On the skeleton method and an application to a quantum scissor

TL;DR

This paper introduces tools for analyzing the spectral structure of the skeleton operator in quantum particle models with delta interactions, demonstrated on a two-dimensional quantum scissor system.

Contribution

It presents new analytical tools for the spectral analysis of the skeleton operator in quantum systems with delta potentials, applied to a quantum scissor model.

Findings

Spectral analysis of the skeleton operator is feasible with the proposed tools.

Application to the quantum scissor model reveals specific spectral properties.

The method provides direct insight into the spectral structure of complex quantum interactions.

Abstract

In the spectral analysis of few one dimensional quantum particles interacting through delta potentials it is well known that one can recast the problem into the spectral analysis of an integral operator (the skeleton) living on the submanifold which supports the delta interactions. We shall present several tools which allow direct insight into the spectral structure of this skeleton. We shall illustrate the method on a model of a two dimensional quantum particle interacting with two infinitely long straight wires which cross one another at a certain angle : the quantum scissor.