On the spectral theory of sign-changing Laplace operators

TL;DR

This paper investigates the spectral properties of Laplace operators that change sign, using semi-classical Dirichlet-to-Neumann maps to identify interface-concentrated modes and derive effective equations.

Contribution

It introduces a novel approach to analyze sign-changing Laplace operators through semi-classical methods, revealing new spectral phenomena.

Findings

Existence of modes concentrated on the interface

Derivation of an effective semi-classical equation for these modes

Advancement in understanding spectral properties of sign-changing operators

Abstract

We study spectral theory of sign-changing Laplace operators using semi-classical Dirichlet-to-Neumann maps. We prove the existence of modesconcentrated on the interface and describe an effective semi-classical equation for them.