A regular version of Smilansky model

TL;DR

This paper introduces a regularized version of Smilansky's model with a shrinking potential, demonstrating a spectral transition and identifying the critical coupling value, extending the analysis to multiple channels.

Contribution

It presents a modified Smilansky model with a regular potential, analyzing spectral transitions and extending results to multiple potential channels.

Findings

Spectral transition occurs at a critical coupling value.

A new spectral branch opens above the critical coupling.

The model generalizes to multiple potential channels.

Abstract

We discuss a modification of Smilansky model in which a singular potential `channel' is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential `channels'.