Geodesic
Series solution of a ten-parameter second-order differential equation with three regular singularities and one irregular singularity
Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 20-33
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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.
Keywords: differential equation, tridiagonal representation, $J$-matrix method, recurrence relation, continuous Hahn polynomial. 
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A. D. Alhaidari. Series solution of a ten-parameter second-order differential equation with three regular singularities and one irregular singularity. Teoretičeskaâ i matematičeskaâ fizika, Tome 202 (2020) no. 1, pp. 20-33. http://geodesic.mathdoc.fr/item/TMF_2020_202_1_a2/

[1] A. D. Alhaidari, “Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications”, J. Math. Phys., 59:6 (2018), 063508, 22 pp., arXiv: 1802.09708 | DOI | MR

[2] A. D. Alhaidari, “Solution of the nonrelativistic wave equation using the tridiagonal representation approach”, J. Math. Phys., 58:7 (2017), 072104, 37 pp., arXiv: 1703.01268 | DOI | MR

[3] E. J. Heller, H. A. Yamani, “New $L^2$ approach to quantum scattering: Theory”, Phys. Rev. A, 9:3 (1974), 1201–1208 | DOI

[4] E. J. Heller, H. A. Yamani, “$J$-matrix method: Application to $S$-wave electron-hydrogen scattering”, Phys. Rev. A, 9:3 (1974), 1209–1214 | DOI

[5] H. A. Yamani, W. P. Reinhardt, “$L^2$ discretizations of the continuum: Radial kinetic energy and Coulomb Hamiltonian”, Phys. Rev. A, 11:4 (1975), 1144–1156 | DOI | MR

[6] H. A. Yamani, L. Fishman, “$J$-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering”, J. Math. Phys., 16 (1975), 410–420 | DOI

[7] M. E. H. Ismail, E. Koelink, “The $J$-matrix method”, Adv. Appl. Math., 46:1–4 (2011), 379–395 | DOI | MR

[8] M. E. H. Ismail, E. Koelink, “Spectral properties of operators using tridiagonalization”, Anal. Appl. (Singap.), 10:3 (2012), 327–343 | DOI | MR

[9] M. E. H. Ismail, E. Koelink, “Spectral analysis of certain Schrödinger operators”, SIGMA, 8 (2012), 061, 19 pp. | DOI | MR

[10] V. X. Genest, M. E. H. Ismail, L. Vinet, A. Zhedanov, “Tridiagonalization of the hypergeometric operator and the Racah–Wilson algebra”, Proc. AMS, 144:10 (2016), 4441–4454 | DOI | MR

[11] A. D. Alhaidari, “Series solutions of Heun-type equation in terms of orthogonal polynomials”, J. Math. Phys., 59:11 (2018), 113507, 12 pp. | DOI | MR

[12] K. Heun, “Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit vier Verzweigungspunkten”, Math. Ann., 33:2 (1889), 161–179 | DOI | MR

[13] A. Ronveaux (ed.), Heun's Differential Equations, Oxford Univ. Press, New York, 1995 | MR | Zbl

[14] S. Yu. Slavyanov, V. Lai, Spetsialnye funktsii: edinaya teoriya, osnovannaya na analize osobennostei, Nevskii dialekt, SPb., 2002 | MR | Zbl

[15] A. D. Alhaidari, M. E. H. Ismail, “Quantum mechanics without potential function”, J. Math. Phys., 56:7 (2015), 072107, 19 pp., arXiv: 1408.4003 | DOI | MR

[16] G. Beitmen, A. Erdeii, Vysshie transtsendentnye funktsii, v. 3, Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mate, Nauka, M., 1967 | MR | Zbl

[17] G. Szegő, Orthogonal Polynomials, American Mathematical Society, Colloquium Publications, XXIII, AMS, Providence, RI, 1975 | MR

[18] T. S. Chihara, An Introduction to Orthogonal Polynomials, Mathematics and its Applications, 13, Gordon and Breach, New York, 1978 | MR

[19] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge Univ. Press, Cambridge, 2009 | MR

[20] R. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their $q$-Analogues, Springer, Berlin, 2010 | MR

[21] W. Van Assche, “Compact Jacobi matrices: from Stieltjes to Krein and $M(a;b)$”, Ann. Fac. Sci. Toulouse Math., Ser. 6, S5 (1996), 195–215 | MR | Zbl

[22] P. Moon, D. E. Spencer, Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions, Springer, Berlin, 1988 | MR