January 9, 2006

Yossi Strauss

Title: Contractive semigroups and model operators in the description of resonances in quantum mechanics

Abstract:

One of the fundamental and important phenomena observed in scattering experiments of quantum particles is that of the formation of resonances. Basic problems in the theoretical study of resonances include:

  1. Identification of resonances as well defined objects within the theoretical framework of quantum scattering theory.
  2. Finding an appropriate quantum state corresponding to a resonance.
  3. Description of the time evolution of a resonance.

The appropriate description of quantum resonances is an old and difficult problem dating back to the early days of quantum mechanics (for example: Gamow, 1928 and Wigner-Weisskopf, 1930). Over the years several successful methods were developed for dealing with the questions raised above. However, certain aspects of the nature of resonances are still not clear.

A new framework appropriate for the study of the problem of resonances, developed in analogy to the structure of the Lax-Phillips scattering theory, has been introduced recently. This formalism involves two basic ingredients:

  1. Resonance evolution is described by model operators for class $ C_{\cdot, 0}$ semigroups of contractions.
  2. The evolution semigroup and the Hilbert space for the model operator, the Hardy space $H^2_+(R)$, are associated with the Hilbert space $\mathcal H$ and unitary evolution $U(t)$ of the physical scattering problem via a contractive quasi-affine map (a nesting map).

I will describe the new framework, present the results of its application to certain simple models and indicate some of the insights provided by the new formalism.