Weak Coupling and Spectral Instability for Neumann Laplacians

We prove an abstract criterion on spectral instability of nonnegative selfadjoint extensions of a symmetric operator and apply this to self-adjoint Neumann Laplacians on bounded Lipschitz domains, intervals, and graphs. Our results can be viewed as variants of the classical weak coupling phenomenon for Schrödinger operators in $L^2({\mathbb{R}}^n)$ for $n=1,2$.