Three formulas for eigenfunctions of integrable Schroedinger operators

TL;DR

This paper presents three explicit formulas for eigenfunctions of integrable N-body Schrödinger operators, including hypergeometric, Bethe ansatz, and trace formulas, with applications to elliptic and q-deformed cases.

Contribution

It introduces three novel formulas for meromorphic eigenfunctions of integrable Schrödinger operators, connecting them to hypergeometric functions, Bethe ansatz, and trace computations, extending previous elliptic results.

Findings

Explicit trace formula for eigenfunctions

Integral hypergeometric representation

Bethe ansatz parametrization of eigenfunctions

Abstract

We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable N-body Schroedinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik-Zamolodchikov-Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a ``Hermite-Bethe'' variety, a generalization of the spectral variety of the Lame' operator. We also give the q-deformed version of our first formula. In the scalar sl_N case, this gives common eigenfunctions of the commuting Macdonald-Rujsenaars difference operators.