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

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 epsilon > 0 tends to zero, namely it becomes almost periodic in the logarithmic scale vertical bar ln epsilon vertical bar, and in this way "wanders" along the real axis at a speed O(epsilon(-1)). (C) 2018 Academie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license.