Refined asymptotics for eigenvalues on domains of infinite measure

TL;DR

This paper investigates the asymptotic behavior of eigenvalues on unbounded one- and two-dimensional domains using elementary methods related to lattice point problems, extending spectral analysis to infinite measure sets.

Contribution

It introduces a new elementary approach to analyze eigenvalue distributions on infinite measure domains, including estimates for spectral counting functions in unbounded 2D regions.

Findings

Asymptotic eigenvalue distribution characterized for infinite measure domains

Elementary proof technique based on Dirichlet lattice points problem

Spectral counting function estimates for unbounded 2D domains

Abstract

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.