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

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 $\mathbf{\text{Mathematica}}$ for the decatic potential.