Spectral asymptotics for manifolds with cylindrical ends (original) (raw)

Séminaire de Théorie spectrale et géométrie

1992

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Some problems of global analysis on asymptotically simple manifolds

Compositio Mathematica, 1979

L'accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Restriction and spectral multiplier theorems on asymptotically conic manifolds

Analysis & PDE, 2013

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure dE(λ) of the square root of the Laplacian on R n is bounded from L p (R n ) to L p ′ (R n ) for 1 ≤ p ≤ 2(n+1)/(n+3), where p ′ is the conjugate exponent to p, with operator norm scaling as λ n(1/p−1/p ′ )−1 . We prove a geometric generalization in which the Laplacian on R n is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete L 2 restriction theorem, which is an O(λ n(1/p−1/p ′ )−1 ) estimate on the L p → L p ′ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for p in the range above.

Spectral Asymptotics of Elliptic Operators on Manifolds

arXiv (Cornell University), 2023

The study of spectral properties of natural geometric elliptic partial differential operators acting on smooth sections of vector bundles over Riemannian manifolds is a central theme in global analysis, differential geometry and mathematical physics. Instead of studying the spectrum of a differential operator L directly one usually studies its spectral functions, that is, spectral traces of some functions of the operator, such as the spectral zeta function ζ(s) = Tr L −s and the heat trace Θ(t) = Tr exp(−tL). The kernel U(t; x, x ′) of the heat semigroup exp(−tL), called the heat kernel, plays a major role in quantum field theory and quantum gravity, index theorems, non-commutative geometry, integrable systems and financial mathematics. We review some recent progress in the study of spectral asymptotics. We study more general spectral functions, such as Tr f (tL), that we call quantum heat traces. Also, we define new invariants of differential operators that depend not only on the their eigenvalues but also on the eigenfunctions, and, therefore, contain much more information about the geometry of the manifold. Furthermore, we study some new invariants, such as Tr exp(−tL +) exp(−sL −), that contain relative spectral information of two differential operators. Finally we show how the convolution of the semigroups of two different operators can be computed by using purely algebraic methods.

Asymptotics of the length spectrum for hyperbolic manifolds of infinite volume

Geometric and Functional Analysis, 2001

W e compute the leading asymptotics of the counting function for closed geodesics on a convex co-compact hyperbolic manifold in terms of spectral data and scattering resonances for the Laplacian. Our result extends classical results of Selberg for compact and nite-volume surfaces to this class of in nite-volume h yperbolic manifolds.

Semiclassical Spectral Asymptotics on Foliated Manifolds

Mathematische Nachrichten, 2002

We consider a (hypo)elliptic pseudodifferential operator A h on a closed foliated manifold (M, F), depending on a parameter h > 0, of the form A h = A + h m B, where A is a formally self-adjoint tangentially elliptic operator of order µ > 0 with the nonnegative principal symbol and B is a formally self-adjoint classical pseudodifferential operator of order m > 0 on M with the holonomy invariant transversal principal symbol such that its principal symbol is positive, if µ < m, and its transversal principal symbol is positive, if µ ≥ m. We prove an asymptotic formula for the eigenvalue distribution function N h (λ) of the operator A h when h tends to 0 and λ is constant.

1 an Approach to Spectral Problems on Riemannian Manifolds

2016

It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that corresponding eigenfunctions can be reconstructed as uniform limits of the same polyharmonic functions with appropriate fixed set of singularities.

The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

The Journal of Geometric Analysis, 2019

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except for the flat spaces. Denote by h > 0 the mean curvature of horospheres in X, and set ρ = h/2. Fixing a basepoint o ∈ X, for ξ ∈ ∂X, denote by B ξ the Busemann function at ξ such that B ξ (o) = 0. then for λ ∈ C the function e (iλ−ρ)B ξ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue −(λ 2 + ρ 2). For a function f on X, we define the Fourier transform of f bỹ f (λ, ξ) := X f (x)e (−iλ−ρ)B ξ (x) dvol(x) for all λ ∈ C, ξ ∈ ∂X for which the integral converges. We prove a Fourier inversion formula f (x) = C 0 ∞ 0 ∂Xf (λ, ξ)e (iλ−ρ)B ξ (x) dλo(ξ)|c(λ)| −2 dλ for f ∈ C ∞ c (X), where c is a certain function on R − {0}, λo is the visibility measure on ∂X with respect to the basepoint o ∈ X and C 0 > 0 is a constant. We also prove a Plancherel theorem, and a version of the Kunze-Stein phenomenon. Contents 1. Introduction 2. Basics about harmonic manifolds 3. Radial and horospherical parts of the Laplacian 4. Analysis of radial functions 4.1. Chebli-Trimeche hypergroups 4.2. The density function of a harmonic manifold 4.3. The spherical Fourier transform 5. Fourier inversion and Plancherel theorem 6. An integral formula for the c-function 7. The convolution algebra of radial functions 8. The Kunze-Stein phenomenon References