Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

TL;DR

This paper provides a comprehensive algebraic and parametric analysis of Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficients, including explicit algebraic characterizations and applications to special functions.

Contribution

It introduces a complete algebraic classification of Picard-Vessiot integrable equations within this family and improves criteria for polynomial solutions.

Findings

The set of integrable equations forms an enumerable union of algebraic subvarieties.

Explicit algebraic equations for the components of integrable equations are computed.

Applications include analysis of Heun equations and algebraically solvable Schrödinger potentials.

Abstract

This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shr\"odinger equations. Also, as an auxiliary tool, we improve a previously known criterium for second order linear differential equations to admit a polynomial solution.