Time Dependent Resonance Theory

Abstract

An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (time-dependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (non-vanishing of the Fermi golden rule) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.

A Time-Dependent Theory of Quantum Resonances

“…This ingenious idea was used first by [SW3]. It yields the following representation for ψ: 9) where B = O(κ) (i.e.…”

“…The timedependent theory was initiated in works of E. Skibsted and W. Hunziker ([Sk1,2,Hu2]), and a space-time and the phase-space-time and variational analysis was given in [GS] and [PF], respectively. A new powerful approach was suggested by A. Soffer and M. Weinstein [SW3], who also obtained a rather detailed space-time description of evolution of metastable (resonance) states in the one-body Schrödinger case and for a Schrödinger particle coupled to a massive quantum field. Our paper generalizes the result of [SW3] to many-body Schrödinger operators (and degenerate eigenvalues).…”

“…A new powerful approach was suggested by A. Soffer and M. Weinstein [SW3], who also obtained a rather detailed space-time description of evolution of metastable (resonance) states in the one-body Schrödinger case and for a Schrödinger particle coupled to a massive quantum field. Our paper generalizes the result of [SW3] to many-body Schrödinger operators (and degenerate eigenvalues). Besides we also treat the case of quasiclassical resonances, not considered in [SW3].…”

“…Besides we also treat the case of quasiclassical resonances, not considered in [SW3]. Though our approach follows the same general line as that of [SW3], we had to introduce some essential changes right at the beginning in order to make it applicable to a considerably wider class of systems.…”

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“…There are a small number of works about the definition of resonances for nonanalytic potentials, for example, [Orth 1990;Gérard and Sigal 1992;Soffer and Weinstein 1998;Cancelier et al 2005;Jensen and Nenciu 2006]. In [Orth 1990;Gérard and Sigal 1992;Soffer and Weinstein 1998;Jensen and Nenciu 2006], the point of view is quite different from ours, while in [Cancelier et al 2005], the definition is based on the use of an almost-analytic extension of the potential and seems to strongly depend both on the choice of this extension and on the complex distortion.…”