Advertisement

Journal of Statistical Physics

, Volume 85, Issue 5–6, pp 575–606 | Cite as

Upper bounds for the energy expectation in time-dependent quantum mechanics

  • Alain Joye
Articles

Abstract

We consider quantum systems driven by Hamiltonians of the formH+W(t), where the spectrum ofH consists of an infinite set of bands andW(t) depends arbitrarily on time. Let 〈H〈φ(t) denote the expectation value ofH with respect to the evolution at timet of an initial state φ. We prove upper bounds of the type 〈H〉φ(t)=O(tδ), δ>0, under conditions on the strength ofW(t) with respect toH. Neither growth of the gaps between the bands nor smoothness ofW(t) is required. Similar estimates are shown for the expectation value of functions ofH. Sufficient conditions to have uniformly bounded expectation values are made explicit and the consequences on other approaches to quantum stability are discussed.

Key Words

Quantum stability energy expectations quantum diffusion 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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).Google Scholar
  2. 2.
    J. M. Combes, Connections between quantum dynamics and spectral properties of time-evolution operators, inDifferential Equations with Applications to Mathematical Physics, W. F. Ames, E. M. Harrell, and J. V. Herod, eds. (Academic Press, Boston, 1993).Google Scholar
  3. 3.
    P. Duclos and P. Stovicek, Floquet Hamiltonians with pure point spectrum, CPT-94/P.3128 preprint (1994).Google Scholar
  4. 4.
    P. Duclos and P. Stovicek, Quantum Fermi accelerators with pure-point quasispectrum, CPT-94/P.3127 preprint (1994).Google Scholar
  5. 5.
    V. Enss and K. Veselić, Bound states and propagating states for time-dependent Hamiltonians,Ann. Inst. H. Poincaré A 39:159–191 (1983).Google Scholar
  6. 6.
    I. Guarneri and G. Mantica, On the asymptotic properties of quantum dynamics in the presence of a fractal spectrum,Ann. Inst. H. Poincaré A 61:369–379 (1994).Google Scholar
  7. 7.
    G. Hagedorn, M. Loss, and J. Slawny, Nonstochasticity of time-dependent quadratic Hamiltonians and spectra of canonical transformations,J. Phys. A 19:521–531 (1986).Google Scholar
  8. 8.
    J. Howland, Floquet operator with singular spectrum I,Ann. Inst. H. Poincaré A 49:309–323 (1989).Google Scholar
  9. 9.
    J. Howland, Floquet operator with singular spectrum II,Ann. Inst. H. Poincaré A 49:325–334 (1989).Google Scholar
  10. 10.
    J. Howland, Floquet operator with singular spectrum III, preprint (1995).Google Scholar
  11. 11.
    H. R. Jauslin, Stability and chaos in classical and quantum Hamiltonian systems, inII Granada Seminar on Computational Physics, P. Garrido and J. Marro, eds. (World Scientific, Singapore, 1983).Google Scholar
  12. 12.
    A. Joye, Absence of absolutely continuous spectrum of Floquet operators,J. Stat. Phys. 75:929–952 (1994).Google Scholar
  13. 13.
    H. R. Jauslin and J. L. Lebowitz, Spectral and stability aspects of quantum chaos,Chaos 1:114–137 (1991).Google Scholar
  14. 14.
    T. Kato,Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966).Google Scholar
  15. 15.
    S. G. Krein,Linear Differential Equations in Banach Spaces (American Mathematical Society, Providence, Rhode Island, 1971).Google Scholar
  16. 16.
    Y. Last, Quantum dynamics and decompositions of singular continuous spectra, preprint (1995).Google Scholar
  17. 17.
    G. Nenciu, Floquet operators without absolutely continuous spectrum,Ann. Inst. H. Poincaré A 59:91–97 (1993).Google Scholar
  18. 18.
    G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, CPT-95/P.3171 preprint (1995).Google Scholar
  19. 19.
    C. R. de Oliveira, Some remarks concerning stability for nonstationary quantum systems,J. Stat. Phys. 78:1055–1066 (1995).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Alain Joye
    • 1
  1. 1.Department of Mathematics and Center for Transport Theory and Mathematical PhysicsVirginia Polytechnic Institute and State UniversityBlacksburg

Personalised recommendations