Solutions of the nonrelativistic wave equation with position-dependent effective mass

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 66, Issue 4, 2002, Pages 421161-421167

  Alhaidari, A D ^a  

^a KING FAHD UNIVERSITY OF PETROLEUM AND MINERALS   (Saudi Arabia)

Author keywords

[No Author keywords available]

Indexed keywords

DIFFERENTIAL EQUATIONS; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; PROBLEM SOLVING; WAVE EQUATIONS;

ENERGY SPECTRUM;

QUANTUM THEORY;

EID: 3843143661     PISSN: 10502947     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevA.66.042116     Document Type: Article

References (76)

1. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
2. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
3. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
4. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
5. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
6. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
7. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
8. See, for example, G.A. Natanzon, Teor. Mat. Fiz. 38, 146 (1979); L.E. Gendenshtein, Zh. Eksp. Teor. Fiz. Pis'ma Red. 38, 299 (1983) [JETP Lett. 38, 356 (1983)]; F. Cooper, J.N. Ginocchi, and A. Khare, Phys. Rev. D 36, 2438 (1987); R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys. 56, 163 (1988); ibid. 59, 723 (1991); G. Lévai, J. Phys. A 22, 689 (1989); 27, 3809 (1994); R. De, R. Dutt, and U. Sukhatme, ibid. 25, L843 (1992).
9. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
10. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
11. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
12. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
13. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
14. See, for example, M.F. Manning, Phys. Rev. 48, 161 (1935); A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Cimento 25, 864 (1962); N.K. Pak and I. Sökmen, Phys. Lett. 103A, 298 (1984); H.G. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1986); R. Montemayer, Phys. Rev. A 36, 1562 (1987); G. Junker, J. Phys. A 23, L881 (1990).
15. See, for example, E. Witten, Nucl. Phys. B 185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); C.V. Sukumar, J. Phys. A 18, 2917 (1985); A. Arai, J. Math. Phys. 30, 1164 (1989); F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
16. See, for example, E. Witten, Nucl. Phys. B 185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); C.V. Sukumar, J. Phys. A 18, 2917 (1985); A. Arai, J. Math. Phys. 30, 1164 (1989); F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
17. See, for example, E. Witten, Nucl. Phys. B 185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); C.V. Sukumar, J. Phys. A 18, 2917 (1985); A. Arai, J. Math. Phys. 30, 1164 (1989); F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
18. See, for example, E. Witten, Nucl. Phys. B 185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); C.V. Sukumar, J. Phys. A 18, 2917 (1985); A. Arai, J. Math. Phys. 30, 1164 (1989); F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
19. See, for example, E. Witten, Nucl. Phys. B 185, 513 (1981); F. Cooper and B. Freedman, Ann. Phys. (N.Y.) 146, 262 (1983); C.V. Sukumar, J. Phys. A 18, 2917 (1985); A. Arai, J. Math. Phys. 30, 1164 (1989); F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
20. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
21. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
22. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
23. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
24. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
25. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
26. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
27. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
28. See, for example, B.G. Wybourne, Classical Groups for Physicists (Wiley-Interscience, New York, 1974), and references therein; W. Miller Jr., Lie Theory and Special Functions (Academic, New York, 1968); Y. Alhassid, F. Gürsey, and F. Iachello, Phys. Rev. Lett. 50, 873 (1983); Ann. Phys. (N.Y.) 148, 346 (1983); 167, 181 (1986); Y. Alhassid, F. Iachello, and R. Levine, Phys. Rev. Lett. 54, 1746 (1985); Y. Alhassid, F. Iachello, and J. Wu, Phys. Rev. Lett. 56, 271 (1986); J. Wu and Y. Alhassid, J. Math. Phys. 31, 557 (1990); M.J. Englefield and C. Quesne, J. Phys. A 24, 3557 (1991).
29. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
30. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
31. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
32. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
33. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
34. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
35. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
36. See, for example, A. de Souza-Dutra, Phys. Rev. A 47, R2435 (1993); N. Nag, R. Roychoudhury, and Y.P. Varshni, ibid. 49, 5098 (1994); R. Dutt, A. Khare, and Y.P. Varshni, J. Phys. A 28, L107 (1995); C. Grosche, ibid. 28, 5889 (1995); 29, 365 (1996); G. Lévai and P. Roy, Phys. Lett. A 270, 155 (1998); G. Junker and P. Roy, Ann. Phys. (N.Y.) 264, 117 (1999); R. Roychoudhury, P. Roy, M. Zonjil, and G. Lévai, J. Math. Phys. 42, 1996 (2001).
37. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
38. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
39. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
40. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
41. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
42. See, for example, A.V. Turbiner, Commun. Math. Phys. 118, 467 (1988); M.A. Shifman, Int. J. Mod. Phys. A 4, 2897 (1989); R. Adhikari, R. Dutt, and Y. Varshni, Phys. Lett. A 141, 1 (1989); J. Math. Phys. 32, 447 (1991); R.K. Roychoudhury, Y.P. Varshni, and M. Sengupta, Phys. Rev. A 42, 184 (1990); L.D. Salem and R. Montemayor, ibid. 43, 1169 (1991); M.W. Lucht and P.D. Jarvis, ibid. 47, 817 (1993); A.G. Ushveridze, Quasi-exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994).
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76. In this case, q(x) is obtained as a solution to the equation: exp[- 2λq(x)] + 2ξλq(x) = - (2√2τ/ρ)μ (x).