Liouvillian solutions for second order linear differential equations with polynomial coefficients

Acosta-Humánez, Primitivo B.; Blázquez-Sanz, David; Venegas-Gómez, Henock

doi:10.1007/s40863-020-00186-0

Cited by 4 publications

(2 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Algebraic Methods. Concerning algebraic aspects considered in this paper, we follow the references [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and also [1,2,3,4,5,6].…”

Section: Saddle-focus-saddle Bifurcationsmentioning

confidence: 99%

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Contreras¹,

Linero²,

Montes³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis of these families are graphed. To properly perform this study it was necessary to use some results of the non-linear systems theory, for this reason vital definitions and theorems were included because of their importance during the study of the multiparametric families. Algebraic aspects are also included.

show abstract

“…Algebraic Methods. Concerning algebraic aspects considered in this paper, we follow the references [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30] and also [1,2,3,4,5,6].…”

Section: Saddle-focus-saddle Bifurcationsmentioning

confidence: 99%

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Contreras¹,

Linero²,

Montes³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following theorem also can be found in [5, §2] and see also [4]. Here we present a quantum mechanics adapted version.…”

Section: Polynomial Potentialsmentioning

confidence: 82%

Galoisian approach to supersymmetric quantum mechanics

Acosta Humánez

View full text Add to dashboard Cite

Esta tesis versa sobre el punto de vista desde la Teoría de Galois Diferencial hacia la mecánica cuántica supersimétrica. El objeto principal considerado aquí es la ecuación deSchrödinger estacionaria no relativista, especialmente los casos integrables en el sentido de la teoría de Picard-Vessiot theory y las principales herramientas algorítmicas utilizadas aquí son el algoritmo de Kovacic y el método de la algebrización para obtener ecuaciones diferenciales lineales con coeficientes racionales. Analizamos las transformaciones de Darboux, la iteración de Crum y la mecánica cuántica supersimétrica con sus versiones algebrizadas desde un acercamiento Galoisiano. Aplicando el método de la algebrización y el algoritmo de Kovacic obtenemos el estado base, las funciones propias, los valores propios los grupos de Galois diferenciales y los anillos propios de algunas ecuaciones de Schrödinger con potenciales tales como exactamente resoluble y potenciales de forma invariante. Finalmente, introducimos una metodología para buscar potenciales exactamente resolubles. Para construir otros potenciales, aplicamos el método de la algebrización en forma inversa, desde ecuaciones diferenciales que tengan polinomios ortogonales y funciones especiales como soluciones

show abstract

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

Acosta-Humánez

Blázquez-Sanz

Venegas-Gómez

2023

São Paulo J. Math. Sci.

View full text Add to dashboard Cite

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 $$\infty$$ ∞ 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.

show abstract

Liouvillian solutions for second order linear differential equations with polynomial coefficients

Cited by 4 publications

References 16 publications

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

Galoisian approach to supersymmetric quantum mechanics

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

Contact Info

Product

Resources

About