Spectral asymptotics for the Dirichlet Laplacian with a Neumann window via a Birman-Schwinger analysis of the Dirichlet-to-Neumann operator

TL;DR

This paper provides a new proof for the asymptotic behavior of the ground state of the Dirichlet Laplacian with a Neumann window, using a pseudo-differential operator approach to analyze the Dirichlet-to-Neumann map.

Contribution

It introduces a novel proof technique based on pseudo-differential operator analysis for spectral asymptotics of the Laplacian with a Neumann window.

Findings

Asymptotic expansion of the ground state as window length decreases

Explicit symbol representation of the Dirichlet-to-Neumann operator

New proof method for spectral asymptotics

Abstract

In the present article we will give a new proof of the ground state asymptotics of the Dirichlet Laplacian with a Neumann window acting on functions which are defined on a two-dimensional infinite strip or a three-dimensional infinite layer. The proof is based on the analysis of the corresponding Dirichlet-to-Neumann operator as a first order classical pseudo-differential operator. Using the explicit representation of its symbol we prove an asymptotic expansion as the window length decreases.