Survival probability in rank-one perturbation problems

A finite complex Borel measure μ on the unit circle or on the real line is called Rajchman if its Fourier coefficients μ(n) tend to 0 as n →∞. In quantum dynamics the self-adjoint operators (Hamiltonians) whose spectral measures are Rajchman correspond to the systems having certain scattering properties. In this paper we study how a small perturbation of the operator can affect the Rajchman property of its spectral measure. Our approach is based on the notion of the local symmetry of measures. (orig.)