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Weak Coupling and Spectral Instability for Neumann Laplacians

Jussi Behrndt, Fritz Gesztesy, Henk de Snoo

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 Schrödinger operators in $L^2(\mathbb R^n)$ for $n=1,2$.

Weak Coupling and Spectral Instability for Neumann Laplacians

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 Schrödinger operators in for .

Paper Structure

This paper contains 3 sections, 5 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Spectral instability of nonnegative self-adjoint extensions
  3. Spectral instability of the Neumann Laplacian

Key Result

Theorem 2.2

Let $A$ and $V$ be as in Hypothesis h1 and assume, in addition, that $\ker(A)\not\subseteq \ker(V)$. Then

Theorems & Definitions (12)

  • Theorem 2.2
  • proof
  • Remark 2.3
  • Corollary 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Corollary 3.4
  • Corollary 3.5
  • ...and 2 more