Spectral analysis of a quantum system with a double line singular interaction

TL;DR

This paper analyzes the spectral properties of a quantum particle interacting with a double-line singular potential, identifying conditions for discrete eigenvalues, embedded eigenvalues, and resonances, with implications for quantum systems with singular interactions.

Contribution

It provides a detailed spectral analysis of a quantum system with a double-line singular interaction, including conditions for eigenvalues and resonances, extending previous single-line models.

Findings

Identified the essential spectrum under asymptotic conditions.

Established criteria for the existence of discrete eigenvalues.

Showed how embedded eigenvalues become resonances under perturbations.

Abstract

We consider a non-relativistic quantum particle interacting with a singular potential supported by two parallel straight lines in the plane. We locate the essential spectrum under the hypothesis that the interaction asymptotically approaches a constant value and find conditions which guarantee either the existence of discrete eigenvalues or Hardy-type inequalities. For a class of our models admitting a mirror symmetry, we also establish the existence of embedded eigenvalues and show that they turn into resonances after introducing a small perturbation.