[Paper Review] Galoisian Approach to Supersymmetric Quantum Mechanics

Primitivo B. Acosta-Humánez|ArXiv.org|Jun 18, 2009

Quantum Mechanics and Non-Hermitian Physics12 references20 citations

TL;DR

This dissertation introduces a Galois-theoretic framework to analyze supersymmetric quantum mechanics (SUSY QM), using differential Galois theory to classify exactly solvable Hamiltonians. By linking the solvability of Schrödinger equations to the structure of Galois groups of associated linear differential equations, the work establishes a systematic method to determine when SUSY QM systems admit closed-form solutions, with key results showing that solvability corresponds precisely to the solvability of the Galois group.

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.

Motivation & Objective

• To develop a Galois-theoretic approach to supersymmetric quantum mechanics using differential Galois theory.
• To address the long-standing problem of determining when Schrödinger equations in SUSY QM admit closed-form solutions.
• To classify exactly solvable Hamiltonians in terms of the algebraic structure of their associated linear differential equations.
• To establish a rigorous connection between the solvability of quantum mechanical systems and the solvability of their Galois groups.

Proposed method

• Apply differential Galois theory to the second-order linear differential equations arising from supersymmetric quantum mechanical systems.
• Construct the Galois group of the Schrödinger equation associated with a given superpotential.
• Use the structure of the Galois group to determine whether the solutions can be expressed in terms of Liouville quadratures.
• Analyze the symmetry and algebraic properties of superpotentials to identify conditions under which the corresponding Hamiltonians are solvable.
• Utilize the Kovacic algorithm as a computational tool to determine the Galois group structure for rational and algebraic superpotentials.
• Map the solvability of the system to the solvability of the Galois group, providing a decision procedure for exact solvability.

Experimental results

Research questions

• RQ1 Which supersymmetric quantum mechanical systems admit exact solutions, and what algebraic conditions determine this solvability?
• RQ2 How can differential Galois theory be systematically applied to classify solvable superpotentials in SUSY QM?
• RQ3 What is the precise relationship between the structure of the Galois group of the Schrödinger equation and the existence of closed-form wavefunctions?
• RQ4 Can the solvability of a superpotential be determined algorithmically using group-theoretic criteria?
• RQ5 To what extent does the Galois group of the differential equation reflect the underlying supersymmetry algebra?

Key findings

• The paper establishes that a supersymmetric quantum mechanical system is exactly solvable if and only if the Galois group of its associated Schrödinger equation is solvable.
• The solvability of the Galois group provides a necessary and sufficient condition for the existence of Liouvillian solutions to the Schrödinger equation.
• The method successfully classifies a broad class of superpotentials—particularly rational and algebraic ones—based on their Galois-theoretic properties.
• The application of the Kovacic algorithm enables explicit computation of the Galois group for many standard SUSY QM potentials, confirming their solvability or non-solvability.
• The framework reveals that certain well-known exactly solvable potentials (e.g., harmonic oscillator, Pöschl-Teller) correspond precisely to systems with solvable Galois groups.
• The work demonstrates that the algebraic structure of the superpotential directly determines the symmetry and solvability of the system through its Galois group.