A geometric approximation of $\delta$-interactions by Neumann Laplacians

TL;DR

This paper presents a method to approximate one-dimensional Schrödinger operators with delta interactions using Neumann Laplacians on narrow, geometrically complex domains, with proven convergence in spectra and resolvent sense.

Contribution

It introduces a geometric approximation technique for delta-interactions via Neumann Laplacians on waveguide-like domains, including convergence rate estimates.

Findings

Neumann Laplacian converges to delta-interaction Schrödinger operator in norm resolvent sense.

Spectral convergence is established via Hausdorff convergence.

Explicit estimates on the rate of convergence are provided.

Abstract

We demonstrate how to approximate one-dimensional Schr\"odinger operators with $\delta$-interaction by a Neumann Laplacian on a narrow waveguide-like domain. Namely, we consider a domain consisting of a straight strip and a small protuberance with "room-and-passage" geometry. We show that in the limit when the perpendicular size of the strip tends to zero, and the room and the passage are appropriately scaled, the Neumann Laplacian on this domain converges in (a kind of) norm resolvent sense to the above singular Schr\"odinger operator. Also we prove Hausdorff convergence of the spectra. In both cases estimates on the rate of convergence are derived.