Liouvillian solutions for second order linear diferential equations with polynomial coefcients
In this paper we present an algebraic study concerning the general second order linear diferential equation with polynomial coefcients. By means of Kovacic’s algorithm and asymptotic iteration method we fnd a degree independent algebraic description of the spectral set: the subset, in the parameter...
- Autores:
-
Acosta‑Humánez, Primitivo B.
Blázquez‑Sanz, David
Venegas‑Gómez, Henock
- Tipo de recurso:
- Fecha de publicación:
- 2020
- Institución:
- Universidad Simón Bolívar
- Repositorio:
- Repositorio Digital USB
- Idioma:
- eng
- OAI Identifier:
- oai:bonga.unisimon.edu.co:20.500.12442/6724
- Acceso en línea:
- https://hdl.handle.net/20.500.12442/6724
https://doi.org/10.1007/s40863-020-00186-0
https://link.springer.com/article/10.1007/s40863-020-00186-0
- Palabra clave:
- Anharmonic oscillators
Asymptotic iteration method
Kovacic algorithm
Liouvillian solutions
Parameter space
Quasi-solvable model
Schrödinger equation
Spectral varieties
- Rights
- openAccess
- License
- Attribution-NonCommercial-NoDerivatives 4.0 Internacional
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Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| dc.title.abbreviated.eng.fl_str_mv |
São Paulo J. Math. Sci. |
| title |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| spellingShingle |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients Anharmonic oscillators Asymptotic iteration method Kovacic algorithm Liouvillian solutions Parameter space Quasi-solvable model Schrödinger equation Spectral varieties |
| title_short |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| title_full |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| title_fullStr |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| title_full_unstemmed |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| title_sort |
Liouvillian solutions for second order linear diferential equations with polynomial coefcients |
| dc.creator.fl_str_mv |
Acosta‑Humánez, Primitivo B. Blázquez‑Sanz, David Venegas‑Gómez, Henock |
| dc.contributor.author.none.fl_str_mv |
Acosta‑Humánez, Primitivo B. Blázquez‑Sanz, David Venegas‑Gómez, Henock |
| dc.subject.eng.fl_str_mv |
Anharmonic oscillators Asymptotic iteration method Kovacic algorithm Liouvillian solutions Parameter space Quasi-solvable model Schrödinger equation Spectral varieties |
| topic |
Anharmonic oscillators Asymptotic iteration method Kovacic algorithm Liouvillian solutions Parameter space Quasi-solvable model Schrödinger equation Spectral varieties |
| description |
In this paper we present an algebraic study concerning the general second order linear diferential equation with polynomial coefcients. By means of Kovacic’s algorithm and asymptotic iteration method we fnd a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouville integrable diferential equations. For each fxed 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. |
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2020 |
| dc.date.accessioned.none.fl_str_mv |
2020-10-20T18:21:17Z |
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2020-10-20T18:21:17Z |
| dc.date.issued.none.fl_str_mv |
2020-09-10 |
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http://purl.org/coar/version/c_970fb48d4fbd8a85 |
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23169028 |
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https://hdl.handle.net/20.500.12442/6724 |
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https://doi.org/10.1007/s40863-020-00186-0 |
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https://link.springer.com/article/10.1007/s40863-020-00186-0 |
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23169028 |
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https://hdl.handle.net/20.500.12442/6724 https://doi.org/10.1007/s40863-020-00186-0 https://link.springer.com/article/10.1007/s40863-020-00186-0 |
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eng |
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eng |
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Attribution-NonCommercial-NoDerivatives 4.0 Internacional |
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openAccess |
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pdf |
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São Paulo Journal of Mathematical Sciences |
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Universidad Simón Bolívar |
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Acosta‑Humánez, Primitivo B.52c4ca90-1a26-42d1-953b-2c003e3fe2daBlázquez‑Sanz, Davidad198f28-6b0e-4bba-819b-32969dfb2d70Venegas‑Gómez, Henockc39908ba-4473-4cdb-bc0e-e6c27782d57f2020-10-20T18:21:17Z2020-10-20T18:21:17Z2020-09-1023169028https://hdl.handle.net/20.500.12442/6724https://doi.org/10.1007/s40863-020-00186-0https://link.springer.com/article/10.1007/s40863-020-00186-0In this paper we present an algebraic study concerning the general second order linear diferential equation with polynomial coefcients. By means of Kovacic’s algorithm and asymptotic iteration method we fnd a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouville integrable diferential equations. For each fxed 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.pdfengSpringerAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2São Paulo Journal of Mathematical SciencesAnharmonic oscillatorsAsymptotic iteration methodKovacic algorithmLiouvillian solutionsParameter spaceQuasi-solvable modelSchrödinger equationSpectral varietiesLiouvillian solutions for second order linear diferential equations with polynomial coefcientsSão Paulo J. Math. Sci.info:eu-repo/semantics/articleArtículo científicohttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_2df8fbb1Acosta-Humánez, P., Blázquez-Sanz, D.: Non-integrability of some Hamiltonian systems with rational potential. Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 265–293 (2008)Acosta-Humánez, P.B.: Galoisian Approach to Supersymmetric Quantum Mechanics. PhD thesis, Universitat Politècnica de Catalunya (2009). https://www.tdx.cat/handle/10803/22723Acosta-Humánez, P.B.: Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Diferential Galois Theory. VDM Verlag, Dr Müller, Saarbrücken, Deutschland (2010)Acosta-Humánez, P.B., Morales-Ruiz, J.J., Weil, J.-A.: Galoisian approach to integrability of schrödinger equation. Rep. Math. Phys. 67(3), 305–374 (2011)Bender, C., Dunne, G.: Quasi-exactly solvable systems and orthogonal polynomials. J. Math. Phys. 37(1), 6–11 (1996)Blázquez-Sanz, David, Yagasaki, Kazuyuki: Galoisian approach for a Sturm–Liouville problem on the infnite interval. Methods Appl. Anal. 19(3), 267–288 (2012)Ciftci, H., Hall, R., Saad, N.: Asymptotic iteration method for eigenvalue problems. J. Phys. A Math. Gen. 36(47), 11807–11816 (2003)Ciftci, H., Hall, R., Saad, N., Dogu, E.: Physical applications of second-order linear diferential equations that admit polynomial solutions. J. Phys. A Math. Theor. 43(41), 415206–415219 (2010)Combot, T.: Integrability of the one dimensional Schrödinger equation. J. Math. Phys. 59(2), 022105 (2018)Duval, A., Loday-Richaud, M.: Kovacic’s algorithm and its application to some families of special functions. Appl. Algebra Eng. Commun. Comput. 3(3), 211–246 (1992)Hall, R., Saad, N., Ciftci, H.: Sextic harmonic oscillators and orthogonal polynomials. J. Phys. A Math. Gen. 39(26), 8477–8486 (2006)Kovacic, J.: An algorithm for solving second order linear homogeneous diferential equations. J. Symb. Comput. 2(1), 3–43 (1986)Martinet, J., Ramis, J.-P.: Theorie de galois diferentielle et resommation. In: Tournier, E. (ed.) Computer Algebra and Diferential Equations, pp. 117–214. Academic Press, London (1989)Natanzon, G.A.: Investigation of a one dimensional Schrödinger equation that is generated by a hypergeometric equation. Vestnik Leningrad. Univ 10, 22–28 (1971). in RussianRainville, E.D.: Necessary conditions for polynomial solutions of certain Riccati equations. Am. Math. Mon. 43(8), 473–476 (1936)Saad, N., Hall, R., Ciftci, H.: Criterion for polynomial solutions to a class of linear diferential equations of second order. J. Phys. A Math. Gen. 39(43), 13445–13454 (2006)Singer, Michael F.: Moduli of linear diferential equations on the Riemann sphere with fxed Galois groups. Pac. J. Math. 160(2), 343–395 (1993)Turbiner, A.V.: Quantum mechanics: problems intermediate between exactly solvable and completely unsolvable. Soviet Phys. JETP 10(2), 230–236 (1988)Venegas-Gómez, H.: Enfoque galoisiano de la ecuación de schrödinger con potenciales polinomiales y polinomios de laurent. 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