A Characteristic Decay Semigroup for the Resonances of Trace Class Perturbations with Analyticity Conditions of Semibounded Hamiltonians

Abstract

To asymptotic complete scattering systems {M++V,M+} on \({\mathcal{H}}_{+}:=L^{2}(\mathbf{R}_{+},{\mathcal{K}}\), dλ), where M+ is the multiplication operator on \({\mathcal{H}}_{+}\) and V is a trace class operator with analyticity conditions, a decay semigroup is associated such that the spectrum of the generator of this semigroup coincides with the set of all resonances (poles of the analytic continuation of the scattering matrix into the lower half plane across the positive half line), i.e. the decay semigroup yields a “time-dependent” characterization of the resonances. As a counterpart a “spectral characterization” is mentioned which is due to the “eigenvalue-like” properties of resonances.