmartín, baixo a miña dirección, e autorizo a súa presentación, considerando que reúne os requisitos esixidos no Regulamento de Estudos de Doutoramento da USC, e que como director desta non incorro nas causas de abstención establecidas na lei 40/2015. En Santiago de Compostela, 2 de maio de 2019 Asdo. Jesús Antonio Álvarez López . Contents Preface v 1 Introduction 1 1.1 Ideal boundary conditions of the de Rham complex . . . . . . . . . 1 1.2 Stratified spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 General adapted metrics . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Relatively Morse functions . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Lefschetz trace formula . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Main theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Applications to intersection homology . . . . . . . . . . . . . . . . 10 1.8 Ideas of the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9 Some open problems . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.10 Dunkl Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 16 2 Witten’s Perturbation on Strata 23 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 Products of cones . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 General adapted metrics . . . . . . . . . . . . . . . . . . . 27 2.1.3 Relatively Morse functions . . . . . . . . . . . . . . . . . . 28 2.1.4 Hilbert and elliptic complexes . . . . . . . . . . . . . . . . 29 2.2 Two simple types of elliptic complexes . . . . . . . . . . . . . . . . 31 2.2.1 An elliptic complex of length one . . . . . . . . . . . . . . 31 2.2.2 An elliptic complex of length two . . . . . . . . . . . . . . 34 2.2.3 The wave operator . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Witten’s perturbation on a cone . . . . . . . . . . . . . . . . . . . . 47 2.3.1 Witten’s perturbation . . . . . . . . . . . . . . . . . . . . . 47 2.3.2 De Rham operators on a cone . . . . . . . . . . . . . . . . 48 2.3.3 Witten’s perturbation on a cone . . . . . . . . . . . . . . . 50 2.4 Splitting of the Witten’s complex on a cone . . . . . . . . . . . . . 51 2.4.1 Spectral decomposition on the link of the cone . . . . . . . 51 i Contents 2.4.2 Subcomplexes of length one . . . . . . . . . . . . . . . . . 51 2.4.3 Subcomplexes of length two . . . . . . . . . . . . . . . . . 53 2.4.4 Splitting into subcomplexes . . . . . . . . . . . . . . . . . 55 2.5 Relatively local model of the Witten’s perturbation . . . . . . . . . 58 2.6 Proof of Theorem 1.6.1 . . . . . . . . . . . . . . . . . . . . . . . . 60 2.7 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.8 The wave operator . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9 Proof of Theorem 1.6.2 . . . . . . . . . . . . . . . . . . . . . . . . 62 3 Lefschetz trace formula 65 3.1 Reduction to the contribution from the fixed points . . . . . . . . . 66 3.2 Lefschetz trace formula on a cone . . . . . . . . . . . . . . . . . . 68 3.3 Contribution from the singular fixed points . . . . . . . . . . . . . . 71 4 Dunkl Harmonic Oscillator 73 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 The sesquilinear form t. . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Scalar products of mixed generalized Hermite functions . . . . . . . 90 4.3.1 Case where σ=θ6=τand τ−σ6∈ −N. . . . . . . . . . 91 4.3.2 Case where σ6=θ6=τand σ−θ, τ −θ6∈ −N. . . . . . . 94 4.4 The sesquilinear form t0. . . . . . . . . . . . . . . . . . . . . . . . 99 4.4.1 Case where σ=θ=τ. . . . . . . . . . . . . . . . . . . . 99 4.4.2 Case where σ=θ6=τand τ−σ6∈ −N. . . . . . . . . . 99 4.4.3 Case where σ6=θ=τand σ−θ6∈ −N. . . . . . . . . . . 100 4.4.4 Case where σ6=θ=τ+ 1 and σ−τ−16∈ −N. . . . . . 101 4.4.5 Case where σ6=θ6=τand σ−θ, τ −θ6∈ −N. . . . . . . 103 4.4.6 Proof of Theorem 1.10.3 . . . . . . . . . . . . . . . . . . . 104 4.5 A preliminary estimate . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.5.2 Proof of Lemma 4.5.1 . . . . . . . . . . . . . . . . . . . . 107 4.6 The main estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.7 Operators induced on R+. . . . . . . . . . . . . . . . . . . . . . . 126 Appendix 132 A Preliminaries on Stratified Spaces and Global Analysis 133 A.1 Stratified Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.1.1 Thom-Mather stratifications . . . . . . . . . . . . . . . . . 133 A.1.2 Products of stratifications . . . . . . . . . . . . . . . . . . . 135 A.2 Some Results of Global Analysis . . . . . . . . . . . . . . . . . . . 136 ii Contents Resumo 139 1 Condicións de fronteira ideal do complexo de de Rham . . . . . . . 140 2 Espazos estratificados . . . . . . . . . . . . . . . . . . . . . . . . . 141 3 Métricas adaptadas xerais . . . . . . . . . . . . . . . . . . . . . . . 144 4 Funcións de rel-Morse . . . . . . . . . . . . . . . . . . . . . . . . 145 5 Fórmula da traza de Lefschetz . . . . . . . . . . . . . . . . . . . . 147 6 Teoremas principais . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7 Aplicacións a homoloxía intersección . . . . . . . . . . . . . . . . 150 8 Ideas principais das demostracións . . . . . . . . . . . . . . . . . . 152 9 Problemas abertos . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 10 Oscilador harmónico de Dunkl . . . . . . . . . . . . . . . . . . . . 154 Bibliography 161 iii Introduction Let xbe a point of a stratum Xof dimension mXin a stratification A. A local trivialization of TXon some open neighborhood Uof xdefines a chart O≡O0of A for some open O0⊂RmX×c(LX). We can assume O0=U0×c(LX), where U0is some open neighborhood of 0in RmXand c(LX)is the subset of c(LX)defined by the condition ρX< , for some  > 0. This chart is centered at xif x≡(0,∗X)∈O0. The corresponding concept of atlas has the obvious meaning. These concepts can be generalized as follows. Any finite product of stratifications has a non-canonical stratified structure (see Appendix A.1.2); in particular, any finite product of cones is isomorphic to a cone [5, Lemma 3.8]. Moreover Aut(P)×Aut(Q)is canonically injected in Aut(P×Q)for stratifications Pand Q. Thus it makes sense to consider a decomposition c(LX)∼ =QaX i=1 c(LX,i)(aX∈N), for compact stratifications LX,i. The vertex and radial function of every c(LX,i)are denoted by ∗X,i and ρX,i. Then we can also consider general tube representatives given by bundles TXwith typical fibers QaX i=1 c(LX,i)and structural groups QaX i=1 c(Aut(LX,i)). This gives rise to a general chart O≡O0around xfor some open O0⊂RmX×QaX i=1 c(LX,i), which is centered at xif x≡(0,∗X,1, . . . , ∗X,aX)∈O0. As above, we can assume O0= U0×QaX i=1 c(LX,i)for some  > 0. Let ρX,0denote the norm function on RmX. The function ρ= (ρ2 X,0+··· +ρ2 X,aX)1/2is called the radial function of RmX× QaX i=1 c(LX,i), even though, when mX= 0,ρis not the radial function of any cone structure on QaX i=1 c(LX,i)[5, Example 3.6 and Proof of Lemma 3.8]. A collection of general charts covering Ais called a general atlas. Example 1.2.1.Figure 1.2.1 (taken from [35]) illustrates a compact stratification with four strata. It is constructed by “choking” a meridian section of a usual torus into a point x(0-dimensional stratum), and covering its hole with a disk (2-dimensional stratum). The border line between the torus and the disk is a 1-dimensional stratum. The regular part of the torus is another bidimensional stratum. There is no dense stratum, so we cannot choose a regular one. Notice that two charts are indicated in the picture, taken around the points xand y, respectively. The chart centered at xhas trivial Euclidean factor; whereas the conic factor of the chart centered at yhas a link consisting on three points. We can suppose that the strata of Aare connected (see Appendix A.1.1). Fix a stratum Mof dimension nin A. Since the stratified structure of Acan be restricted to M, we can also assume without loss of generality that M=A(any other stratum is < M); in particular, depth A= depth Mand dim A=n. It is said that Ais an orientable stratification if Mis an orientable manifold. With the above notation, for a chart O≡O0centered at x, we get M∩O≡M0∩O0, where M0=RmX×N×R+ for some dense stratum Non LX. In the case of a general chart O≡O0centered at x, we have M∩O≡M0∩O0for M0=RmX×QaX i=1(Ni×R+), where every Ni 4 1.3 General adapted metrics Figure 1.2.1: Charts on a compact stratification is some dense stratum of LX,i. We will use the notation kX,i = dim Ni+ 1. 1.3 General adapted metrics Ageneral adapted metric gon Mis defined by induction on the depth of M. It is any (Riemannian) metric if depth M= 0. Now, assume that depth M > 0and general adapted metrics are defined for lower depth. Given any general chart O≡O0 as above, take any general adapted metric ˜gion every Ni(depth Ni<depth M), and let gi=ρ2uX,i X,i ˜gi+ (dρX,i)2on Ni×R+for some uX,i >0. Let also g0 be the Euclidean metric on RmX. Then gis a general adapted metric if, via any such general chart, g|Ois quasi-isometric to (PaX i=0 gi)|O0. In this case, the mapping X7→ uX:= (uX,1, . . . , uX,aX)∈RaX +(X < M) is called the general type of g. Such a general chart is called compatible with g, or with its general type. Let us point out that a general metric does not completely determine its general type. For instance, suppose uX,i =uX,j = 1 for indices i6=j. Write c(LX,i)× c(LX,j)≡c(L), with radial function ρ, for some stratification L. Then Ni×R+× Nj×R+≡N×R+for some dense stratum Nof L. Moreover there is a general adapted metric ˜gon Nsuch that gi+gjis quasi-isometric to ρ2˜g+ (dρ)2via the above identity. Therefore we can omit uX,i or uX,j in uX, obtaining a different type of g. This cannot be done if uX,i =uX,j 6= 1 (Proposition 2.1.1). If the above definition of general adapted metric is modified by requiring that, at every inductive step, the general type satisfies uX,i ≤1for all X < M and i= 1, . . . , aX, then the general adapted metric is called good for the scope of this thesis. On the other hand, if the definition is modified by requiring at every inductive step that aX= 1 and uXdepends only on k:= kX,1= codim Xfor all X < M, 5 Introduction then we get the adapted metrics considered in [13,53,54]. In this case, the general charts compatible with the general type are indeed charts. Writing uk=uX≡ uX,1∈R+, the condition on an adapted metric gto be good becomes uk≤1for all k, at every inductive step of its definition. In [13, 53, 54], it is assumed that Ais a stratified pseudomanifold, and then ˆu= (u2, . . . , un)stands for the type of g. This ˆuis determined by g. In particular, if the definition is modified by taking uk= 1 for all kat every inductive step, we get the adapted metrics of conic type considered in [18–20]. Be alerted about the three slightly different terms used for the scope of this thesis: general adapted metrics, adapted metrics and adapted metrics of conic type. The class of (good) general adapted metrics is preserved by products, as well as the class of adapted metrics of conic type, but the class of adapted metrics does not have this property. The existence of general adapted metrics with any possible general type can be shown like in the case of adapted metrics [53, Lemma 4.3], [13, Appendix]. Like in [5], the term “relative(ly)” (or simply “rel-”) usually means that some condition is required in the intersection of Mwith small neighborhoods of the points in M, or that some concept can be described using those intersections. Let Mbe equipped with a general adapted metric g, with a general type X7→ uX as above. The rel-local metric completion c Mof Mconsists of the points in the metric completion represented by Cauchy sequences that converge in M(c Mis the metric completion of Mif Mis compact). Figure 1.3.1 illustrates this concept. The limits of Cauchy sequences define a continuous map lim : c M→M. The following properties can be proved like in the case of conic metrics [5, Proposition 3.20-(i),(ii)]. c Mhas a unique stratified structure with connected strata so that lim : c M→Mis a morphism whose restrictions to the strata are local diffeomorphisms. Moreover gis also a general adapted metric with respect to c M. 1.4 Relatively Morse functions A smooth function fon Mis called rel-admissible when the functions f,|df|and |Hess f|are rel-bounded. In this case, fmay not have any continuous extension to M, but it has a continuous extension to c M. So it makes sense to say that x∈c M is a rel-critical point of fwhen lim inf |df(y)|= 0 as y→xin c Mwith y∈M. The set of rel-critical points of fis denoted by Critrel(f). It is said that fis a relMorse function if it is rel-admissible and has the following description around every x∈Critrel(f): •there is a general chart O≡O0of c M, centered at xand compatible with g, such that M∩O≡M0∩O0for M0=RmX×QaX i=1(Ni×R+), where Xis 6 1.4 Relatively Morse functions (a) M(b) c M Figure 1.3.1: The stratified space c M. the stratum of c Mcontaining x; and •f|M∩O≡f(x)+ 1 2(ρ2 +−ρ2 −)|M0∩O0, where ρ±is the radial function of Rm±× Qi∈I±c(LX,i)for some expression mX=m++m−(m±∈N) and some partition of {1, . . . , aX}into sets I±. This local condition is used instead of requiring that Hess fis “rel-non-degenerate” at the rel-critical points because a “rel-Morse lemma” is missing in this context. Moreover, for r= 0, . . . , n, let νr x,max/min =X (r1,...,raX) aX Y i=1 βri max/min(Ni),(1.4.1) where (r1, . . . , raX)runs in the subset of NaXdetermined by r=m−+PaX i=1 ri+|I−|, ri<kX,i−1 2+1 2uX,i if i∈I+ ri≥kX,i−1 2+1 2uX,i if i∈I−  for νr x,max, ri≤kX,i−1 2−1 2uX,i if i∈I+ ri>kX,i−1 2−1 2uX,i if i∈I−  for νr x,min.                    (1.4.2) When aX= 0 in (1.4.1), the singleton N0consists of the empty sequence, obtaining1 νr x,max/min =δr,m−with the convention that the value of empty products is 1. Finally, 1Kronecker’s delta symbol is used. 7 Introduction let νr max/min =Pxνr x,max/min with xrunning in Critrel(f). The notation νr x,max/min(f) and νr max/min(f)may be used if necessary. The existence of rel-Morse functions for general adapted metrics holds like in the case of adapted metrics [5, Proposition 4.9]. 1.5 Lefschetz trace formula Let Abe a compact orientable stratification of dimension nwith isolated singularities and equipped with a good adapted metric gover its regular part reg(A) = M. In particular, if n > 1,Ais an orientable stratified pseudomanifold with depth A= 1. Consider a morphism of stratifications ψ:A→A. Recall that we assume that ψ|Mis a smooth map by definition of morphism (Section 1.2). Let ψ∗be the endomorphism of (Ω(M), d)induced by ψ. Let also Fix(ψ)denote the set of fixed points of ψ. Every singular point is a zero dimensional stratum of A. Then, for every q∈ sing(A) = A\M, there exists a chart Oq≡O0 q={0}×cq(Lq)≡cq(Lq) centered at qsuch that depth Lq= 0 (see Section 1.2). This means that every link Lqis a compact smooth manifold. Moreover, this local structure around the singular points implies that, for every q∈Fix(ψ)∩sing(A), the morphism ψsatisfies ψ(ρ, x) = (ρFq(ρ, x), Gq(ρ, x)) (1.5.1) in any such conic chart Oq≡cq(Lq). Here, the maps Fq:Lq×[0, q)→[0, q) and Gq:Lq×[0, q)→Lqare smooth up to zero. It is said that: •a point q∈Fix(ψ)∩Mis simple if det(1 −Tqψ)6= 0. •a point q∈Fix(ψ)∩sing(A)is simple if for every x∈Lqit happens that Fq(0, x)6= 1 or Gq(0, x)6=x. The required conditions in the previous definition imply that the simple fixed points are isolated in Fix(ψ)(see an explanation for the singular case in [9, § 3]). The following hypotheses will be assumed on ψ: (a) The morphism ψfixes every singular point of A. (b) The set Fix(ψ)contains only simple fixed points. (c) For every q∈sing(A), the maps Fqand Gqdo not depend on ρaround q. 8 1.6 Main theorems In particular, hypothesis (a) implies that ψpreserves the regular and the singular parts of A. Then, on every conic chart Oq≡cq(Lq), we have Fq(ρ, x)>0for all ρ > 0. By (1.5.1), hypothesis (c) means that ψ(ρ, x) = (ρFq(x), Gq(x)) on every small enough conic chart Oq≡cq(Lq). The adapted metric g|Oqis quasi-isometric to the metric ρ2u˜gq+ (dρ)2on the manifold Lq×(0, q), where ˜gqis a Riemannian metric on Lqand 0< u ≤1. There exists a maximum/minimum Lefschetz number associated to the morphism ψ, defined by Lmax/min(ψ) = n X r=0 (−1)rtr(ψ∗on Hr max/min(M)).(1.5.2) 1.6 Main theorems The results stated in this section and the next one were published in [7], with the exception of the results about Leftchetz trace formula. The following is the first main theorem of this thesis, where property (ii) is a weak version of the Weyl’s asymptotic formula. Theorem 1.6.1. The following properties hold on any stratum of a compact stratification with a good general adapted metric: (i)∆max/min has a discrete spectrum, 0≤λmax/min,0≤λmax/min,1≤ ···, where every eigenvalue is repeated according to its multiplicity. (ii)lim infkλmax/min,k k−θ>0for some θ > 0. The second main result is the following version of Morse inequalities for relMorse functions. Theorem 1.6.2. For any rel-Morse function on a stratum of dimension nof a compact stratification, equipped with a good general adapted metric, we have k X r=0 (−1)k−rβr max/min ≤ k X r=0 (−1)k−rνr max/min (0 ≤k < n), χmax/min = n X r=0 (−1)rνr max/min. The third main result is the following version of Lefschetz trace formula for stratified spaces with isolated singularities. 9 Introduction Theorem 1.6.3. Consider an orientable compact stratification Awith isolated singularities and equipped with a good adapted metric over its regular stratum M. Then, for any morphism ψ:A→Asatisfying hypothesis (a)–(c), we have Lmax/min(ψ) = X q∈Fix(ψ)∩M sign det(1 −Tqψ) + X q∈sing(A) L(Gq). Observe that Lmax/min(ψ)does not depend on the chosen (maximum or minimum) ideal boundary condition. In the case of adapted metrics of conic type, the previous results have been already obtained: •Theorem 1.6.1-(i) is essentially due to Cheeger [18,19] (see also [1,2,5]). •Theorem 1.6.1-(ii) was proved by Álvarez López and Calaza [5]. •Theorem 1.6.2 was proved by Álvarez López and Calaza [5], and also by Ludwig [48] (with more restrictive conditions but stronger consequences). •A version of Theorem 1.6.3 was proved by Francesco Bei [9] (with more restrictive conditions). Other developments of elliptic theory on strata were made in [1,2,16,23,40,42,65], all of them using adapted metrics of conic type. The main novelty of this thesis is the extension of the elliptic theory on strata to the wider class of good general adapted metrics, including good adapted metrics. 1.7 Applications to intersection homology Consider now the case where Ais a stratified pseudomanifold, and therefore Mis its regular stratum. Let I¯pH∗(A)denote its intersection homology with perversity ¯p[32, 33], taking real coefficients. Let β¯p r=β¯p r(A)and χ¯p=χ¯p(A)denote the versions of Betti numbers and Euler characteristic for I¯pH∗(A). Every perversity can be considered as a sequence ¯p= (p2, p3, . . . )in Nsatisfying p2= 0 and pk≤pk+1 ≤ pk+ 1. For example, the zero perversity is ¯ 0 = (0,0, . . . ), the top perversity is ¯ t= (0,1,2, . . . )(tk=k−2), the lower middle perversity is ¯m= (0,0,1,1,2,2,3, . . . ) (mk=bk 2c − 1), and the upper middle perversity is ¯n= (0,1,1,2,2,3,3, . . . ) (nk=dk 2e−1). Recall also that two perversities ¯pand ¯qare called complementary if ¯p+ ¯q=¯ t. Write ¯p≤¯qif pk≤qkfor all k. Let gbe an adapted metric on Mof type ˆu= (u2, . . . , un). If ˆuis associated with a perversity ¯p≤¯min the sense 1 k−1−2pk≤uk<1 k−3−2pkif 2pk≤k−3, 1≤uk<∞if 2pk=k−2,)(1.7.1) 10 1.7 Applications to intersection homology then (see [13,53,54]) I¯pHr(A)∗∼ =Hr (2)(M)∼ =Hr max(M), and therefore β¯p r=βr max. In particular, Hr (2)(M)∼ =I¯mHr(A)∗if gis an adapted metric of conic type [20]. Thus the incompatibility of adapted metrics with products is related to the subtleties of the versions of the Künneth theorem for intersection homology [22,29]. For instance, for arbitrary pseudomanifolds Pand Q, the isomorphism I¯pH∗(P×Q)∼ =I¯pH∗(P)⊗I¯pH∗(Q)only holds with some special perversities ¯p, including ¯p= ¯m. According to (1.7.1), there exist good adapted metrics on Mwhose type is associated with any given perversity ≤¯m. In (1.7.1), only the choices 2pk=k−2, k −4, . . . are possible if kis even, and only the choices 2pk=k−3, k −5, . . . are possible if kis odd. Thus, for every k, (1.7.1) establishes a bijection between the possibilities for pkand the elements of a partition of [1 k−1,∞)into semi-open intervals, where ukis taken. Let fbe a rel-Morse function on M, let x∈Critrel(f), let Xbe the stratum of c Mcontaining x, and let k= codim X. With the above notation for a chart O≡O0of c Mcentered at x, there is an adapted metric ˜gon Nso that, via the chart, g|Ois quasi-isometric to the restriction of g0+ρ2uk X˜g+ (dρX)2to M0∩O0. Then the type of ˜gis also associated with ¯p. Moreover there is some expression, mX=m++m−(m±∈N), and some decomposition, c(LX)≡c(L+)×c(L−), so that M0≡Rm+×N+×R+×Rm−×N−×R+for dense strata N±of L±, and f|O≡f(x) + 1 2(ρ2 +−ρ2 −)|O0, where ρ±is the radial function of Rm±×c(L±). Let k±= dim N±+ 1; thus k=k++k−. Here, some of the stratifications L±may be empty; in fact, L+6=∅ 6=L−only can happen if uk= 1 (Section 1.3). From (1.4.1) and (1.4.2), it follows that the numbers νr x,max are independent of the choice of ˆu associated with ¯p, and therefore the notation ν¯p x,r =ν¯p x,r(f)will be used. Precisely, they have the following expressions: •If L+6=∅ 6=L−(only if uk= 1), then ν¯p x,r =X (r+,r−) β¯p r+(L+)β¯p r−(L−), where (r+, r−)runs in the subset of N2determined by the conditions r=m−+r++r−+ 1, r+<k+ 2, r−≥k− 2. •If LX=L+6=∅(L−=∅), then ν¯p x,r =X r+ β¯p r+(LX), 11 Introduction where r+runs in the subset of Ndetermined by the conditions r=m−+r+, r+<(k−1−pkif uk<1 k 2if uk= 1. •If LX=L−6=∅(L+=∅), then ν¯p x,r =X r− β¯p r−(LX), where r−runs in the subset of Ndetermined by the conditions r=m−+r−+ 1, r−≥(k−1−pkif uk<1 k 2if uk= 1. •If LX=∅, then ν¯p x,r =δr,m−. Finally, let ν¯p r=ν¯p r(f) = Pxν¯p x,r (x∈Critrel(f)), which equals νr max. Suppose now that Ais oriented (Mis oriented) and compact. We have βr min = βn−r max for all rbecause ∆min corresponds to ∆max by the Hodge star operator. On the other hand, for any perversity ¯q≥¯n, if ¯p≤¯mis complementary of ¯q, then I¯qHr(A)∼ =I¯pHn−r(A)∗[32,33], and therefore β¯q r=β¯p n−r, obtaining β¯q r=βr min. As before, it follows from (1.4.1) and (1.4.2) that the numbers νr x,min are independent of the choice of ˆuassociated with ¯p. Precisely, with the notation ν¯q x,r =ν¯q x,r(f) = νr x,min, they have the following expressions: •If L+6=∅ 6=L−(only if uk= 1), then ν¯q x,r =X (r+,r−) β¯q r+(L+)β¯q r−(L−), where (r+, r−)runs in the subset of N2determined by the conditions r=m−+r++r−+ 1, r+≤k+ 2−1, r−>k− 2−1. •If LX=L+6=∅(L−=∅), then ν¯q x,r =X r+ β¯q r+(LX), where r+runs in the subset of Ndetermined by the conditions r=m−+r+, r+≤(k−2−qkif uk<1 k 2−1if uk= 1. 12 1.7 Applications to intersection homology •If LX=L−6=∅(L+=∅), then ν¯q x,r =X r− β¯q r−(LX), where r−runs in the subset of Ndetermined by the conditions r=m−+r−+ 1, r−>(k−2−qkif uk<1 k 2−1if uk= 1. •If LX=∅, then ν¯q x,r =δr,m−. Like ν¯p r, we also define ν¯q r=ν¯q r(f) = Pxν¯q x,r (x∈Critrel(f)), which equals νr min. Theorem 1.6.2 has the following direct consequence. Corollary 1.7.1. Let Abe a compact pseudomanifold of dimension n, let Mbe its regular stratum, and let ¯pbe a perversity. If ¯p≤¯m, or if Ais oriented and ¯p≥¯n, then, for any rel-Morse function on M(with respect to any good adapted metric), we have k X r=0 (−1)k−rβ¯p r≤ k X r=0 (−1)k−rν¯p r(0 ≤k < n), χ¯p= n X r=0 (−1)rν¯p r. Stratified Morse theory was introduced by Goresky and MacPherson [35], and has a great wealth of applications. In particular, Goresky and MacPherson have proved Morse inequalities on complex analytic varieties with Whitney stratifications, involving the intersection homology with perversity ¯m[35, Chapter 6, Section 6.12]. Ludwig also gave an analytic interpretation of Morse theory in the spirit of Goresky and MacPherson for conformally conic manifolds [44–47]. In this thesis, the version of Morse functions, critical points and associated numbers is different from those used in [35], even in the case of perversity ¯m. To the author’s knowledge, Corollary 1.7.1 is the first version of Morse inequalities given for intersection homology with perversity 6= ¯m. Consider now the notation of Section 1.5. The intersection Lefschetz number of a morphism ψ, with respect to the perversity ¯p, is defined [34] as I¯pL(ψ) = n X r=0 (−1)rtr(ψ∗on I¯pHr(A)∗). Theorem 1.6.3 has the following direct consequence. 13 Introduction (ii) Taking θ0=θ−1>−3 2, since hxφ, ψiθ0=hφ, x−1ψiθ for all φ∈ Sev and ψ∈ Sodd, we can write (1.10.9) as hV1/2φ, V1/2ψiσ,τ =hJσ,τ φ, ψiσ,τ +ξh|x|−uφ, |x|−uψiσ,τ +η(hφodd, xψeviθ0+hxφev, ψoddiθ0), for all φ, ψ ∈ S, and, correspondingly, V=Uσ,ev η|x|2(θ0−σ)x η|x|2(θ0−τ)x Uτ,odd . (iii) The conditions (1.10.5), (1.10.6) and (1.10.7) describe three convex open subsets of R2(Figure 1.10.1). The condition (1.10.8) describes a convex open subset of R3(Figure 1.10.2), which is symmetric with respect to the plane defined by σ=τ+ 1. It is a “semi-infinite bar” with 4lateral faces, and 5faces at the “bounded end.” (iv) In Theorem 1.10.3-(iii), the condition (1.10.12) means that (1.10.4) also holds with ˜uand v+ 1 −˜uinstead of u. There exists ˜usatisfying (1.10.12) just when 0, v, τ −2θ+1 2, σ −2θ−1 2<1, v + 1, σ +1 2, τ +3 2.(1.10.15) This property holds in the cases (b) and (d) by (1.10.4), (1.10.6) and (1.10.8); in particular, we can take ˜u=v+1 2. In the case (a), if τ < 3σ, then (1.10.15) holds by (1.10.4) and (1.10.5). In the case (c), if σ < 3τ+ 4, then (1.10.15) holds by (1.10.4) and (1.10.7). The main arguments of the proofs are given in Sections 4.2–4.4. But some needed estimates are postponed to Sections 4.5 and 4.6 because they are of rather independent nature, and with rather long and tedious proofs. 20 -4-2 0 2 4 -4 -2 0 2 4 s t (a) Set defined by (1.10.5). -4-2 0 2 4 -4 -2 0 2 4 s t (b) Set defined by (1.10.6). -4-2 0 2 4 -4 -2 0 2 4 s t (c) Set defined by (1.10.7). Figure 1.10.1: Sets in Theorem 1.10.3-(a),(b),(c). Figure 1.10.2: Set defined by (1.10.8) in Theorem 1.10.3-(d). Chapter 2 Witten’s Perturbation on Strata Contents 2.1 Preliminaries........................... 25 2.1.1 Products of cones . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 General adapted metrics . . . . . . . . . . . . . . . . . 27 2.1.3 Relatively Morse functions . . . . . . . . . . . . . . . . 28 2.1.4 Hilbert and elliptic complexes . . . . . . . . . . . . . . 29 2.2 Two simple types of elliptic complexes . . . . . . . . . . . . . 31 2.2.1 An elliptic complex of length one . . . . . . . . . . . . 31 2.2.2 An elliptic complex of length two . . . . . . . . . . . . 34 2.2.3 The wave operator . . . . . . . . . . . . . . . . . . . . 46 2.3 Witten’s perturbation on a cone . . . . . . . . . . . . . . . . 47 2.3.1 Witten’s perturbation . . . . . . . . . . . . . . . . . . . 47 2.3.2 De Rham operators on a cone . . . . . . . . . . . . . . 48 2.3.3 Witten’s perturbation on a cone . . . . . . . . . . . . . 50 2.4 Splitting of the Witten’s complex on a cone . . . . . . . . . . 51 2.4.1 Spectral decomposition on the link of the cone . . . . . 51 2.4.2 Subcomplexes of length one . . . . . . . . . . . . . . . 51 2.4.3 Subcomplexes of length two . . . . . . . . . . . . . . . 53 2.4.4 Splitting into subcomplexes . . . . . . . . . . . . . . . 55 2.5 Relatively local model of the Witten’s perturbation . . . . . . 58 2.6 Proof of Theorem 1.6.1 . . . . . . . . . . . . . . . . . . . . . 60 2.7 Functional calculus . . . . . . . . . . . . . . . . . . . . . . . 60 2.8 Thewaveoperator........................ 60 2.9 Proof of Theorem 1.6.2 . . . . . . . . . . . . . . . . . . . . . 62 23 Witten’s Perturbation on Strata In this chapter, consider the notation of Sections 1.1, 1.2, 1.3 and 1.4. Then for a Riemannian manifold M, let Ω0(M)be the space of compactly supported differential forms, and L2Ω(M)the graded Hilbert space of square integrable differential forms. Let dand δbe the de Rham derivative and coderivative acting on Ω0(M), and let D=d+δand ∆ = D2=dδ+δd (the Laplacian). Every Hilbert complex extension dof din L2Ω(M)is called an ideal boundary condition (i.b.c.), giving rise to selfadjoint extensions Dand ∆of Dand ∆in L2Ω(M). There exists a minimum/maximum i.b.c., dmin =dand dmax =δ∗, inducing self-adjoint extensions Dmax/min and ∆max/min of Dand ∆. If Mis oriented, then ∆max corresponds to ∆min by the Hodge star operator. The corresponding cohomologies, Hmax/min(M), are quasi-isometric invariants of M; for instance, Hmax(M)is the usual L2-cohomology H(2)(M). They give rise to versions of Betti numbers and Euler characteristic, βr max/min(M)and χmax/min(M)(assuming finite dimension of the corresponding cohomologies). In this chapter, the spaces under consideration are Thom-Mather stratifications. They are Hausdorff, locally compact and second countable spaces equipped with a partition into C∞manifolds called strata, and satisfying certain “gluing” conditions. Given a stratification A, an order relation on the family of strata is defined by declaring X≤Ywhen X⊂Y. With respect to this ordering, the maximum length of chains of strata less or equal than a stratum Xis called the depth of X. Recall that the cone with link a stratification Lis defined by c(L) = (L×[0,∞))/(L×{0}). A conic bundle over a manifold is a bundle that has a cone as typical fiber. Stratified spaces are locally described by charts constructed from conic bundles over their strata, called tube representatives. The possible decomposition of a cone into a product of cones leads to the concept of general tube representative of X. Then a general chart centered at a point of Xis an open subset of a product space of the type RmX×QaX i=1 c(LX,i). It can be assumed without loss of generality the existence of a dense stratum M, called the regular stratum. Moreover, if at every depth inductive step of the construction only stratifications with no strata of codimension 1are admitted, then Ais called a pseudomanifold. It can be defined a general adapted metric gon Mby induction on the depth. It is any (Riemannian) metric if depth M= 0. Now, assume depth M > 0and that general adapted metrics are defined for lower depth. Take any general adapted metric ˜gon every dense stratum Nof a compact stratification Lsuch that depth L < depth M. Then, associated to a general chart, gis given by PaX i=0 gi, where g0is the Euclidean metric on RmXand gi=ρ2uX,i X,i ˜gi+ (dρX,i)2on Ni×R+, for some uX,i >0. If every uX,i ≤1g, then gis called good. This is the kind of general adapted metrics considered in this chapter. Other restrictions of this metrical concept define adapted metrics and adapted metrics of conic type. 24 2.1 Preliminaries A smooth function fon Mis called rel-admissible when the functions f,|df|and |Hess f|are rel-bounded. In this case, fmay not have any continuous extension to M=A, but it has a continuous extension to the rel-local metric completion c M. So it makes sense to say that x∈c Mis a rel-critical point of fwhen lim inf |df(y)|= 0 as y→xin c M, with y∈M. It is said that fis a rel-Morse function if it is reladmissible and verifies certain local description around every rel-critical point. The aim of this chapter is to present the proof of Theorems 1.6.1 and 1.6.2. The first one gives the discrete character of the spectrum of ∆max/min and a version of Weyl’s asymptotic formula for its eigenvalues. The second theorem consists on a version of Morse inequalities for rel-Morse functions. An application of this result to the context of intersection homology appears in Section 1.7. The contents of this chapter are included in [7]. 2.1 Preliminaries 2.1.1 Products of cones Let Land L0be compact stratifications, and let ∗and ρ, and ∗0and ρ0be the vertices and radial functions of c(L)and c(L0). Any morphism ψ:c(L)→c(L0)is of the form c(φ)around ∗for some morphism φ:L→L0. In particular, ψ(∗) = ∗0, and ψ∗ρ0=ρaround ∗. The product of two stratifications, A×A0, has a stratification structure whose strata are the products of strata of Aand A0. However the tubes in A×A0depend on the choice of certain funtion h: [0,∞)2→[0,∞)(see Appendix A.1.2). Thus the stratification structure of A×A0is not unique. In the case of two cones, c(L)×c(L0)can be described as another cone in the following way [5, Lemma 3.8]. The function h(ρ×ρ0) : c(L)×c(L0)→[0,∞) satisfies that L00 = (h(ρ×ρ0))−1(1) is a compact saturated substratification of c(L)× c(L0). Then the map φ:c(L00)→c(L)×c(L0),[([x, r],[x0, r0]), s]7→ ([x, rs],[x0, r0s]), is an isomorphism of stratifications. The vertex of c(L00)is ∗00 =φ−1(∗,∗0), and its radial function is ρ00 =φ∗(h(ρ×ρ0)). Thus the radial function of c(L)×c(L0), (ρ2+ρ02)1/2, does not correspond to ρ00 via φif L6=∅ 6=L0. Assume that L6=∅ 6=L0. Let Nand N0be strata of Land L0, and let M= N×R+and M0=N0×R+be the corresponding strata of c(L)and c(L0). Take general adapted metrics ˜gand ˜g0on Nand N0, and fix any u > 0. We get general adapted metrics g=ρ2u˜g+ (dρ)2and g0=ρ02u˜g0+ (dρ0)2on Mand M0. On the other hand, with the above notation, we have φ−1(M×M0) = N00 ×R+=: M00, 25 Witten’s Perturbation on Strata where N00 = (M×M0)∩L00 (a stratum of L00). Let ˜g00 be any general adapted metric on N00 so that N00 ,→M×M0is quasi-isometric; for instance, we may take ˜g00 = (g+g0)|N00 . We get the general adapted metric g00 =ρ002u˜g00 + (dρ00)2on M00. Equip M×M0with g+g0and M00 with g00. Proposition 2.1.1. (i)If u= 1, then φ:M00 →M×M0is a quasi-isometry. (ii)If u < 1, then φ:M00 ∩O→(M×M0)∩φ(O)is not quasi-isometric for any neighborhood Oof ∗00 in c(L00). Proof. Without lost of generality, we can assume ˜g00 = (g+g0)|N00 . We have M00 =N00 ×R+⊂M×M0×R+=N×R+×N0×R+×R+. According to this expression, an arbitrary point p∈M00 can be written in the form p= (x, r, x0, r0, r00)≡(¯p, r00), obtaining φ(p)=(x, rr00, x0, r0r00)∈M×M0=N×R+×N0×R+. Thus we can canonically consider T¯pN00 ⊂TxN⊕R⊕Tx0N0⊕R, TpM00 ⊂TxN⊕R⊕Tx0N0⊕R⊕R, Tφ(p)(M×M0) = TxN⊕R⊕Tx0N0⊕R. We easily get φ∗(∂ρ00 (p)) = (0, r∂ρ(rr00),0, r0∂ρ0(r0r00)), φ∗(X, 0) = (Y, cr00∂ρ(rr00), Y 0, c0r00∂ρ0(r0r00)), for X= (Y, c∂ρ(r), Y 0, c0∂ρ0(r0)) ∈T¯pN00. Hence k∂ρ00 (p)k2 g00 = 1,(2.1.1) kφ∗(∂ρ00 (p))k2 g+g0=r2+r02,(2.1.2) k(X, 0)k2 g00 =r002ukXk2 ˜g+˜g0 =r002ukYk2 ˜g+c2+kY0k2 ˜g0+c02,(2.1.3) kφ∗(X, 0)k2 g+g0=r002ukYk2 ˜g+c2r002+r002ukY0k2 ˜g0+c02r002 =r002ukYk2 ˜g+c2r002(1−u)+kY0k2 ˜g0+c02r002(1−u),(2.1.4) where every metric is added as subindex of the corresponding norm. 26 2.1 Preliminaries Observe that C0:= minN00 (ρ2+ρ02)>0and C1:= maxN00 (ρ2+ρ02)<∞by the properties of h. So, by (2.1.1) and (2.1.2), C0k∂ρ00 (p)k2 g00 ≤ kφ∗(∂ρ00 (p))k2 g+g0≤C1k∂ρ00 (p)k2 g00 . Moreover, if u= 1, then kφ∗(X, 0)k2 g+g0=k(X, 0)k2 g00 by (2.1.3) and (2.1.4), obtaining (i). Now, suppose that u < 1. With the above notation, by the conditions satisfied by h, we can take ¯p= (x, r, x0,1) ∈N00 and X= (0, ∂ρ(r),0,0) ∈T¯pN00 for all r small enough. By (2.1.3) and (2.1.4), it follows that kφ∗(X, 0)k2 g+g0 k(X, 0)k2 g00 =r002(1−u)→0 as r00 →0, giving (ii). Similar observations apply to the product of any finite number of cones. 2.1.2 General adapted metrics Consider the notation of Section 1.3. Remark 2.1.2.For m∈Z+, there is a canonical homeomorphism c(Sm−1)≈Rm, [x, ρ]7→ ρx, so that the radial function ρcorresponds to the norm on Rm[5, Example 3.7]. This is not an isomorphism of stratifications: c(Sm−1)has two strata and Rmonly one; the stratum Sm−1×R+of c(Sm−1)corresponds to Rmr{0}. If ˜g denotes the standard metric on Sm−1, then ρ2˜g+(dρ)2on Sm−1×R+corresponds to the Euclidean metric on Rmr{0}. Thus, with the notation of Chapter 1, the factors RmXor Rm±could be also described as cones, or as strata of cones after removing one point. Remark 2.1.3.By taking charts and using induction on the depth, we get the following (cf. [5, Remark 7]): (i) If two general adapted metrics on Mhave the same type with respect to the same general tubes, then they are rel-locally quasi-isometric. In particular, they are quasi-isometric if Mis compact. (ii) Any point in Mhas a countable base {Om|m∈N}of open neighborhoods such that, with respect to any general adapted metric, vol(M∩Om)→0and max{diam P|P∈π0(M∩Om)} → 0as m→ ∞. Thus, if Mis compact, then vol M < ∞and diam P < ∞for all P∈π0(M). 27 Witten’s Perturbation on Strata Remark 2.1.4.The argument of [13, Appendix] also shows the following. Let {Oa} be a locally finite open covering of M, let {λa}be a smooth partition of unity of M subordinated to the open covering {M∩Oa}, and let gabe a general adapted metric on every M∩Oa. Suppose that the metrics gahave the same general type with respect to restrictions to the sets Oaof the same general tubes. Then the metric Paλagais general adapted on Mand has the same general type with respect to those general tubes. When Mis not connected, c Mis defined as the disjoint union of the rel-local completion of the connected components of M(Section 1.3), making use of [5, Remark 1-(v)]. Remark 2.1.5.(i) By Remark 2.1.3-(i), c Mis independent of the choice of the general adapted metric of a given general type. In fact, by Remark 2.1.3-(ii) and [5, Example 3.19], c Mis also independent of the general type. (ii) For any open O⊂A, we have \ M∩O≡lim−1(M∩O)⊂c M. Remark 2.1.6.The following is a direct consequence of Remark 2.1.5-(i) and [5, Remark 9-(i),(ii) and Proposition 3.20-(iii)]: (i) lim : c M→Mis surjective with finite fibers. (ii) Mis rel-locally connected with respect to c M. (iii) Let M0be a connected stratum of another stratification A0equipped with a general adapted metric, and let φ:A→A0be a morphism with φ(M)⊂M0. Then the restriction φ:M→M0extends to a morphism ˆ φ:c M→c M0. Moreover ˆ φis an isomorphism if φis an isomorphism. 2.1.3 Relatively Morse functions Consider the notation of Section 1.4. Besides the observations given in that section, the following holds like in the case of adapted metrics of conic type [5, Section 4]. Remark 2.1.7.(i) The rel-local boundedness of |df|is invariant by rel-local quasiisometries, and therefore it depends only on the general type of g. Similarly, the definition of rel-critical point depends only on the general type of g. But the rel-local boundedness of |Hess f|depends on the choice of g. However it follows from (iv) and (v) below that the existence of gso that fis rel-admissible with respect to gis a rel-local property. (ii) If depth M= 0, then any smooth function is admissible, and its rel-critical points are its critical points. 28 2.1 Preliminaries (iii) With the notation of Section 2.1.1, let h∈C∞(R+)with h0∈C∞ 0(R+). Then the function h(ρ)is rel-admissible on the stratum Mof c(L)with respect to any general adapted metric. (iv) Let {Oa|a∈ A} be a locally finite covering of Mby open subsets of A. Then there is a C∞partition of unity {λa}on Msubordinated to {M∩Oa} such that |dλa|is rel-locally bounded for all general adapted metrics on Mof any fixed general type. (v) Suppose that {λa}and {ga}satisfy the conditions of Remark 2.1.4 and (iv). Let f∈C∞(M)such that every f|M∩Oais rel-admissible with respect to ga. Then fis rel-admissible with respect to the general adapted metric g=Paλagaon M. (vi) Let F ⊂ C∞(M)denote the subset of functions with continuous extensions to Mthat restrict to rel-Morse functions with respect to all general adapted metrics of all possible general types on all strata ≤M. Then Fis dense in C∞(M) with the weak C∞topology. 2.1.4 Hilbert and elliptic complexes Consider the notation of Section 1.1. Hilbert complexes with a discrete positive spectrum Let (D,d)be a Hilbert complex in a graded separable Hilbert space H, defining self-adjoint operators Dand ∆according to Section 1.1. The direct sum of homogeneous subspaces of even/odd degree are denoted with the subindex “ev/odd”. The same subindex is used to denote the restriction of homogeneous operators to such subspaces. Lemma 2.1.8. The positive spectrum of ∆ev is discrete1and bounded away from zero if and only if the positive spectrum of ∆odd is discrete and bounded away from zero. In this case, both operators have the same positive eigenvalues, with the same multiplicity. Proof. For instance, suppose that the positive spectrum of ∆ev is discrete and bounded away from zero. It follows from the spectral theorem that D∞(∆ev/odd) = ker ∆ev/odd ⊕∆(D∞(∆ev/odd)), 1Recall that a complex number is in the discrete spectrum of a normal operator in a Hilbert space when it is an eigenvalue of finite multiplicity. 29 Witten’s Perturbation on Strata Then, by Corollaries 4.7.3, 4.7.4 and 4.7.5, and Remark 4.7.6-(v), ∆0,∆2and ∆1 define the self-adjoint operators Piand Qjin L2 +, and Wi,j in L2 +⊕L2 +, indicated in Table 2.4, where the conditions come from (4.7.9), (4.7.11), (4.7.13) and (1.10.5)– (1.10.8). The notation P± i,Q± jand W± i,j may be used as well to specify that these operators are defined by ∆± 0,∆± 2and ∆± 1. Note that v=ufor all Wi,j. The cores of P1/2 i,Q1/2 jand W1/2 i,j , given by Corollaries 4.7.3, 4.7.4 and 4.7.5, will be denoted by F0 i,F2 jand F1 i,j =F1,1 i⊕F1,2 j, respectively. Remark 2.2.3.In contrast to Eiin Section 2.2.1, note that the graded subspace F0 i⊕ F1 i,j ⊕F2 jof C∞(F)∩L2(F), whenever defined, is not preserved by D=d+δ. For instance, it is preserved by dbut not by δwhen i=j= 1, and it is preserved by δbut not by dwhen i=j= 2. σ τ θ Condition ∆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 2.4: Self-adjoint operators defined by ∆0,∆2and ∆1. Let us explain the contents of Table 2.4. Since c1=d1= 0, we have σ=a and τ=b, which are given by (4.7.3) and (4.7.6). Moreover σ,τand udetermine θin Table 2.4 so that Uis of the form (4.7.16) because 2θ−σ−τ=−u. Let us check the conditions written in this table, which are given by the hypothesis of Corollaries 4.7.3–4.7.5. For Piand Qj, only (4.7.9) and (4.7.11) are required. For Wi,j, we also require (4.7.13), and the hypothesis (a)–(d) of Theorem 1.10.3, obtaining the following: •For W1,1, we have σ=θ6=τand τ−σ=u6∈ −N. Thus (a) applies in this case. Note that (4.7.9), (4.7.11) and (4.7.13) mean κ>u−1 2. Then (1.10.5) holds because 0<u<1and κ>u−1 2. So (a) is satisfied. 36 2.2 Two simple types of elliptic complexes •For W2,2, we have σ6=θ=τ+ 1 and σ−τ−1 = 1 + u6∈ −N. Thus (c) applies in this case. Now, (4.7.9), (4.7.11) and (4.7.13) mean κ < 1 2−2u. Then (1.10.7) holds because 0< u < 1and κ < 1 2−2u. So (c) is satisfied. •There is no W1,2because θ < −1 2in that case. •For W2,1, (4.7.9), (4.7.11) and (4.7.13) mean −3 2<κ<3 2−u, and we have the following possibilities: –The case σ=θ=τis not possible because u6= 0. –The case σ=θ6=τhappens when κ=1 2. Then σ=1 2and τ=1 2+u, obtaining τ−σ=u6∈ −N. Thus (a) applies in this case. Moreover (1.10.5) holds because 0<u<1. So (a) is satisfied. –The case σ6=θ=τhappens when κ=1 2−u. Then σ=1 2+u and τ=1 2, obtaining σ−τ=u6∈ −N. Thus (b) applies in this case. Moreover (1.10.6) holds because 0< u < 1. Hence (b) is satisfied. –The case σ6=θ=τ+ 1 happens when κ=−1 2−u. Then σ=3 2+u and τ=−1 2, obtaining σ−τ−1 = 1 +u6∈ −N. Thus (c) applies in this case. Moreover (1.10.7) holds because 0< u < 1. Hence (c) is satisfied. –Finally, assume that σ6=θ6=τ. The condition σ−θ, τ −θ6∈ −N means that κ6∈ (1 2+N)∪(1 2−u−N), which in turn means that κ6= 1 2,1 2−u, −1 2−ubecause −3 2<κ<3 2−u. But σ=θif κ=1 2,τ=θif κ=1 2−u, and θ=τ+1 if κ=−1 2−u, as we have seen in the previous cases. So σ−θ, τ −θ6∈ −N, and (d) applies in this case. Moreover, since 0< u < 1, (1.10.8) holds just when −1−u 2< κ < 1−u 2. Thus (d) is satisfied assuming the stated conditions on κ. Therefore W2,1is defined in one of the above ways if −1−u 2< κ < 1−u 2. There are the following overlaps of the conditions in Table 2.4: •Both P1and P2are defined for −1 2<κ<3 2−2u, and P1=P2just when κ=1 2−u. •Both Q1and Q2are defined for u−3 2< κ < 1 2−u, and Q1=Q2just when κ=−1 2. •Both W1,1and W2,2are defined for u−1 2< κ < 1 2−2u(if u < 1 3), but W1,16=W2,2for all such κ. •Both W1,1and W2,1are defined for u−1 2< κ < 1−u 2, and W1,1=W2,1 just when κ=1 2. 37 Witten’s Perturbation on Strata •Both W2,2and W2,1are defined for −1−u 2< κ < 1 2−2u, and W2,2=W2,1 just when κ=−1 2−u. Corollaries 4.7.3, 4.7.4 and 4.7.5 also give the following spectral estimates, for all  > 0: •The spectrum of P1consists of eigenvalues λ0≤λ2≤ ···, taking multiplicity into account, such that there are D=D(κ, u)>0and C=C(, κ, u)>0so that, for all k∈2N, λk≥(2k+ (1 ∓1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(2.2.9) λk≤(2k+ (1 ∓1)(1 + 2(κ+u)))s + (2k+ 1 + 2(κ+u))µ2su+µ2Csu.(2.2.10) The first term of the right-hand side of (2.2.9) and (2.2.10) for P+ 1and P− 1is 2ks and 2(k+ 1 + 2(κ+u))s, respectively. •The spectrum of P2consists of eigenvalues λ0≤λ2≤ ···, taking multiplicity into account, such that there are D=D(κ, u)>0and C=C(, κ, u)>0so that, for all k∈2N, λk≥(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(2.2.11) λk≤(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s + (2k+ 3 −2(κ+u))µ2su+µ2Csu.(2.2.12) The first term of the right-hand side of (2.2.11) and (2.2.12) for P+ 2and P− 2 becomes 2(k+ 1 −2(κ+u))sand 2(k+ 2)s, respectively. •The spectrum of Q1consists of eigenvalues λ1≤λ3≤ ···, taking multiplicity into account, such that there are D=D(κ, u)>0and C=C(, κ, u)>0so that, for all k∈2N+ 1, λk≥(2k+ 2 −(1 ∓1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(2.2.13) λk≤(2k+ 2 −(1 ∓1)(1 −2κ))s + (2k+ 1 + 2κ)µ2su+µ2Csu.(2.2.14) The first term of the right-hand side of (2.2.13) and (2.2.14) for Q+ 1and Q− 1is 2(k+ 1)sand 2(k+ 2κ)s, respectively. •The spectrum of Q2consists of eigenvalues λ1≤λ3≤ ···, taking multiplicity into account, such that there are D=D(κ, u)>0and C=C(, κ, u)>0so 38 2.2 Two simple types of elliptic complexes that, for all k∈2N+ 1, λk≥(2k−2 + (1 ±1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(2.2.15) λk≤(2k−2 + (1 ±1)(1 −2κ))s + (2k−1−2κ)µ2su+µ2Csu.(2.2.16) The first term of the right-hand side of (2.2.15) and (2.2.16) for Q+ 2and Q− 2is 2(k−2κ)sand 2(k−1)s, respectively. •For W2,1, we can take ˜u=u+1 2satisfying (1.10.12). Moreover the maximum eigenvalue of ∓sV is s(1∓(2κ+u)−u). Thus the spectrum of W2,1consists of two groups of eigenvalues, λ0≤λ2≤ ··· and λ1≤λ3≤ ···, repeated according to multiplicity, such that there are some D=D(κ, u)>0,C= C(, κ, u)>0,e C=e C(, κ, u)>0and E=E(, κ)>0so that, for 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, (2.2.17) λk≤(2k+ 4 −(1 ±1)(2κ+u))s + (2k+ 3 −2κ)(µ2su+ 4µusu+1 2) +µ2Csu+ 4µuEsu+1 2,(2.2.18) and, for all k∈2N+ 1, λk≥1−2µusu−1 2(2k+ 1 + 2(κ+u))s +µ2Dsu(k+ 1)−u−2µu e Csu+1 2±(1 + 2(κ+u))s, (2.2.19) λk≤(2k+ 2 + (1 ∓1)(2κ+u))s + (2k+ 1 + 2(κ+u))(µ2su+ 4µusu+1 2) +µ2Csu+ 4µuEsu+1 2.(2.2.20) • W1,1and W2,2also have a discrete spectrum, which has the lower bound given by (1.10.10) and Corollary 4.7.5-(v). We omit its explicit expression because it will not be used. The lower estimate of Corollary 4.7.5-(iii) may not be possible for W1,1and W2,2in general. In fact, according to Remark 1.10.4- (iv), the existence of ˜ufor W1,1(respectively, W2,2) is characterized by the additional condition 2κ>u(respectively, 2κ < −3u), which is an additional restriction. Table 2.5 contains the information about the sign of the eigenvalues of Pi,Qjand Wi,j given by the above spectral estimates. 39 Witten’s Perturbation on Strata Sign of eigenvalues P1+∀k∈2N P+ 2 κ > 1 2−u? if k < 2(κ+u)−1even +if k≥2(κ+u)−1even κ≤1 2−u+∀k∈2N P− 2+∀k∈2N Q+ 1+∀k∈2N+ 1 Q− 1 κ≥ −1 2+∀k∈2N+ 1 κ < −1 2 ? if k < −2κodd +if k≥ −2κodd Q2+∀k∈2N+ 1 Wi,j +if k0 Table 2.5: Sign of the eigenvalues of Pi,Qiand Wi,j. Laplacians of the maximum/minimum i.b.c. Proposition 2.2.4. Tables 2.6, 2.7 and 2.8 describe ∆max/min for the stated values of κ. Proof. The operators d0,2,δ0,2,d1,1and δ1,1are of the same type as dand δin Section 2.2.1. Then, applying Proposition 2.2.1 and Remark 2.2.2-(ii), we obtain the inclusions D(d0,2,max)⊃(F0 1if κ > −1 2−u F0 2if κ≤ −1 2−u, (2.2.21) D(d0,2,min)⊃(F0 1if κ≥1 2−u F0 2if κ < 1 2−u, (2.2.22) 40 2.2 Two simple types of elliptic complexes ∆max,0∆min,0 κ > −1 2P1κ≥1 2−uP1 −1 2−u < κ ≤ −1 2?κ < 1 2−uP2 κ≤ −1 2−uP2 Table 2.6: Description of ∆max/min,0. ∆max,2∆min,2 κ > −1 2Q1κ≥1 2Q1 κ≤ −1 2Q21 2−u≤κ < 1 2? κ < 1 2−uQ2 Table 2.7: Description of ∆max/min,2. D(δ0,2,max)⊃(F1,2 1if κ≥1 2−u F1,2 2if κ < 1 2−u, (2.2.23) D(δ0,2,min)⊃(F1,2 1if κ > −1 2−u F1,2 2if κ≤ −1 2−u, (2.2.24) D(d1,1,max)⊃(F1,1 1if κ > −1 2 F1,1 2if κ≤ −1 2,(2.2.25) D(d1,1,min)⊃(F1,1 1if κ≥1 2 F1,1 2if κ < 1 2,(2.2.26) D(δ1,1,max)⊃(F2 1if κ≥1 2 F2 2if κ < 1 2,(2.2.27) D(δ1,1,min)⊃(F2 1if κ > −1 2 F2 2if κ≤ −1 2,(2.2.28) and the equalities d0,2,max =d0,2,min, δ0,2,max =δ0,2,min if |κ+u| ≥ 1 2, d1,1,max =d1,1,min, δ1,1,max =δ1,1,min if |κ| ≥ 1 2. 41 Witten’s Perturbation on Strata ∆max,1∆min,1 κ>u−1 2W1,1κ≥1 2W1,1 −1 2< κ ≤u−1 2?1 2−u≤κ < 1 2W2,1 −1 2−u<κ≤ −1 2W2,11 2−2u≤κ < 1 2−u? κ≤ −1 2−uW2,2κ < 1 2−2uW2,2 Table 2.8: Description of ∆max/min,1. On the other hand, since d0,1,δ0,1,d1,2and δ1,2are multiplication operators, we have d0,1,max =d0,1,min, δ0,1,max =δ0,1,min, d1,2,max =d1,2,min, δ1,2,max =δ1,2,min. These are maximal multiplication operators [41, Examples III-2.2 and V-3.22]. They satisfy the following: D(d0,1,max/min)⊃(F0 1if κ > −1 2 F0 2if κ < 3 2−2u, (2.2.29) D(δ0,1,max/min)⊃(F1,1 1if κ > u −1 2 F1,1 2if κ < 3 2−u, (2.2.30) D(d1,2,max/min)⊃(F1,2 1if κ > −3 2 F1,2 2if κ < 1 2−2u, (2.2.31) D(δ1,2,max/min)⊃(F2 1if κ > u −3 2 F2 2if κ < 1 2−u. (2.2.32) By Remark 2.2.2-(i), we also get D(dmin,0) = D(d0,1,min)∩D(d0,2,min), dmin,0=d0,1,min|D(dmin,0) d0,2,min|D(dmin,0),(2.2.33) D(δmin,1) = D(δ1,1,min)∩D(δ1,2,min), δmin,1=δ1,1,min|D(δmin,1) δ1,2,min|D(δmin,1),(2.2.34) complementing Lemma 2.1.9 in this case. From (2.2.21)–(2.2.34), Lemmas 2.1.9 and 2.1.10, and [72, Chapter XI-12, p. 338, 42 2.2 Two simple types of elliptic complexes Eq. (1)], it follows that D(∆1/2 max,0) = D(dmax,0) = D(d0,1,max)∩D(d0,2,max)⊃(F0 1if κ > −1 2 F0 2if κ≤ −1 2−u, D(∆1/2 min,0) = D(dmin,0) = D(d0,1,min)∩D(d0,2,min)⊃(F0 1if κ≥1 2−u F0 2if κ < 1 2−u, D(∆1/2 max,2) = D(δmin,1) = D(δ1,1,min)∩D(δ1,2,min)⊃(F2 1if κ > −1 2 F2 2if κ≤ −1 2, D(∆1/2 min,2) = D(δmax,1) = D(δ1,1,max)∩D(δ1,2,max)⊃(F2 1if κ≥1 2 F2 2if κ < 1 2−u, D(∆1/2 max,1) = D(δmin,0+dmax,1) = D(δmin,0)∩D(dmax,1) ⊃(D(δ0,1,min)⊕D(δ0,2,min)) ∩(D(d1,1,max ⊕D(d1,2,max)) ⊃     F1 1,1if κ > u −1 2 F1 2,1if −1 2−u < κ ≤ −1 2 F1 2,2if κ≤ −1 2−u, D(∆1/2 min,1) = D(δmax,0+dmin,1) = D(δmax,0)∩D(dmin,1) ⊃(D(δ0,1,max)⊕D(δ0,2,max)) ∩(D(d1,1,min ⊕D(d1,2,min)) ⊃     F1 1,1if κ≥1 2 F1 2,1if 1 2−u≤κ < 1 2 F1 2,2if κ < 1 2−2u. Since F0 i,F2 jand F1 i,j are cores of P1/2 i,Q1/2 jand W1/2 i,j , respectively, and taking into account Table 2.4, it follows that ∆1/2 max,0⊃(P1/2 1if κ > −1 2 P1/2 2if κ≤ −1 2−u, ∆1/2 min,0⊃(P1/2 1if κ≥1 2−u P1/2 2if κ < 1 2−u, ∆1/2 max,2⊃(Q1/2 1if κ > −1 2 Q1/2 2if κ≤ −1 2,∆1/2 min,2⊃(Q1/2 1if κ≥1 2 Q1/2 2if κ < 1 2−u, 43 Witten’s Perturbation on Strata ∆1/2 max,1⊃     W1/2 1,1if κ>u−1 2 W1/2 2,1if −1 2−u < κ ≤ −1 2 W1/2 2,2if κ≤ −1 2−u, ∆1/2 min,1⊃     W1/2 1,1if κ≥1 2 W1/2 2,1if 1 2−u≤κ < 1 2 W1/2 2,2if κ < 1 2−2u. But these inclusions are equalities because they involve self-adjoint operators. Proposition 2.2.5. We have ker ∆max/min = 0. Proof. We have ker ∆max/min,ev = 0 because ker dmax/min,0= 0 and ker δmax/min,1= 0 by Lemma 2.1.9, (2.2.33) and (2.2.34), since d0,1,max/min and δ1,2,max/min are maximal multiplication operators in L2 +by continuous non-vanishing functions.4 Since σ(∆max/min,ev)is bounded away from 0, we get R(∆max/min,0) = L2 += R(∆max/min,2)by the spectral theorem. The maximal multiplication operator by ρ±u in L2 +will be also denoted by ρ±u. Let φ∈D(∆max/min,0)such that ∆max/min,0φ∈ D(ρu). By (2.2.7), ψ:= 1 µρud0,2,max/minφ∈D(δ0,2,max/min ρ−u)∩D(ρuδ0,2,max/min ρ−u) =D(ρ−uδ1,1,max/min)∩D(δ1,1,max/min). Then ψ∈D(δmax/min,1)by (2.2.34) since ρ−uψ∈L2 +and δ1,2,max/min is the maximal multiplication operator by −µρ−u. In the following, for the sake of simplicity, the notation d0,2,δ1,1,δ0,2and ∆0is used for d0,2,max/min,δ1,1,max/min,δ0,2,max/min and ∆max/min,0, respectively. It also follows from (2.2.7) that dmax/min,0(φ) + δmax/min,1(ψ) = µρ−uφ+δ1,1ψ d0,2φ−µρ−uψ =µρ−uφ+1 µδ1,1ρud0,2φ 0=µρ−uφ+1 µρuδ0,2d0,2φ 0=1 µρu∆0φ 0. Since R(∆max/min,0) = L2 +, we get R(ρu)⊕0⊂R(dmax/min,0) + R(δmax/min,1). With an analogous argument, using Lemma 2.1.9 instead of (2.2.34), we get 0⊕R(ρu)⊂R(dmax/min,0) + R(δmax/min,1). 4We may also use Table 2.5 and Proposition 2.2.4 for some values of κ(Tables 2.6 and 2.7). 44 2.2 Two simple types of elliptic complexes Therefore R(ρu)⊕R(ρu)⊂R(dmax/min,0) + R(δmax/min,1), obtaining that R(dmax/min,0) + R(δmax/min,1)is dense in L2 +⊕L2 +because R(ρu)is dense in L2 +. Thus ker ∆max/min,1= 0 [15, Lemma 2.1]. Corollary 2.2.6. ∆max/min,ev and ∆max/min,1have the same eigenvalues, with the same multiplicity. Proof. This is a direct consequence of Proposition 2.2.5 and Lemma 2.1.8. Remark 2.2.7.Some general properties of this complex of length two hold for all u > 0, like (2.2.21)–(2.2.34), Proposition 2.2.5 and Corollary 2.2.6. But the main results require 0< u < 1. Concerning the spectrum, the following corollary fills the gaps in Tables 2.6–2.8. Corollary 2.2.8. Tables 2.9 and 2.10 describe the spectra of ∆max/min,ev and ∆max/min,1 in terms of the spectra of Pi,Qjand Wi,j for the stated values of κ. σ(∆max,ev)σ(∆min,ev) κ > −1 2σ(P1⊕Q1)κ≥1 2σ(P1⊕Q1) −1 2−u<κ≤ −1 2σ(W2,1)1 2−u≤κ < 1 2σ(W2,1) κ≤ −1 2−u σ(P2⊕Q2)κ < 1 2−u σ(P2⊕Q2) Table 2.9: Spectrum of ∆max/min,ev. σ(∆max,1)σ(∆min,1) κ > u −1 2σ(W1,1)κ≥1 2σ(W1,1) −1 2< κ ≤u−1 2σ(P1⊕Q1)1 2−u≤κ < 1 2σ(W2,1) −1 2−u < κ ≤ −1 2σ(W2,1)1 2−2u≤κ < 1 2−u σ(P2⊕Q2) κ≤ −1 2−u σ(W2,2)κ < 1 2−2u σ(W2,2) Table 2.10: Spectrum of ∆max/min,1. Proof. This is a direct consequence of Proposition 2.2.4 and Corollary 2.2.6. 45 Witten’s Perturbation on Strata Moreover, using (2.4.2), ds,r =d dρ ±sρ, δs,r =−d dρ −2κρ−1±sρ. Let Eγ,0denote the subcomplex of length one of (Ω(M), ds)defined by Er γ,0=C∞ +,0γ≡C∞ +,0, Er+1 γ,0=C∞ +,0dρ ∧γ≡C∞ +,0. The closure of Eγ,0in L2Ω(M)is denoted by L2Eγ. By (2.3.13), L2Er γ=L2 κ,+γ≡L2 κ,+, L2Er+1 γ=L2 κ,+dρ ∧γ≡L2 κ,+. Assume now that s > 0. With the notation of Section 2.2.1, consider the real version of the elliptic complex (E, d)determined by sand κ(given by (2.3.9)). Using Lemma 2.4.1 and (4.7.1), like in [5, Proposition 12.3], we get the following. Proposition 2.4.2. The operator ρκ:L2 κ,+→L2 +defines a unitary isomorphism L2Eγ→L2(E), which restricts to an isomorphism of complexes, (Eγ,0, ds)→ (C∞ 0(E), d), up to a shift of degree. By Proposition 2.4.2, (Eγ,0, ds)has a maximum/minimum Hilbert complex extension in L2Eγ. Let (Dγ,ds,γ)be the maximum/minimum Hilbert complex extension of (Eγ,0, ds)if γ∈e Hr max/min, and ∆s,γ the corresponding Laplacian. Let Hs,γ =Hr s,γ ⊕ Hr+1 s,γ = ker ∆s,γ, with the induced grading. The more explicit notation d± s,γ,∆± s,γ and H± s,γ =H±,r s,γ ⊕H±,r+1 s,γ may be also used. Corollary 2.4.3. (i)∆s,γ has a discrete spectrum. (ii)The dimensions of H±,r s,γ and H±,r+1 s,γ are given in Table 2.11. (iii)If es∈ Hs,γ with norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)All nonzero eigenvalues of ∆s,γ are positive and in O(s)as s→ ∞. Proof. This follows from Propositions 2.4.2 and 2.2.1, Corollary 4.7.8, Section 2.2.1, and the choice made to define ds,γ. 52 2.4 Splitting of the Witten’s complex on a cone γ∈e Hr max γ∈e Hr min H+,r s,γ H+,r+1 s,γ H−,r s,γ H−,r+1 s,γ H+,r s,γ H+,r+1 s,γ H−,r s,γ H−,r+1 s,γ κ≥1 21001 0 0 |κ|<1 20 1 κ≤ −1 20 1 Table 2.11: Dimensions of H±,r s,γ and H±,r+1 s,γ . 2.4.3 Subcomplexes of length two Let µ=p˜ λfor an eigenvalue ˜ λof the restriction of e ∆max/min to e Rmax/min,r−1. According to [5, Section 5.1], there are nonzero differential forms, α∈e Rmax/min,r−1,˜ λ⊂Ωr(N), β∈e R∗ max/min,r−1,˜ λ⊂Ωr−1(N), such that ˜ dβ =µα and ˜ δα =µβ. Consider the canonical identities C∞ +≡C∞ +β⊂Ωr−1(M), C∞ +≡C∞ +dρ ∧α⊂Ωr+1(M),(2.4.3) C∞ +⊕C∞ +≡C∞ +α+C∞ +dρ ∧β⊂Ωr(M).(2.4.4) The following result follows from (2.3.18) and (2.3.19). Lemma 2.4.4. For s≥0,dsand δsdefine maps 0C∞ +β C∞ +α+C∞ +dρ ∧β C∞ +dρ ∧α0. ds,r−2 δs,r−2 ds,r−1 δs,r−1 ds,r δs,r ds,r+1 δs,r+1 - -  - -  53 Witten’s Perturbation on Strata Moreover, according to (2.4.3) and (2.4.4), ds,r−1=µ d dρ ±sρ, δs,r−1=µρ−2u−d dρ −2(κ+u)ρ−1±sρ, ds,r =d dρ ±sρ −µ, δs,r =−d dρ −2κρ−1±sρ −µρ−2u. Let Fα,β,0=Fr−1 α,β,0⊕Fr α,β,0⊕Fr+1 α,β,0denote the subcomplex of length two of (Ω(M), ds)defined by Fr−1 α,β,0=C∞ +,0β≡C∞ +,0, Fr+1 α,β,0=C∞ +,0dρ ∧α≡C∞ +,0, Fr α,β,0=C∞ +,0α+C∞ +,0dρ ∧β≡C∞ +,0⊕C∞ +,0. The closure of Fα,β,0in L2Ω(M)is denoted by L2Fα,β. By (2.3.13), L2Fr−1 α,β =L2 κ+u,+β≡L2 κ+u,+, L2Fr+1 α,β =L2 κ,+dρ ∧α≡L2 κ,+, L2Fr α,β =L2 κ,+α+L2 κ+u,+dρ ∧β≡L2 κ,+⊕L2 κ+u,+. Assume now that s > 0. With the notation of Section 2.2.2, consider the real version of the elliptic complex (F, d)determined by sand κ(given by (2.3.9)). Using Lemma 2.4.4 and (4.7.1), we get the following (cf. [5, Proposition 12.9]). Proposition 2.4.5. If u < 1, then ρκ:L2 κ,+→L2 +and ρκ+u:L2 κ+u,+→L2 + define a unitary isomorphism L2Fα,β →L2(F), which restricts to an isomorphism of complexes, (Fα,β,0, ds)→(C∞ 0(F), d), up to a shift of degree. By Proposition 2.4.5, (Fα,β,0, ds)has a maximum/minimum Hilbert complex extension in L2Fα,β. Let (Dα,β,ds,α,β)be the maximum/minimum Hilbert complex extension of (Fα,β,0, ds)if α∈e Rmax/min,r−1,˜ λand β∈e R∗ max/min,r−1,˜ λ. Let ∆s,α,β denote the corresponding Laplacian. The more explicit notation d± s,α,β and ∆± s,α,β may be used. Corollary 2.4.6. (i)∆s,α,β has a discrete spectrum. (ii)The eigenvalues of ∆s,α,β are positive and in O(s)as s→ ∞. 54 2.4 Splitting of the Witten’s complex on a cone Proof. In the case u < 1, this follows from Proposition 2.4.5 and Corollary 2.2.8. In the case u= 1, this is the content of [5, Proposition 12.11]. Remark 2.4.7.According to (2.3.21)–(2.3.24), we have ∆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, where 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 the results of Section ?? could be applied to these expressions. We opted for analyzing first the complexes of Section 2.2 for the sake of simplicity because we have a=b= 0,L2 +is used instead of L2 κ,+or L2 κ+u,+, and Remark 2.2.2 is directly applied. 2.4.4 Splitting into subcomplexes Let Cmax/min,0denote an orthonormal frame of e Hmax/min consisting of homogeneous differential forms. For every positive eigenvalue µof e Dmax/min, let Cmax/min,µ be an orthonormal frame of the µ-eigenspace of e Dmax/min consisting of differential forms α+βlike in Section 2.4.3. Then let ds,max/min =M γ ds,γ ⊕d M µM α+β ds,α,β, where γruns in Cmax/min,0,µruns in the positive spectrum of e Dmax/min, and α+β runs in Cmax/min,µ. The notation d± s,max/min may be also used when d± s,γ and d± s,α,β are considered. Proposition 2.4.8. We have ds,max/min =ds,max/min. Proof. This follows like [5, Proposition 12.12], using [5, Lemma 5.2], [15, Lemma 3.6 and (2.38b)], (2.3.6) and (2.4.1). 55 Witten’s Perturbation on Strata Let Hs,max/min =LrHr s,max/min = ker ∆s,max/min, with the induced grading. The superindex “±” may be added to this notation to indicate that we are referring to ∆± s,max/min. Corollary 2.4.9. (i)∆s,max/min has a discrete spectrum. (ii)Table 2.12 describes the isomorphism class of H±,∗ s,max/min. (iii)If es∈ Hs,max/min has norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)Let 0≤λs,max/min,0≤λs,max/min,1≤ ··· be the eigenvalues of ∆s,max/min, repeated according to their multiplicities. Given k∈N, if λs,max/min,k >0for some s, then λs,max/min,k >0for all s, and λs,max/min,k ∈O(s)as s→ ∞. (v)There is some θ > 0such that lim infkλs,max/min,kk−θ>0. H+,r s,max H−,r+1 s,max H+,r s,min H−,r+1 s,min κ≥1 2Hr max(N) 0 Hr min(N) 0 |κ|<1 20Hr min(N) κ≤ −1 20Hr max(N) Table 2.12: Spaces isomorphic to H±,∗ s,max/min. Proof. In the case u= 1, this result was already shown in [5, Corollary 12.13]. So we consider only the case 0< u < 1. For all γ,µand α+βas above, ∆s,γ and ∆s,α,β have a discrete spectrum by Corollaries 2.4.3-(i) and 2.4.6-(i). Moreover the union of their spectra has no accumulation points according to Section 2.2 and since e ∆max/min is discrete. Then (i) follows by Proposition 2.4.8. Now, properties (ii)–(iv) follow directly from Corollaries 2.4.3 and 2.4.6, and Proposition 2.4.8. To prove (v), let 0≤˜ λmax/min,0≤˜ λmax/min,1≤ ··· denote the eigenvalues of e ∆max/min, repeated according to their multiplicities. Since, by induction hypothesis, Nsatisfies Theorem 1.6.1-(ii) with ˜g, there is some C0, θ0>0such that ˜ λmax/min,` ≥C0`θ0(2.4.5) 56 2.4 Splitting of the Witten’s complex on a cone for all `large enough. Consider the counting function N± s,max/min(λ) = # nk∈N|λ± s,max/min,k < λ o(λ > 0). From Proposition 2.2.5, Corollary 2.2.8, (2.2.3)–(2.2.6), (2.2.9), (2.2.11), (2.2.13), (2.2.15), (2.2.17), (2.2.19) and (2.4.5), and the choices made to define dγand dα,β (Sections 2.4.2 and 2.4.3), it follows that there are some C1, C2>0and C3, C0 3∈R such that 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). Consider the function f:−1, a := λ−C3 C1→[0,∞), f(x) = λ−C3−C1x C2C01 θ0(x+ 1) u θ0. Elementary calculus shows that fvanishes at x=−1, a, it reaches its maximum at x=b:= λu −C3u−C1 C1(1 + u), and it is strictly increasing (respectively, decreasing) on [−1, b](respectively, [b, a]). It follows that7 N± s,max/min(λ)≤Za 0 f(x)dx + 2f(b) + a+ 1. But f(b) = λ−C3+C1 (1 + u)C2C01 θ0u(λ−C3+C1) (1 + u)C1u θ0, 7A similar argument is made in the proof of [5, Corollary 12.13-(viii)]. In that case, the authors use a strictly decreasing function f: (−∞, a]→[0,∞). The resulting estimate should be N± s,max/min(λ)≤Za 0 f(x)dx +f(0) + a+ 1, but the terms f(0) + a+ 1 were missing in that publication. This correction does not affect the final estimate of N± s,max/min(λ)obtained there. 57 Witten’s Perturbation on Strata and Za 0 f(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 . So N± s,max/min(λ)≤Cλ 1+u θ0+1 for some C > 0and all large enough λ, giving (v) with θ=1+u θ0+ 1−1>0. Table 2.13 describes the above conditions on κin terms of r. κ≥1 2r≤n−1 2−1 2u |κ|<1 2|r−n−1 2|<1 2u κ≤ −1 2r≥n−1 2+1 2u Table 2.13: Correspondence between conditions on κand r. 2.5 Relatively local model of the Witten’s perturbation Let m∈N, and let L1, . . . , Labe compact stratifications. For each i= 1, . . . , a, let Nibe a dense stratum of Li, let ki= dim Ni+ 1, and let ∗iand ρibe the vertex and radial function of c(Li). Then M:= Rm×Qa i=1(Ni×R+)is a dense stratum of A:= Rm×Qa i=1 c(Li). For any relatively compact open neighborhood Oof x:= (0,∗1, . . . , ∗a), all general adapted metrics on Mare quasi-isometric on M∩O to a metric of the form g=g0+Pa i=1 ρ2ui i˜gi+ (dρi)2, where g0is the Euclidean metric on Rm, every ˜giis a general adapted metric on Ni, and ui>0. Suppose that gis good; i.e., the metrics ˜giare good, and ui≤1. We can assume that every Niis connected, which means that the fiber of lim : c M→Mover xconsists of a unique point, which can be identified to x(see [5, Proof of Proposition 3.20]). According to Section 1.4, the rel-local model of a rel-Morse function around a relcritical point is of the form f=1 2(ρ2 +−ρ2 −), where ρ±is the radial function of 58 2.5 Relatively local model of the Witten’s perturbation Rm±×Qi∈I±c(Li), for some decomposition m=m++m−(m±∈N), and some partition of {1, . . . , a}into sets I±. The rel-critical set of fconsists only of x. Let ds,δs,Dsand ∆sbe the Witten’s perturbations of d,δ,Dand ∆on Ω(M)induced by f. Let Hs,max/min =LrHr s,max/min = ker ∆s,max/min, with the induced grading. The following result is a direct consequence of Corollary 2.4.9 and [5, Example 9.1 and Lemma 5.1], taking also into account Table 2.13. Corollary 2.5.1. (i)∆s,max/min has a discrete spectrum. (ii)We have Hr s,max/min ∼ =M (r1,...,ra) a O i=1 Hri max/min(Ni), where (r1, . . . , ra)runs in the subset of Nadefined by the conditions r=m−+ a X i=1 ri+|I−|, ri<ki−1 2+1 2uiif i∈I+ ri≥ki−1 2+1 2uiif i∈I−)for Hr s,max, ri≤ki−1 2−1 2uiif i∈I+ ri>ki−1 2−1 2uiif i∈I−)for Hr s,min. (iii)If es∈ Hs,max/min with norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)Let 0≤λs,max/min,0≤λs,max/min,1≤ ··· be the eigenvalues of ∆s,max/min, repeated according to their multiplicities. Given k∈N, if λs,max/min,k >0for some s, then λs,max/min,k >0for all sand λs,max/min,k ∈O(s)as s→ ∞. (v)There is some θ > 0such that lim infkλs,max/min,k k−θ>0. For every ρ > 0, let Bρbe the open ball of center 0and radius ρin Rm, and let Ux,ρ =Bρ× a Y i=1 (Ni×(0, ρ)) ⊂M. Taking complex coefficients, by Propositions 2.4.2, 2.4.5 and 2.4.8, the following result clearly boils down to the case of Proposition 2.2.9. Proposition 2.5.2. For α∈L2Ω(M), let αt= exp(itDs,max/min)α. If supp α⊂ Ux,a for some a > 0, then supp αt⊂Ux,a+|t|for all t∈R. 59 Witten’s Perturbation on Strata 2.6 Proof of Theorem 1.6.1 This theorem follows from Corollary 2.5.1-(i),(v) with the same arguments as [5, Theorem 1.1]. More precisely, Propositions A.2.1 and A.2.2 are used to globalize the properties of the rel-local model, the min-max principle [58, Theorem XIII.1] is used to show that the properties of the statement are invariant by taking Witten’s perturbation defined by rel-admissible functions, and Remark 2.1.7-(iii),(iv) is used to produce rel-admissible cutoff functions and partitions of unity with bounded differential. These functions are needed for the Witten’s perturbation and to apply Propositions A.2.1 and A.2.2. 2.7 Functional calculus Let Mbe a stratum of a compact stratified space, equipped with a good general adapted metric g. Let fbe any rel-admissible function on M, and let ds,δs,Dsand ∆sbe the corresponding Witten’s perturbations of d,δ,Dand ∆. Since fis reladmissible, for every s,∆s−∆is a homomorphism with uniformly bounded norm by (2.3.4). From (2.3.4) and the min-max principle (see e.g. [58, Theorem XIII.1]), it also follows that D(∆s,max/min) = D(∆max/min),D∞(∆s,max/min) = D∞(∆max/min), and that the properties stated in Theorem 1.6.1 can be extended to the perturbation ∆s,max/min. For any rapidly decaying function φon R,φ(∆s,max/min)is a Hilbert-Schmidt operator on L2Ω(M)by the version of Theorem 1.6.1-(ii) for ∆s,max/min. In fact, φ(∆s,max/min)is of trace class because φcan be given as the product of two rapidly decaying functions, |φ|1/2and sign(φ)|φ|1/2, where sign(φ)(x) = sign φ((x)) ∈ {±1}if φ(x)6= 0. Like in the case of closed manifolds (see e.g. [59, Chapters 5 and 8]), the operator φ(∆s,max/min)is given by a Schwartz kernel Ks, and Tr φ(∆s,max/min)equals the integral of the pointwise trace of Kson the diagonal. But we do not know whether Ksis uniformly bounded because a “rel-Sobolev embedding theorem” is missing [5, Section 19]. Theorem 1.6.1-(ii) becomes important in the arguments exposed here to make up for this lack. 2.8 The wave operator With the notation of Section 2.7, suppose that fis a rel-Morse function. Take a general chart O≡O0around every x∈Critrel(f), like in Section 1.4. Let us add the subindex “x” to the notation of M0,Ni,m±and I±in this case. Take a good adapted metric g0 xon M0 xof the form used in Section 2.5. Consider the 60 2.8 The wave operator Witten’s perturbed operators d0 x,s,δ0 x,s,D0 x,s and ∆0 x,s on Ω(M0 x)determined by the function f0:= 1 2(ρ2 +−ρ2 −)(a prime and the subindex xis added to their notation). Add also a prime to the notation of the sets Ux,ρ of Section 2.5, considered in M0 x. Let ρ0>0such that U0 x,ρ0⊂O0. Then, for 0< ρ ≤ρ0, there is some open Ux,ρ ⊂Mso that Ux,ρ ≡U0 x,ρ. Moreover, according to Remark 2.1.4, we can assume g|Ux,ρ0≡g0 x|U0 x,ρ0. Consider the wave equation dαt dt −iDsαt= 0,(2.8.1) where αt∈Ω(M)depends smoothly on t. Given any α∈D∞(∆s,max/min), its solution with the initial condition α0=αis given by αt= exp(itDs,max/min)α. Moreover a usual energy estimate shows that such a solution is unique (see e.g. [59, Proposition 7.4]); in fact, given any c > 0, it is also unique for |t| ≤ c. Proposition 2.8.1. Let 0< a < b < ρ0and α∈L2Ω(M). The following properties hold for αt= exp(itDs,max/min)α: (i)If supp α⊂MrUx,a, then supp αt⊂MrUx,a−|t|for 0<|t| ≤ a. (ii)If supp α⊂Ux,a, then supp αt⊂Ux,a+|t|for 0<|t| ≤ b−a. Proof. First, let us prove (ii). We can assume that α∈D∞(∆s,max/min)because exp(itDs,max/min)is bounded. Since supp α⊂Ux,a, we have α|Ux,ρ0≡α0|U0 x,ρ0for a unique α0∈Ω(M0 x)supported in U0 x,a. We get α0∈D∞(∆0 x,s,max/min)because α∈D∞(∆s,max/min). Let α0 t= exp(itD0 x,s,max/min)α0. By Proposition 2.5.2, we have supp α0 t⊂U0 x,a+|t|for 0<|t| ≤ b−a. Then α0 t|U0 x,ρ0≡βt|Ux,ρ0for a unique βt∈Ω(M)supported in Ux,a+|t|. Now, βt∈D∞(∆s,max/min)because α0 t∈ D∞(∆0 x,s,max/min). Moreover βtsatisfies (2.8.1) for |t| ≤ b−awith initial condition β0=α. So βt=αtby the uniqueness of the solution of (2.8.1), obtaining supp αt⊂ Ux,a+|t|. Finally, (i) follows from (ii) in the following way. For any β∈Ω0(M)with supp β⊂Ux,a−|t|, let βτ= exp(iτDs,max/min)βfor τ∈R. By (ii), we get supp β−t⊂Ux,a, and therefore hαt, βi=hα, β−ti= 0. This shows that supp αt⊂ MrUx,a−|t|. Remark 2.8.2.The steps given to achieve Proposition 2.8.1 are simpler here than in [5]. In fact, it would be difficult to adapt the arguments of [5] since an expression of D∞(∆max/min)is missing in Section 2.2.2. 61 Lefschetz trace formula singular fixed points, lim t→0 X q∈Fix(ψ)∩M n X r=0 (−1)rZOq tr(ψ∗Kr t(ψ(p), p)) vol(p)!(3.1.7) + lim t→0 X q∈sing(A) n X r=0 (−1)rZOq\{q} tr(ψ∗Kr t(ψ(p), p)) vol(p)!.(3.1.8) Moreover, by Corollary A.2.5 (and the asymptotic ideas used in proof of Theorem A.2.4), it follows that (3.1.7) is equal to X q∈Fix(ψ)∩M sign det(1 −Tqψ).(3.1.9) The computation of (3.1.8) is included in Section 3.3. 3.2 Lefschetz trace formula on a cone Let Lbe a non-empty compact stratified space and consider the dense stratum N of L. Let f=1 2ρ2be the model rel-Morse function on the stratum N×R+of the cone c(L), whose only rel-critical point is the vertex. Consider a good adapted metric ρ2u˜g+ (dρ)2on N×R+, where ˜ga is good adapted metric on Nand 0< u ≤1. In this section, let ψ:c(L)→c(L)be a morphism of (non-compact) stratifications, smooth on N×R+and satisfying hypotheses (a)–(c). In particular, the vertex is a simple fixed point of ψ, and ψ(ρ, x) = (ρF(x), G(x)) for all (ρ, x)∈N×R+. For s > 0, let ds,δs,Dsand ∆sbe the corresponding Witten’s perturbations of d,δ,Dand ∆on N×R+(see (2.3.1)–(2.3.3)). Let also ψ∗ s=e−sf ψ∗esf , which is called the Witten’s perturbation of the endomorphism ψ∗. Then esf : (D(ds,max/min), ds,max/min)→(D(dmax/min), dmax/min) is a Hilbert complex isomorphism [5, § 18.1]. Hence, Hr max/min = ker ∆max/min,r ≡ker ∆s,max/min,r =Hr s,max/min, for r= 0, . . . , n. As a consequence, given an orthonormal frame {ei}of the Hilbert space L2Ωr(N×R+), we get Tr(ψ∗Pr) = X ihψ∗Prei, eii=X ihψ∗ sPs,r ei, eii= Tr(ψ∗ sPs,r),(3.2.1) 68 3.2 Lefschetz trace formula on a cone where Ps,r :L2Ωr(N×R+)→ Hr s,max/min is the orthogonal projection. Moreover, given an orthonormal frame {ξj}of Hr s,max/min, we have Tr(ψ∗ sPs,r) = X jhψ∗ sξj, ξji.(3.2.2) From Proposition 2.4.8, it follows that Hs,max/min = ker ∆s,max/min = ker ∆s,max/min = ker M γ ∆s,γ ⊕d M µM α+β ∆s,α,β =M γ ker ∆s,γ ⊕d M µM α+β ker ∆s,α,β =M γ ker ∆s,γ =M γHs,γ =M γHr s,γ ⊕Hr+1 s,γ , where γruns in an orthonormal frame Cmax/min,0of e Hmax/min = ker e ∆max/min,µ runs in the positive spectrum of e Dmax/min, and α+βruns in an orthonormal frame Cmax/min,µ of the µ-eigenspace of e Dmax/min. Then, applying Proposition 2.4.9-(ii) and standard arguments from Hodge Theory (see Tables 2.11 and 2.12, where only the positive sign cases must be considered), for r= 0, . . . , n −1, we obtain Hr s,max =M γ∈Cr max,0 Hr s,γ ≡M γ∈Cr max,0 hγi=e Hr max ≡Hr max(N) if κ > −1/2, and Hr s,max = 0 if κ≤ −1/2. Analogously, for r= 0, . . . , n −1, Hr s,min =M γ∈Cr min,0 Hr s,γ ≡M γ∈Cr min,0 hγi=e Hr min ≡Hr min(N) if κ≥1/2, and Hr s,min = 0 if κ < 1/2. In addition, Hn s,max/min = 0, for all κ∈R. Remember that hγi ⊂ L2Er γ=L2 κ,+γ≡L2 κ,+for all the harmonic forms γ∈ Cr max/min,0. By Proposition 2.4.2, the Laplacian ∆s,γ =∆s,γ,r ⊕∆s,γ,r+1 is associated (up to a shift of degree) to the maximum/minimum extension of the operator ∆=∆0⊕∆1studied in Section 2.2.1. So Hr s,γ = ker ∆s,γ,r corresponds to ∆max/min,0by the unitary isomorphism L2Er γ→L2(E0)defined by the multiplication operator ρκ:L2 κ,+→L2 +. The non-trivial case for Hr s,max/min determines a condition on κyielding ∆max/min,0=A1, according to Proposition 2.2.1 (see Table 2.3). So we must consider σ=a=κ= (n−2r−1)u 2 69 Lefschetz trace formula in Proposition 4.7.1, obtaining that ρκSev,+⊂L2 +is the smooth core of ∆max/min,0, and χ0(ρ) = χs,κ,κ,0(ρ) = √2ρκφs,κ,0,+(ρ) =√2ρκs(2κ+1)/4Γ(κ+1 2)−1 2e−sρ2/2 is a normalized generator of the space of harmonic eigenfunctions (i.e., associated to the eigenvalue 0of multiplicity one, given for A+ 1by the integer k= 0 in Table 2.2). Therefore Sev,+γ⊂L2Er γis the smooth core of ∆s,γ,r, and ξ(ρ, x) = √2φs,κ,0,+(ρ)γ(x) = √2s(2κ+1)/4Γ(κ+1 2)−1 2e−sρ2/2γ(x) is a normalized harmonic eigenform of Hr s,max/min. By hypotheses (a)–(c), ψ∗ξ(ρ, x) = √2s(2κ+1)/4Γ(κ+1 2)−1 2e−s(ρF(x))2/2(G∗γ)(x). Thus es(ψ∗f−f)ψ∗ξ=√2s(2κ+1)/4Γ(κ+1 2)−1 2e−sρ2/2G∗γ =√2φs,κ,0,+(ρ)G∗γ because (ψ∗f−f)(ρ, x) = f(ψ(ρ, x)) −f(ρ, x) = 1 2(ρF(x))2−1 2ρ2. Then hψ∗ sξ, ξi=hes(ψ∗f−f)ψ∗ξ, ξi =h√2φs,κ,0,+(ρ)G∗γ, √2φs,κ,0,+(ρ)γi =k√2φs,κ,0,+(ρ)k2 L2 κ,+hG∗γ, γi =k√2ρκφs,κ,0,+(ρ)k2 L2 +hG∗γ, γi =hG∗γ, γi.(3.2.3) Combining (3.2.2) and (3.2.3), we obtain Tr(ψ∗ sPs,r) = X γ∈Cr max/min,0 hG∗γ, γi(3.2.4) 70 3.3 Contribution from the singular fixed points for r= 0, . . . , n −1; whereas Tr(ψ∗ sPs,n)=0. By (3.1.2), (3.2.1) and (3.2.4), we deduce Lmax/min(ψ) = n−1 X r=0 (−1)rX γ∈Cr max/min,0 hG∗γ, γi = n−1 X r=0 (−1)rtr(G∗on e Hr max/min) = n−1 X r=0 (−1)rtr(G∗on Hr max/min(N)) = Lmax/min(G).(3.2.5) In particular, if N=Lis a compact smooth manifold, it happens that Lmax/min(ψ) = Lmax/min(G) = L(G),(3.2.6) since by completness there is only one i.b.c. (see Section 1.1). Here L(G)denotes the Lefschetz number associated to the smooth map G:L→L(see [59, Equation (10.1)]). 3.3 Contribution from the singular fixed points Consider the orthogonal projections Πand e Πintroduced in Section 2.9, but constructed using charts around the singular points of A(instead of around the rel-critical points of the chosen rel-Morse function f). Applying [59, Theorem 8.12 and Proposition 10.7], the Hilbert complex isomorphism given by esρ2/2and Lemmma 2.9.5, it follows that (3.1.8) is equal to lim t→0 n X r=0 (−1)rTr(ψ∗e−t∆max/min,r e Π)! = lim t→∞ n X r=0 (−1)rTr(ψ∗e−t∆max/min,r e Π)! = n X r=0 (−1)rlim t→∞ Tr(ψ∗e−t∆max/min,r e Π) = n X r=0 (−1)rlim t→∞ Tr(ψ∗ se−t∆s,max/min,r e Π) 71 Lefschetz trace formula =X q∈sing(A) n X r=0 (−1)rlim t→∞ Tr(ψ∗ se−t∆s,max/min,r e Πq) =X q∈sing(A) n X r=0 (−1)rlim t→∞ Tr(ψ∗ se−t∆0 q,s,max/min,r e Πq).(3.3.1) Observe that (3.3.1) is independent of the Witten’s parameter s. The heat operator e−t∆0 q,s,max/min,r is of trace class, and ψ∗ sis a bounded operator. Then |Tr(ψ∗ se−t∆0 q,s,max/min,r Πq)| ≤ kψ∗ sk|Tr( e−t∆0 q,s,max/min,r Πq)| by the general properties of trace class operators. But Lemma 2.9.3 gives Tr( e−t∆0 q,s,max/min,r Πq)→0as s→ ∞. Hence Tr(ψ∗ se−t∆0 q,s,max/min,r Πq)→0as s→ ∞. Consequently, since 1 = Πq+e Πq, Tr( ψ∗ se−t∆0 q,s,max/min,r ) = Tr(ψ∗ se−t∆0 q,s,max/min,r Πq) + Tr(ψ∗ se−t∆0 q,s,max/min,r e Πq) →Tr(ψ∗ se−t∆0 q,s,max/min,r e Πq)as s→ ∞. From (3.3.1), (3.2.4), (3.2.5) and (3.2.6), it follows that (3.1.8) is equal to X q∈sing(A) n X r=0 (−1)rlim t→∞ Tr(ψ∗ se−t∆0 q,s,max/min,r ) =X q∈sing(A) n X r=0 (−1)rTr(ψ∗ sPq,s,r) =X q∈sing(A) n−1 X r=0 (−1)rX γ∈Cr q,max/min,0 hG∗ qγ, γi =X q∈sing(A) L(Gq),(3.3.2) because (3.3.1) is independent of s. Finally, applying (3.1.9) and (3.3.2), we get Lmax/min(ψ) = X q∈Fix(ψ)∩M sign det(1 −Tqψ) + X q∈sing(A) L(Gq), which is the statement of Theorem 1.6.3. 72 Chapter 4 Dunkl Harmonic Oscillator Contents 4.1 Preliminaries........................... 74 4.2 The sesquilinear form t..................... 76 4.3 Scalar products of mixed generalized Hermite functions . . . 90 4.3.1 Case where σ=θ6=τand τ−σ6∈ −N........ 91 4.3.2 Case where σ6=θ6=τand σ−θ, τ −θ6∈ −N..... 94 4.4 The sesquilinear form t0..................... 99 4.4.1 Case where σ=θ=τ.................. 99 4.4.2 Case where σ=θ6=τand τ−σ6∈ −N........ 99 4.4.3 Case where σ6=θ=τand σ−θ6∈ −N......... 100 4.4.4 Case where σ6=θ=τ+ 1 and σ−τ−16∈ −N. . . . 101 4.4.5 Case where σ6=θ6=τand σ−θ, τ −θ6∈ −N. . . . . 103 4.4.6 Proof of Theorem 1.10.3 . . . . . . . . . . . . . . . . . 104 4.5 A preliminary estimate . . . . . . . . . . . . . . . . . . . . . 106 4.5.1 Statement......................... 106 4.5.2 Proof of Lemma 4.5.1 . . . . . . . . . . . . . . . . . . 107 4.6 The main estimates . . . . . . . . . . . . . . . . . . . . . . . 121 4.7 Operators induced on R+.................... 126 In this chapter, consider the notation of Section 1.10. Then let S=Sev ⊕Sodd be the Schwartz space on R, considered with its Fréchet topology and decomposed as direct sum of subspaces of even and odd functions. Let xdenote the standard coordinate of R. The multiplication operator xinterchanges the components, because Sodd =xSev and the function |x|2σis even. Let L2 σ=L2(R,|x|2σdx)(σ∈R), whose scalar product and norm are denoted by h,iσand k kσ. The above decomposition of Sextends to an orthogonal decomposition L2 σ=L2 σ,ev ⊕L2 σ,odd. The 73 Dunkl Harmonic Oscillator subspace Sis dense in L2 σif σ > −1 2, and Sodd is dense in L2 τ,odd if τ > −3 2. So we assume σ > −1 2and τ > −3 2, unless otherwise stated. The harmonic oscillator is the operator H=−d2 dx2+s2x2(s > 0) in L2 0with domain D(H) = S. The Dunkl operator on Ris the operator Tin L2 σ, with D(T) = S, determined by T=d dx on Sev and T=d dx +2σx−1on Sodd. The Dunkl harmonic oscillator on Ris J=−T2+s2x2in L2 σwith D(J) = S. Thus Jpreserves the above decomposition of S, being Jev =H−2σx−1d dx and Jodd =H−2σd dx x−1. This Jis essentially self-adjoint, and the spectrum of its closure Jis well known; in particular, J > 0. In fact, even for τ > −3 2, the operator Jτ,odd is defined in L2 τ,odd with D(Jτ,odd) = Sodd because it is a conjugation of Jτ+1,ev by a unitary operator (Section 4.1). Let also Jσ,τ =Jσ,ev ⊕Jτ,odd in L2 σ,τ =L2 σ,ev ⊕L2 τ,odd, with D(Jσ,τ ) = S. The aim of this chapter is to use different analytic techniques in order to prove Theorems 1.10.1 and 1.10.3, which describe certain perturbations of Jσand Jσ,τ , respectively. The contents of this chapter are included in [6]. 4.1 Preliminaries The Dunkl annihilation and creation operators are B=sx +Tand B0=sx −T (s > 0). Like J, the operators Band B0are considered in L2 σwith domain S. They are perturbations of the usual annihilation and creation operators. The operators T, B,B0and Jare continuous on S. The following properties hold [4,60]: •B0is adjoint of B, and Jis essentially self-adjoint. •The spectrum of Jconsists of the eigenvalues1(2k+ 1 + 2σ)s(k∈N), of multiplicity one. •The corresponding normalized eigenfunctions φkare inductively defined by φ0=s(2σ+1)/4Γ(σ+1 2)−1 2e−sx2/2,(4.1.1) φk=((2ks)−1 2B0φk−1if kis even (2(k+ 2σ)s)−1 2B0φk−1if kis odd (k≥1).(4.1.2) •The eigenfunctions φkalso satisfy Bφ0= 0,(4.1.3) Bφk=((2ks)1 2φk−1if kis even (2(k+ 2σ)s)1 2φk−1if kis odd (k≥1).(4.1.4) 1It is assumed that 0∈N. 74 4.1 Preliminaries •T∞ m=0 D(Jm) = S. By (4.1.1) and (4.1.2), 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−Tpk−1)if kis even (2(k+ 2σ)s)−1 2(2sxpk−1−Tpk−1)if kis odd (k≥1). Up to normalization, pkis the sequence of generalized Hermite polynomials [66, p. 380, Problem 25], and φkis the sequence of generalized Hermite functions. Each pkis of degree k, even/odd if kis even/odd, and with positive leading coefficient. They satisfy the recursion formula [4, Eq. (13)] 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.(4.1.5) When k= 2m+ 1 (m∈N), we have [4, Eq. (14)] x−1pk= m X i=0 (−1)m−ism!Γ(i+1 2+σ)s i!Γ(m+3 2+σ)p2i.(4.1.6) The Pochhammer symbol could be used to simplify this expression, as well as many other expressions in Sections 4.2 and 4.3. However there are quotients of gamma functions in Sections 4.3 and 4.4 that can not be simplified in this way (see e.g. Proposition 4.3.7). Thus, for the sake of uniformity, we use gamma functions in all quotients of this type. Let jbe the positive definite symmetric sesquilinear form in L2 σ, with D(j) = S, given by j(φ, ψ) = hJφ, ψiσ. Like in the case of J, the subindex σwill be added to the notation T,B,B0and φkand jif necessary. Observe that Bσ=(Bτon Sev Bτ+ 2(σ−τ)x−1on Sodd,(4.1.7) B0 σ=(B0 τon Sev B0 τ+ 2(τ−σ)x−1on Sodd.(4.1.8) The operator x:Sev → Sodd is a homeomorphism [4], which extends to a unitary operator x:L2 σ,ev →L2 σ−1,odd. We get x Jσ,ev x−1=Jσ−1,odd because x[d2 dx2, x−1] = −2d dx x−1. Thus, even for any τ > −3 2, the operator Jτ,odd is densely defined in L2 τ,odd, with D(Jτ,odd) = Sodd, and has the same spectral properties as 75 Dunkl Harmonic Oscillator Jτ+1,ev; in particular, the eigenvalues of Jτ,odd are (2k+ 1 + 2τ)s(k∈2N+ 1), and φτ,k =xφτ+1,k−1. To prove the results of this chapter, alternative arguments could be given by using the expression of the generalized Hermite polynomials in terms of the Laguerre ones (see e.g. [61, p. 525] or [62, p. 23]). In particular, certain asymptotic estimates of Laguerre functions [28,51] (see also [8,52]), yield the following asymptotic estimates of the generalized Hermite functions [3, Section 2.4]: there are some C, c > 0, depending only on σ, such that |φk(x)xσ| ≤                Cs ¯σ 2+1 4x¯σν¯σ 2−1 4if 0< x ≤q1 sν Cs1 4ν−1 4if q1 sν < x ≤pν 2s Cs1 4(ν1 3+|sx2−ν|)−1 4if pν 2s< x ≤q3ν 2s C(sx)1 2e−csx2if q3ν 2s< x , (4.1.9) where ¯σ= ¯σk=σ+1−(−1)k 2and ν=νk= 2k+1+2σ, with the proviso that we must take ν= 2 if k= 0 and σ < 1 2. 4.2 The sesquilinear form t Let 0<u<1such that σ > u −1 2. Then |x|−uS ⊂ L2 σ, and therefore a positive definite symmetric sesquilinear form tin L2 σ, with D(t) = S, is defined by t(φ, ψ) = h|x|−uφ, |x|−uψiσ=hφ, ψiσ−u. The notation tσmay be also used. The goal of this section is to study tand apply it to prove Theorem 1.10.1. Precisely, an estimation of the values t(φk, φ`)is needed. Lemma 4.2.1. For all φ∈ Sodd and ψ∈ Sev, t(B0φ, ψ)−t(φ, Bψ) = t(φ, B0ψ)−t(Bφ, ψ) = −2ut(x−1φ, ψ). Proof. By (4.1.7) and (4.1.8), for all φ∈ Sodd and ψ∈ Sev, t(B0 σφ, ψ)−t(φ, Bσψ) =hB0 σ−uφ, ψiσ−u−2uhx−1φ, ψiσ−u−hφ, Bσ−uψiσ−u =−2ut(x−1φ, ψ), t(φ, B0 σψ)−t(Bσφ, ψ) =hφ, B0 σ−uψiσ−u−hBσ−uφ, ψiσ−u−2uhx−1φ, ψiσ−u =−2ut(x−1φ, ψ). 76 4.2 The sesquilinear form t In the whole of this section, k,`,m,n,i,j,pand qwill be natural numbers. Let ck,` =t(φk, φ`)and dk,` =ck,`/c0,0. Thus dk,` =d`,k, and dk,` = 0 when k+`is odd. Since R∞ −∞ e−sx2|x|2κdx =s−(2κ+1)/2Γ(κ+1 2)(4.2.1) for κ > −1 2, we get c0,0= Γ(σ−u+1 2)Γ(σ+1 2)−1su.(4.2.2) Lemma 4.2.2. If k= 2m > 0, then dk,0=u √m m−1 X j=0 (−1)m−js(m−1)!Γ(j+1 2+σ) j!Γ(m+1 2+σ)d2j,0. Proof. By (4.1.2), (4.1.3), (4.1.6) and Lemma 4.2.1, ck,0=1 √2sk t(B0φk−1, φ0) =1 √2sk t(φk−1, Bφ0)−2u √2sk t(x−1φk−1, φ0) =−2u √2sk t(x−1φk−1, φ0) =u √m m−1 X j=0 (−1)m−js(m−1)!Γ(j+1 2+σ) j!Γ(m+1 2+σ)c2j,0. Lemma 4.2.3. If k= 2m > 0and `= 2n > 0, then dk,` =rm ndk−1,`−1+u √n n−1 X j=0 (−1)n−js(n−1)!Γ(j+1 2+σ) j!Γ(n+1 2+σ)dk,2j. Proof. By (4.1.2), (4.1.4), (4.1.6) and Lemma 4.2.1, ck,` =1 √2s` t(φk, B0φ`−1) =1 √2`s t(Bφk, φ`−1)−2u √2`s t(φk, x−1φ`−1) =rm nck−1,`−1+u √n n−1 X j=0 (−1)n−js(n−1)!Γ(j+1 2+σ) j!Γ(n+1 2+σ)ck,2j. 77 Dunkl Harmonic Oscillator Then the result follows in this case from Lemma 4.2.10. When k= 2m+ 1 ≥`= 2n+ 1, the result follows from the above case and Corollary 4.2.9. Lemma 4.2.12. For each t∈R\(−N), there is some C1=C1(t)≥1such that, for all p, C−1 1(p+ 1)1−t≤Γ(p+ 1) |Γ(p+t)|≤C1(p+ 1)1−t. Proof. We can assume that p≥1. Write t=q+r, where q=btc. If q= 0, then 0< r < 1and the the result follows from the Gautschi’s inequality, stating that x1−r≤Γ(x+ 1) Γ(x+r)≤(x+ 1)1−r(4.2.15) for 0< r < 1and x > 0, because x1−r≥2r−1(x+ 1)1−rfor x≥1. If q≥1and r= 0, then Γ(p+ 1) Γ(p+t)=p! (p+q−1)! ≤1 (p+ 1)q−1= (p+ 1)1−t, Γ(p+ 1) Γ(p+t)=p! (p+q−1)! ≥1 (p+q−1)q−1≥1 (qp)q−1 ≥t1−t(p+ 1)1−t. If q≥1and r > 0, then, by (4.2.15), Γ(p+ 1) Γ(p+t)≤Γ(p+ 1) (p+ 1)q−1(p+r)Γ(p+r)≤(p+ 1)2−q−r p+r≤2(p+ 1)1−t, Γ(p+ 1) Γ(p+t)≥Γ(p+ 1) (p+t−1)qΓ(p+r)≥p1−r (p+t−1)q≥(p+ 1)1−r 21−r(p+t−1)q ≥min{1,(t−1)−q}2r−1(p+ 1)1−t, because (p+t−1)−q≥((p+ 1)−qif 0< t ≤2 (t−1)−q(p+ 1)−qif t > 2. In the case q < 0(t < 0), apply reverse induction on q: with C1=C1(t+ 1), we get Γ(p+ 1) |Γ(p+t)|=|p+t|Γ(p+ 1) |Γ(p+t+ 1)|≤ |p+t|C1(p+ 1)−t≤C1|q|(p+ 1)1−t, Γ(p+ 1) |Γ(p+t)|=|p+t|Γ(p+ 1) |Γ(p+t+ 1)|≥ |p+t|C−1 1(p+ 1)−t =|p+t| p+ 1 C−1 1(p+ 1)1−t, 84 4.2 The sesquilinear form t where |p+t|/(p+ 1) is bounded uniformly on p. Corollary 4.2.13. There is some C00 =C00(σ)>0such that Πk,` ≤(C00(n+1 m+1)σ 2−1 4if k= 2m≥`= 2n C00(n+1 m+1)σ 2+1 4if k= 2m+ 1 ≥`= 2n+ 1. Proof. This follows from (4.2.3), (4.2.4) and Lemma 4.2.12. For the sake of simplicity, let us use the following notation. For real valued functions fand gof (m, n), for (m, n)in some subset of N×N, write f4gif there is some C > 0such that f(m, n)≤C g(m, n)for all (m, n). The same notation is used for functions depending also on other variables, s, σ, u, . . . , taking Cindependent of m,nand s, but possibly depending on the rest of variables. Lemma 4.2.14. For α, β, γ ∈R, if α+β, α +γ, α +β+γ < 0, then there is some ω > 0such that, for all naturals m≥n, (m+ 1)α(n+ 1)β(m−n+ 1)γ4(m+ 1)−ω(n+ 1)−ω. Proof. We consider the following cases: 1. If α, β, γ < 0, then (m+ 1)α(n+ 1)β(m−n+ 1)γ≤(m+ 1)α(n+ 1)β. 2. If β≥0and γ < 0, then (m+ 1)α(n+ 1)β(m−n+ 1)γ≤(m+ 1)α+β ≤(m+ 1)α+β 2(n+ 1)α+β 2. 3. If α≥0and m+ 1 ≤2(n+ 1), then β, γ < 0and (m+ 1)α(n+ 1)β(m−n+ 1)γ≤2−β(m+ 1)α+β ≤2−β(m+ 1)α+β 2(n+ 1)α+β 2. 4. If α≥0and m+ 1 >2(n+ 1), then β, γ < 0and m−n+ 1 >(m+ 1)/2, and therefore (m+ 1)α(n+ 1)β(m−n+ 1)γ≤2−γ(m+ 1)α+γ(n+ 1)β. 85 Dunkl Harmonic Oscillator 5. If β < 0and γ≥0, then (m+ 1)α(n+ 1)β(m−n+ 1)γ≤(m+ 1)α+γ(n+ 1)β. 6. If β≥0and γ≥0, then (m+ 1)α(n+ 1)β(m−n+ 1)γ≤(m+ 1)α+β+γ ≤(m+ 1)α+β+γ 2(n+ 1)α+β+γ 2. Proposition 4.2.15. There is some ω=ω(σ, u)>0such that |dk,`|4(m+ 1)−ω(n+ 1)−ω for k= 2mand `= 2n, or for k= 2m+ 1 and `= 2n+ 1. Proof. We can assume k≥`because dk,` =d`,k. If k= 2m+1 ≥`= 2n+1, then, according to Proposition 4.2.5, Lemma 4.2.11 and Corollary 4.2.13, |dk,`|4(m+ 1)−σ 2−1 4−u(1−u)(n+ 1)σ 2+1 4(m−n+ 1)−(1−u)2. Thus the result follows by Lemma 4.2.14 since −σ 2−1 4−u(1 −u)−(1 −u)2=−σ 2+u−5 4<u 2−1<0. If k= 2m≥`= 2n, then, according to Proposition 4.2.5, Lemma 4.2.11 and Corollary 4.2.13, |dk,`|4(m+ 1)−σ 2+1 4−u(1−u)(n+ 1)σ 2−1 4(m−n+ 1)−(1−u)2. Thus the result follows by Lemma 4.2.14 since −σ 2+1 4−u(1 −u)−(1 −u)2=−σ 2+u−3 4<u 2−1 2<0. Corollary 4.2.16. There is some ω=ω(σ, u)>0such that, for k= 2mand `= 2n, or for k= 2m+ 1 and `= 2n+ 1, |ck,`|4su(m+ 1)−ω(n+ 1)−ω. Proof. This follows from Proposition 4.2.15 and (4.2.2). 86 4.2 The sesquilinear form t Proposition 4.2.17. For any  > 0, there is some C=C(, σ, u)>0such that, for all φ∈ S, t(φ)≤su−1j(φ) + Csukφk2 σ. Proof. For each k, let νk= 2k+ 1 + 2σ. By Corollary 4.2.16, there are some K0=K0(σ, u)>0and ω=ω(σ, u)>0such that |ck,`| ≤ K0suν−ω kν−ω `(4.2.16) for all kand `. Since S=S(σ, u) := Pkν−1−2ω k<∞, given  > 0, there is some k0=k0(, σ, u)so that S0=S0(, σ, u) := X k>k0 ν−1−2ω k<2 4K2 0S. Let S1=S1(, σ, u) = Pk≤k0ν−ω k. For φ=Pktkφk∈ S, by (4.2.16) and the Schwartz inequality, we have t(φ) = X k,` tkt`ck,` ≤X k,` |tk||t`||ck,`| ≤K0su−1 2X k≤k0 |tk| νω kX ` |t`|(ν`s)1 2 ν 1 2+ω ` +K0su−1X k>k0 |tk|(νks)1 2 ν 1 2+ω kX ` |t`|(ν`s)1 2 ν 1 2+ω ` ≤K0S1S1 2su−1 2kφkσj(φ)1 2+K0S 1 2 0S1 2su−1j(φ) ≤K0S1S1 2su−1 2kφkσj(φ)1 2+su−1 2j(φ) ≤K2 0S2 1Ssu 2kφk2 σ+su−1j(φ). Proposition 4.2.18. There is some D=D(σ, u)>0such that, for all k∈Nand φ in the linear span of φ0, . . . , φk, t(φ)≥Dsu(k+ 1)−ukφk2 σ. Proof. Let φ=Pk i=0 tiφi(ti∈C) and ν=νk= 2k+ 1 + 2σ. Let K≥3, which 87 Dunkl Harmonic Oscillator will be fixed later. By (4.1.9), Z|x|≥qKν 2s|φ(x)|2|x|2σdx = k X i,j=0 titjZ|x|≥qKν 2s φi(x)φj(x)|x|2σdx ≤2 k X i,j=0 |ti||tj|Z∞ qKν 2s|φi(x)||φj(x)|x2σdx ≤ k X i,j=0 |ti|2+|tj|2C2sZ∞ qKν 2s xe−2csx2dx = 2(k+ 1)kφk2 σC2sZ∞ qKν 2s xe−2csx2dx =C2(k+ 1) 2ce−Kcνkφk2 σ, where C, c > 0depend only on σ. We can choose some K=K(σ)≥3and D=D(σ, u)>0such that 2s Kν u1−C2(k+ 1) 2ce−Kcν≥Dsu(k+ 1)−u for all s > 0and k∈N, obtaining t(φ)≥Z|x|≤qKν 2s|φ(x)|2|x|2σ−2udx ≥2s Kν uZ|x|≤qKν 2s|φ(x)|2|x|2σdx ≥2s Kν ukφk2 σ1−C2(k+ 1) 2ce−Kcν ≥Dsu(k+ 1)−ukφk2 σ. Remark 4.2.19.For φ=φk, we can also use the following argument. By Proposition 4.2.5 and (4.2.2), and since Πk,k = 1, it is enough to prove that there is some D0=D0(σ, u)>0so that Σk,k ≥D0(k+ 1)−u. Moreover we can assume that k= 2m+1 by Corollary 4.2.8. We have p0:= b1 2+σc ≥ 0because 1 2+σ > u. According to Corollary 4.2.9, Lemma 4.2.10 and (4.2.9), there is some C0=C0(u)≥1 88 4.2 The sesquilinear form t such that Σk,k ≥ m Y i=0 1−u i+1 2+σ≥1−u 1 2+σm+p0 Y p=1+p01−u p =1−u 1 2+σm+p0 Y p=1 1−u pp0 Y p=1 1−u p−1 ≥1−u 1 2+σC−2 0(m+p0+ 1)−u(p0+ 1)u ≥1−u 1 2+σC−2 0(k+ 1)−u. Remark 4.2.20.If 0<u<1 2, then limmt(φ2m+1)=0. To check it, we use that there is some K=K(σ, s)>0so that |x|2σφ2 k(x)≤Kk−1 6for all x∈Rand all odd k∈N[3, Theorem 1.1-(ii)] (this also follows from (4.1.9)). For any  > 0, take some x0>0and k0∈Nsuch that x−2u 0< /2and Kk−1 6 0x1−2u 0< (1 −2u)/4. Then, for all odd natural k≥k0, t(φk) = 2 Zx0 0 φ2 k(x)x2(σ−u)dx + 2 Z∞ x0 φ2 k(x)x2(σ−u)dx ≤2Kk−1 6Zx0 0 x−2udx + 2x−2u 0Z∞ x0 φ2 k(x)x2σdx ≤2Kk−1 6x1−2u 0 1−2u+x−2u 0<  because 1−2u > 0and kφkkσ= 1. In the case where σ≥0, this argument is also valid when kis even. We do not know if infkt(φk)>0when 1 2≤u < 1. Proof of Theorem 1.10.1. The positive definite sesquilinear form jof Section 4.1 is closable by [41, Theorems VI-2.1 and VI-2.7]. Then, taking  > 0such that ξsu−1<1, it follows from [41, Theorem VI-1.33] and Proposition 4.2.17 that the positive definite sesquilinear form u:= j+ξtis also closable, and D(¯ u) = D(j). By [41, Theorems VI-2.1, VI-2.6 and VI-2.7], there is a unique positive definite selfadjoint operator Usuch that D(U)is a core of D(¯ u), which consists of the elements φ∈D(¯ u)so that, for some χ∈L2 σ, we have ¯ u(φ, ψ) = hχ, ψiσfor all ψin some core of ¯ u(in this case, U(φ) = χ). By [41, Theorem VI-2.23], we have D(U1/2) = D(¯ u), Sis a core of U1/2(since it is a core of u), and (1.10.1) is satisfied. By Proposition 4.2.18, there is some D(σ, u)so that, for all s > 0and k∈N, and every φis in the linear span of φ0, . . . , φk, we have t(φ)≥Dsu(k+ 1)−ukφk2 σ. Moreover we 89 Dunkl Harmonic Oscillator can assume that the sequence (2k+ 1 + 2σ)s+ξDsu(k+ 1)−uis strictly increasing after reducing Dif necessary. So u(φ)≥(2k+ 1 + 2σ)s+ξDsu(k+ 1)−ukφk2 σ if φ∈ S is orthogonal in L2 σto the linear span of φ0, . . . , φk−1(assuming that this span is 0when k= 0). Therefore Uhas a discrete spectrum satisfying (1.10.2) by the form version of the min-max principle [58, Theorem XIII.2]. The inequality (1.10.3) holds because ¯ u(φ)≤1 + ξsu−1¯ j(φ) + ξCsukφk2 σ for all φ∈D(¯ u)by Proposition 4.2.17 and [41, Theorem VI-1.18], since Sis a core of ¯ uand¯ j. Remark 4.2.21.In the above proof, note that ¯ u=¯ j+ξ¯ tand D(¯ j) = DJ1/2. Thus (1.10.1) can be extended to φ, ψ ∈DU1/2using J1/2φ, J1/2ψσinstead of hJφ, ψiσ. Remark 4.2.22.Extend the definition of the above forms and operators to the case of ξ∈C. Then |¯ t(φ)| ≤ su−1<¯ j(φ) + Csukφk2 σfor all φ∈D(¯ j), like in the proof of Theorem 1.10.1. Thus the family ¯ u=¯ u(ξ)becomes holomorphic of type (a) by Remark 4.2.21 and [41, Theorem VII-4.8], and therefore U=U(ξ)is a self-adjoint holomorphic family of type (B). So the functions λk=λk(ξ)(ξ∈R) are continuous and piecewise holomorphic [41, Remark VII-4.22, Theorem VII-3.9, and VII-§ 3.4], with λk(0) = (2k+1+2σ)s. Moreover [41, Theorem VII-4.21] gives an exponential estimate of |λk(ξ)−λk(0)|in terms of ξ. But (1.10.2) and (1.10.3) are a better estimate for the eigenvalues. 4.3 Scalar products of mixed generalized Hermite functions Let σ, τ, θ > −1 2, and write v=σ+τ−2θ. This section is devoted to describe the scalar products ˆck,` = ˆcσ,τ,θ,k,` =hφσ,k, φτ,`iθ, which will be needed to prove Theorem 1.10.3. Note that ˆck,` = 0 if k+`is odd, and ˆcσ,τ,θ,k,` = ˆcτ,σ,θ,`,k (4.3.1) for all kand `. Of course, ˆck,` =δk,` if σ=τ=θ. According to Section 4.1, if kand `are odd, then ˆcσ,τ,θ,k,` is also defined when σ, τ, θ > −3 2, and we have ˆcσ,τ,θ,k,` =hxφσ+1,k−1, xφτ+1,`−1iθ= ˆcσ+1,τ+1,θ+1,k−1,`−1.(4.3.2) 90 4.3 Scalar products of mixed generalized Hermite functions 4.3.1 Case where σ=θ6=τand τ−σ6∈ −N In this case, we have v=τ−σ. By (4.1.1) and (4.2.1), ˆc0,0=sv 2Γ(σ+1 2)1 2Γ(τ+1 2)−1 2.(4.3.3) Lemma 4.3.1. If k > 0is even, then ˆck,0= 0. Proof. By (4.1.2), (4.1.3) and (4.1.7), ˆck,0=1 √2kshB0 σφσ,k−1, φτ,0iσ=1 √2kshφσ,k−1, Bτφτ,0iσ= 0. Lemma 4.3.2. If `= 2n > 0, then ˆc0,` =v √n n−1 X j=0 (−1)n−js(n−1)!Γ(j+1 2+τ) j!Γ(n+1 2+τ)ˆc0,2j. Proof. By (4.1.2), (4.1.3), (4.1.6) and (4.1.8), ˆc0,` =1 √2`s hφσ,0, B0 τφτ,`−1iσ =1 √2`s hφσ,0,(B0 σ−2vx−1)φτ,`−1iσ =1 √2`s hBσφσ,0, φτ,`−1iσ −2v √2` n−1 X j=0 (−1)n−1−js(n−1)!Γ(j+1 2+τ) j!Γ(n+1 2+τ)ˆc0,2j =v √n n−1 X j=0 (−1)n−js(n−1)!Γ(j+1 2+τ) j!Γ(n+1 2+τ)ˆc0,2j. Lemma 4.3.3. If k= 2m > 0and `= 2n > 0, then ˆck,` =pn/m ˆck−1,`−1. Proof. By (4.1.2), (4.1.4) and (4.1.7), ˆck,` =1 √2ks hB0 σφσ,k−1, φτ,`iσ=1 √2ks hφσ,k−1, Bσφτ,`iσ =1 √2ks hφσ,k−1, Bτφτ,`iσ=r2`s 2ks ˆck−1,`−1=rn mˆck−1,`−1. 91 Dunkl Harmonic Oscillator Lemma 4.3.4. If k= 2m+ 1 and `= 2n+ 1, then ˆck,` =n+1 2+σ q(m+1 2+σ)(n+1 2+τ) ˆck−1,`−1 −v qm+1 2+σ n−1 X j=0 (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ)ˆck−1,2j. Proof. By (4.1.2), (4.1.4), (4.1.6) and (4.1.7), ˆck,` =1 p2(k+ 2σ)shB0 σφσ,k−1, φτ,`iσ =1 p2(k+ 2σ)shφσ,k−1,(Bτ−2vx−1)φτ,`iσ =sn+1 2+τ m+1 2+σˆck−1,`−1 −v qm+1 2+σ n X j=0 (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ)ˆck−1,2j =n+1 2+σ q(m+1 2+σ)(n+1 2+τ) ˆck−1,`−1 −v qm+1 2+σ n−1 X j=0 (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ)ˆck−1,2j. Corollary 4.3.5. If k > `, then ˆck,` = 0. Proof. This follows by induction on `using Lemmas 4.3.1, 4.3.3 and 4.3.4. Remark 4.3.6.By Corollary 4.3.5, in Lemma 4.3.4, it is enough to consider the sum with jrunning from mto n−1. Proposition 4.3.7. If k= 2m≤`= 2n, then ˆck,` = (−1)m+nsv 2sn!Γ(m+1 2+σ) m!Γ(n+1 2+τ) Γ(n−m+v) (n−m)!Γ(v), and, if k= 2m+ 1 ≤`= 2n+ 1, then ˆck,` = (−1)m+nsv 2sn!Γ(m+3 2+σ) m!Γ(n+3 2+τ) Γ(n−m+v) (n−m)!Γ(v). 92 4.3 Scalar products of mixed generalized Hermite functions Proof. This is proved by induction on k. In turn, the case k= 0, ˆc0,` = (−1)nsv 2sΓ(1 2+σ) n!Γ(n+1 2+τ) Γ(n+v) Γ(v),(4.3.4) is proved by induction on `. If k=`= 0, (4.3.4) coincides with (4.3.3). Given `= 2n > 0, assume that the result holds for k= 0 and all `0= 2n0< `. Then, by Lemma 4.3.2, ˆc0,` =v √n n−1 X j=0 (−1)n−js(n−1)!Γ(j+1 2+τ) j!Γ(n+1 2+τ) ×(−1)jsv 2sΓ(1 2+σ) j!Γ(j+1 2+τ) Γ(j+v) Γ(v) = (−1)nsv 2s(n−1)!Γ(1 2+σ) nΓ(n+1 2+τ) v Γ(v) n−1 X j=0 Γ(j+v) j!, obtaining (4.3.4) because Γ(p+1+t) p!=t p X i=0 Γ(i+t) i!(4.3.5) for all p∈Nand t∈R r (−N), as can be easily checked by induction on p. Given k > 0, assume that the result holds for all k0< k. If kis even, the statement follows directly from Lemma 4.3.3. If kis odd, by Lemma 4.3.4, Remark 4.3.6 and (4.3.5), ˆck,` =n+1 2+σ q(m+1 2+σ)(n+1 2+τ) ×(−1)m+nsv 2sn!Γ(m+1 2+σ) m!Γ(n+1 2+τ) Γ(n−m+v) (n−m)!Γ(v) −v qm+1 2+σ n−1 X j=m (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ) ×(−1)m+jsv 2sj!Γ(m+1 2+σ) m!Γ(j+1 2+τ) Γ(j−m+v) (j−m)!Γ(v) 93 Dunkl Harmonic Oscillator Proof. By (4.1.6), Corollary 4.3.5, Proposition 4.3.7 and (4.3.5), c0 k,` =s1 2 n X j=m (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ) ×(−1)m+jsv 2sj!Γ(m+1 2+σ) m!Γ(j+1 2+τ) Γ(j−m+v) (j−m)!Γ(v) = (−1)m+ns1+v 2sn!Γ(m+1 2+σ) m!Γ(n+3 2+τ) 1 Γ(v) n−m X i=0 Γ(i+v) i! = (−1)m+ns1+v 2sn!Γ(m+1 2+σ) m!Γ(n+3 2+τ) Γ(n−m+1+v) (n−m)!Γ(1 + v). Proposition 4.4.4. If (σ, τ)satisfies (1.10.5), then there is some ω=ω(σ, τ)>0so that, for k= 2m < ` = 2n+ 1, |c0 k,`|4s1+v 2(m+ 1)−ω(n+ 1)−ω. Proof. By Proposition 4.4.3 and Lemma 4.2.12, |c0 k,`|4s1+v 2(m+ 1)σ 2−1 4(n+ 1)−τ 2−1 4(n−m+ 1)v. Then the result follows by Lemma 4.2.14, interchanging the roles of mand n, using the condition of Theorem 1.10.3-(a). 4.4.3 Case where σ6=θ=τand σ−θ6∈ −N Recall that v=σ−τin this case. Proposition 4.4.5. For k= 2mand `= 2n+ 1, c0 k,` = (−1)m+ns1+v 2sm!n! Γ(m+1 2+σ)Γ(n+3 2+τ) × min{m,n} X j=0 Γ(j+1 2+τ)Γ(m−j+v) j!(m−j)!Γ(v). 100 4.4 The sesquilinear form t0 Proof. By (4.1.6), Corollary 4.3.5, Proposition 4.3.7 and (4.3.1), c0 k,` =s1 2 min{m,n} X j=0 (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ) ×(−1)j+msv 2sm!Γ(j+1 2+τ) j!Γ(m+1 2+σ) Γ(m−j+v) (m−j)!Γ(v) = (−1)m+ns1+v 2sm!n! Γ(m+1 2+σ)Γ(n+3 2+τ) × min{m,n} X j=0 Γ(j+1 2+τ)Γ(m−j+v) j!(m−j)!Γ(v). Proposition 4.4.6. If (σ, τ)satisfies (1.10.6), then there is some ω=ω(σ, τ)>0so that, for k= 2mand `= 2n+ 1, |c0 k,`|4s1+v 2(m+ 1)−ω(n+ 1)−ω. Proof. By Proposition 4.4.5 and Lemma 4.2.12, |c0 k,`|4s1+v 2(m+ 1)1 4−σ 2(n+ 1)−1 4−τ 2 min{m,n} X j=0 (m−j+ 1)v−1(j+ 1)τ−1 2. Then the result follows by Corollary 4.6.4, proved in Section 4.6, since (σ, τ)satisfies (1.10.6). 4.4.4 Case where σ6=θ=τ+ 1 and σ−τ−16∈ −N Note that v=σ−τ−2in this case. Moreover c0 k,` =hφσ,k, x−1φτ,`iτ+1 =hxφσ,k, φτ,`iτ=hφτ,`, xφσ,kiτ(4.4.1) for k= 2mand `= 2n+ 1 (Remark 1.10.4-(ii)). Proposition 4.4.7. Let k= 2mand `= 2n+1. If k+1 < ` (m < n), then c0 k,` = 0. If k+ 1 ≥`(m≥n), then c0 k,` = (−1)m+nsv+1 2sm!Γ(n+3 2+τ) n!Γ(m+1 2+σ) Γ(m−n+v+ 1) (m−n)!Γ(v+ 1) . 101 Dunkl Harmonic Oscillator Proof. By (4.1.5) and (4.4.1), c0 k,` =sm+1 2+σ sˆcτ,σ,τ,`,k+1 +rm sˆcτ,σ,τ,`,k−1.(4.4.2) So c0 k,` = 0 if k+ 1 < ` by Corollary 4.3.5. When k+ 1 = `(m=n), by (4.4.2) and Proposition 4.3.7, c0 k,` =sm+1 2+σ ssv+2 2sΓ(n+3 2+τ) Γ(m+3 2+σ)=sv+1 2sΓ(n+3 2+τ) Γ(m+1 2+σ). When k−1≥`(m>n), by (4.4.2) and Proposition 4.3.7, c0 k,` =sm+1 2+σ s(−1)m+nsv+2 2sm!Γ(n+3 2+τ) n!Γ(m+3 2+σ) Γ(m−n+v+ 2) (m−n)!Γ(v+ 2) +rm s(−1)m+n−1sv+2 2s(m−1)!Γ(n+3 2+τ) n!Γ(m+1 2+σ) Γ(m−n+v+ 1) (m−1−n)!Γ(v+ 2) = (−1)m+nsv+1 2sm!Γ(n+3 2+τ) n!Γ(m+1 2+σ) Γ(m−n+v+ 1) (m−1−n)!Γ(v+ 2) ×m−n+v+ 1 m−n−1 = (−1)m+nsv+1 2sm!Γ(n+3 2+τ) n!Γ(m+1 2+σ) Γ(m−n+v+ 1) (m−n)!Γ(v+ 1) . Proposition 4.4.8. If (σ, τ)satisfies (1.10.7), then there is some ω=ω(σ, τ)>0so that, for k= 2mand `= 2n+ 1, |c0 k,`|4sv+1 2(m+ 1)−ω(n+ 1)−ω. Proof. By Proposition 4.4.7, we can assume that k+ 1 ≥`(m≥n), and, in this case, using also Lemma 4.2.12, we get |c0 k,`|4sv+1 2(m+ 1)1 4−σ 2(n+ 1)1 4+τ 2(m−n+ 1)v. Then the result follows using Lemma 4.2.14. 102 4.4 The sesquilinear form t0 4.4.5 Case where σ6=θ6=τand σ−θ, τ −θ6∈ −N Proposition 4.4.9. For k= 2mand `= 2n+ 1, c0 k,` = (−1)m+ns1+v 2sm!n! Γ(m+1 2+σ)Γ(n+3 2+τ) × min{m,n} X p=0 Γ(p+1 2+θ)Γ(m−p+σ−θ)Γ(n−p+1+τ−θ) p!(m−p)!(n−p)!Γ(σ−θ)Γ(1 + τ−θ). Proof. By (4.1.6) and Proposition 4.3.12, c0 k,` =s1 2 n X j=0 (−1)n−jsn!Γ(j+1 2+τ) j!Γ(n+3 2+τ) ×(−1)m+jsv 2sm!j! Γ(m+1 2+σ)Γ(j+1 2+τ) × min{m,j} X p=0 Γ(p+1 2+θ)Γ(m−p+σ−θ)Γ(j−p+τ−θ) p!(m−p)!(j−p)!Γ(σ−θ)Γ(τ−θ) = (−1)m+ns1+v 2sm!n! Γ(m+1 2+σ)Γ(n+3 2+τ) × n X j=0 min{m,j} X p=0 Γ(p+1 2+θ)Γ(m−p+σ−θ)Γ(j−p+τ−θ) p!(m−p)!(j−p)!Γ(σ−θ)Γ(τ−θ). But, by (4.3.5), n X j=0 min{m,j} X p=0 Γ(m−p+σ−θ)Γ(j−p+τ−θ) (m−p)!(j−p)!Γ(σ−θ)Γ(τ−θ) = min{m,n} X p=0 n X j=p Γ(m−p+σ−θ)Γ(j−p+τ−θ) (m−p)!(j−p)!Γ(σ−θ)Γ(τ−θ) = min{m,n} X p=0 n−p X i=0 Γ(m−p+σ−θ)Γ(i+τ−θ) (m−p)!i!Γ(σ−θ)Γ(τ−θ) = min{m,n} X p=0 Γ(m−p+σ−θ)Γ(n−p+1+τ−θ) (m−p)!(n−p)!Γ(σ−θ)Γ(1 + τ−θ). 103 Dunkl Harmonic Oscillator Proposition 4.4.10. If (σ, τ, θ)satisfies (1.10.8), then there exists ω=ω(σ, τ, θ)>0 so that, for k= 2mand `= 2n+ 1, |c0 k,`|4s1+v 2(m+ 1)−ω(n+ 1)−ω. Proof. Let prun from 0to min{m, n}. By Proposition 4.4.9 and Lemma 4.2.12, |c0 k,`|4s1+v 2(m+ 1)1 4−σ 2(n+ 1)−1 4−τ 2 ×X p (m−p+ 1)σ−θ−1(n−p+ 1)τ−θ(p+ 1)θ−1 2. Then the result follows by Corollary 4.6.2, proved in Section 4.6, since (σ, τ, θ)satisfies (1.10.8). 4.4.6 Proof of Theorem 1.10.3 Assume the conditions of Theorem 1.10.3. Let jσ,τ be the positive definite symmetric sesquilinear form in L2 σ,τ , with domain S, defined by jσ,τ (φ, ψ) = hJσ,τ φ, ψiσ,τ . Proposition 4.4.11. For any  > 0, there is some E=E(, σ, τ, θ)>0such that, for all φ∈ S, |t0(φ)| ≤ sv−1 2jσ,τ (φ) + Es1+v 2kφk2 σ,τ . Proof. This follows from Propositions 4.4.2, 4.4.4, 4.4.6, 4.4.8 and 4.4.10 using the arguments of the proof of Proposition 4.2.17. Proof of Theorem 1.10.3. This is analogous to the proof of Theorem 1.10.1. Thus some details and the bibliographic references are omitted. Let tσ,τ be the positive definite symmetric sesquilinear form in L2 σ,τ , with domain D(tσ,τ ) = S, defined by tσon Sev and tτon Sodd, and vanishing on Sev ×Sodd (and therefore also on Sodd ×Sev). Let sbe the symmetric sesquilinear form in L2 σ,τ , with D(s) = S, defined by s(φ, ψ) = t0(φ, ψ)+t0(ψ, φ). Then the symmetric sesquilinear form v=jσ,τ +ξtσ,τ +ηsin L2 σ,τ , with D(v) = S, is given by the right hand side of (1.10.9). Using Propositions 4.2.17 and 4.4.11, for any  > 0, there are some C=C(, σ, τ, u)>0and E=E(, σ, τ, θ)>0such that, for all φ∈ S, |(ξtσ,τ +ηs)(φ)| ≤ξsu−1+ 2|η|sv−1 2jσ,τ (φ) + ξCsu+ 2|η|Es1+v 2kφk2 σ,τ .(4.4.3) Then, taking so that (ξsu−1+ 2|η|sv−1 2)<1, since jσ,τ is closable and positive definite, it follows that vis sectorial and closable, and D(¯ v) = D(jσ,τ ); in particular, v 104 4.4 The sesquilinear form t0 is bounded from below because it is also symmetric. So ¯ vis induced by a self-adjoint operator Vin L2 σ,τ with D(V1/2) = D(¯ v). Thus Sis a core of ¯ vand V1/2. For all φ∈ S, v(φ)≥jσ,τ (φ) + ξtσ,τ (φ)−|η|s(φ)≥jσ,τ (φ) + ξtσ,τ (φ)−2|η||t0(φ)|.(4.4.4) Since Sis a core of ¯ vand jσ,τ , using Propositions 4.2.18 and 4.4.11 like in the proof of Theorem 1.10.1, it follows from (4.4.4) that Vhas a discrete spectrum, which consists of two groups of eigenvalues, λ0≤λ2≤ ··· and λ1≤λ3≤ ···, repeated according to their multiplicity, satisfying (1.10.10). On the other hand, by (4.4.3), for all φ∈ S, v(φ)≤1 + ξsu−1+ 2|η|sv−1 2jσ,τ (φ) + ξCsu+ 2|η|Es1+v 2kφk2 σ,τ ,(4.4.5) obtaining (1.10.11) because Sis a core of ¯ vand jσ,τ . With the notation of (iii), let ˜ tσ(respectively, ˜ tτ) be the symmetric sesquilinear form in L2 σ(respectively, L2 τ), with D(˜ tσ) = S(respectively, D(˜ tτ) = S), defined like tσ(respectively, tτ), using ˜u(respectively, v−˜u+1) instead of u. Let˜ tσ,τ be the positive definite symmetric sesquilinear form in L2 σ,τ , with D(˜ tσ,τ ) = S, defined by ˜ tσon Sev and ˜ tτon Sodd, and vanishing on Sev ×Sodd. By the Schwartz inequality, we deduce 2|t0(φ)|= 2φev|x|−˜u+σ−θ, x−1φodd|x|˜u−σ+θθ ≤2kφev|x|−˜u+σ−θkθ·kφodd|x|˜u−σ+θ−1kθ = 2kφev|x|−˜ukσ·kφodd|x|˜u−v−1kτ ≤ kφev|x|−˜uk2 σ+kφodd|x|˜u−v−1k2 τ=˜ tσ,τ (φ).(4.4.6) Hence (1.10.13) follows like in the proof of Theorem 1.10.1, using Propositions 4.2.17 and 4.2.18. If u=v+1 2, then we can take ˜u=u=v−˜u+ 1 in (iii), yielding v(φ)≥jσ,τ (φ) + (ξ−|η|)tσ,τ (φ) by (4.4.4) and (4.4.6). Thus (1.10.14) follows if |η| ≤ ξ, like in the proof of Theorem 1.10.1, using Proposition 4.2.18. If we add the term ξ0hφev, ψeviσ+ξ00hφodd, ψoddiτto the right hand side of (1.10.9), for some ξ0, ξ00 ∈R, then the same argument can be used by adding the term ξ0kφevk2 σ +ξ00kφoddk2 τto jσ,τ , obtaining (v). 105 Dunkl Harmonic Oscillator 4.5 A preliminary estimate 4.5.1 Statement The standard coordinates of R5are denoted by (α, β, γ, δ, κ). Consider the partition of Rinto the following intervals: I1= (−∞,−1],I2= (−1,−1 2],I3= (−1 2,−1 3], I4= (−1 3,0) and I5= [0,∞). Let Qijk =Ii×Ij×Ik, and consider the following subsets of R5: S515:This is the subset of R2×Q515 defined by α+γ, α +β+γ+κ<0.(4.5.1) S522:This is the subset of R2×Q522 defined by α+γ, α +β+γ < 0.(4.5.2) S252:This is the subset of R2×Q252 defined by 0≤γ+δ⇒α+γ, α +β+γ+δ+κ+ 1 <0,(4.5.3) γ+δ < 0⇒         α+γ+δ, α +β+κ+ 1 <0,or α+γ+1 2, α +β+δ < 0,or α+γ+1 3, α +β+δ+1 3<0,or α+γ, α +β+δ+κ+ 1 <0. (4.5.4) S155:This is the subset of R2×Q155 defined by (4.5.3) and γ+δ < 0⇒             α+γ+δ, α +β+κ+ 1 <0,or α+γ+ 1, α +β+δ+κ<0,or α+γ+1 2, α +β+δ+κ+1 2<0,or α+γ+1 3, α +β+δ+κ+2 3<0,or α+γ, α +β+δ+κ+ 1 <0. (4.5.5) S212:This is the subset of R2×Q212 defined by γ≤κ⇒α+γ, α +β+κ<0,(4.5.6) κ≤γ⇒α+γ, α +β+γ < 0.(4.5.7) 106 4.5 A preliminary estimate Let ˇ S=S515 ∪S522 ∪S252 ∪S155 ∪S212. On the other hand, consider the linear isomorphism of R5defined by (α, β, γ, δ, κ)7→ (β, α, δ, γ, κ).(4.5.8) This is the reflection with respect to the linear subspace defined by α=βand γ=δ. The image of any subset X⊂R5by the mapping (4.5.8) is denoted by X0, and let Xconv be the convex hull of X. Thus X0 conv := (X0)conv = (Xconv)0. Lemma 4.5.1. If (α, β, γ, δ, κ)∈ˇ Sconv ∩ˇ S0 conv, then there is some ω > 0such that, for all m, n ∈N, (m+ 1)α(n+ 1)β min{m,n} X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ 4(m+ 1)−ω(n+ 1)−ω.(4.5.9) 4.5.2 Proof of Lemma 4.5.1 Since the roles of mand nin Lemma 4.5.1 are interchanged by the mapping (4.5.8), we can assume that m≥n. Then Lemma 4.2.14 gives (4.5.9) once n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ is appropriately estimated. Estimates of this expression are achieved with several strategies explained in Sections 4.5.2–4.5.2, giving rise to several lists of conditions that guarantee (4.5.9) when m≥n. Then, for the chosen subindices ijk equal to 515,522,252,155,212, every Sijk is defined by the most general of those conditions on R2×Qijk. This will show that (4.5.9) holds for m≥nand (α, β, γ, δ, κ)∈ˇ S. In Section 4.5.2, it will be shown that this property can be extended to the convex hull ˇ Sconv, completing the proof of Lemma 4.5.1. First list of conditions For all  > 0, n X p=0 (p+ 1)κ= n+1 X q=1 qκ≤(Rn+2 1xκdx if κ≥0 1 + Rn+1 1xκdx if κ<0 4     (n+ 1)κ+1 if κ>−1 1 + ln(n+ 1) if κ=−1 1if κ<−1 4     (n+ 1)κ+1 if κ>−1 (n+ 1)if κ=−1 1if κ<−1. (4.5.10) 107 Dunkl Harmonic Oscillator On the other hand, we claim that (m−p+1)γ(n−p+1)δ4                      (m+ 1)γ(n+ 1)δif δ≥ −γ, 0 (m−n+ 1)γ+δand (m−n+ 1)γ(n+ 1)δ)if 0≤δ < −γ (m−n+ 1)γif δ≤ −γ, 0 (m−n+ 1)γor (m+ 1)γ(n+ 1)δ)if −γ < δ < 0 (4.5.11) for all p= 0, . . . , n. Combining (4.5.10) and (4.5.11), it follows that (m+ 1)α(n+ 1)β n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ4A1,(4.5.12) where A1=A1(m, n, α, β, γ, δ, κ)can be taken to be equal to: (m+ 1)α+γ(n+ 1)β+δ+κ+1 if −γ, 0≤δ,−1<κ, (m+ 1)α(n+ 1)β+κ+1(m−n+ 1)γ+δand (m+ 1)α(n+ 1)β+δ+κ+1(m−n+ 1)γ)if 0≤δ < −γ,−1<κ. In the definition of A1, the other cases of γ, δ, κare omitted because they will not be used. We will continue omitting such cases, often without further comment. Resulting tautologies will be also removed without further comment. By (4.5.12), applying Lemma 4.2.14 to the above list, we get the first list of conditions that guarantee (4.5.9) when m≥n: −γ, 0≤δ, −1<κ⇒α+γ, α +β+γ+δ+κ+ 1 <0,(4.5.13) 0≤δ < −γ, −1<κ⇒α+γ+δ, α +β+κ+ 1 <0,or α+γ, α +β+δ+κ+ 1 <0.(4.5.14) To prove (4.5.11), it is enough to study the maximum of the C∞function f(x)=(m−x+ 1)γ(n−x+ 1)δ on [0, n](the natural domain of fcontains (−∞, n + 1)). We have f0(x)=(m−x+ 1)γ−1(n−x+ 1)δ−1h(x), where h(x) = (γ+δ)x−γ(n+ 1) −δ(m+ 1). 108 4.5 A preliminary estimate Observe that this expression is valid even when γ= 0 or δ= 0. Since f0and hhave the same zero set on [0, n], and they have the same sign on the complement of the zero set in [0, n], it is enough to analyze hto know where freaches its maximum on [0, n]. We consider several cases. Case where γ+δ= 0.Then h≡γ(m−n). If m > n and γ6= 0, then h6= 0 and sign h= sign γ. If m=nor γ= 0, then h≡0. Hence: max 0≤x≤nf(x) = (f(n) = (m−n+ 1)γif γ=−δ≥0 f(0) = (m+ 1)γ(n+ 1)δif γ=−δ≤0.(4.5.15) Case where γ+δ6= 0.Then hvanishes just at the point x0:= γ(n+ 1) + δ(m+ 1) γ+δ. Case where γ+δ < 0.We have h > 0on (−∞, x0)and h < 0on (x0,∞), yielding max 0≤x≤nf(x) =      f(0) = (m+ 1)γ(n+ 1)δif x0≤0 f(x0)if 0≤x0≤n f(n) = (m−n+ 1)γif x0≥n. (4.5.16) Case where γ+δ < 0and δ≤0.Then x0≥n+ 1, and therefore, by (4.5.16), γ+δ < 0, δ ≤0⇒max 0≤x≤nf(x)=(m−n+ 1)γ.(4.5.17) Case where γ+δ < 0and δ > 0; i.e., 0< δ < −γ.We may have x0≤0, 0≤x0≤nor n≤x0. Moreover f(x0) = (−γ)γδδ (−γ−δ)γ+δ(m−n)γ+δ4(m−n+ 1)γ+δ. Therefore 0< δ < −γ⇒max 0≤x≤nf(x)4(m−n+ 1)γ+δ(4.5.18) by (4.5.16) and since (m−n+ 1)γ,(m+ 1)γ(n+ 1)δ<(m−n+ 1)γ+δ, which follows using that γ < γ +δand n≥m 2⇒m−n+ 1 n+ 1 ≤1⇒m−n+ 1 m+ 1 <m−n+ 1 n+ 1 ≤m−n+ 1 n+ 1 −δ γ, n < m 2⇒m−n+ 1 n+ 1 >1⇒m−n+ 1 m+ 1 ≤1<m−n+ 1 n+ 1 −δ γ, 109 Dunkl Harmonic Oscillator Fourth list of conditions We have (p+ 1)κ≤((n+ 1)κif κ≥0 1if κ≤0(4.5.46) for p= 0, . . . , n. Moreover, by (4.5.10), (4.5.38) and (4.5.39), for all  > 0, n X p=0 (n−p+ 1)2δ= n+1 X q=1 q2δ4     (n+ 1)2δ+1 if δ > −1 2 (n+ 1)if δ=−1 2 1if δ < −1 2, (4.5.47) n X p=0 (m−p+ 1)2γ4     (m+ 1)2γ+1 if γ > −1 2 (m+ 1)if γ=−1 2 (m−n+ 1)2γ+1 if γ < −1 2, (4.5.48) n X p=0 (m−p+ 1)2γ≤((m+ 1)2γ(n+ 1) if γ≥0 (m−n+ 1)2γ(n+ 1) if γ < 0.(4.5.49) The estimate (4.5.49) is better than (4.5.48) when γ≥0, and it may be better or worse than (4.5.48) when γ < 0, depending on the values of mand n. By the Cauchy-Schwartz inequality, n X p=0 (m−p+ 1)γ(n−p+ 1)δ≤ n X p=0 (m−p+ 1)2γ!1 2 n X p=0 (n−p+ 1)2δ!1 2 . Therefore, by (4.5.46)–(4.5.49), (m+ 1)α(n+ 1)β n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ4A4,(4.5.50) where A4=A4(m, n, α, β, γ, δ, κ)can be taken to be equal to: (m+ 1)α(n+ 1)β+δ+κ+1 2(m−n+ 1)γ+1 2and (m+ 1)α(n+ 1)β+δ+κ+1(m−n+ 1)γ)if γ < −1 2< δ,0≤κ. By (4.5.50), applying Lemma 4.2.14 to the above list, we get the fourth list of conditions that guarantee (4.5.9) when m≥n: γ < −1 2< δ, 0≤κ⇒(α+γ+1 2, α +β+δ+κ+1 2<0,or α+γ, α +β+δ+κ+ 1 <0.(4.5.51) 116 4.5 A preliminary estimate Fifth list of conditions This is analogous to the estimates of Section 4.5.2, interchanging the roles of δand κ. We have (n−p+ 1)δ≤((n+ 1)δif δ≥0 1if δ≤0(4.5.52) for p= 0, . . . , n. Moreover, by (4.5.10), for all  > 0, n X p=0 (p+ 1)2κ4     (n+ 1)2κ+1 if κ>−1 2 (n+ 1)if κ=−1 2 1if κ<−1 2. (4.5.53) Applying the Cauchy-Schwartz inequality, we get n X p=0 (m−p+ 1)γ(p+ 1)κ≤ n X p=0 (m−p+ 1)2γ!1 2 n X p=0 (p+ 1)2κ!1 2 . Therefore, by (4.5.52), (4.5.53), (4.5.48) and (4.5.49), for all  > 0, (m+ 1)α(n+ 1)β n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ4A5,(4.5.54) where A5=A5(m, n, α, β, γ, δ, κ, )can be taken to be equal to: (m+ 1)α+(n+ 1)β+δ+and (m+ 1)α(n+ 1)β+δ+1 2+(m−n+ 1)γ)if γ=κ=−1 2,0≤δ, (m+ 1)α+(n+ 1)β+δand (m+ 1)α(n+ 1)β+δ+1 2(m−n+ 1)γ)if κ< γ =−1 2,0≤δ, (m+ 1)α(n+ 1)β+δ+(m−n+ 1)γ+1 2and (m+ 1)α(n+ 1)β+δ+1 2+(m−n+ 1)γ)if γ < κ=−1 2,0≤δ, (m+ 1)α(n+ 1)β+δ(m−n+ 1)γ+1 2and (m+ 1)α(n+ 1)β+δ+1 2(m−n+ 1)γ)if γ, κ<−1 2,0≤δ. By (4.5.54), applying Lemma 4.2.14 to the above list, we get the fifth list of conditions that guarantee (4.5.9) when m≥n: γ, κ≤ −1 2,0≤δ⇒(α+γ+1 2, α +β+δ < 0,or α+γ, α +β+δ+1 2<0.(4.5.55) 117 Dunkl Harmonic Oscillator Sixth list of conditions We have (m−p+ 1)γ≤((m+ 1)γif γ≥0 (m−n+ 1)γif γ≤0(4.5.56) for p= 0, . . . , n. Moreover, by (4.5.10), for all  > 0, n X p=0 (n−p+ 1)2δ= n+1 X q=1 q2δ4     (n+ 1)2δ+1 if δ > −1 2 (n+ 1)if δ=−1 2 1if δ < −1 2. (4.5.57) Applying the Cauchy-Schwartz inequality, we get n X p=0 (n−p+ 1)δ(p+ 1)κ≤ n X p=0 (n−p+ 1)2δ!1 2 n X p=0 (p+ 1)2κ!1 2 . Therefore, by (4.5.56), (4.5.57) and (4.5.53), for all  > 0, (m+ 1)α(n+ 1)β n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ4A6,(4.5.58) where A6=A6(m, n, α, β, γ, δ, κ, )can be taken to be equal to: (m+ 1)α+γ(n+ 1)β+if (κ≤δ=−1 2,0≤γ, or δ≤κ=−1 2,0≤γ, (m+ 1)α+γ(n+ 1)βif δ, κ<−1 2,0≤γ. By (4.5.58), applying Lemma 4.2.14 to the above list, we get the sixth list of conditions that guarantee (4.5.9) when m≥n: δ, κ≤ −1 2,0≤γ⇒α+γ, α +β+γ < 0.(4.5.59) 118 4.5 A preliminary estimate Seventh list of conditions By (4.5.10), (4.5.38) and (4.5.39), for all  > 0, n X p=0 (n−p+ 1)3δ= n+1 X q=1 q3δ4     (n+ 1)3δ+1 if δ > −1 3 (n+ 1)if δ=−1 3 1if δ < −1 3, (4.5.60) n X p=0 (p+ 1)3κ4     (n+ 1)3κ+1 if κ>−1 3 (n+ 1)if κ=−1 3 1if κ<−1 3, (4.5.61) n X p=0 (m−p+ 1)3γ4     (m+ 1)3γ+1 if γ > −1 3 (m+ 1)if γ=−1 3 (m−n+ 1)3γ+1 if γ < −1 3, (4.5.62) n X p=0 (m−p+ 1)3γ≤((m+ 1)3γ(n+ 1) if γ≥0 (m−n+ 1)3γ(n+ 1) if γ < 0.(4.5.63) Note that (4.5.63) is better than (4.5.62) for γ≥0, and it is an alternative estimate for γ < 0. Applying the generalized Hölder inequality [21], we get n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ ≤ n X p=0 (m−p+ 1)3γ!1 3 n X p=0 (n−p+ 1)3δ!1 3 n X p=0 (p+ 1)3κ!1 3 . Therefore, by (4.5.60)–(4.5.63), (m+ 1)α(n+ 1)β n X p=0 (m−p+ 1)γ(n−p+ 1)δ(p+ 1)κ4A7,(4.5.64) where A7=A7(m, n, α, β, γ, δ, κ)can be taken to be equal to: (m+ 1)α(n+ 1)β+δ+κ+2 3(m−n+ 1)γ+1 3and (m+ 1)α(n+ 1)β+δ+κ+1(m−n+ 1)γ)if γ < −1 3< δ, κ, (m+ 1)α(n+ 1)β+δ+1 3(m−n+ 1)γ+1 3and (m+ 1)α(n+ 1)β+δ+2 3(m−n+ 1)γ)if γ, κ<−1 3< δ. 119 Dunkl Harmonic Oscillator By (4.5.64), applying Lemma 4.2.14 to the above list, we get the seventh list of conditions that guarantee (4.5.9) when m≥n: γ < −1 3< δ, κ⇒(α+γ+1 3, α +β+δ+κ+2 3<0,or α+γ, α +β+δ+κ+ 1 <0,(4.5.65) γ, κ<−1 3< δ ⇒(α+γ+1 3, α +β+δ+1 3<0,or α+γ, α +β+δ+2 3<0.(4.5.66) Obtaining the sets Sijk from the lists of conditions The left hand side of the conditions from the lists of Sections 4.5.2–4.5.2 involve only (γ, δ, κ). Now, we indicate which of them define sets covering every Qijk, for the chosen subindices ijk equal to 515,522,252,155,212. Those conditions will produce the definition of Sijk ⊂R2×Qijk so that (4.5.9) holds for m≥n. The set Q515 is given by the left hand side of (4.5.25), whose right hand side is (4.5.1), defining S515. Any (γ, δ, κ)∈Q522 satisfies the left hand side of (4.5.59), whose right hand side is (4.5.2), defining S522. Any (γ, δ, κ)∈Q252 satisfies the left hand side of (4.5.13) or (4.5.14), and satisfies the left hand side of (4.5.55) and (4.5.66). So, when m≥n, the estimate (4.5.9) is guaranteed for any (α, β, γ, δ, κ)∈R2×Q252 satisfying both (4.5.13) and (4.5.14), or any of (4.5.55) or (4.5.66). On R2×Q252, these conditions mean that (α, β, γ, δ, κ) satisfies (4.5.3) and (4.5.4), defining S252. Any (γ, δ, κ)∈Q155 satisfies the left hand side of (4.5.13) or (4.5.14), and satisfies the left hand side of (4.5.51), (4.5.42) and (4.5.65). So, when m≥n, the estimate (4.5.9) is guaranteed for any (α, β, γ, δ, κ)∈R2×Q155 satisfying both (4.5.13) and (4.5.14), or any of (4.5.51), (4.5.42) or (4.5.65). On R2×Q155, these conditions mean that (α, β, γ, δ, κ)satisfies (4.5.3) and (4.5.5), defining S155. Any (γ, δ, κ)∈Q212 satisfies the left hand side of (4.5.26) or (4.5.27). So, when m≥n, the estimate (4.5.9) is guaranteed for any (α, β, γ, δ, κ)∈R2×Q212 satisfying both (4.5.26) and (4.5.27). On R2×Q212, these conditions become (4.5.6) and (4.5.7), defining S212. The preliminary estimate is satisfied on a convex set Let us show the convexity of the set of elements x= (α, β, γ, δ, κ)∈R5satisfying (4.5.9) for m≥n, with ω=ω(x)>0. For i= 0,1, suppose that xi= (αi, βi, γi, δi,κi)satisfies (4.5.9) for m≥nwith ωi=ω(xi)>0. Recall that the case where m≤nfollows from the case where m≥nby using the mapping (4.5.8). 120 4.6 The main estimates For 0<t<1, let xt= (αt, βt, γt, δt,κt) = (1 −t)x0+tx1, ωt= (1 −t)ω0+tω1>0. By Hölder inequality, for all m≥n, n X p=0 (m−p+ 1)γt(n−p+ 1)δt(p+ 1)κt = n X p=0 (m−p+ 1)γ0(n−p+ 1)δ0(p+ 1)κ01−t × n X p=0 ×(m−p+ 1)γ1(n−p+ 1)δ1(p+ 1)κ1t ≤n X p=0 (m−p+ 1)γ0(n−p+ 1)δ0(p+ 1)κ01−t ×n X p=0 (m−p+ 1)γ1(n−p+ 1)δ1(p+ 1)κ1t. So (m+ 1)αt(n+ 1)βt n X p=0 (m−p+ 1)γt(n−p+ 1)δt(p+ 1)κt 4(m+ 1)−ω0(n+ 1)−ω01−t(m+ 1)−ω1(n+ 1)−ω1t = (m+ 1)−ωt(n+ 1)−ωt. Thus xtsatisfies (4.5.9) for m≥nwith ωt. This completes the proof of Lemma 4.5.1. 4.6 The main estimates Here, we show the estimates used in the proofs of Propositions 4.4.6 and 4.4.10. We continue with the notation of Section 4.5. Moreover let (σ, τ, θ)denote the standard coordinates of R3. Consider the affine injection R3→R5and the affine isomorphism of R3defined by (σ, τ, θ)7→ 1 4−σ 2,−1 4−τ 2, σ −θ−1, τ −θ, θ −1 2,(4.6.1) (σ, τ, θ)7→ (τ+ 1, σ −1, θ).(4.6.2) 121 Dunkl Harmonic Oscillator The mapping (4.6.2) is the reflection with respect to the plane defined by σ=τ+ 1, and it corresponds to the mapping (4.5.8) via (4.6.1). Let ˇ K,ˇ K0⊂R3be the inverse images of ˇ S,ˇ S0by (4.6.1), and let ˇ Kconv,ˇ K0 conv be their convex hulls. So ˇ Kconv,ˇ K0 conv are contained in the inverse images of ˇ Sconv,ˇ S0 conv by (4.6.1), and ˇ K0,ˇ K0 conv are the images of ˇ K,ˇ Kconv by (4.6.2). Thus ˇ Kconv ∩ˇ K0 conv is symmetric with respect to the plane σ=τ+ 1. We will show the following. Lemma 4.6.1. ˇ Kconv∩ˇ K0 conv consists of the elements (σ, τ, θ)∈R3that satisfy (1.10.8). The following is a direct consequence of Lemmas 4.5.1 and 4.6.1. Corollary 4.6.2. If (σ, τ, θ)∈R3satisfies (1.10.8), then there is some ω > 0such that (4.5.9) holds with the image (α, β, γ, δ, κ)of (σ, τ, θ)by (4.6.1). Let ˇ J⊂R2be the inverse image of ˇ Kconv ∩ˇ K0 conv by the affine injection R2→R3, (σ, τ)7→ (σ, τ, τ). The following is a direct consequence of Lemma 4.6.1. Lemma 4.6.3. ˇ Jconsists of the elements (σ, τ)∈R2that satisfy (1.10.6). Lemma 4.6.3 and Corollary 4.6.2 have the following direct consequence. Corollary 4.6.4. If (σ, τ)∈R2satisfies (1.10.6), then there is some ω > 0such that (4.5.9) holds with the image (α, β, γ, δ, κ)of (σ, τ, τ)by (4.6.1). Let us prove Lemma 4.6.1. For the subindices ijk equal to 515,522,252,155 and 212, let Kijk and Rijk be the inverse images of Sijk and R2×Qijk by the mapping (4.6.1). Thus ˇ K=K515 ∪K522 ∪K252 ∪K155 ∪K212. Moreover, for every θ∈R, let I1 1(θ) = (−∞, θ], I2 1(θ) = (−∞, θ −1], I3 1=−∞,−1 2, I1 2(θ) = θ, θ +1 2, I2 2(θ) = θ−1, θ −1 2, I3 2=−1 2,0, I1 3(θ) = θ+1 2, θ +2 3, I2 3(θ) = θ−1 2, θ −1 3, I3 3=0,1 6, I1 4(θ) = θ+2 3, θ + 1, I2 4(θ) = θ−1 3, θ, I3 4=1 6,1 2, I1 5(θ)=[θ+ 1,∞), I2 5(θ)=[θ, ∞), I3 5=1 2,∞. It can be directly checked that Rijk =(σ, τ, θ)∈R3|(σ, τ)∈I1 i(θ)×I2 j(θ), θ ∈I3 k. Simple computations show that, via (4.6.1), the conditions defining the sets Sijk (Section 4.5.1) become the following descriptions of the sets Kijk (Figure 4.6.1-(a)): 122 4.6 The main estimates K515:This is the subset of R515 defined by σ 2−3 4< θ, σ −τ−3<0. K522:This is the subset of R522 defined by σ 2−3 4,σ−τ 2−1< θ. K252:This is the subset of R252 defined by σ+τ−1 2< θ, (4.6.3)      σ 2−1 4,τ−σ 2< θ, or σ 2−5 12,τ−σ 2+1 3< θ, or σ 2−3 4< θ, τ −σ+ 1 <0. K155:This is the subset of R155 defined by (4.6.3) and          σ 2+1 4< θ, τ −σ−1<0,or σ 2−1 4< θ, τ −σ < 0,or σ 2−5 12 < θ, τ −σ+1 3<0,or σ 2−3 4< θ, τ −σ+ 1 <0. K212:This is the subset of R212 defined by σ 2−1 4≤θ⇒θ < σ+τ+1 2, θ≤σ 2−1 4⇒σ 2−3 4,σ−τ 2−1< θ. With tedious computations assisted by graphics produced with Mathematica, it follows that ˇ Kconv is the open subset of R3defined by (Figure 4.6.1-(b)) σ−τ 2−1,τ−σ 2,σ+τ−1 4,σ+3τ−2 14 ,σ+τ−1 2< θ < σ+τ+1 2, τ−1< σ < τ + 3.)(4.6.4) This is a “semi-infinite bar” with 4 lateral faces, and 4 faces at the “bounded end”. Applying the affine transformation (4.6.2) to this description, we get that ˇ K0 conv consists of the triples (σ, τ, θ)∈R3satisfying the following conditions: σ−τ 2−1,τ−σ 2,σ+τ−1 4,3σ+τ−4 14 ,σ+τ−1 2< θ < σ+τ+1 2, τ−1< σ < τ + 3.(4.6.5) Combining (4.6.4) and (4.6.5), it follows that ˇ Kconv ∩ˇ K0 conv is given by (1.10.8) (Figure 1.10.2), completing the proof of Lemma 4.6.1. 123 Dunkl Harmonic Oscillator (a) ˇ K(b) ˇ Kconv Figure 4.6.1: The sets ˇ Kand ˇ Kconv. Remark 4.6.5.In Sections 4.5.2–4.5.2, we have only written the cases that provide the most general conditions to define S515,S522,S252,S155,S212. But indeed much more hidden work was needed to produce this shorter proof: 1. We have computed all cases in Sections 4.5.2–4.5.2, giving rise to seven long lists of conditions that guarantee (4.5.9) when m≥n. 2. We have studied which of those conditions are the most general ones on every subset R2×Qijk, for all ijk = 1, . . . , 5. This produces 125 sets Sijk, whose inverse images by (4.6.1) give 125 sets Kijk. The corresponding unions are denoted by Sand K, and their convex hulls by Sconv and Kconv. 3. We got that 41 sets Kijk are empty, including the 25 sets of the form Kij1, and the remaining 84 sets Kijk fit together forming a “semi-infinite bar” (Figures 4.6.2 and 4.6.3). 4. With tedious computations, we have shown that Kconv is given by (4.6.4). 5. We have chosen the most simple family, K515,K522,K252,K155,K212, defining the same convex hull (ˇ Kconv =Kconv). 6. Finally, we have made some attempts to improve the estimates of Section 4.5.2 by using more general versions of the Hölder inequality [21]. Some better estimates were obtained in this way, but they produce the same set Kconv after taking the convex hull. 124 4.6 The main estimates Remark 4.6.6.The set Sconv may have a simple expression, like Kconv, but its computation became too involved. This is the reason we have used Kconv, obtaining the conditions of Theorem 1.10.3, which are general enough for the applications contained in Chapter 2. But, of course, the inverse image of Sconv by (4.6.1) is possibly larger than Kconv. Therefore a simple expression of Sconv would possibly give a better version of Theorem 1.10.3. Even a simple expression of ˇ Sconv would possibly give a better version of Theorem 1.10.3. (a) Si,j Kij2(b) Si,j Kij3 (c) Si,j Kij4(d) Si,j Kij5 Figure 4.6.2: Construction of K. 125 Appendix A Preliminaries on Stratified Spaces and Global Analysis A.1 Stratified Spaces For the reader’s convenience, we recall some basics about Thom-Mather stratifications. Here we mainly follow [5, Section 3], which is based on [69]. A.1.1 Thom-Mather stratifications Let Abe a Hausdorff, locally compact and second countable topological space. It is said that Y, Z ⊂Aare equal near a locally closed subset X⊂Awhen Y∩U= Z∩Ufor some neighborhood Uof Xin A. Then two maps, f:Y→Band g:Z→B, are equal near Xif moreover the restrictions of fand gto Y∩Uare equal. Consider triples (T, π, ρ), where Tis an open neighborhood of X,π:T→X is a continuous retraction, and ρ:T→[0,∞)is a continuous function satisfying ρ−1(0) = X. Two such triples, (T, π, ρ)and (T0, π0, ρ0), are said to be equal near X when T=T0,π=π0and ρ=ρ0near X. This defines an equivalence relation whose equivalence classes are called tubes of Xin A. If Xis open in A, then [(X, idX,0)] is its unique tube, called the trivial tube. According to [69, Definiton 1.2.1], a Thom-Mather stratification is given by triple (A, S, τ), where: (i) Ais a Hausdorff, locally compact and second countable space, (ii) Sis a partition of Ainto locally closed subspaces with the additional structure of smooth (C∞) manifolds, called strata, and (iii) τis the assignment of a tube τXof each X∈ S in A, such that the following conditions are satisfied with some choice of (TX, πX, ρX)∈ τXfor each X∈ S: 133 Appendix (iv) For all X, Y ∈ S, if X∩Y6=∅, then X⊂Y. The notation X≤Yis used in this case, and this defines a partial order relation on S. As usual, X < Y means that X≤Ybut X6=Y. (v) If Y6=Xin Sand TX∩Y6=∅, then X < Y and (πX, ρX) : TX∩Y→ X×R+is a smooth submersion; in particular, dim X < dim Y. (vi) If X < Y in S, then πY(TX∩TY)⊂TX, and πXπY=πXand ρXπY=ρX on TX∩TY. Some important considerations about stratified spaces are the following: •Ais paracompact and normal. •By the normality of A, we can also assume that, if X, Y ∈ S and TX∩TY6=∅, then X≤Yor Y≤X. •The frontier of a stratum Xequals the union of the strata Y < X. •The connected components of each stratum may have different dimensions. •The connected components of the strata, with the restrictions of the tubes, define an induced Thom-Mather stratification Acon ≡(A, Scon, τcon). •Aweak Thom-Mather stratification is defined by removing ρXπY=ρXfrom the condition (vi). We introduce now some examples of stratified spaces: (1) Any smooth manifold is a Thom-Mather stratification with one stratum and the trivial tube. (2) Any smooth manifold with boundary is a stratification with two strata, the interior and the boundary. It can be equipped with a Thom-Mather structure by using a collar of the boundary. (3) Any subanalytic subset of Rmhas primary and secondary stratifications [37–39, 43,50]. (4) J. Mather [49] has proved that the so called Whitney stratified subspaces of any smooth manifold admit a Thom-Mather structure. Let B⊂Abe a locally closed subset. Suppose that, for all X∈ S,X∩Bis a smooth submanifold of X, and that B∩π−1 X(X∩B)defines a tube τX∩Bof X∩B in B. Let S|B={X∩B|X∈ S}, and let τ|Bbe defined by the assignment of 134 A.1 Stratified Spaces τX∩Bto each X∩B∈ S|B. If (B, S|B, τ|B)satisfies conditions (i)–(vi), it is said that B≡(B, S|B, τ|B)is a Thom-Mather substratification of A. Let A0≡(A0,S0, τ0)be another Thom-Mather stratification. A continuous map f:A→A0is called a (smooth)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 X0satisfying f(TX)⊂T0 X0, fπX=π0 X0fand ρX=ρ0 X0f. The continuity of a morphism follows from the other conditions. Morphisms between stratifications form a category with the composition operation; in particular, we have the corresponding concepts of isomorphism and automorphism. Example A.1.1.Let Gbe a compact Lie group acting smoothly on a closed manifold M. Consider the orbit type stratifications of Mand G\M[14]. The cocient space G\Madmits a Thom-Mather structure [69, Introduction] because G\Mis locally isomorphic to a semi-algebraic subset of an Euclidean space whose primary and secondary stratifications are equal [10]. Thus, 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. A.1.2 Products of stratifications The product of two weak Thom-Mather stratifications, Aand A0, is a weak ThomMather stratification A×A0≡(A×A0,S00, τ00)with S00 ={X×X0|X∈ S, X0∈ S0}, τ00 X×X0= [T00 X×X0, π00 X×X0, ρ00 X×X0], where T00 X×X0=TX×T0 X0, ρ00 X×X0(x, x0) = ρX(x) + ρ0 X0(x0). If Aand A0are Thom-Mather stratifications and the complexity of at least one of them is zero, then A×A0is a Thom-Mather stratification, but this is not true when both complexities are positive [69, Section 1.2.9]. Example A.1.2.Let A=A0= [0,∞), with the strata X={0}< Y = (0,∞), taking TX= [0,∞),TY=Y,πX(x) = 0,πY(y) = y,ρX(x) = xand ρY(y)=0. Then the second equality of condition (vi) fails for the strata X×X < X ×Yof A×A0: ρ00 X×Xπ00 X×Y(x, x0) = ρ00 X×X(0, x0) = x06=x+x0=ρ00 X×X(x, x0) 135 Appendix for all (x, x0)∈(0,∞)2, which is an open dense subset of T00 X×X∩T00 X×Y=T00 X×Y= [0,∞)×(0,∞), contradicting (vi). Thus another choice of ρ00 X×X0is needed to get the second equality of condition (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 ThomMather stratification by setting ρ00 X×X0(x, x0) = h(ρX(x), ρ0 X0(x0)); it will be called a product of Aand A0. A.2 Some Results of Global Analysis Here we recall some results of Global Analysis on manifolds that play a fundamental role in the arguments used to prove the main theorems (see Section 1.6) presented in this thesis. Just the statements, without proofs, are included. Proposition A.2.1. [5, Proposition 14.2] Let (E, d)be an elliptic complex on a Riemannian manifold M. Let {Ua}be a finite open covering of M, and let {fa}be a smooth partition of unity on Msubordinated to {Ua}such that each |[d, fa]|is bounded. Assume also that there is another family {˜ fa} ⊂ C∞(M)such that ˜ fa and |[d, ˜ fa]|are bounded, ˜ fa= 1 on supp fa, and supp ˜ fa⊂Ua. For each a, let (Ea, da)be an elliptic complex on a Riemannian manifold Ma, let Va⊂Mabe an open subset, and let ζa: (E|Ua, d)→(Ea|Va, da)be a quasi-isometric isomorphism of elliptic complexes over ξa:Ua→Va. Then the following properties hold: (i)D(dmin/max) = {u∈L2(E)|ζa(fau)∈ D(da min/max)∀a}. (ii)If da min/max is discrete for all a, then dmin/max is discrete. Proposition A.2.2. [5, Proposition 14.3] With the notation of Proposition A.2.1, suppose that every da min/max is discrete, and therefore dmin/max is also discrete. Let 0≤λa min/max,0≤λa min/max,1≤ ··· ,0≤λmin/max,0≤λmin/max,1≤ ··· denote the eigenvalues, repeated according to their multiplicities, of the Laplacians ∆a min/max and ∆min/max defined by da min/max and dmin/max, respectively. Suppose that, for all a, there is some1θa>0such that lim infkλa min/max,kk−θa>0. Then we have lim infkλmin/max,k k−θ>0with θ= minaθa. 1The notation θa,min/max would be more correct, but, for the sake of simplicity, reference to the maximum/minimum i.b.c. is omitted here. 136 A.2 Some Results of Global Analysis Theorem A.2.3. [9, Theorem 2] Let (M, g)be an open oriented Riemannian manifold and Ea vector bundle over M. Let P0:C∞ 0(M, E)→C∞ 0(M, E)be a non-negative symmetric differential operator, and P:L2(M, E)→L2(M, E)a non-negative self-adjoint extension of P0. Then the heat operator e−tP satisfies the following properties: (i)e−tP has a C∞-kernel in C∞((0,∞)×M×M, EE∗), denoted by kP(t, p, q). (ii)If K1and K2are compact subsets of Msuch that K1∩K2=∅, then kkP(t, p, q)kCr(K1×K2,EE∗)=O(tn)as t→0, for all r, n ∈N. Theorem A.2.4. (Atiyah-Bott) [59, Theorem 10.12] Let (ζ, ψ)be a geometric endomorphism of a Dirac complex over a compact oriented manifold of dimension n. Then the Lefschetz number of (ζ, ψ)is given by L(ζ, ψ) = X q∈Fix(ψ) n X r=0 (−1)rtr(ζr(q)) |det(1 −Tqψ)|. Corollary A.2.5. (Lefschetz) [59, Example 10.14] Let ψ:M→Mbe a smooth map on a compact oriented manifold M. Then the Lefschetz number of ψis given by L(ψ) = X q∈Fix(ψ) sign det(1 −Tqψ). 137 Resumo Os obxectos estudados nesta tese son os espazos estratificados de Thom-Mather. Tal concepto foi introducido por René Thom e John Mather arredor de 1970. Posteriormente, as estratificacións foron profundamente estudadas por Mark Goresky e Robert MacPherson, empregando a homoloxía intersección. Por definición, os espazos estratificados de Thom-Mather admiten unha partición en variedades C∞chamadas estratos, que en xeral poderán ter diferentes dimensións. Os estratos péganse entre si baixo certas condicións técnicas que involucran fibrados cónicos. Isto dá lugar a unha descrición local destes espazos usando cartas locais cónicas, que xeneralizan ás cartas usuais en variedades. As estratificacións de ThomMather tamén admiten diversos tipos de métricas: métricas adaptadas xerais, métricas adaptadas e métricas adaptadas de tipo cónico. Os obxectivos principais da investigación en estratificacións de Thom-Mather consisten en demostrar resultados xeométricos e topolóxicos, xeneralizando ou adaptando a este contexto teoremas e propiedades clásicas de variedades con borde. Nos estratos pódense considerar certos operadores diferenciais actuando nos correspondentes espazos de formas diferenciais, ou sobre as seccións diferenciables doutros fibrados vectoriais. O seu estudo é unha técnica moi potente para a obtención de moitas propiedades dos espazos estratificados. Polo tanto, a Análise Funcional e as Ecuacións en Derivadas Parciais, particularmente a ecuación do calor e a ecuación de onda, son ferramentas fundamentais neste campo. A área das Matemáticas que aplica a Teoría de Operadores para obter resultados xeométricos e topolóxicos en variedades e outros obxectos relacionados denomínase Análise Global. Introdúcese a continuación o contexto xeral que se considera ao longo desta tese, explicado detalladamente no capítulo 1. Sexa Mun estrato dunha estratificación compacta Aequipado cunha métrica adaptada xeral g. Tal noción é lixeiramente máis xeral ca das métricas adaptadas de Nagase e Brasselet-Hector-Saralegi. En particular, gten un tipo xeral, que é unha extensión do tipo das métricas adaptadas. Asumirase certa condición no tipo xeral, e entón dirase que gé boa. Considerarase a condición de fronteira ideal máxima/mínima dmax/min do subcomplexo de de Rham das formas diferenciais en Mcon soporte compacto, no sentido de Brüning-Lesch. A 139 Resumo cohomoloxía e o laplaciano de dmax/min denótanse por H∗ max/min(M)e∆max/min, respectivamente. O primeiro dos teoremas principais desta tese establece que ∆max/min ten espectro discreto, que ademais satisfai unha versión débil da fórmula asintótica de Weyl. O segundo teorema principal é unha versión das desigualdades de Morse, no que se utiliza H∗ max/min(M)e o que se chamarán funcións de rel-Morse. Un ingrediente fundamental para a demostración de ambos teoremas é a versión para dmax/min da perturbación de Witten do complexo de de Rham, que é un método moi potente para a obtención das desigualdades de Morse en variedades mediante un punto de vista analítico e físico. Os argumentos usados para probar ambos teoremas están incluídos no capítulo 2. O terceiro teorema importante desta tese é unha versión da fórmula da traza de Lefschetz en espazos estratificados con singularidades illadas, demostrado no capítulo 3 e onde a perturbación de Witten xoga tamén un papel fundamental. Outro ingrediente esencial para a obtención de todos estes resultados é o estudo de certa perturbación do oscilador harmónico de Dunkl, sobre o que trata o capítulo 4. A condición de que gsexa boa é suficientemente xeral no sentido indicado a continuación. Sexa Aunha pseudovariedade estratificada con estrato regular M. Considérese a súa homoloxía intersección I¯pH∗(A)con perversidade ¯p; en particular, as perversidades intermedia inferior e superior denótanse por ¯me¯n, respectivamente. Entón para toda perversidade ¯p≤¯mexiste en Munha boa métrica adaptada asociada a ela que satisfai o isomorfismo de Nagase Hr max(M)∼ =I¯pHr(A)∗(r∈N). Se Mé orientable e ¯p≥¯n, tamén se obtén que Hr min(M)∼ =I¯pHr(A). Polo tanto, as versións das desigualdades de Morse e da fórmula da traza de Lefschetz que se presentan nesta tese poden ser descritas en termos de I¯pH∗(A). 1 Condicións de fronteira ideal do complexo de de Rham A seguinte notación empregarase en referencia a un operador linear Tdensamente definido nun espazo de Hilbert separable. O seu dominio e rango denótanse por D(T)eR(T), respectivamente. Se Té esencialmente autoadxunto a súa clausura escríbese T. Se Té autoadxunto o seu smooth core é D∞(T) := T∞ m=1 D(Tm), e o seu espectro denótase por σ(T). Un complexo de Hilbert (D,d)é un complexo diferencial de lonxitude finita determinado por un operador pechado e densamente definido dnun espazo de Hilbert separable graduado H[15]. Logo o operador D=d+d∗, con D(D) = D(d)∩D(d∗), é autoadxunto en He, polo tanto, o laplaciano ∆=D2=dd∗+d∗dé tamén autoadxunto. Ademais D∞(∆)é un subcomplexo de (D,d)coa mesma homoloxía [15, Teorema 2.12]. Dise que D∞(∆)é o smooth core de d. Dada unha variedade riemanniana M, sexa Ω0(M)o espazo de formas diferenciables con soporte compacto, e L2Ω(M)o espazo de hilbert graduado das formas 140 2 Espazos estratificados diferenciais de cadrado integrable. Sexan deδa diferencial e a codiferencial de de Rham actuando en Ω0(M), e considérense D=d+δe∆ = D2=dδ +δd (o laplaciano). Toda extensión como complexo de Hilbert dde den L2Ω(M)denomínase condición de fronteira ideal (i.b.c. segundo as siglas en inglés) [15], dando lugar a extensións autoadxuntas De∆de De∆en L2Ω(M). Existe unha máxima/mínima i.b.c., sendo dmax =δ∗edmin =d, que induce extensións autoadxuntas Dmax/min e∆max/min de De∆. Se Mé orientable, ∆max correspóndese con ∆min mediante o operador estrela de Hodge. As cohomoloxías correspondentes Hmax/min(M)son invariantes cuasi-isométricos de M; de feito, Hmax(M)é a L2cohomoloxía usual H(2)(M)[18]. Isto permite definir versións dos números de Betti e da característica de Euler, βr max/min =βr max/min(M)and χmax/min =χmax/min(M) (asumindo dimensión finita nas cohomoloxías correspondentes). Estes conceptos poden ser definidos para complexos elípticos arbitrarios [15]. Ademais é ben coñecido que dmin =dmax se Mé variedade completa. Logo as i.b.c. son interesantes no caso en que Mnon é completa. Por exemplo, se Mé o interior dunha variedade riemanniana compacta Ncon ∂N 6=∅, defínese dmax/min tomando condicións de fronteira absolutas/relativas. En xeral, asumiremos que Mé estrato dunha estratificación compacta A[49,50,67,69], equipado cunha xeneralización do concepto de métricas adaptadas considerado en [13,53,54]. Poderase asumir tamén que M=A, e dirase entón que Mé o estrato regular de A. 2 Espazos estratificados Pódese dicir, dun xeito non preciso, que unha estratificación de Thom-Mather é un espazo Hausdorff, localmente compacto e segundo numerable Aprovisto dunha partición en variedades C∞(chamadas estratos) tal que se verifican certas condicións [49,67]. En particular, establecendo que X≤Yse X⊂Y, tense unha relación de orde na familia de estratos. Con respecto a esta orde, a profundidade dun estrato X é a lonxitude máxima de cadeas de estratos menores ou iguais ca X. A profundidade de Aé o supremo das profundidades dos estratos. Nesta sección indícase como os estratos de Aestán pegados entre si, describindo tamén os morfismos/isomorfismos de estratificacións e, particularmente, o grupo de automorfismos Aut(A). Os procedementos xerais fanse habitualmente mediante indución na profundidade. Así, se depth A= 0, tense que Aé unha variedade C∞eAut(A)consiste no seu grupo de difeomorfismos. Dado k∈Z+, asúmase que todo espazo estratificado Lcon depth L < k está ben descrito, así como Aut(L). Se Lé compacto, o cono con enlace Lconsiste en c(L) = (L×[0,∞))/(L×{0}), que ten por vértice a ∗=L×{0} ∈ c(L). Sexan L0 outra estratificación compacta de profundidade < k, e φ:L→L0un morfismo. Sexa 141