Weyl Law on Asymptotically Euclidean Manifolds

2019/12/31 by Coriasco, Sandro, Doll, Moritz · 4 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1912.13402

Abstract

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-Δ) on ℝ^d.

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• Global Solvability for Involutive Systems on Non-Compact Manifolds
• Representation formula, regularity, and decay of solutions for sub-diffusion equations
• Global hypoellipticity for a class of time-periodic operators on asymptotically Euclidean manifolds
• Perturbations of globally hypoelliptic pseudo-differential operators on ℝⁿ

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