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Journal of Statistical Physics

, Volume 93, Issue 1–2, pp 359–391 | Cite as

Nonautonomous Hamiltonians

  • A. Soffer
  • M. I. Weinstein
Article

Abstract

We present a theory of resonances for a class of nonautonomous Hamiltonians to treat the structural instability of spatially localized and time-periodic solutions associated with an unperturbed autonomous Hamiltonian. The mechanism of instability is radiative decay, due to resonant coupling of the discrete modes to the continuum modes by the time-dependent perturbation. This results in a slow transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. The ideas are closely related to the authors' work on (i) a time-dependent approach to the instability of eigenvalues embedded in the continuous spectra, and (ii) resonances, radiation damping, and instability in Hamiltonian nonlinear wave equations. The theory is applied to a general class of Schrödinger equations. The phenomenon of ionization may be viewed as a resonance problem of the type we consider and we apply our theory to find the rate of ionization, spectral line shift, and local decay estimates for such Hamiltonians.

Resonance lifetime ionization breather 

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REFERENCES

  1. 1.
    L. Alan and J. H. Eberley, Optical Resonance and Two Level Atoms (Dover, 1987).Google Scholar
  2. 2.
    V. Bach, J. Fröhlich, I. M. Sigal, and A. Soffer, Positive commutators and the spectrum of Pauli-Fierz Hamiltonians of atoms and molecules, 1997, preprint.Google Scholar
  3. 3.
    J. Carr, Applications of Center Manifold Theory (Springer-Verlag, New York, 1981).Google Scholar
  4. 4.
    J. Cooper and W. Strauss, Time periodic scattering of symmetric hyperbolic systems, J. Math. Anal. Appl. 122:444–452 (1987).Google Scholar
  5. 5.
    H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry (Springer-Verlag, Berlin/Heidelberg/New York, 1987).Google Scholar
  6. 6.
    V. Enss and K. Veselic, Bound states and propagation states for time dependent hamiltonians, Ann. Inst. H. Poincaré A 39:159–191 (1983).Google Scholar
  7. 7.
    A. Galindo and P. Pascual, Quantum Mechanics II (Springer, 1991).Google Scholar
  8. 8.
    C. Gérard and I. M. Sigal, Space-time picture of semiclassical resonances, Commun. Math. Phys. 145:281–328 (1992).Google Scholar
  9. 9.
    J. S. Howland, Stationary scattering theory for time-dependent Hamiltonians, Math. Ann. 207:315–335 (1974).Google Scholar
  10. 10.
    J. S. Howland, Scattering theory for Hamiltonians periodic in time, Indiana Math. J. 28:471–494 (1979).Google Scholar
  11. 11.
    W. Hunziker and I. M. Sigal, Time dependent scattering theory for N-body quantum systems, ETH-Zürich preprint (1997).Google Scholar
  12. 12.
    W. Hunziker, I. M. Sigal, and A. Soffer, Minimal escape velocities, 1998, preprint.Google Scholar
  13. 13.
    A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of wave functions, Duke Math. J. 46:583–611 (1979).Google Scholar
  14. 14.
    H. R. Jauslin and J. L. Lebowitz, Generalized Floquet operator for quasiperiodically driven quantum systems, in Mathematical Physics X, K. Schmudgen, ed. (Springer, 1991, and cited ref.).Google Scholar
  15. 15.
    J.-L. Journé, A. Soffer, and C. Sogge, L pL p' estimates for the time dependent Schrödinger equation, Bull. AMS 23 (1990).Google Scholar
  16. 16.
    C.-A. Pillet and C. E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations, J. Diff. Eq. 141:310–326 (1997).Google Scholar
  17. 17.
    H. Kitada and K. Yajima, A scattering theory for time dependent long range potentials, Duke Math. J. 49:341–376 (1982).Google Scholar
  18. 18.
    H. Lamb, on a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc. 32:208–211 (1900).Google Scholar
  19. 19.
    L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-relativistic Theory, in Course of Theoretical Physics (Pergamon Press, Oxford, 1965), Vol. 3.Google Scholar
  20. 20.
    J. L. Lebowitz, Macroscopic laws, microscopic dynamics, time's arrow and Boltzmann's entropy, Phys. A 194(1–4):1–27 (1993).Google Scholar
  21. 21.
    P. D. Miller, A. Soffer, and M. I. Weinstein, Metastability of breather modes of time dependent potentials, in preparation.Google Scholar
  22. 22.
    E. Mourre, Absence of singular continuous spectrum for certain self-adjoint operators, Commun. Math. Phys. 78:391 (1981).Google Scholar
  23. 23.
    M. Murata, Rate of decay of local energy and spectral properties of elliptic operators, Jpn. J. Math. 6:77–127 (1980).Google Scholar
  24. 24.
    R. M. Pyke and I. M. Sigal, Nonlinear wave equations: Constraints on periods and exponential bounds for periodic solutions, Duke Math. J. 88 (1997).Google Scholar
  25. 25.
    P. Perry, I. M. Sigal, and B. Simon, Spectral analysis of N-body Schrödinger operators, Ann. Math. 114:519–567 (1981).Google Scholar
  26. 26.
    R. Pyke, A. Soffer, and M. I. Weinstein, Stability of Kinks, in progress.Google Scholar
  27. 27.
    J. Rauch, Local decay of scattering solutions to Schrödinger's equation, Comm. Math. Phys. 61:149–168 (1978).Google Scholar
  28. 28.
    M. Reed and B. Simon, Methods in Modern Mathematical Physics, I. Functional Analysis (Academic Press, 1972).Google Scholar
  29. 29.
    M. Reed and B. Simon, Methods in Modern Mathematical Physics, IV. Analysis of Operator (Academic Press, 1978).Google Scholar
  30. 30.
    J. J. Sakurai, Advanced Quantum Mechanics (Addison Wesley, 1967).Google Scholar
  31. 31.
    T. Schonbek, Decay of solutions of Schrödinger equations, Duke Math. J. 46:203–213 (1979).Google Scholar
  32. 32.
    I. M. Sigal, Nonlinear wave and Schrödinger equations I. Instability of time-periodic and quasiperiodic solutions, Commun. Math. Phys. 153:297 (1993).Google Scholar
  33. 33.
    I. M. Sigal, General characteristics of nonlinear dynamics, in Spectral and Scattering Theory; Proceedings of the Taniguchi International Workshop, M. Ikawa, ed. (Marcel Dekker, New York/Basel/Hong Kong, 1994).Google Scholar
  34. 34.
    I. M. Sigal and A. Soffer, Local decay and velocity bounds for Quantum propagation, preprint (Princeton, 1988) (ftp://www.math.rutgers.edu/pub/soffer).Google Scholar
  35. 35.
    I. M. Sigal and A. Soffer, Asymptotic completeness for short range many body systems, Ann. Math. 126:35–108 (1987).Google Scholar
  36. 36.
    E. Skibsted, Propagation estimates for N-body Schrödinger operators, Commun. Math. Phys. 142:67–98 (1991).Google Scholar
  37. 37.
    A. Soffer and M. I. Weinstein, Dynamic theory of quantum resonances and perturbation theory of embedded eigenvalues, in Proceedings of Conference on Partial Differential Equations and Applications, University of Toronto, June, 1995, CRM Lecture Notes, P. Greiner, V. Ivrii, L. Seco, and C. Sulem, eds.Google Scholar
  38. 38.
    A. Soffer and M. I Weinstein, Time dependent resonance theory, to appear in Geometric and Functional Analysis.Google Scholar
  39. 39.
    A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, to appear in Inventiones Mathematicae.Google Scholar
  40. 40.
    B. Vainberg, Scattering of waves in a medium depending periodically on time, Asterisque 210:327–340 (1992).Google Scholar
  41. 41.
    A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, Dynamics Reported 2 (1990).Google Scholar
  42. 42.
    J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 77:2863 (1996) and cited ref.Google Scholar
  43. 43.
    K. Yajima, Scattering theory for Schrödinger operators with potentials periodic in time, J. Math. Soc. Japan 29:729–743 (1977).Google Scholar
  44. 44.
    K. Yajima, Resonances for the AC-Stark effect, Commun. Math. Phys. 78:331–352 (1982).Google Scholar
  45. 45.
    K. Yajima, A multichannel scattering theory for some time dependent hamiltonians, charge transfer problem, Commun. Math. Phys. 75:153–178 (1980).Google Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • A. Soffer
    • 1
  • M. I. Weinstein
    • 2
    • 3
  1. 1.Department of MathematicsRutgers UniversityNew Brunswick
  2. 2.Department of MathematicsUniversity of MichiganAnn Arbor
  3. 3.Mathematical Sciences ResearchMurray Hill

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