Regularity for Eigenfunctions of Schrödinger Operators

Ammann, Bernd; Carvalho, Catarina; Nistor, Victor

doi:10.1007/s11005-012-0551-z

Cited by 23 publications

(84 citation statements)

References 54 publications

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“…See [6] for the definition of Lie manifolds with boundary. General blow-up procedures for Lie manifolds were studied in [5]. It can be shown that the class of Lie manifolds satisfying Theorem 4.14 is closed under blow-ups with respect to tame Lie submanifolds.…”

Section: 4mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.Comment: 43 pages, based on my four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 201

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Section: 4mentioning

confidence: 99%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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“…The set of inward pointing vectors in v ∈ T x (M ) will form a closed cone denoted T + x (M ). If, close to x, our manifold with corners is given by the conditions {f i (y) ≥ 0} with df i linearly independent at x, then the cone T + x (M ) is given by (1) T…”

Section: Preliminaries On Lie Algebroidsmentioning

confidence: 99%

“…Proof. The proof follows the lines of the proof of Theorem 3.10 in [1], using the A-tameness of L in order to construct the Lie algebra structure on Γ(M ; A). We include the details for the benefit of the reader, taking also advantage of the results in Subsection 1.2.…”

Section: 1mentioning

confidence: 99%

“…More precisely, we prove that the reduction of G to U is isomorphic to π ↓↓ (G L L ), the fibered pull-back groupoid to U of the reduction of G to L. This allows us to define the desingularization first for this type of fibered pull-back groupoids. We identify the Lie algebroid of the desingularization as the desingularization of its Lie algebroid (the desingularization of a Lie algebroid was introduced in [1]). We also introduce an anisotropic version of the desingularized groupoid and determine its Lie algebroid as well.…”

Section: Introductionmentioning

confidence: 99%

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Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Nistor

2019

Communications in Analysis and Geometry

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We introduce and study a "desingularization" of a Lie groupoid G along an "A(G)-tame" submanifold L of the space of units M . An A(G)-tame submanifold L ⊂ M is one that has, by definition, a tubular neighborhood on which A(G) becomes a thick pull-back Lie algebroid. The construction of the desingularization [[G : L]] of G along L is based on a canonical fibered pull-back groupoid structure result for G in a neighborhood of the tame A(G)submanifold L ⊂ M . This local structure result is obtained by integrating a certain groupoid morphism, using results of Moerdijk and Mrcun (Amer. J. Math. 2002). Locally, the desingularization [[G : L]] is defined using a construction of Debord and Skandalis (Advances in Math., 2014). The space of units of the desingularization [[G : L]] is [M : L], the blow up of M along L. The space of units and the desingularization groupoid [[G : L]] are constructed using a gluing construction of Gualtieri and Li (IMRN 2014). We provide an explicit description of the structure of the desingularized groupoid and we identify its Lie algebroid, which is important in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example relating our constructions to the so called "edge pseudodifferential calculus." The paper is written such that it also provides an introduction to Lie groupoids designed for applications to analysis on singular spaces.

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“…6. A interesting alternate approach to understanding the Kato cusp conditions in terms of singularities at corners is found in Ammann et al [11].…”

Section: As Withmentioning

confidence: 99%