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

Abstract

We study the Laplacian in a 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 a 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 its spectrum by the method of matched asymptotic expansions. A part of the spectrum acquires a strange behavior when the small perturbation parameter ε>0 tends to zero, namely it becomes almost periodic in the logarithmic scale |ln⁡ε|, and in this way “wanders” along the real axis at a speed O(ε−1).