On the negative spectrum of the Robin Laplacian in corner domains

Abstract

For a bounded corner domain $\Omega$, we consider the attractive 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 bottom 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 Schrödinger operator in ${ℝ}^n$ with a strong attractive $\delta$-interaction supported on $\partial \Omega$. Applications to some Ehrling-type estimates and the analysis of the critical temperature of some superconductors are also provided.