Nonautonomous Hamiltonians

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.