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

Denis Ivanovich Borisov ()
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Denis Ivanovich Borisov: Institute of Mathematics, Ufa Federal Research Center of Russian Academy of Sciences, 450008 Ufa, Russia

Mathematics, 2023, vol. 11, issue 4, 1-23

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.

Keywords: point interaction; small cavity; nonlocal Robin condition; norm resolvent convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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