Letters in Mathematical Physics

, Volume 48, Issue 4, pp 339–352 | Cite as

Ionization and Scattering for Short-Lived Potentials

  • A. Soffer
  • M. I. Weinstein


We consider perturbations of a model quantum system consisting of a single bound state and continuum radiation modes. In many problems involving the interaction of matter and radiation, one is interested in the effect of time-dependent perturbations. A time-dependent perturbation will couple the bound and continuum modes causing ‘radiative transitions’. Using techniques of time-dependent resonance theory, developed in earlier work on resonances in linear and nonlinear Hamiltonian dispersive systems, we develop the scattering theory of short-lived (\( (\mathcal{O}(t^{ - 1 - {\varepsilon }} ))\)(t−1−ε)) spatially localized perturbations. For weak pertubations, we compute (to second order) the ionization probability, the probability of transition from the bound state to the continuum states. These results can also be interpreted as a calculation, in the paraxial approximation, of the energy loss resulting from wave propagation in a waveguide in the presence of a localized defect.

perturbation time-dependent resonance theory scattering theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ben-Artzi, M. and Klainerman, S.: Decay and regularity for the Schrödinger equation, J. Anal. Math. 58 (1992), 25–37.Google Scholar
  2. 2.
    Cohen-Tannoudji, C., Dupont-Roc, J. and Grynberg, G.: Atom-Photon Interactions: Basic Processes and Applications, Wiley, New York, 1992.Google Scholar
  3. 3.
    Costin, O., Lebowitz, J. L. and Rokhlenko, A.: Exact results for the ionization of a model atom, Rutgers University preprint, 1999.Google Scholar
  4. 4.
    Galindo, A. and Pascual, P.: Quantum Mechanics II, Springer-Verlag, New York, 1991.Google Scholar
  5. 5.
    Howland, J. S.: Stationary scattering theory for time dependent Hamiltonians, Math. Ann. 207 (1974), 315–335.Google Scholar
  6. 6.
    Kirr, E. and Weinstein, M. I.: Parametrically excited Hamiltonian partial differential equations, In preparation.Google Scholar
  7. 7.
    Marcuse, D.: Theory of Dielectric Optical Waveguides, Quantum Electronics-Theory and Applications, Academic Press, New York, 1974.Google Scholar
  8. 8.
    Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, Volume 3: Scattering Theory, Academic Press, New York, 1978.Google Scholar
  9. 9.
    Rokhlenko, A. and Lebowitz, J. L.: Ionization of a model atom by perturbations of the potential, Rutgers University preprint, 1998.Google Scholar
  10. 10.
    Soffer, A. and Weinstein, M. I.: Time dependent resonance theory, Geom. Funct. Anal. 8 (1998), 1086–1128.Google Scholar
  11. 11.
    Soffer, A. and Weinstein, M. I.: Nonautonomous Hamiltonians, J. Statist. Phys. 93 (1998), 359–391.Google Scholar
  12. 12.
    Soffer, A. and Weinstein, M. I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9–74.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • A. Soffer
    • 1
  • M. I. Weinstein
    • 2
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickU.S.A
  2. 2.Mathematical Sciences Research, Bell Laboratories – Lucent Technologies, and Department of MathematicsUniversity of MichiganAnn ArborU.S.A

Personalised recommendations