The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions

TL;DR

This paper studies the limit behavior of a non-Hermitian Laplacian in thin layers with Robin boundary conditions, showing it converges to a self-adjoint Schrödinger operator, clarifying spectral properties.

Contribution

It demonstrates the norm resolvent convergence of the non-Hermitian Laplacian to a self-adjoint Schrödinger operator with boundary-derived potential.

Findings

Laplacian converges to a self-adjoint Schrödinger operator as layer width diminishes

Spectral properties of the non-Hermitian Laplacian are explained via known Schrödinger results

Boundary coupling function determines the potential in the limiting operator

Abstract

The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schroedinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schroedinger operators.