Liouvillian Solutions of Schrödinger Equation with Polynomial Potentials using Gröbner Basis

Primitivo Belén Acosta-Humánez, Henock Venegas-Gómez

Published 2018-10-22Version 1

The main aim of this paper is the presentation of a new methodology to obtain Liouvillian solutions of Schr\"odinger equation with quasi-solvable polynomial potentials through the using of differential Galois theory and Gr\"obner basis. We illustrate these results by the computing of polynomial potentials of degree 4, 6, 8, 10, 12, 14. We start the paper with the analysis of some transformations for polynomial and differential equations. The paper ends with the appendix that contains some tables to illustrate the completing squares in polynomials of degree 4, 6, 8, 10, 12 and 14.