Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction

Abstract

The repulsive $\delta$ interaction problem in one dimension for $N$ particles is reduced, through the use of Bethe's hypothesis, to an eigenvalue problem of matrices of the same sizes as the irreducible representations $R$ of the permutation group $S_N$. For some $R\text{'}\mathrm{s}$ this eigenvalue problem itself is solved by a second use of Bethe's hypothesis, in a generalized form. In particular, the ground-state problem of spin-½ fermions is reduced to a generalized Fredholm equation.