Quantum mechanicalN-body systems with dilatation analytic interactions are investigated. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [λe, ∞), where λe is the lowest many body threshold of the system. In the complement of the set of thresholds the point spectrum is discrete; corresponding bound state wave-functions are analytic with respect to the dilatation group.

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Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions

We give a sufficient condition for a self-adjoint operator to have the following properties in a neighborhood of a pointE of its spectrum:

a)

its point spectrum is finite;




b)

its singular continuous spectrum is empty;




c)

its resolvent satisfies a class of a priori estimates.

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Absence of singular continuous spectrum for certain self-adjoint operators

The method of complex coordinate rotation and its applications to atomic collision processes

Using dilation invariance and dilation analytic techniques, and with the help of a new virial theorem, we give a detailed description of the spectral properties of the operator (p2+m2)1/2−Ze2/r. In the process the norm of the operator ∣x∣α∣p∣−α is calculated explicitly inLp(® N ).

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Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$

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

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

We study a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group inRn. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [0, ∞). The point spectrum consists in R−{0} of finite multiplicity isolated energy bound-states standing in a bounded domain. Bound-state wave functions are analytic with respect to the dilatation group. Some properties of resonance poles are investigated.

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J. Aguilar

Papers

A class of analytic perturbations for one-body Schrödinger Hamiltonians