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The scattering problem for the Schrödinger equation with a potential linear in time and in space. II. Correctness, smoothness, behavior of the solution at infinity

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Abstract

We study the scattering problem associated with the behavior of whispering gallery waves near the inflection point of the boundary. In order to solve the scattering problem, we prove the theorems of existence, uniqueness and smoothness of the solution. The formal asymptotic behavior is justified for t→−∞ and superexponential smallness of the wave field in the shadow zone is proved.

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Literature cited

  1. V. M. Babich and V. P. Smyshlyaev, “The scattering problem for the Schrodinger equation with potential linear in time and in space,” Dokl. Akad. Nauk SSSR,280, No. 6, 1330–1333 (1985).

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  2. M. M. Popov, “The problem of whispering gallery waves in the neighborhood of a simple zero of effective curvature of the boundary,” J. Sov. Math.,11, No. 5 (1979).

  3. V. M. Babich and V. P. Smyshlyaev, “The scattering problem for the Schrodinger equation with potential linear in time and in space. I. Shadow zone asymptotics,” J. Sov. Math.,32, No. 2 (1986).

  4. M. M. Popov, “A whispering gallery in the neighborhood of the boundary inflection point. Wave field asymptotics for t→∞,” J. Sov. Math.,32, No. 2 (1986).

  5. M. M. Popov and V. G. Krasavin, “The radiation directional diagram in a problem with boundary inflection point,” J. Sov. Math.,32, No. 2 (1986).

  6. O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Embedding Theorems [in Russian], Moscow (1975).

  7. O. A. Ladyzhenskaya (ed.), Boundary Value Problems of Mathematical Physics, Amer. Math. Soc. (1977).

  8. M. C. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Academic Press (1972).

  9. T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag (1966).

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Authors
  1. V. M. Babich
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  2. V. P. Smyshlyaev
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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 13–29, 1985.

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Babich, V.M., Smyshlyaev, V.P. The scattering problem for the Schrödinger equation with a potential linear in time and in space. II. Correctness, smoothness, behavior of the solution at infinity. J Math Sci 38, 1562–1576 (1987). https://doi.org/10.1007/BF01100135

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  • DOI: https://doi.org/10.1007/BF01100135

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