Galoisian Approach to integrability of Schr\"odinger Equation

TL;DR

This paper applies differential Galois theory to analyze the integrability of the Schrödinger equation, providing a systematic approach to identify solvable cases and compute spectral data.

Contribution

It introduces a Galoisian framework for studying Schrödinger equations, including algorithms for algebrization, analysis of transformations, and criteria for exact solvability.

Findings

Determined Galois groups for a class of Schrödinger equations.

Computed eigenvalues and eigenfunctions for shape invariant potentials.

Developed a method to assess when exact solutions are possible.

Abstract

In this paper, we examine the non-relativistic stationary Schr\"odinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schr\"odinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.