Abstract
Methods of generating exactly integrable potentials for the Schrödinger equation are consolidated within the framework of a simple construction. The Abraham-Moses method is generalized to the case of the nonstationary Schrödinger equation. An algorithm is proposed for solving the Schrödinger equation based on nonlocal symmetry operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
V. G. Bagrov, A. V. Shapovalov, and I. V. Shirokov, Phys. Lett. A,147, No. 7, 348–350 (1990).
V. G. Bagrov, A. V. Shapovalov, and I. V. Shirokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 11, 114–116 (1989).
G. Darboux, C. R. Acad. Sci., Paris,94, 1456 (1982).
P. B. Abraham and H. E. Moses, Phys. Rev. A,22, No. 4, 1333–1340 (1980).
M. Luban and D. L. Pursey, Phys. Rev. D,33, No. 2, 431–436 (1986).
D. L. Pursey, Phys. Rev. D,33, No. 4, 1048–1055 (1986).
D. Levi, Inverse Problems, No. 4, 165–172 (1988).
Humi Mayer, J. Phys. A: Math. Gen.,21, 2075–2084 (1988).
O. V. Kaptsov, Dokl. Akad. Nauk SSSR,262, No. 5, 1056–1059 (1982).
A. M. Vinogradov and I. S. Krasil'shchik, Dokl. Akad. Nauk SSSR,275, No. 5, 1044–1049 (1984).
V. I. Fushchich, Dokl. Akad. Nauk SSSR,246, No. 4, 846–850 (1979).
K. Kiso, Hokkaido Math. J.,18, No. 1, 125–136 (1989).
N. Kh. Ibraginov, Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983).
A. V. Mikhailov, A. B. Shabat, and R. I. Yamilov, Usp. Mat. Nauk,44, No. 4, 3–53 (1987).
G. Neugebauer and R. Meinel, Phys. Lett.,A100, No. 9, 467–470 (1984).
V. B. Matveev, M. A. Salle, and A. V. Rybin, Inverse Problems, No. 4, 173–183 (1988).
V. E. Zakharov, “Method of inverse scattering problems,” in: Solitons, R. K. Bullough and P. J. Caudrey (eds.), Springer-Verlag, New York (1980).
V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977).
B. M. Levitan, Inverse Sturm-Liouville Problems [in Russian], Nauka, Moscow (1984).
V. I. Malkin and V. I. Man'ko, Pis'ma Zh. Éksp. Teor. Fiz.,2, 230–234 (1965).
V. N. Shapovalov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 6, 57–64, 64–70 (1977).
W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading, MA (1977).
A. V. Ovsyannikov (Ovsiannikov), Group Analysis of Differential Equations, Academic Press, New York (1982).
P. Oliver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986).
V. N. Shapovalov and N. B. Sukhomlin, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 12, 100–105 (1974).
Author information
Authors and Affiliations
Additional information
Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 19–25, September, 1991.
Rights and permissions
About this article
Cite this article
Bagrov, V.G., Shapovalov, A.V. & Shirokov, I.V. Methods of generating integrable potentials for the Sochrödinger equation and nonlocal symmetries. Soviet Physics Journal 34, 755–761 (1991). https://doi.org/10.1007/BF00896705
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00896705