Asymptotic Resonance Properties of the Finite-Dimensional Friedrichs Model

Abstract

The Friedrichs model H=H_o+Г +Г1* with an embedded eigen-value λ_o of H_o of finite multiplicity is considered. Put B(λ)≔Г*E_o(dλ)Г/dλ, where E_o(•) denotes the spectral measure of H_o. Let B(λ) > 0. Further put σ(λ;B) = ||T(λ)|| ²₂ ,_λ (б scattering cross-section, T(λ) scattering amplitude). Sufficient conditions on B are given for “asymptotic resonance”, that is for б(λ;B)➙0 if λ≠λ_o and lim inf б(λ_o;B) > 0, if B tends to zero in some sense. If B=B_ε, the result can be interpreted in the sense of yielding the existence of a simple “resonance trajectory” in the {λ,ε}-plane tending to {λ_o,0}. This result is supplemented by an example which shows that already in very simple cases other “trajectories” must be used in order to obtain the resonance property at{λ_o,0}.