<<

. 25
( 42 .)



>>

a tanh2 g ’ b sechg ’ tanh4 g
V [f (g)] = (5.138)
2 4


© 2001 by Chapman & Hall/CRC
which will induce

1
cosech2 g
VT = 2VT = ’b cosechg cothg ’ En + (5.139)
4

Since from Table 5.1, (5.139) corresponds to the Rosen-Morse II po-
tential, its associated eigenfunctions and energy levels are known. As
before, these determine the corresponding eigenfunctions and eigen-
values of (5.138). Thus we have another CES potential given by

be’x
a 3
V (x) = ’ ’ (5.140)
1 + e’2x (1 + e’2x )1/2 4(1 + e’2x )2

whose parameters a and b are restricted by

1/2
3 1 1
+ b)1/2 ’ ( ’ b)1/2 ’ n +
n+a’ = ( (5.141)
n n
4 2 2

The potential (5.139) also a¬ords a superpotential W (x) in the form

1 b
W (x) = a + ’ (5.142)
2(1 + e2x ) (1 + e2x )1/2

Other examples of CES potentials include the singular one [96]

a b 3
V (x) = + 1/2 ’ (5.143)
32r2
rr

It turns out that the restriction required on the parameters a
and b for (5.143) to be a CES potential is the same as the one for
which this potential can be put in a supersymmetric form [98]. The
list of CES potentials has been expanded to include Natanzan po-
tentials also [99,100]. By ¬xing the free parameters they give rise to
fractional power, long-range and strongly anharmonic terms. Apart
from SUSY, CES potentials have also appeared as ordinary quan-
tum mechanical problems as well. One such instance is a class of
partially solvable rational potentials with known zero-energy solu-
tions. For some of them the zero-energy wave function has been
found to be normalizable and to describe a bound state [94].


© 2001 by Chapman & Hall/CRC
5.10 References
[1] F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep., 251, 267,
1995.

[2] C.V. Sukumar, J. Phys. A. Math. Gen., 18, L57, 1985.

[3] C.V. Sukumar, J. Phys. A. Math. Gen., 18, 2917, 1985.

[4] L.E. Gendenshtein, JETP Lett., 38, 356, 1983.

[5] R. Dutt, A. Khare, and U.P. Sukhatme, Am. J. Phys., 56, 163,
1987.

[6] F. Cooper, J.N. Ginocchio, and A. Khare, Phys. Rev., D36,
2458, 1987.

[7] L. Infeld and T.E. Hull, Rev. Mod. Phys., 23, 21, 1951.

[8] L. Infeld and T.E. Hull, Phys. Rev., 74, 905, 1948.

[9] D. L. Pursey, Phys. Rev., D33, 1048, 1986.

[10] D. L. Pursey, Phys. Rev., D33, 2267, 1986.

[11] M. Luban and D.L. Pursey, Phys. Rev., D33, 431, 1986.

[12] R. Montemayor, Phys. Rev., A36, 1562, 1987.

[13] A. Stahlhofen and K. Bleuler Nuovo Cim, B104, 447, 1989.

[14] V.G. Bagrov and B.F. Samsonov, Theor. Math. Phys., 104,
1051, 1945.

[15] L.J. Boya, Eur. J. Phys., 9, 139, 1988.

[16] R. Montemayor and L.D. Salem, Phys. Rev., A40, 2170, 1989.

[17] E. Schroedinger, Proc. Roy Irish Acad., A46, 9, 1940.

[18] E. Schroedinger, Proc. Roy Irish Acad., A46, 183, 1941.

[19] E. Schroedinger, Proc. Roy Irish Acad., A47, 53, 1941.

[20] H. Weyl, The Theory of Groups and Quantum Mechanics, E.P.
Dutton and Co., New York, 1931.


© 2001 by Chapman & Hall/CRC
[21] P.A.M. Dirac, Principles of Quantum Mechanics, 2nd ed., Claren-
don Press, Oxford, 1947.

[22] A.F.C. Stevenson, Phys. Rev., 59, 842, 1941.

[23] A. Lahiri, P.K. Roy, and B. Bagchi, Int. J. Mod. Phys., A5,
1383, 1990.

[24] A. Khare and U.P. Sukhatme, J. Phys. A. Math. Gen., 26,
L901, 1993.

[25] A.B. Shabat and R.I. Yamilov, Leningrad Math. J., 2, 377,
1991.

[26] V. De. Alfaro, S. Fubini, and G. Furlan, Nuovo Cim., A34,
569, 1976.

[27] A.P. Veslov and A.B. Shabat, Funct. Anal. Appl., 27, n2,1,
1993.

[28] A. Shabat, Inverse Prob., 8, 303, 1992.

[29] V. Spiridonov, Mod. Phys. Lett., A7, 1241, 1992.

[30] V. Spiridonov, Comm. Theor. Phys., 2, 149, 1993.

[31] V. Spiridonov, Phys. Rev. Lett., 69, 398, 1992.

[32] T. Fukui, Phys. Lett., A189, 7, 1994.

[33] A.B. Balantekin, M.A. Cˆndido Riberio, and A.N.F. Aleixo, J.
a
Phys. A. Math. Gen., 32, 2785, 1999.

[34] E.P. Wigner, Phys. Rev., 77, 711, 1950.

[35] N. Mukunda, E.C.G. Sudarshan, J. Sharma, and C.L. Mehta,
J. Math. Phys., 21, 2386, 1980.

[36] J. Jayraman and R. de Lima Rodrigues, J. Phys. A. Math.
Gen., 23, 3123, 1990.

[37] M.A. Vasiliev, Int. J. Mod. Phys., A6, 1115, 1991.

[38] L. Brink, T.H. Hansson, and M.A. Vasiliev, Phys. Lett., B286,
109, 1992.


© 2001 by Chapman & Hall/CRC
[39] L. Brink, T.H. Hansson, S. Konstein, and M.A. Vasilev, Nucl.
Phys., B401, 591, 1993.

[40] T. Brzezi´ski, I.L. Egusquiza, and A.J. Macfarlane, Phys. Lett.,
n
B311, 202, 1993.

[41] B. Bagchi, Phys. Lett., A189, 439, 1994.

[42] A. Jevicki and J.P. Rodrigues, Phys. Lett., B146, 55, 1984.

[43] L.O™ Raifeartaigh and C. Ryan, Proc. Roy. Irish Acad., 62,
93, 1963.

[44] L.M. Yang, Phys. Rev., 84, 788, 1951.

[45] Y. Ohnuki and S. Kamefuchi, J. Math. Phys., 19, 67, 1978.

[46] S. Watanabe, Prog Theor. Phys., 80, 947, 1988.

[47] B. Mielnik, J. Math. Phys., 25, 3387, 1984.

[48] A. Mitra, A. Lahiri, P.K. Roy, and B. Bagchi, Int. J. Theor.
Phys., 28, 911, 1989.

[49] D.J. Fernandez, Lett. Math. Phys., 8, 337, 1984.

[50] P.G. Leach, Physica, D17, 331, 1985.

[51] V. Bargmann, Rev. Mod. Phys., 21, 488, 1949.

[52] C.V. Sukumar, J. Phys. A. Math. Gen., 18, 2937, 1985.

[53] D. Baye, Phys. Rev. Lett., 58, 2738, 1987.

[54] D. Baye and J-M Sparenberg, Phys. Rev. Lett., 73, 2789, 1994.

[55] D. Baye, Phys. Rev., A48, 2040, 1993.

[56] L.U. Ancarani and D. Baye, Phys. Rev., A46, 206, 1992.

[57] J-M Sparenberg, Supersymmetric transformations and the in-
verse problem in quantum mechanics, Ph.D. Thesis, University
of Brussels, Brussels, 1999.

[58] N. Levinson, Phys. Rev., 75, 1445, 1949.


© 2001 by Chapman & Hall/CRC
[59] R.D. Amado, Phys. Rev., A37, 2277, 1988.

[60] G. Levai, D. Baye, and J-M Sparenberg, J. Phys. A. Math.
Gen., 30, 8257, 1997.

[61] B. Talukdar, U. Das, C. Bhattacharyya, and K. Bera, J. Phys.
A. Math. Gen., 25, 4073, 1992.

[62] N. Nag, and R. Roychoudhury, J. Phys. A. Math. Gen., 28,
3525, 1995.

[63] R.D. Amado, F. Cannata, and J.P. Dedonder, Int. J. Mod.
Phys., A5, 3401, 1990.

[64] F. Cannata and M.V. Io¬e, Phys. Lett., B278, 399, 1992.

[65] C. Eckart, Phys. Rev., 35, 1303, 1930.

[66] B. Bagchi and R. Roychoudhury, Mod. Phys. Lett., A12, 65,
1997.

[67] Q.K.K. Liu, Nucl. Phys., A550, 263, 1992.

[68] L.F. Urrutia and E. Hernandez, Phys. Rev. Lett., 51, 755,
1983.

[69] V.A. Kostelecky and M.M. Nieto, Phys. Rev., A32, 1293, 1985.

[70] G. Levai, Lecture Notes in Physics, Springer, Berlin, 427, 127,
1993.

[71] A. Bhattacharjie and E.C.G. Sudarshan, Nuovo Gim, 25, 864,
1962.

[72] A.K. Bose, Nuovo. Cim., 32, 679, 1964.

[73] W. Miller Jr., Lie Theory of Special Functions, Academic Press,
New York, 1968.

[74] G.A. Natanzon, Theor. Math. Phys., 38, 146, 1979.

[75] J.N. Ginocchio, Ann. Phys., 152, 203, 1984.

[76] J.W. Dabrowska, A. Khare, and U. Sukhatme, J. Phys. A.
Math. Gen., 21, L195, 1988.


© 2001 by Chapman & Hall/CRC
[77] A.V. Turbiner, Commun. Math. Phys., 118, 467, 1988.

[78] M.A. Shifman, Int. J. Mod. Phys., A4, 3305, 1989.

[79] O.B. Zaslavskii, J. Phys. A. Math. Gen., 26, 6563, 1993.

[80] G. Levai, J. Phys. A. Math. Gen., 22, 689, 1989.

[81] B.W. Williams and D.P. Poulios, Eur. J. Phys., 14, 222, 1993.

[82] P. Roy, B. Roy, and R. Roychoudhuy, Phys. Lett., A144, 55,
1990.

[83] J.M. Cervaro, Phys. Lett., A153, 1, 1991.

[84] B.W. Williams, J. Phys. A. Math. Gen., 24, L667, 1991.

<<

. 25
( 42 .)



>>