Weyl Law on Asymptotically Euclidean Manifolds

TL;DR

This paper establishes precise asymptotic formulas for eigenvalue counts of elliptic operators on asymptotically Euclidean manifolds, improving remainder estimates and deriving refined three-term Weyl asymptotics under geometric conditions.

Contribution

It proves a two-term Weyl law with improved remainders and derives a three-term asymptotic expansion under specific geometric assumptions.

Findings

Two-term Weyl law with improved remainder estimates

Refined three-term Weyl asymptotics under geometric conditions

Application to the operator Q=(1+|x|^2)(1-Δ) on ℝ^d

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 $Q=(1+|x|^2)(1-\Delta)$ on $\mathbb{R}^d$.