Effective operators for Robin eigenvalues in domains with corners

TL;DR

This paper derives precise asymptotics for Robin Laplacian eigenvalues in polygonal domains with corners, linking them to effective boundary Schrödinger operators, advancing understanding of spectral properties in such geometries.

Contribution

It introduces a method to obtain detailed asymptotics for all eigenvalues of Robin Laplacians in domains with corners, extending previous rough estimates.

Findings

Asymptotics of eigenvalues are explicitly linked to boundary Schrödinger operators.

New asymptotic formulas are provided for eigenvalues beyond the critical index.

The approach applies to domains with specific geometric assumptions.

Abstract

We study the eigenvalues of the Laplacian with a strong attractive Robin boundary condition in curvilinear polygons. It was known from previous works that the asymptotics of several first eigenvalues is essentially determined by the corner openings, while only rough estimates were available for the next eigenvalues. Under some geometric assumptions, we go beyond the critical eigenvalue number and give a precise asymptotics of any individual eigenvalue by establishing a link with an effective Schr\"odinger-type operator on the boundary of the domain with boundary conditions at the corners.