Perturbation of an eigen-value from a dense point spectrum : a general Floquet hamiltonian
Duclos, P. ; Šťovíček, P. ; Vittot, M.
Annales de l'I.H.P. Physique théorique, Tome 71 (1999), p. 241-301 / Harvested from Numdam
Publié le : 1999-01-01
@article{AIHPA_1999__71_3_241_0,
     author = {Duclos, Pierre and \v S\v tov\'\i \v cek, P. and Vittot, M.},
     title = {Perturbation of an eigen-value from a dense point spectrum : a general Floquet hamiltonian},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {71},
     year = {1999},
     pages = {241-301},
     mrnumber = {1714346},
     zbl = {0972.81041},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1999__71_3_241_0}
}
Duclos, P.; Šťovíček, P.; Vittot, M. Perturbation of an eigen-value from a dense point spectrum : a general Floquet hamiltonian. Annales de l'I.H.P. Physique théorique, Tome 71 (1999) pp. 241-301. http://gdmltest.u-ga.fr/item/AIHPA_1999__71_3_241_0/

[1] V.I. Arnold, Small divisors II. Proof of the A.N. Kolmogorov Theorem on conservation of conditionally periodic motions under small perturbations of the Hamiltonian function, Usp. Mat. Nauk 118 (1963) 13-40. | MR 163025 | Zbl 0129.16606

[2] J. Bellissard, Stability and instability in quantum mechanics, in: Albeverio and Blanchard, eds., Trends and Developments in the Eighties, Word Scientific, Singapore, 1985, pp. 1-106. | MR 853743 | Zbl 0584.35024

[3] P.M. Bleher, H.R. Jauslin and J.L. Lebowitz, Floquet spectrum for two-level systems in quasi-periodic time dependent fields, J. Stat. Phys. 68 (1992) 271. | Zbl 0925.58074

[4] K.L. Chung, A Course in Probability Theory, Academic Press, 1970. | MR 1796326

[5] M. Combescure, The quantum stability problem for time-periodic perturbations of the harmonic oscillator, Ann. Inst. Henri Poincaré 47 (1987) 62-82; Erratum, Ann. Inst. Henri Poincaré 47 (1987) 451-454. | Numdam | Numdam | MR 933686 | Zbl 0628.70017

[6] P. Duclos and P. Šťovíček, Floquet Hamiltonians with pure point spectrum, Commun. Math. Phys. 177 (1996) 327-347. | MR 1384138 | Zbl 0848.34072

[7] P. Duclos, P. Šťovíček and M. Vittot, Perturbation of an eigen-value from a dense point spectrum: an example, J. Phys. A: Math. Gen. 30 (1997) 7167-7185. | MR 1601924 | Zbl 0932.81007

[8] L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV (15) (1988) 115-147. | Numdam | MR 1001032 | Zbl 0685.58024

[9] V. Enss and K. Veselić, Bound states and propagating states for time-dependent Hamiltonians, Ann. Inst. Henri Poincaré 39 (1983) 159-191. | Numdam | MR 722684 | Zbl 0532.47007

[10] J.S. Howland, Scattering theory for Hamiltonians periodic in time, Indiana J. Math. 28 (1979) 471-494. | MR 529679 | Zbl 0444.47010

[11] J.S. Howland, Floquet operators with singular spectrum I, Ann. Inst. Henri Poincaré 49 (1989) 309-323; Floquet operators with singular spectrum II, Ann. Inst. Henri Poincaré 49 (1989) 325-334. | Numdam | Numdam | MR 1017967 | Zbl 0689.34022

[12] A. Joye, Absence of absolutely continuous spectrum of Floquet operators, J. Stat. Phys. 75 (1994) 929-952. | MR 1285294 | Zbl 0835.47010

[13] T. Kato, Perturbation Theory of Linear Operators, Springer, New York, 1966. | MR 203473 | Zbl 0148.12601

[14] A.N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbations of the Hamiltonian function, Dokl. Akad. Nauk SSSR 98 (1954) 527-530. | MR 68687

[15] A. Messiah, Mécanique Quantique II, Dunod, Paris, 1964. | MR 129304

[16] J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, II. Math. Phys. Kl. 11a (1962) 1-20. | MR 147741 | Zbl 0107.29301

[17] G. Nenciu, Floquet operators without absolutely continuous spectrum, Ann. Inst. Henri Poincaré 59 (1993) 91-97. | Numdam | MR 1244183 | Zbl 0795.47032

[18] C.R. De Oliviera, I. Guarneri and G. Casati, From power-localization to extended quasi-energy eigenstates in a quantum periodically driven system, Europhys. Lett. 27 (1994) 187-192.

[19] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978. | MR 751959 | Zbl 0401.47001

[20] F. Rellich, Störungstheorie der Spektralzerlegung, I, Math. Ann. 113 (1937) 600- 619. | JFM 62.0452.04 | MR 1513109

[21] H. Sambe, Steady states and quasienergies of a quantum-mechanical system in a oscillating field, Phys. Rev. A 17 (1973) 2203-2213.

[22] J.H. Shirley, Solution of the Schrödinger equation with a Hamiltonian periodic in time, Phys. Rev. B 138 (1965) 979.

[23] C.L. Siegel, Iterations of analytic functions, Ann. Math. 143 (1942) 607-612. | Zbl 0061.14904

[24] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, NJ, 1970. | MR 290095 | Zbl 0207.13501

[25] K. Yajima, Scattering theory for Schrödinger equations with potential periodic in time, J. Math. Soc. Japan 29 (1977) 729-743. | MR 470525 | Zbl 0356.47010