Galoisian Approach to integrability of Schr\"odinger Equation

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.Comment: 62 page

Galoisian Approach to Supersymmetric Quantum Mechanics

This thesis is concerning to the Differential Galois Theory point of view of the Supersymmetric Quantum Mechanics. The main object considered here is the non-relativistic stationary Schr\"odinger equation, specially the integrable cases in the sense of the Picard-Vessiot theory and the main algorithmic tools used here are the Kovacic algorithm and the \emph{algebrization method} to obtain linear differential equations with rational coefficients. We analyze the Darboux transformations, Crum iterations and supersymmetric quantum mechanics with their \emph{algebrized} versions from a Galoisian approach. Applying the algebrization method and the Kovacic's algorithm we obtain the ground state, the set of eigenvalues, eigenfunctions, the differential Galois groups and eigenrings of some Schr\"odinger equation with potentials such as exactly solvable and shape invariant potentials. Finally, we introduce one methodology to find exactly solvable potentials: to construct other potentials, we apply the algebrization algorithm in an inverse way since differential equations with orthogonal polynomials and special functions as solutions.Comment: Phd Dissertation, Universitat Politecnica de Catalunya, 200