Sojourn time for rank one perturbations.

We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨χ,u⟩χ such that for β0 Hβ0≔M+β0P has a simple eigenvalue E0 embedded in its absolutely continuous spectrum, with corresponding eigenvector ψ. Let Hω be a rank one perturbation of the operator Hβ0, namely, Hω =M+(β0+ω)P. Under suitable conditions, the operator Hω has no point spectrum in a neighborhood of Eω, for ω small. Here, we study the evolution of the state χ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)= ʃ∞-∞ ∣ ⟨ψ,e-iHωtψ⟩∣² dt. By perturbation theory, we prove that τω(ψ) is is finite for ω≠0, and that for ω small it is of order ω-2. Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in ℝn. [ABSTRACT FROM AUTHOR]