Analysis of geometric operators on open manifolds: A groupoid approach

Lauter, Robert; Nistor, Victor

doi:10.1007/978-3-0348-8364-1_8

Cited by 63 publications

(156 citation statements)

References 36 publications

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“…[17]. A similar result for type (1, 0) operators is proved in the same way as in [17]. This proves (a) because…”

Section: Proposition 43 (I) Let X Be a Defining Function Of Some Hysupporting

confidence: 77%

“…Since the map π M respects adjoints: π M (P * ) = π M (P ) * , [17], we obtain the following corollary. We end this section with three remarks.…”

Section: Theorem 32 Let M 0 Be a Manifold With A Lie Structure At Imentioning

confidence: 80%

“…In particular, we obtain the following definition of the algebra of G-equivariant pseudodifferential operators associated to (A, M, G). For example, the generalizations of the morphisms considered in [17] are of the form (47). However, we do not know exactly what the conditions are under which the morphism R N above is defined.…”

Section: Group Actionsmentioning

confidence: 99%

“…See [39], [40] and the references therein. Other important cases in which this program was completed can be found in [15], [16], [17], [35], [37]. An earlier important motivation for the construction of these algebras was the method of layer potentials for boundary value problems and questions in analysis on locally symmetric spaces.…”

Section: Introductionmentioning

confidence: 99%

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Pseudodifferential operators on manifolds with a Lie structure at infinity

2007

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We define and study an algebra Ψ ∞ 1,0,V (M 0 ) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M 0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M 0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M 0 ). We also consider the algebra Diff * V (M 0 ) of differential operators on M 0 generated by V and C ∞ (M ), and show that. Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra Ψ ∞ 1,0,V (M 0 ).

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“…[17]. A similar result for type (1, 0) operators is proved in the same way as in [17]. This proves (a) because…”

Section: Proposition 43 (I) Let X Be a Defining Function Of Some Hysupporting

confidence: 77%

“…Since the map π M respects adjoints: π M (P * ) = π M (P ) * , [17], we obtain the following corollary. We end this section with three remarks.…”

Section: Theorem 32 Let M 0 Be a Manifold With A Lie Structure At Imentioning

confidence: 80%

Section: Group Actionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pseudodifferential operators on manifolds with a Lie structure at infinity

2007

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“…It is a special case of the fibered-cusp calculus [33] and can be obtained using the groupoid techniques of [29].…”

Section: Magnetic Fields and Cohomologymentioning

confidence: 99%

Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds

Golenia

Moroianu²

2008

Ann. Henri Poincaré

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We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When B is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect.In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle.We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.

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On the η‐function for bisingular pseudodifferential operators

Bohlen

2016

Mathematische Nachrichten

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In this work we consider the η‐invariant for pseudodifferential operators of tensor product type, also called bisingular pseudodifferential operators. We study complex powers of classical bisingular operators. We prove the trace property for the Wodzicki residue of bisingular operators and show how the residues of the η‐function can be expressed in terms of the Wodzicki trace of a projection operator. Then we calculate the K‐theory of the algebra of 0‐order (global) bisingular operators. With these preparations we establish the regularity properties of the η‐function at the origin for global bisingular operators which are self‐adjoint, elliptic and of positive orders.

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Analysis of geometric operators on open manifolds: A groupoid approach

Cited by 63 publications

References 36 publications

Pseudodifferential operators on manifolds with a Lie structure at infinity

Pseudodifferential operators on manifolds with a Lie structure at infinity

Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds

On the η‐function for bisingular pseudodifferential operators

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