Abstract:
We consider a ten-parameter second-order ordinary linear differential equation with four singular points. Three of them are finite and regular, while the fourth is irregular at infinity. We use the tridiagonal representation approach to obtain a solution of the equation as a bounded infinite series of square-integrable functions written in terms of Jacobi polynomials. The expansion coefficients of the series satisfy a three-term recurrence relation, which is solved in terms of a modified version of the continuous Hahn orthogonal polynomial. We present a physical application in which we identify the quantum mechanical systems that could be described by the differential equation, give the corresponding class of potential functions and energy in terms of the equation parameters, and write the system wave function.
Citation:
A. D. Alhaidari, “Series solution of a ten-parameter second-order differential equation with three regular singularities and one irregular singularity”, TMF, 202:1 (2020), 20–33; Theoret. and Math. Phys., 202:1 (2020), 17–29
\Bibitem{Alh20}
\by A.~D.~Alhaidari
\paper Series solution of a~ten-parameter second-order differential equation with three regular singularities and one irregular singularity
\jour TMF
\yr 2020
\vol 202
\issue 1
\pages 20--33
\mathnet{http://mi.mathnet.ru/tmf9707}
\crossref{https://doi.org/10.4213/tmf9707}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4045702}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2020TMP...202...17A}
\transl
\jour Theoret. and Math. Phys.
\yr 2020
\vol 202
\issue 1
\pages 17--29
\crossref{https://doi.org/10.1134/S0040577920010031}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000521153500003}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85082034555}
Linking options:
https://www.mathnet.ru/eng/tmf9707
https://doi.org/10.4213/tmf9707
https://www.mathnet.ru/eng/tmf/v202/i1/p20
This publication is cited in the following 7 articles:
A. G. M. Schmidt, M. E. Pereira, “A quantum analog model for a scalar particle interacting with a Kerr–de Sitter black hole”, Annals of Physics, 458 (2023), 169465
A. D. Alhaidari, “Progressive approximation of bound states by finite series of square-integrable functions”, Journal of Mathematical Physics, 63:8 (2022)
I. A. Assi, A. D. Alhaidari, H. Bahlouli, “Deformed Morse-like potential”, J. Math. Phys., 62:9 (2021), 093501
A. J. Sous, “Studying novel 1D potential via the aim”, Mod. Phys. Lett. A, 36:20 (2021), 2150141
A. D. Alhaidari, H. Bahlouli, “Solutions of a Bessel-type differential equation using the tridiagonal representation approach”, Rep. Math. Phys., 87:3 (2021), 313–327
Alhaidari A.D., “Open Problem in Orthogonal Polynomials”, Rep. Math. Phys., 84:3 (2019), 393–405