Liouvillian Solutions of Schr\"odinger Equation with Polynomial Potentials using Gr\"obner Basis

TL;DR

This paper introduces a novel method combining differential Galois theory and Gr"obner basis to find Liouvillian solutions of the Schr"odinger equation with polynomial potentials, demonstrated through computational examples.

Contribution

It presents a new approach for solving the Schr"odinger equation with polynomial potentials using differential Galois theory and Gr"obner basis, including implementation details.

Findings

Successfully computed Liouvillian solutions for polynomial potentials of degrees 4, 6, 8, 10, 12, 14.

Developed a Mathematica implementation for decatic potential.

Demonstrated the effectiveness of the method on quasi-solvable polynomial potentials.

Abstract

The main aim of this paper is the presentation of a new methodology to obtain Liouvillian solutions of stationary one dimensional 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. Moreover, we show an implementation in \textbf{Mathematica} for the decatic potential.