Liouvillian solutions for second order linear differential equations with polynomial coefficients
- PDF / 1,622,657 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 67 Downloads / 222 Views
ouvillian solutions for second order linear differential equations with polynomial coefficients Primitivo B. Acosta‑Humánez1,2 · David Blázquez‑Sanz3 · Henock Venegas‑Gómez3 Accepted: 10 September 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic’s algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable potentials in the Schrödinger equation. Keywords Anharmonic oscillators · Asymptotic iteration method · Kovacic algorithm · Liouvillian solutions · Parameter space · Quasi-solvable model · Schrödinger equation · Spectral varieties Mathematics Subject Classification Primary 34M15 · Secondary 81Q35
1 Introduction Let us consider the family of second order linear differential equations,
Communicated by Pedro Hernandez Rizzo. * David Blázquez‑Sanz [email protected] 1
Instituto Superior de Formación Docente Salomé Ureña, Santiago, Dominican Republic
2
Fac. Ciencias Básicas y Biomédicas, Universidad Simón Bolívar, Barranquilla, Colombia
3
Escuela de Matemáticas, Facultad de Ciencias, Universidad Nacional de Colombia - Sede Medellín, Medellín, Colombia
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
u�� + P(x)u� + Q(x)u = 0,
(1)
with polynomial coefficients of bounded degree. This family is parameterized by the coefficients of P and Q and therefore endowed of an structure of affine algebraic variety. We are interested in characterizing the moduli of Liouville integrable differentential equations in (1) and describing how the Liouvillian solutions of those integrable equations depend on the coefficients. From a result of Singer [17], we expect that this moduli to be enumerable union of constructible set corresponding to possible choices of local exponents at infinity of Liouvillian solutions. With this purpose we explore the application of Kovacic’s algorithm (see [10, 12]) to the family (1). Some steps of the algorithm, dealing with polynomial solutions of auxiliar equations, are very sensitive to changes of the parameters. However, the Asymptotic Iteration Method (see [7]) allows us to describe the algebraic conditions on the parameters giving rise to the existence of Liouvillian solutions. The structure of the paper is as follows. Section 2 is devoted to the definitions of parameter space ℙ2n , spectral set 𝕃2n , spectral varieties 𝕃2n,d and the statement of our first main result, Theorem 2.3. Section 3 is devoted to the definition of polynomial-hyperexponential solutions, the reduction of the parameter space
Data Loading...
