The Spectrum of Resonances and the Trace Formula in a Potential Scattering Problem

For the wave equation with a potential perturbation, the localization and the asymptotic distribution of resonances, i.e., poles of the scattering matrix, are studied. To this end, it proves useful to exploit the relation between these poles and the spectrum of the corresponding Lax–Phillips semigroup. An explicit description of the generator of this semigroup is given. The regularized trace formula for the operators specifying the evolution of the initial data is applied to estimate how rarefied the spectrum of resonances can be.