Juan J. Morales-Ruiz

Galoisian approach to integrability of Schrödinger equation

In this paper, we examine the nonrelativistic stationary Schrodinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second-order ordinary linear differential operators, so as to achieve rational function coefficients (“algebrization”), and Kovacics algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrodinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.

Integrable Systems and Difierential Galois Theory

At the end of the nineteenth century, Picard [25, 26], [27, Chapter XVII] and, in a clearer way, Vessiot in his PhD Thesis [30], created and developed a Galois theory for linear differential equations. This field of study, henceforth called Picard–Vessiot theory, was continued from the forties to the sixties of the twentieth century by Kolchin, through the introduction of the modern algebraic abstract terminology and the obtention of new important results, see [12] and references therein. Today, the standard reference of this theory is the monograph [29].

Differential Galois Theory and Lie Symmetries

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.

Symmetries and Related Topics in Differential and Difference Equations

This volume represents the 2009 Jairo Charris Seminar in Symmetries of Differential and Difference Equations, which was held at the Universidad Sergio Arboleda in Bogota, Colombia. The papers include topics such as Lie symmetries, equivalence transformations and differential invariants, group theoretical methods in linear equations, namely differential Galois theory and Stokes phenomenon, and the development of some geometrical methods in theoretical physics The reader will find new interesting results in symmetries of differential and difference equations, applications in classical and quantum mechanics, two fundamental problems of theoretical mechanics, the mathematical nature of time in Lagrangian mechanics and the preservation of the equations of motion by changes of frame, and discrete Hamiltonian systems arising in geometrical optics and analogous to those of finite quantum mechanics. This book is published in cooperation with Instituto de Matematicas y sus Aplicaciones (IMA).Abstract. Let k be a differential field and let [A] : Y ′ = AY be a linear differential system where A ∈ Mat(n , k). We say that A is in a reduced form if A ∈ g(k̄) where g is the Lie algebra of [A] and k̄ denotes the algebraic closure of k. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic [Kov71]. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system X. Using a previous result [AW], we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order m+1 if the variational equations at order m are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of X. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in [WN63]). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the Hénon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the MoralesRamis-Simó theorem.