Liouvillian solutions for second order linear diferential equations with polynomial coefcients

Abstract

In this paper we present an algebraic study concerning the general second order linear diferential equation with polynomial coefcients. By means of Kovacic’s algorithm and asymptotic iteration method we fnd a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouville integrable diferential equations. For each fxed 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 Schrödinger equation.