# On the number of isolated eigenvalues of a pair of particles in a quantum wire

## TL;DR

This paper analyzes the spectral properties of a two-particle quantum wire model with a hard-wall potential, proving that the Hamiltonian generally has only one isolated eigenvalue.

## Contribution

It establishes that, for the model of two particles in a quantum wire with a hard-wall potential, the Hamiltonian typically has a single isolated eigenvalue, extending previous special-case results.

## Key findings

- The discrete spectrum of the Hamiltonian generally consists of a single eigenvalue.
- The model's spectral analysis confirms the uniqueness of the isolated eigenvalue.
- The result applies broadly to the studied quantum wire system.

## 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.