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

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 $O(\eps^{-1})$.