Two interacting particles on the half-line

Joachim Kerner, Tobias Mühlenbruch

Published 2015-04-30Version 1

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 at most one such eigenstate. As a final result, it is shown that the ground state is non-degenerate and decays exponentially as $\sqrt{x^2+y^2} \to \infty$.