Geometric Approximation of Point Interactions in Two-Dimensional Domains for Non-Self-Adjoint Operators

Abstract

We define the notion of a point interaction for general non-self-adjoint elliptic operators in planar domains. We show that such operators can be approximated in a geometric way by cutting out a small cavity around the point, at which the interaction is concentrated. On the boundary of the cavity, we impose a special Robin-type boundary condition with a nonlocal term. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting one with a point interaction containing an arbitrary prescribed complex-valued coupling constant. The mentioned convergence holds in a few operator norms, and for each of these norms we establish an estimate for the convergence rate. As a corollary of the norm resolvent convergence, we prove the convergence of the spectrum.