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

TL;DR

This paper introduces new classes of one-dimensional N-body quantum systems that are exactly or conditionally exactly solvable, with specific potentials, providing explicit spectra and scattering phase shifts, expanding the solvable models in quantum many-body physics.

Contribution

It identifies specific potential forms in Calogero-Sutherland type models that are exactly or conditionally exactly solvable for arbitrary or fixed parameters, respectively.

Findings

Exact solutions for certain potential choices and arbitrary g'

Non-degenerate spectra with energy-dependent phase shifts when g' ≠ 0

Conditional solvability with phase shifts for specific partial waves

Abstract

We study a class of Calogero-Sutherland type one dimensional N-body quantum mechanical systems, with potentials given by $$ V( x_1, x_2, \cdots x_N) = \sum_{i <j} {g \over {(x_i - x_j)^2}} - \frac{g^{\prime}}{\sum_{i<j}(x_i - x_j)^2} + U(\sqrt{\sum_{i<j}(x_i - x_j)^2}),$$ 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.