Stability of driven systems with growing gaps, quantum rings, and Wannier ladders

Journal of Statistical Physics

Volumn 92, Issue 5-6, 1998, Pages 1053-1070

  Asch, Joachim ^a,b   Duclos, Pierre ^a,b   Exner, Pavel ^c,d  

^a CENTRE DE PHYSIQUE THÉORIQUE   (France)

^b UNIVERSITÉ DE TOULON   (France)

^c INSTITUTE OF MATHEMATICS   (Czech Republic)

^d CZECH TECHNICAL UNIVERSITY IN PRAGUE   (Czech Republic)

Author keywords

Driven rings; Energy expectations; Quantum stability

Indexed keywords

EID: 0032163923     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1023000828437     Document Type: Article

References (27)

1. Ping Ao, Absence of localization in energy space of a Bloch electron driven by a constant electric force. Phys. Rev. B 41:3998-4001 (1990).
2. J. Asch, P. Duclos, and P. Exner. Stark-Wannier hamiltonians with pure point spectrum, in Partial Differential Operators and Mathematical Physics, M. Demuth and B.-W. Schultze, eds. (Birkhäuser, 1997).
3. J. E. Avron, P. Exner, and Y. Last, Periodic Schrödinger operators with large gaps and Wannier-Stark ladders, Phys. Rev. Lett. 72:896-899 (1994).
4. J. E. Avron and J. Nemirovsky, Quasienergies, stark hamiltonians and energy growth for driven quantum rings, Phys. Rev. Lett. 68:2212-2215 (1992).
5. J. Barbaroux and A. Joye, Expectation values of observables in time-dependent quantum mechanics, J. Stat. Phys. 90:1225-1251 (1998).
6. J. Bellissard, Stability and instability in quantum mechanics, in Trends and Developments in the Eighties, S. Albeverio and P. Blanchard, eds. (Word Scientific, 1985).
7. F. Bentosela, R. Carmona, P. Duclos, B. Simon, B. Souillard, and R. Weder, Schrödinger operators with an electric field and random or deterministic potentials, Commun. Math. Phys. 88:387-397 (1983).
8. F. Bentosela, V. Grecchi, and F. Zironi, Approximate ladder of resonances in a semi-infinite crystal, J. Phys. C 15:7119-7131 (1982).
9. A. M. Berezhkovski and A. A. Ovchinnikov, True width of electron levels in a crystal in a static electric field, Sov. Phys.: Solid State 18:1908-1911 (1976).
10. L. Bunimovich, H. R. Jauslin, J. L. Lebowitz, A. Pellegrinotti, and P. Nielaba, Diffusive energy growth in classical and quantum driven oscillators, J. Stat. Phys. 62:793-817 (1991).
11. M. Combescure, The quantum stability problem for time-periodic perturbations of the harmonic oscillator, Ann. Inst. H. Poincaré 47:62-82, 451-454 (1987).
12. S. De Bièvre and G. Forni, Transport properties of kicked and quasi-periodic hamiltonians, J. Stat. Phys. 90:1201-1223 (1998).
13. C. R. de Oliveira, Some remarks concerning stability for nonstationary quantum systems, J.Stat.Phys. 78:1055-1066 (1995).
14. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon. Operators with singular continuous spectrum, IV, Hausdorff dimensions, rank one perturbations, and localization, J. d'Analyse Math. 69:153-200 (1996).
15. P. Duclos and P. Šťovíček, Floquet hamiltonians with pure point spectrum. Commun. Math. Phys. 177:327-374 (1996).
16. P. Duclos, P. Šťovíček, and M. Vittot, Pertubation of an eigenvalue from a dense point spectrum: A general Floquet hamiltonian. Preprint CPT, (P.3559), 1997.
17. V. Enss and K. Veselić, Bound states and propagating states for time-dependent hamiltonians, Ann. Inst. H. Poincaré, Sect. A 39:159-191, 1983.
18. P. Exner, The absence of the absolutely continuous spectrum for δ′ Wannier-Stark ladders, J. Math. Phys. 36:4561-4570 (1995).
19. V. Grecchi and A. Sacchetti, Lifetime of the Wannier-Stark resonances and perturbation theory, Comm. Math. Phys. 185:359-378 (1993).
20. G. Hagedorn, M. Loss, and J. Slawny, Non stochasticity of time-dependent quadratic hamiltonians and spectra of canonical transformations, J. Phys. A 19:521-531 (1986).
21. J. Howland, Floquet operator with singular spectrum i, ii, Ann. Inst. H. Poincaré, Sect. A 49:309-334 (1989).
22. A. Joye, Upper bounds for the energy expectation in time-dependent quantum mechanics, J. Stat. Phys. 85:575-606 (1996).
23. A. Kiselev, Some examples in one-dimensional geometric scattering on manifolds, J. Math. Anal. Appl. 212:263 ff. (1997).
24. M. Maioli and A. Sacchetti, Absence of the absolutely continuous spectrum for Stark-Bloch operators with strongly singular periodic potentials, erratum, J. Phys. A 31:1115-1119 (1998).
25. E. E. Mendez and G. Bastard, Wannier-Stark resonances. Physics Today 46(6):34-42 (1993).
26. G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincaré 67(4):411-424 (1997).
27. B. Simon, Spectral analysis of rank one perturbations and applications, in CRM Lecture Notes, Vol. 8, J. Feldman, R. Froese, and L. Rosen, eds. (AMS, 1995).