2010/08/20 by Primitivo B. Acosta-Humánez, Acosta-Humánez, Primitivo B., Juan J. Morales-Ruiz +3 · 1 citation
Physics and Astronomy · #81Q60 #81T60 #FOS: Physical sciences #Nonlinear Waves and Solitons #Primary 12H05 #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #Secondary 65L80
paper · pdf · doi:10.48550/arxiv.1008.3445
openalex publication_date 2010/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we examine the non-relativistic 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 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 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.