Title: Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition,
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 $O(\eps^{-1})$.