Spectral asymptotics for eigenvalues and resonances in the presence of a change of boundary conditions

TL;DR

This paper analyzes how eigenvalues and resonances behave asymptotically when boundary conditions change from Dirichlet to Neumann on a shrinking set in a domain with cylindrical ends.

Contribution

It provides the first detailed asymptotic analysis of eigenvalues and resonances under localized boundary condition changes for elliptic operators.

Findings

Derived asymptotic formulas for eigenvalues and resonances

Identified the influence of boundary condition changes on spectral properties

Extended spectral analysis to domains with cylindrical ends

Abstract

We consider a general second-order elliptic differential operator on a domain with a cylindrical end. We impose Dirichlet boundary conditions on the boundary with the exception of a small set, where we impose Neumann boundary conditions. Shrinking this set to a point we calculate the asymptotic behaviour of eigenvalues and resonances.