Eigenvalues for the Robin Laplacian in domains with variable curvature

TL;DR

This paper derives precise asymptotic formulas for the low-lying eigenvalues of the Robin Laplacian in domains with variable curvature as the Robin parameter tends to negative infinity, highlighting curvature effects.

Contribution

It extends previous asymptotic results to variable curvature domains and introduces a WKB construction for the ground state energy.

Findings

Asymptotic expansion includes curvature influence.

Eigenvalue splitting analyzed in variable curvature domains.

WKB method proposed for ground state energy approximation.

Abstract

We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $-\infty$. The two first terms in the expansion have been obtained by K. Pankrashkin in the $2D$-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the scalar curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as a candidate for the ground state energy.