Galoisian Approach to Supersymmetric Quantum Mechanics

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 Schrödinger 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 Schrödinger 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.

Phd Dissertation, Universitat Politecnica de Catalunya, 2009

References in corpus (2)

• Universal Superpositions of Coherent States and Self-Similar Potentials
• The factorization of the hypergeometric equation

Cited by in corpus (6)

• Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations
• Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory
• On the integrability of a system describing the stationary solutions in Bose--Fermi mixtures
• Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory
• Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory
• Liouvillian Solutions of Schrödinger Equation with Polynomial Potentials using Gröbner Basis