On the Spectral Asymptotics of Operators on Manifolds with Ends

Coriasco, Sandro; Maniccia, Lidia

doi:10.1155/2013/909782

Cited by 14 publications

(17 citation statements)

References 25 publications

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“…Let us assume that M has a smooth boundary ∂M with defining function x = x ∂M and let V sc := rV b . The resulting differential operators Diff(V sc ) and the associated pseudodifferential ooperators are the SG-operators of [19,74,88,90,89] (called "scattering operators" in [59]). They can be obtained by considering the groupoid (14) G…”

mentioning

confidence: 99%

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

Nistor

Qiao

2018

Operator Theory, Operator Algebras, and Matrix Theory

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We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

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confidence: 99%

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

Nistor

Qiao

2018

Operator Theory, Operator Algebras, and Matrix Theory

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“…However, it was not possible, through the aforementioned method, to give a good estimate of the remainder term. We notice that the asymptotic behavior of the counting function in the bisingular case has some similarities with the Weyl law in the setting of SG-classical operators on manifolds with ends [BC11,CM13].…”

Section: Introductionmentioning

confidence: 69%

“…Similar formulae can be obtained in many other different settings, see [SV97] and [ANPS09] for a detailed analysis and several developments. To mention a few specific situations, see [Shu87,HR81] for the case of the Shubin calculus on R n , [BN03] for the anisotropic Shubin calculus, [BC11,CM13,Nic03] for the SG-operators on R n and the manifolds with ends, [GL02] for operators on conic manifolds, [Mor08] for operators 1 on cusp manifolds, [DD13] for operators on asymptotic hyperbolic manifolds, [Bat12,BGRP13] for bisingular operators.…”

Section: Introductionmentioning

confidence: 99%

Sharp Weyl estimates for tensor products of pseudodifferential operators

Battisti

Borsero

Coriasco

2015

Annali di Matematica

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Abstract. We study the asymptotic behavior of the counting function of tensor products of operators, in the cases where the factors are either pseudodifferential operators on closed manifolds, or pseudodifferential operators of Shubin type on R n , respectively. We obtain, in particular, the sharpness of the remainder term in the corresponding Weyl formulae, which we prove by means of the analysis of some explicit examples.

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“…The results in [22] can be essentially regarded as a particular case of those from [5]. Expansions of the type (1.5) appear also in the recent paper of Coriasco and Maniccia [10] concerning the spectrum of the so-called SG-operators. Summing up, the results mentioned above cover the case of products of two operators, P = P 1 ⊗ P 2 , except for the computation of lower order terms in the expansions, cf.…”

Section: Introductionmentioning

confidence: 89%