Galoisian approach to integrability of Schrِdinger equation

In this paper, we examine the nonrelativistic stationary Schrödinger 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 Kovacicʹs 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 Schrödinger 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.