Liouvillian solutions for second order linear differential equations with polynomial coefficients

TL;DR

This paper provides an algebraic characterization of Liouvillian solutions for second order linear differential equations with polynomial coefficients, using Kovacic's algorithm and asymptotic iteration, and explores the spectral set structure.

Contribution

It introduces a degree-independent algebraic description of the spectral set for Liouvillian solutions, revealing its structure as a union of algebraic varieties and bounding eigenvalues.

Findings

Spectral set is a countable union of algebraic varieties.

Algebraic description bounds the number of eigenvalues.

Method applies to algebraically quasi-solvable Schrödinger potentials.

Abstract

In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouiville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable potentials in the Schr\"odinger equation.