Time-dependent Floquet theory and absence of an adiabatic limit

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 56, Issue 5, 1997, Pages 4045-4054

  Hone, Daniel W ^a,b   Ketzmerick, Roland ^b,c   Kohn, Walter ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001147148     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.56.4045     Document Type: Article

References (14)

1. For a review, see S.-I Chu, Adv. Chem. Phys. 73, 739 (1989).ADCPAA
2. There are many such papers, including those found as references in 1 above. A particularly relevant example, demonstrating the effect of choice of size and nature of the finite set of basis states, is H. P. Breuer and M. Holthaus, Z. Phys. D 11, 1 (1989).ZDACE2
3. J. S. Howland, Ann. Inst. Henri Poincaré Phys. Theor. 49, 309 (1989); AIPTEO
4. 49, 325 (1989).
5. H. Weyl, Math. Ann. 77, 313 (1916).
6. See, e.g., H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory (Princeton University Press, Princeton, 1981), p. 69.
7. H. Sambé, Phys. Rev. A 7, 2203 (1973).PLRAAN
8. A similarly useful analysis of energy as a function of complex quasimomentum, for electrons in crystalline solids, was made many years ago by one of us: W. Kohn, Phys. Rev. 115, 809 (1959).PHRVAO
9. M. Holthaus, Phys. Rev. Lett. 69, 1596 (1992).PRLTAO
10. See, e.g., L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Addison-Wesley, Reading, MA, 1958), Sec. 87.
11. The fundamental difference between periodically driven systems with increasing and those with decreasing spacings between successive energy levels has been emphasized by N. Brenner and S. Fishman, J. Phys. A 28, 5973 (1995); JPHAC5
12. 29, 7199 (1996).
13. See, e.g., P. R. Halmos, Measure Theory (Van Nostrand, Princeton, 1950), Sec. 47.
14. This nonconvergence implies 13 that the corresponding quasienergy spectrum is not a point spectrum for these values of λ. But Howland has shown 3 that the spectrum is also not absolutely continuous. It follows that the quasienergy spectrum is singular continuous for a set of λ that is everywhere dense but of total measure zero. See also the conclusion of G. Casati and I. Guarneri, Commun. Math. Phys. 95, 121 (1984), that the quantum kicked rotor has singular continuous quasienergy spectra for a nonempty set of driving frequencies.CMPHAY