On the negative spectrum of the Robin Laplacian in corner domains

TL;DR

This paper analyzes the asymptotic behavior of the ground state and essential spectrum of the Robin Laplacian in corner domains with large parameters, using multiscale and recursive methods, with applications to Schrödinger operators and superconductivity.

Contribution

It provides a precise description of the ground state mechanism and spectrum asymptotics for the Robin Laplacian in corner domains, extending to Schrödinger operators with delta interactions.

Findings

Asymptotic eigenvalue behavior in corner domains

Description of the ground state mechanism

Applications to superconductivity models

Abstract

For a bounded corner domain $\Omega$, we consider the Robin Laplacian in $\Omega$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the ground state of the spectrum. It allows also the study of the bottom of the essential spectrum on the associated tangent structures given by cones. Then we obtain the asymptotic behavior of the principal eigenvalue for this singular limit in any dimension, with remainder estimates. The same method works for the Schr\"odinger operator in $\mathbb{R}^n$ with a strong attractive delta-interaction supported on $\partial\Omega$. Applications to some Erhling's type estimates and the analysis of the critical temperature of some superconductors are also provided.