Nonexistence of usual wave operators for fractional Laplacian and slowly decaying potentials

Abstract

We consider quantum systems described by the fractional powers of the negative Laplacian and the interaction potentials. When a slowly decaying potential function is given, we prove the nonexistence of the wave operators, under the assumption that the Dollard-type modified wave operators exist and that they are asymptotically complete. This nonexistence indicates the borderline between short-range and long-range behavior.