Two interacting particles on the half-line

TL;DR

This paper investigates two-particle quantum systems on the half-line with singular interactions, analyzing their spectral properties, eigenstate bounds, and ground state uniqueness and decay behavior.

Contribution

It provides a detailed spectral analysis and conditions for eigenstate uniqueness in two-particle systems with singular interactions on the half-line.

Findings

Spectrum description of the two-particle Hamiltonian

Upper bounds on the number of eigenstates below the essential spectrum

Proof of the exponential decay and uniqueness of the ground state

Abstract

In the case of compact quantum graphs, many-particle models with singular two-particle interactions where introduced in [arXiv:1207.5648, arXiv:1112.4751] to provide a paradigm for further studies on many-particle quantum chaos. In this note, we discuss various aspects of such singular interactions in a two-particle system restricted to the half-line $\mathbb{R}_+$. Among others, we give a description of the spectrum of the two-particle Hamiltonian and obtain upper bounds on the number of eigenstates below the essential spectrum. We also specify conditions under which there is exactly one such eigenstate. As a final result, it is shown that the ground state is unique and decays exponentially as $\sqrt{x^2+y^2} \to \infty$.