7 research outputs found
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
Liouvillian solutions of third order differential equations
AbstractThe Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations
Liouvillian solutions of third order differential equations
The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for 4 finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations