Schrödinger equations on elliptic curves: symmetries, solutions and eigenvalue problem (Q2006822)

In this paper, the authors describe a method of finding integrable potentials and solutions for Schrödinger type equations. They show that having a symmetry one can obtain an infinite sequence of integrable potentials and corresponding solutions. The main study are devoted to Schrödinger equations on elliptic curves, the so-called generalized Lamé equations. Such curves are parameterized by Weierstrass \(\wp\)-functions and the authors analyze the case when both potential and symmetry are polynomials in Weierstrass \(\wp\)-function and linear with respect to its first derivative. They apply this method to the Lamé equations and provide a sequence of integrable potentials for which the eigenvalue problem is solved explicitly.