Two interacting particles on the half-line

Abstract

In the case of general compact quantum graphs, many-particle models with singular two-particle interactions were introduced by Bolte and Kerner [J. Phys. A: Math. Theor. 46, 045206 (2013); 46, 045207 (2013)] in order 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 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 x2+y2→∞.