×

Trapping of quantum particles for a class of time-periodic potentials. A semi-classical approach. (English) Zbl 0634.35062

Summary: We study the time evolution for Schrödinger operators with time- periodic potentials when the classical equations of motion possess periodic orbits. We exhibit a class of time-periodic potentials such that for initial states suitably localized around these periodic orbits, then at the dominant order of the semi-classical approximation, the system is trapped forever at sufficiently large frequency. An estimation of the correction to the semi-classical approximation is given, which yields a minimum “trapping time” for these systems.

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
PDF BibTeX XML Cite

References:

[1] J. Bellissard, Stability and instability in quantum mechanics, Marseille Luminy, preprint. · Zbl 0584.35024
[2] Benilan, M. N.; Audoin, C.: Int. J. Mass spectrom. Ion phys.. 11, 421 (1973)
[3] Bonner, R. F.; Fulford, J. E.; March, R. E.: Int. J. Mass spectrom. Ion phys.. 24, 255 (1977)
[4] Borg, G.: Amer. J. Math.. 71, 67 (1949)
[5] Clarke, F.; Ekeland, I.: Commun. pure appl. Math.. 33, 103 (1980)
[6] Combescure, M.: Ann. inst. Henri Poincaré. 44, 293 (1986)
[7] Cook, R. J.; Shamkland, D. G.; Wells, A. L.: Phys. rev. A. 31, 564 (1985)
[8] Fröhlich, J.; Spencer, T.; Wayne, C. E.: J. stat. Phys.. 42, 247 (1986)
[9] Hagedorn, G.: Commun. math. Phys.. 71, 77 (1980)
[10] Hagedorn, G.: Ann. inst. Henri Poincaré. 42, 363 (1985)
[11] Kato, T.: J. fac. Sci. univ. Tokyo, sect. 1. 17, 241 (1970)
[12] Kato, T.: Israel J. Math.. 13, 135 (1972)
[13] Krein, M. G.: Amer. math. Soc. transl.. 1, 171 (1951)
[14] L. Lassoued, private communication.
[15] W. Magnus and S. Winkler, Tracts in Pure and Applied Mathematics (L. Bers, R. Courant, and J. J. Stokes, Eds.) Interscience, New York/London/Sydney.
[16] Paul, W.; Raether, M.: Z. phys.. 140, 262 (1955)
[17] Rabinowitz, P.: Comm. pure appl. Math.. 31, 157 (1978)
[18] Weinstein, A.: Ann. math.. 108, 507 (1978)
[19] Wuerker, R. F.; Shelton, H.; Langmuir, R. V.: J. appl. Phys.. 30, 342 (1959)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.