On Singular Perturbations of Order S, S 2, of the Free Dynamics: Existence and Completeness of the Wave Operators

For a large class of operators H a formula for the resolvent will be derived and suucient conditions will be given in order that the resolvent difference (H?z) ?1 ?(H 0 ?z) ?1 is compact resp. the wave operators (H; H 0) exist and are complete; here H 0 denotes the free quantum mechanical Hamil-tonian. The mentioned class of operators contains, a.o., the generator of a Brownian motion with killing, the generator of the superposition of a Brownian motion and a diiusion process on a submanifold and Hamiltonians describing the interaction of a quantum mechanical particle with a potential supported by a set with classical capacity zero.