Semi-classical edge states for the Robin Laplacian

TL;DR

This paper investigates the semi-classical Robin Laplacian, deriving an effective operator for eigenvalue distribution in 2D and proving eigenfunction decay near the boundary across all dimensions.

Contribution

It introduces an effective operator for the asymptotic distribution of eigenvalues and establishes decay properties of eigenfunctions for the semi-classical Robin Laplacian.

Findings

Effective operator describing eigenvalue distribution in 2D

Eigenfunctions decay away from the boundary in all dimensions

Insights relevant to high energy Steklov eigenfunctions

Abstract

Motivated by the study of high energy Steklov eigenfunctions, we examine the semi-classical Robin Laplacian. In the two dimensional situation, we determine an effective operator describing the asymptotic distribution of the negative eigenvalues, and we prove that the corresponding eigenfunctions decay away from the boundary, for all dimensions.