Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order \((1,1)\), on an asymptotically Euclidean manifold. We first prove a two term Weyl formula, improving previously known remainder estima...

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Bibliographic Details
Published inarXiv.org
Main Authors Coriasco, Sandro, Doll, Moritz
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 31.12.2019
Subjects
Asymptotic properties
Eigenvalues
Euclidean geometry
Linear operators
Manifolds (mathematics)
Operators (mathematics)
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Summary:We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order \((1,1)\), on an asymptotically Euclidean manifold. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator \(Q=(1+|x|^2)(1-\Delta)\) on \(\mathbb{R}^d\).
ISSN:2331-8422