2020/12/22 by Primitivo B. Acosta-Humánez, Acosta-Humánez, Primitivo B., David Blázquez-Sanz +3
Mathematics · Physics and Astronomy · #34M15 #81Q35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP #msc:34M15 #msc:81Q35
paper · pdf · doi:10.48550/arxiv.2012.11795
40 pages
arxiv created 2021/09/25 · arxiv updated 2021/09/28
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 proof 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 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 second order linear differential equations to admit a polynomial solution.