Local Time Decay for Fractional Schrödinger Operators with Slowly Decaying Potentials

A local time decay estimate of fractional Schr & ouml;dinger operators with slowly decaying positive potentials is studied. It is shown that the resolvent is smooth near zero, and the time propagator exhibits fast local time decay, which is very different from very short-range cases. The key element of the proof is to establish a weaker Agmon estimate for a classically forbidden region using exotic symbol calculus. As a byproduct, we prove that the Riesz operator is a pseudodifferential operator with an exotic symbol.