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Resonances in n-Body Quantum Systems With Dilatation Analytic Potentials and the Foundations of Time-Dependent Perturbation Theory

It is our goal in this paper to give a precise mathematical definition of the notion of "resonance" in a class of n-body non-relativistic quantum systems and to begin a systematic development of the theory of such resonances. While this class of n-body systems is rather small when viewed in relation to the class of systems [54] for which most of the standard quantum mechanical lore can be developed, it is large enough to include systems with two-body Coulomb, Yukawa or Yukawian interactions. The class thus includes the systems of greatest importance to physics and, in particular, it includes the standard non-relativistic model of the atom. The principal line of development in this paper and the major new results which we wish to prove concern the so-called "time-dependent perturbation theory", one of the two standard perturbation theories developed during the earliest days of quantum mechanics. The other standard theory, known as "time-independent" or Rayleigh-Schrodinger perturbation theory, has been on a firm mathematical footing since the work of Rellich [46]. (Important refinements of Rellich's theory are due to Kato [35] and Sz-Nagy [59].) The time-dependent theory on the other hand has resisted a general mathematical formulation for over forty years although there has been some partly successful work on the subject which we will review later in this introduction. To avoid the natural confusion between "time-dependent" and "time-independent" we will generally avoid the use of the latter term, employing "Rayleigh-Schrodinger" and "Kato-Rellich" instead. The lowest order terms in the time-dependent perturbation series were developed in the 1920's as a means of computing radiative lifetimes of excited states of atoms. The quantity which is supposed to be approximated by this series is the inverse of the lifetime, z, which was assumed to be