Weak Coupling and Spectral Instability for Neumann Laplacians

TL;DR

This paper establishes an abstract criterion for spectral instability of Neumann Laplacians on various domains, revealing weak coupling phenomena similar to those in Schrödinger operators in low dimensions.

Contribution

It introduces a new abstract criterion for spectral instability and applies it to Neumann Laplacians on domains, intervals, and graphs, highlighting weak coupling effects.

Findings

Spectral instability criteria for Neumann Laplacians

Application to Lipschitz domains, intervals, and graphs

Connection to weak coupling phenomena in Schrödinger operators

Abstract

We prove an abstract criterion on spectral instability of nonnegative selfadjoint extensions of a symmetric operator and apply this to self-adjoint Neumann Laplacians on bounded Lipschitz domains, intervals, and graphs. Our results can be viewed as variants of the classical weak coupling phenomenon for Schr\"odinger operators in $L^2(\mathbb R^n)$ for $n=1,2$.