Let U be a unitary operator defined on a infinite-dimensional separable complex Hilbert space H. Assume there exists a self-adjoint operator A on H such thatfor some positive constant c and compact operator K. Then, assuming the commutators U * AU − A and [A, U * AU ] admit a bounded extension over H, we prove the spectrum of the operator U has no singular continuous component and only a finite number of eigenvalues of finite multiplicity. We give a localized version of this result and apply it to study the spectrum of the Floquet operator of periodic time-dependent kicked quantum systems.
We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨φ,u⟩φ 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 E0, for ω small. Here, we study the evolution of the state ψ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)=∫−∞∞∣⟨ψ,e−iHωtψ⟩∣2dt. By perturbation theory, we prove that τω(ψ) 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.
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