Conditions for the Existence of the Generalized Wave Operators

Abstract

If E0(G) is the spectral projection operator associated with the free Hamiltonian H0, corresponding to a bounded measurable subset G of R, and E1(G) is associated with the total Hamiltonian H = H0 + V, where the operator E1(G)VE0(G) is of trace class, it is proved that the element g = E0(G)f belongs to the domain of the generalized wave operators Ω± if and only if lim lim t→±∞‖(1−E1(G))e−iH0tg‖=0. A stronger version of this result is also proved, from the theory of time-dependent scattering, and is applicable to scattering systems for which families {G1}, {G2} of measurable sets may be found such that E1(G2)VE0(G1) is of trace class.