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Semi-classical bounds on scattering cross sections in two dimensional magnetic fields

Published online by Cambridge University Press:  22 January 2016

Hideo Tamura
Hideo Tamura*
Affiliation:
Department of Mathematics, Ibaraki University, Mito, Ibaraki 310, Japanv, [email protected]
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Abstract

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We prove the uniform boundedness of averaged total cross sections or of quantities related to scattering into cones in the semi-classical limit for scattering by two dimensional magnetic fields. We do not necessarily assume that the energy under consideration is in a non-trapping energy range in the sense of classical dynamics.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 147 , September 1997 , pp. 25 - 61
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[1] Agmon, S., Some new results in spectral and scattering theory of differential operators on Rn , Seminaire Goulaouic-Schwartz, 1978.Google Scholar
[2] Amrein, W. O., Jauch, J. M. and Sinha, K. B., Scattering Theory in Quantum Mechanics, W. A. Benjamin. Inc., 1977.Google Scholar
[3] Avron, J., Herbst, I. and Simon, B., Schrödinger operators with magnetic fields. I, General interactions, Duke Math. J., 45 (1978), 847–883.Google Scholar
[4] Enss, V. and Simon, B., Finite total cross sections in nonrelativistic quantum mechanics, Commun. Math. Phys., 76 (1980), 177–209.Google Scholar
[5] Erdélyi, A., Higher Transcendental Functions, Vol. II, Robert E. Krieger Publ. Company Inc., 1953.Google Scholar
[6] Gérard, C. and Martinez, A., Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris, 306 (1988), 121–123.Google Scholar
[7] Gérard, C., Martinez, A. and Robert, D., Breit-Wigner formulas for the scattering phase and the total scattering cross-section in the semi-classical limit, Commun. Math. Phys., 121 (1989), 323–336.Google Scholar
[8] Gohberg, I. C. and Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, A. M. S., 1969.Google Scholar
[9] Ikebe, T. and Saitō, Y., Limiting absorption method and absolute continuity for the Schrödinger operators, J. Math. Kyoto Univ., 7 (1972), 513–542.Google Scholar
[10] Isozaki, H. and Kitada, H., Scattering matrices for two-body Schrödinger operators, Sci. Papers College Arts Sci. Univ. Tokyo, 35 (1985), 81–107.Google Scholar
[11] Loss, M. and Thaller, B., Scattering of particles by long-range magnetic fields, Ann. of Phys., 176 (1987), 159–180.CrossRefGoogle Scholar
[12] Perry, P. A., Scattering Theory by the Enss Method, Mathematical Reports 1, Harwood Academic, 1983.Google Scholar
[13] Robert, D., Autour de l’approximation Semi-classique, Birkhäuser, 1987.Google Scholar
[14] Robert, D. and Tamura, H., Semi-classical estimates for resolvents and asymptotics for total scattering cross-sections, Ann. Inst. Henri Poincaré, 46 (1987), 415–442.Google Scholar
[15] Ruijsenaars, S. N. M., The Aharonov-Bohm effect and scattering theory, Ann. of Phys., 146 (1983), 1–34.CrossRefGoogle Scholar
[16] Seeley, R., An estimate near the boundary for the spectral function of the Laplace operator, Amer. J. Math., 102 (1980), 869–902.Google Scholar
[17] Sobolev, A. V., On the total scattering cross section for a finite-range potential, Leningrad Math. J., 1 (1990), 1015–1026.Google Scholar
[18] Sobolev, A. V. and Yafaev, D. R., On the quasi-classical limit of the total scattering cross-section in nonrelativistic quantum mechanics, Ann. Inst. Henri Poincaré, 44 (1986), 195–210.Google Scholar
[19] Tamura, H., Semi-classical analysis for total cross sections of magnetic Schrödinger operators in two dimensions, Rev. Math. Phys., 7 (1995), 443–480.Google Scholar
[20] Tamura, H., Shadow scattering by magnetic fields in two dimensions, Ann. Inst. Henri Poincaré, 63 (1995), 253–276.Google Scholar