Gamov vectors for Resonances: a Lax-Phillips point of view

Results from the Lax-Phillips Scattering Theory are used to analyze quantum mechanical scattering systems, in particular to obtain spectral properties of their resonances which are defined to be the poles of the scattering matrix. For this approach the interplay between the positive energy projection and the Hardy-space projections is decisive. Among other things it turns out that the spectral properties of these poles can be described by the (discrete) eigenvalue spectrum of a so-called truncated evolution, whose eigenvectors can be considered as the Gamov vectors corresponding to these poles. Further an expansion theorem of the positive Hardy-space part of vectors $Sg$ ($S$ scattering operator) into a series of Gamov vectors is presented.