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

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.

Automated summary

We introduce the concept of point interaction for general non-self-adjoint elliptic operators in planar domains. By cutting out a small cavity around the point, we show that these operators can be geometrically approximated. A special Robin-type boundary condition with a nonlocal term is imposed on the boundary of the cavity. As the cavity shrinks to a point, the perturbed operator converges in the norm resolvent sense to a limiting operator with a point interaction containing an arbitrary complex-valued coupling constant. We establish convergence rates for several operator norms. The convergence of the spectrum is proven as a corollary of the norm resolvent convergence.