Liouvillian solutions for second order linear differential equations with Laurent plynomial coefficient
Liouvillian solutions for second order linear differential equations with Laurent plynomial coefficient
Date
2023-05-09
Authors
Primitivo Acosta
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Abstract
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.
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Citation
Acosta-Humánez, P.B., Blázquez-Sanz, D. & Venegas-Gómez, H. Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient. São Paulo J. Math. Sci. 17, 638–670 (2023).