Galoisian Approach to integrability of Schrödinger Equation

Details Galoisian Approach to integrability of Schrödinger Equation Galoisian Approach to integrability of Schrödinger Equation

2010-08-20

arxiv.org/abs/1008.3445v2

Physics

Quantum Physics

62 pages

Scientific paper

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.

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