$\delta$-function spin-1/2 fermions in a one-dimensional potential well

TL;DR

This paper solves the quantum problem of multiple spin-1/2 fermions with delta-function interactions in a finite potential well, providing exact wave functions and spectral equations for cases involving tunneling.

Contribution

It introduces an exact Bethe-ansatz solution for N fermions in a finite well with tunneling, extending previous models to finite-depth potentials.

Findings

Exact Bethe-ansatz wave functions derived

Secular equations for energy spectrum obtained

Solution includes tunneling outside the well

Abstract

The quantum-mechanical problem of $N$ fermions with $\delta$-function interaction in a one-dimensional potential well of finite depth is solved. It is shown that there exists exact wave function of Bethe-ansatz form in the case that a single particle tunnels outside of the well. The Bethe-ansatz like secular equations for the spectrum are obtained.