Spectral transition model with the general contact interaction

TL;DR

This paper extends the spectral transition model by incorporating a full family of contact interactions with position-dependent coupling, providing a more general framework for analyzing spectral transitions.

Contribution

It introduces a generalized spectral transition model with a four-parameter family of contact interactions, including position dependence, expanding previous models.

Findings

Generalized the spectral transition model to include position-dependent couplings.

Applied Naboko's technique to analyze the extended model.

Provided insights into the spectral properties of the generalized interaction.

Abstract

Using a technique introduced by Sergei Naboko, we analyze a generalization of the parameter-controlled model of spectral transition, originally proposed by Smilansky and Solomyak, to the situation where the singular interaction responsible for the effect is characterized by the `full' family of four real numbers with the `diagonal' part of the coupling being position-dependent.