# Ionization and Scattering for Short-Lived Potentials

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## Abstract

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.

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