Movement and interplay of the bound state, resonance, and shadow poles of the scattering amplitude in multiphoton processes

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Abstract
We examine, in a heuristic fashion, the analytic structure of the scattering amplitude for an electron interacting with an atomic potential in the presence of a radiation field. For each resonance pole of the amplitude there are infinitely many shadow poles lying on different unphysical energy sheets of the infinitely many sheeted Riemann surface. The pole having the dominant influence on the scattering amplitude is the one closest to the physical energy axis. All poles undergo significant movement when the field intensity is varied, and when a resonance pole which is dominant passes by a multiphoton ionization threshold a shadow pole usually moves closer to the physical energy axis and hence becomes the dominant pole. When this happens, an initially bound electron may jump eigenvalue curves, and in this way the electron can undergo a very large (ponderomotive) shift in its energy and still maintain the correct physical character of its wave function. The movement of resonance poles near thresholds has implications for the fate of autoionizing states and for population trapping. Consideration of threshold effects might also shed light on a puzzling result of a recent energy-shift measurement. We address the problem of how to determine, in numerical calculations, which sheet the energy eigenvalue is on so that the dominant pole can be identified. We illustrate some of our remarks by results of numerical calculations, and we also touch on the question of the completeness of a basis set in numerical calculations of resonance eigenvalues.