Solutions for confluent and double-confluent Heun equations

Journal of Mathematical Physics

Volumn 49, Issue 8, 2008, Pages

  El Jaick, Ĺa Jaccoud ^a   Figueiredo, Bartolomeu D B ^a  

^a CENTRO BRASILEIRO DE PESQUISAS FÍSICAS   (Brazil)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 50849091257     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2970150     Document Type: Article

References (39)

1. Handbook of Mathematical Functions, edited by, Abramowitz, M., and, Stegun, I. A., (Dover, New York, 1965).
2. Arscott, F. M., Periodic Differential Equations (Pergamon, Oxford, 1964).
3. Barber, W. G. and Hasś, H. R., " The two centre problem in wave mechanics.," Proc. Cambridge Philos. Soc. 25, 564 (1935). 0008-1981
4. Birrel, N. D. and Davies, P. C. W., Quantum Fields in Curved Spaces (Cambridge University Press, Cambridge, 1982).
5. Bühring, W., " Schrödinger equation with inverse fourth-power potential, a differential equation with two irregular singular points.," J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1666831 15, 1451 (1974).
6. Cho, H. T. and Ho, C. L., " A novel quasi-exactly solvable model with total transmission modes.," J. Phys. A: Math. Theor. ZZZZZZ 1751-8113 41, 172002 (2008); e-print arXiv:quant-ph/0606144v3.
7. Darboux, G., " Sur une ́quation lińaire.," Compt. Rend. COREAF 0001-4036 94, 1645 (1882).
8. Decarreau, A., Maroni, P., and Robert, A., " Sur les ́quations confluentes de l'́quation de Heun.," Ann. Soc. Sci. Bruxelles, Ser. 1 ASSBAH 0037-959X T92, 151 (1978).
9. Decarreau, A., Dumont-Lepage, M. C., Maroni, P., Robert, A. and Ronveaux, A., " Formes canoniques des ́quations confluentes de l'́quation de Heun.," Ann. Soc. Sci. Bruxelles, Ser. 1 ASSBAH 0037-959X T92, 53 (1978).
10. Erd́lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 1.
11. Erd́lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G., Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2.
12. Figueiredo, B. D. B., " On some solutions to generalized spheroidal wave equations and applications.," J. Phys. A JPHAC5 0305-4470 10.1088/0305-4470/35/12/312 35, 2877 (2002).
13. Figueiredo, B. D. B., " Ince's limits for confluent and double-confluent Heun equations.," J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2104267 46, 113503 (2005).
14. Figueiredo, B. D. B., " Generalized spheroidal wave equation and limiting cases.," J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2406057 48, 013503 (2007).
15. Frank, W. M., Land, D. J., and Spector, R. M., " Singular potentials.," Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.43.36 43, 36 (1971).
16. González-López, A., Kamran, N., and Olver, P. J., " Normalizability of one-dimensional quasi-exactly solvable Schrödinger operators.," Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF02099042 153, 117 (1993).
17. Humbert, P., Fonctions de Laḿ et fonctions de Mathieu, Ḿmorial des Sciences Math́matiques (Gauthier-Villards, Paris, 1926), Vol. X.
18. Hylleraas, E. A., " Über die Elektronenterme des Wasserstoffmoleküls.," Z. Phys. ZEPYAA 0044-3328 10.1007/BF01344443 71, 739 (1931).
19. Ince, E. L., " A linear differential equation with periodic coefficients.," Proc. London Math. Soc. 23, 56 (1925). 0024-6115
20. Jaff́, G., " Zur Theorie des Wasserstoffmolekülions., " Z. Phys. ZEPYAA 0044-3328 10.1007/BF01333263 87, 535 (1934).
21. Kalnins, E. G., Miller, W., and Pogosyan, G. S., " Exact and quasiexact solvability of second-order superintegrable quantum systems: I. Euclidian space preliminaries.," J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.2174237 47, 033502 (2006).
22. Kleinman, C. J., Hahn, Y., and Spruch, L., " Dominant Nonadiabatic Contribution to the Long-Range Electron-Atom Interaction.," Phys. Rev. 10.1103/PhysRev.165.53 165, 53 (1968). 0031-899X
23. Leaver, E. W., " Solutions to a generalized spheroidal wave equation: Teukolsky equations in general relativity, and the two-center problem in molecular quantum mechanics.," J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.527130 27, 1238 (1986).
24. Lemieux, A. and Bose, A. K., " Construction de potentiels pour lesquels l'́quation de Schrödinger est soluble.," Ann. Inst. Henri Poincare, Sect. A 10, 259 (1969). 0020-2339
25. Mano, S., Suzuki, H., and Takasugi, E., " Analytic solutions of the Teukolsky equation and their low frequency expansions.," Prog. Theor. Phys. PTPKAV 0033-068X 10.1143/PTP.95.1079 95, 1079 (1996).
26. Mano, S., Suzuki, H., and Takasugi, E., " Analytic solutions of the Regge-Wheeler equation and the post-Minkowskian expansion.," Prog. Theor. Phys. PTPKAV 0033-068X 10.1143/PTP.96.549 96, 549 (1996).
27. McLachlan, E. W., Theory and Application of Mathieu Functions (Dover, New York, 1964).
28. Meixner, J. and Schäfke, F. W., Mathieusche Funktionen und Sphäroidfunctionen (Springer-Verlag, Berlin, 1954).
29. Olver, F. M. J., Asymptotics and Special Functions (Academic, New York, 1974).
30. Heun's Differential Equations, edited by, Ronveaux, A., (Oxford University Press, New York, 1995).
31. Schrödinger, E., " Eigenschwingungen des sphärischen Raumes.," Commentationes Pontificiae Academiae Scientiarum 2, 321 (1938).
32. Schrödinger, E., " Maxwell's and Dirac's equations in the expanding universe.," Proc. R. Ir. Acad., Sect. B 2, 25 (1940). 0035-8983
33. Turbiner, A. V., " Quantum mechanics: problems intermediate between exactly solvable and completely unsolvable.," Sov. Phys. JETP 67, 230 (1988). 0038-5646
34. Turbiner, A. V., " Quasi-exactly-solvable problems and sl (2) algebra.," Commun. Math. Phys. CMPHAY 0010-3616 10.1007/BF01466727 118, 467 (1988).
35. Ushveridze, A. G., " Quasi-exactly solvable models in quantum mechanics.," Sov. J. Part. Nucl. SJPNA3 0090-4759 20, 504 (1989).
36. Ushveridze, A. G., Quasi-Exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
37. Wilson, A. H., " A generalised spheroidal wave equation.," Proc. R. Soc. London, Ser. A PRLAAZ 0950-1207 10.1098/rspa.1928.0073 118, 617 (1928).
38. Wilson, A. H., " The ionised hydrogen molecule.," Proc. R. Soc. London, Ser. A 118, 635 (1928). 0950-1207
39. Zaslavskii, O. B. and Ulyanov, V. V., " New classes of exact solutions of the Schrödinger equation and the potential-field description of spin systems.," Sov. Phys. JETP 60, 991 (1984). 0038-5646