Schroedinger operators with singular interactions: a model of tunneling resonances

TL;DR

This paper analyzes a generalized Schrödinger operator with singular interactions supported on hyperplanes and points, modeling quantum wires, dots, and surface waves, and explicitly solves the resonance problem using the Birman-Schwinger method.

Contribution

It introduces a solvable model of Schrödinger operators with singular interactions, providing explicit analysis of the discrete spectrum and resonance phenomena.

Findings

Explicit characterization of the discrete spectrum.

Resonance problem reduced to a Friedrichs-like model.

Application to quantum wires, dots, and surface waves.

Abstract

We discuss a generalized Schr\"odinger operator in $L^2(\mathbb{R}^d), d=2,3$, with an attractive singular interaction supported by a $(d-1)$-dimensional hyperplane and a finite family of points. It can be regarded as a model of a leaky quantum wire and a family of quantum dots if $d=2$, or surface waves in presence of a finite number of impurities if $d=3$. We analyze the discrete spectrum, and furthermore, we show that the resonance problem in this setting can be explicitly solved; by Birman-Schwinger method it is cast into a form similar to the Friedrichs model.