SIGMA 1 (2005), 020, 7 pages      quant-ph/0511238      https://doi.org/10.3842/SIGMA.2005.020

Exact Propagators for Soliton Potentials

Andrey M. Pupasov and Boris F. Samsonov
Department of Physics, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia

Received October 01, 2005, in final form November 21, 2005; Published online November 24, 2005

Abstract
Using the method of Darboux transformations (or equivalently supersymmetric quantum mechanics) we obtain an explicit expression for the propagator for the one-dimensional Schrödinger equation with a multi-soliton potential.

Key words: Darboux transformations; SUSY QM; soliton potentials; propagator.

pdf (176 kb)   ps (137 kb)   tex (10 kb)

References

1. Witten E., Dynamical breaking of supersymmetry, Nucl. Phys. B, 1981, V.188, 513-554.
2. Samsonov B.F., SUSY transformations between diagonalizable and non-diagonalizable Hamiltonians, J. Phys. A: Math. Gen., 2005, V.38, L395-L403, quant-ph/0503075.
   Samsonov B.F., Spectral singularities of non-Hermitian Hamiltonians and SUSY transformations, J. Phys. A: Math. Gen., 2005, V.38, L571-L579, quant-ph/0507085.
3. Andrianov A.A., Borisov N.V., Ioffe M.V., Eides M.I., SUSY QM: new look at the equivalence of quantum systems, Theor. Math. Phys., 1984, V.61, 17-28.
4. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, Deuxiéme partie, Paris, Gautier-Villar at Fils, 1889.
5. Matveev V.B., Salle M.A., Darboux transformations and solitons, Berlin, Springer, 1991.
6. Rajaraman R., Solitons and instantons, North Holland Publ., 1982.
7. Jauslin H.R., Exact propagator and eigenfunctions for multistable models with arbitrarily prescribed N lowest eigenvalues, J. Phys. A: Math. Gen., 1988, V.21, 2337-2350.
8. Samsonov B.F., Sukumar C.V., Pupasov A.M., SUSY transformation of the Green function and a trace formula, J. Phys. A: Math. Gen., 2005, V.38, 7557-7565, quant-ph/0507160.
9. Bagrov V.G., Samsonov B.F., Darboux transformations, factorization, supersymmetry in the one-dimensional quantum mechanics, Theor. Math. Phys., 1995, V.104, 356-367.
   Bagrov V.G., Samsonov B.F., Darboux transformations in quantum mechanics, Phys. Part. Nucl., 1997, V.28, 374-397.
10. Mielnik B., Rosas-Ortiz O., Factorization: little or great algorithm?, J. Phys. A: Math. Gen., 2004, V.43, 10007-10035.
    Fernandez D.J.C., Fernandez-Garcia N., Higher-order supersymmetric quantum mechanics, AIP Conf. Proc., 2005, V.744, 236-273, quant-ph/0502098.
11. Crum M., Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2), 1955, V.6, 121-127.
    Krein M.G., On a continual analogue of the Christoffel formula from the theory of orthogonal polynomials, Dokl. Akad. Nauk SSSR, 1957, V.113, 970-973.
12. Sukumar C.V., Supersymmetry and potentials with bound states at arbitrary energies and multisolitons configurations, J. Phys. A: Math. Gen., 1986, V.19, 2297-2316.
    Sukumar C.V., Supersymmetry and potentials with bound states at arbitrary energies, J. Phys. A: Math. Gen., 1987, V.20, 2461-2481.
13. Cohen-Tannoudji C., Diu B., Laloë F., Mécanique quantique, tome 1, Paris, Hermann, 1977.
14. Man'ko V.I., Chikhachev A.S., Integrals of motion and exact solutions of the problem of two dispersing delta-wells, JETP, 1998, V.86, 335-343.