New Exactly and Conditionally Exactly Solvable N-Body Problems in One Dimension

Abstract

We study a class of Calogero-Sutherland type one dimensional N-body quantum mechanical systems, with potentials given by $$V(x_1,x_2,\cdots x_N)=\sum _{i<j}\frac{g}{(x_i-x_j{)}^2}-\frac{g^{\prime }}{{\sum }_{i<j}(x_i-x_j{)}^2}+U(\sqrt{\sum _{i<j}(x_i-x_j{)}^2}),$$ where $U(\sqrt{{\sum }_{i<j}(x_i-x_j{)}^2})$'s are of specific form. It is shown that, only for a few choices of $U$, the eigenvalue problems can be solved {\it exactly}, for arbitrary $g^{\prime }$. The eigen spectra of these Hamiltonians, when $g^{\prime }\ne 0$, are non-degenerate and the scattering phase shifts are found to be energy dependent. It is further pointed out that, the eigenvalue problems are amenable to solution for wider choices of $U$, if $g^{\prime }$ is conveniently fixed. These conditionally exactly solvable problems also do not exhibit energy degeneracy and the scattering phase shifts can be computed {\it only} for a specific partial wave.

10 pages, latex, no figures

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