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Generation of new exactly solvable potentials of a nonstationary Schrödinger equation. (English. Russian original) Zbl 0745.35032
Theor. Math. Phys. 87, No. 3, 635-640 (1991); translation from Teor. Mat. Fiz. 87, No. 3, 426-433 (1991).
Summary: A method for generating integrable potentials of a nonstationary Schrödinger equation (i.e., with time-dependent potential) is developed on the basis of the method of “dressing” of linear differential operators. Potentials that admit separation of variables generate classes of nonseparating potentials for which the Schrödinger equation has nonlocal symmetry operators.

MSC:
35Q40 PDEs in connection with quantum mechanics
35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
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[1] V. A. Marchenko, Sturm-Liouville Operators and Their Applications [in Russian], Naukova Dumka, Kiev (1977). · Zbl 0399.34022
[2] B. M. Levitan, Invertible Sturm-Liouville Problems [in Russian], Nauka, Moscow (1984).
[3] V. E. Zakharov and A. B. Shabat, Funktsional. Analiz i Ego Prilozhen.,8, 43 (1974).
[4] V. E. Zakharov, ?The inverse scattering method,? in: Solitons [Russian translation], (eds. R. K. Bullough and P. J. Caudrey), Mir, Moscow (1983), pp. 270-309. (Originally published by Springer, Berlin (1980).)
[5] V. N. Shapovalov, Differents. Uravneniya.,16, 1864 (1980).
[6] B. G. Konopel’chenko and V. G. Mokhnachev, Yad. Fiz.,30, 559 (1979).
[7] O. V. Kaptsov, Dokl. Akad. Nauk SSSR,262, 1056 (1982).
[8] I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, in: Modern Problems of Mathematics. Latest Developments. (Reviews of Science and Technology), Vol. 34 [in Russian], VINITI, Moscow (1989), pp. 3-83.
[9] V. I. Fushchich and A. G. Nikitin, Fiz. Elem. Chastits At. Yadra,14, 5 (1983).
[10] V. G. Bagrov, D. M. Gitman, I. M. Ternov, et al., Exact Solutions of Relativistic Wave Equations [in Russian], Nauka, Novosibirsk (1982).
[11] P. B. Abraham and H. E. Moses, Phys. Rev. A,22, 1333 (1980).
[12] D. L. Pursey, Phys. Rev. D,33, 1048 (1986).
[13] D. L. Pursey, Phys. Rev. D,36, 1103 (1987).
[14] V. G. Bagrov, A. V. Hapovalov, and I. V. Shirokov, Izv. Vyssh, Uchebn. Zaved. Fiz., No. 11, 114 (1989).
[15] V. G. Bagrov, A. V. Shapovalov, and I. V. Shirokov, Phys. Lett. A,147, 348 (1990).
[16] V. N. Shapovalov and N. B. Sukhomlin, Izv. Vyssh. Uchebn. Zaved. Fiz., No. 12, 100 (1974).
[17] E. Schrödinger, Naturwissenschaften,14, 664 (1926). · JFM 52.0967.01
[18] I. A. Malkin and V. I. Man’ko, Dynamical Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).
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