On the number of isolated eigenvalues of a pair of particles on the half-line

Abstract

In this note we consider a pair of particles moving on the positive half-line with the pairing generated by a hard-wall potential. This model was first introduced in [arXiv:1604.06693] and later applied to investigate condensation of pairs of electrons in a quantum wire [arXiv:1708.03753,arXiv:1801.00696]. For this, a detailed spectral analysis proved necessary and as a part of this it was shown in [arXiv:1708.03753] that, in a special case, the discrete spectrum of the Hamiltonian consists of a single eigenvalue only. It is the aim of this note to prove that this is generally the case.