On the spectrum of a finite-volume negatively-curved manifold

Lott, John

doi:10.1353/ajm.2001.0012

Cited by 19 publications

(17 citation statements)

References 14 publications

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“…Theorem 2. 16. Let (A, τ ) be a unital C * -algebra acting on a Hilbert space H, τ a tracial state on A, then A U is norm closed, therefore Theorem 2.4 holds for A.…”

Section: Finite Trace On a Unital C * -Algebramentioning

confidence: 98%

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Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Guido

Isola

2000

Journal of Functional Analysis

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Given a C*-algebra A with a semicontinuous semifinite trace { acting on the Hilbert space H, we define the family A R of bounded Riemann measurable elements w.r.t. { as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that A R is a C*-algebra, and { extends to a semicontinuous semifinite trace on A R . Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A" and can be approximated in measure by operators in A R , in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a {-a.e. bimodule on A R , denoted by A R , and such a bimodule contains the functional calculi of selfadjoint elements of A R under unbounded Riemann measurable functions. Besides, { extends to a bimodule trace on A R . Type II 1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration and singular traces for C*-algebras are then used to define Novikov Shubin numbers for amenable open manifolds, to show their invariance under quasiisometries, and to prove that they are (noncommutative) asymptotic dimensions. Academic Press

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“…Theorem 2. 16. Let (A, τ ) be a unital C * -algebra acting on a Hilbert space H, τ a tracial state on A, then A U is norm closed, therefore Theorem 2.4 holds for A.…”

Section: Finite Trace On a Unital C * -Algebramentioning

confidence: 98%

“…the function ϕ α given by ϕ α (0) = 0 and ϕ α (t) = t −α when t > 0. Conditions implying the vanishing of L 2 -Betti numbers are given in [16].…”

Section: Novikov-shubin Numbers As Asymptotic Spectral Dimensionsmentioning

confidence: 99%

Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Guido

Isola

2000

Journal of Functional Analysis

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“…This kind of decomposition can be found in various places in the literature, see for instance [32] for applications to finite-volume negatively-curved manifolds.…”

Section: The High and Low Energy Functions Decompositionmentioning

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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“…It is much more difficult to construct complete manifolds of bounded geometry with an infinite number of gaps in the essential spectrum. J. Lott [12] constructed a nonperiodic, negatively curved, finite area 2-dimensional surface with an infinite number of gaps in its essential spectrum. Some intuition for the behavior of the spectrum of Riemannian manifolds comes from spectral results for Schrödinger operators on R n .…”

Section: Introductionmentioning

confidence: 99%

Complete manifolds with bounded curvature and spectral gaps

Schoen

Tran

2016

Journal of Differential Equations

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Abstract. We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the base so that the lifted metric has an arbitrarily large finite number of gaps in its essential spectrum. Also, for any complete noncompact manifold with bounded curvature and positive injectivity radius we construct a metric uniformly equivalent to the given one (also of bounded curvature and positive injectivity radius) with an arbitrarily large finite number of gaps in its essential spectrum.

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On the spectrum of a finite-volume negatively-curved manifold

Cited by 19 publications

References 14 publications

Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Noncommutative Riemann Integration and Novikov–Shubin Invariants for Open Manifolds

Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds

Complete manifolds with bounded curvature and spectral gaps

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