Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition

TL;DR

This paper investigates the spectral properties of the Laplacian with a singular Robin boundary condition, describing its extensions, eigenvalue behavior, and physical relevance, revealing complex asymptotic phenomena as perturbations vanish.

Contribution

It characterizes self-adjoint and skew-symmetric extensions of the Laplacian with singular boundary conditions and analyzes eigenvalue asymptotics under boundary perturbations.

Findings

The Laplacian's residual spectrum covers the entire complex plane.

Eigenvalues exhibit almost periodic behavior in the logarithmic scale as perturbations tend to zero.

A physically relevant skew-symmetric extension is identified.

Abstract

We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behaviour when the small perturbation parameter $\varepsilon>0$ tends to zero, namely they become almost periodic in the logarithmic scale $|\ln \epsilon|$ and, in this way, "wander" along the real axis at a speed $O(\eps^{-1})$.