δ-function spin-1/2 fermions in a one-dimensional potential well

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.