# Galoisian Approach to Supersymmetric Quantum Mechanics - Quantum Physics

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Abstract: This thesis is concerning to the Differential Galois Theory point of view ofthe Supersymmetric Quantum Mechanics. The main object considered here is thenon-relativistic stationary Schr\-odinger equation, specially the integrablecases in the sense of the Picard-Vessiot theory and the main algorithmic toolsused here are the Kovacic algorithm and the \emph{algebrization method} toobtain linear differential equations with rational coefficients. We analyze theDarboux transformations, Crum iterations and supersymmetric quantum mechanicswith their \emph{algebrized} versions from a Galoisian approach. Applying thealgebrization method and the Kovacic-s algorithm we obtain the ground state,the set of eigenvalues, eigenfunctions, the differential Galois groups andeigenrings of some Schr\-odinger equation with potentials such as exactlysolvable and shape invariant potentials. Finally, we introduce one methodologyto find exactly solvable potentials: to construct other potentials, we applythe algebrization algorithm in an inverse way since differential equations withorthogonal polynomials and special functions as solutions.

Author: ** Primitivo B. Acosta-Humanez**

Source: https://arxiv.org/