Solutions of Heun's general equation and elliptic Darboux equation

TL;DR

This paper develops new solutions for the elliptic Darboux equation by deriving various series expansions from Heun's general equation, including finite and infinite series, with applications to quantum potentials and band theory.

Contribution

It introduces novel series solutions for the elliptic Darboux equation, including hypergeometric series, and explores their applications to Schrödinger equations with quasi-exactly solvable potentials.

Findings

Finite-series solutions for associated Lamé potentials

Convergent hypergeometric series for Darboux equation

Transformations generating new solutions from existing series

Abstract

New solutions for the elliptic Darboux equation are obtained as particular cases of solutions constructed for Heun's general equation. We consider two groups of power series expansions and two new groups of expansions in series of Gauss hypergeometric functions. The convergence of one group in power series is determined by means of ratio tests for infinite series, while the other groups are designed to solve problems which admit finite-series solutions. Actually, we envisage periodic quasi-exactly solvable potentials for which the stationary one-dimensional Schr\"odinger equation is reduced to the Darboux equation. In general, finite- and infinite-series solutions are obtained from power series expansions for Heun's equation. However, we show that the Schr\"odinger equation admits additional finite-series expansions in terms of hypergeometric functions for a family of associated Lam\'e potentials used in band theory of solids. For each finite-series solution we find as well four infinite-series expansions which are bounded and convergent for all values of the independent variable. Finally we find that it is possible to use transformations of variables in order to generate new solutions for the Darboux equation out of other expansions in series of hypergeometric functions for the Heun equation.