Articulos FC INSMAT

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Browsing Articulos FC INSMAT by Author "Primitivo Acosta"

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    Darboux transformations for orthogonal differential systems and differential Galois theory
    (2023-03-31) Primitivo Acosta
    Darboux developed an ingenious algebraic mechanism to construct infinite chains of “integrable’’ second-order differential equations as well as their solutions. After a surprisingly long time, Darboux’s results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems (so(3,CK) systems) as well as a framework to extend Darboux transformations to any symmetric power of SL(2,C) -systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.
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    Liouvillian solutions for second order linear differential equations with Laurent plynomial coefficient
    (2023-05-09) Primitivo Acosta
    This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at 0 and ∞ of the Laurent polynomial, is seen as an affine algebraic variety. We prove that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies, such as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for a second order linear differential equations to admit a polynomial solution.