Galoisian Approach to Supersymmetric Quantum Mechanics: The integrability analysis of the Schrödinger equation by means of differential Galois theory

This book is concerning to a Differential Galois (Picard-Vessiot) Theory point of view of the Supersymmetric Quantum Mechanics. The main object is the non-relativistic stationary Schr dinger equation , where are introduced the concepts of Algebraic Spectrum and Hamiltonian Algebrization. Using the Kovacic's Algorithm and the Hamiltonian Algebrization are analyzed Darboux transformations, Crum iterations and supersymmetric quantum mechanics, including their Algebrized Versions from a Galoisian approach. In particular are obtained the ground state, eigenvalues, eigenfunctions, the differential Galois groups and eigenrings of some Schr dinger equations with potentials such as exactly solvable, quasi-exactly solvable and shape invariant potentials. Finally is introduced one methodology to find Algebraically Solvable and Algebraically Quasi- Solvable Potentials. It consists in to apply the Hamiltonian Algebrization, as inverse process, over families of second order linear differential equations integrables in the Picard-Vessiot sense for a set of parameters, in particular, involving orthogonal polynomials and special functions.