Galoisian Approach to Supersymmetric Quantum Mechanics

TL;DR

This thesis applies Differential Galois Theory, Kovacic's algorithm, and algebrization to analyze and solve supersymmetric quantum mechanics problems, focusing on integrable Schrödinger equations and constructing exactly solvable potentials.

Contribution

It introduces a Galoisian framework for analyzing supersymmetric quantum mechanics, utilizing algebrization and Kovacic's algorithm to find solutions and construct solvable potentials.

Findings

Determined eigenvalues and eigenfunctions for specific Schrödinger equations.

Identified differential Galois groups and eigenrings of quantum systems.

Developed a methodology to construct new exactly solvable potentials.

Abstract

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.