 98502 Soffer, A., Weinstein, M.I.
 Nonautonomous Hamiltonians
(325K, LATeX 2e)
Jul 12, 98

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Abstract. We present a theory of resonances for a class of
nonautonomous Hamiltonians
to treat the structural instability
of spatially localized and timeperiodic
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
timedependent
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 Schr\"odinger
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.
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