Liouvillian Solutions of Schrödinger Equation with Polynomial Potentials using Gröbner Basis

Abstract

The main aim of this paper is the presentation of a new methodology to obtain Liouvillian solutions of stationary one dimensional Schrödinger equation with quasi-solvable polynomial potentials through the using of differential Galois theory and Gröbner basis. We illustrate these results by the computing of polynomial potentials of degree 4, 6, 8, 10, 12, 14. Moreover, we show an implementation in \textbf{Mathematica} for the decatic potential.

17 pages, 7 figures. 15 tables