Exact Smoothing Properties of Schr odinger Semigroups

We study Schr odinger semigroups in the scale of Sobolev spaces and show that for Kato class potentials the range of such semigroups in Lp has exactly two more derivatives than the potential this proves a conjecture of B Simon We show that eigenfunctions of Schr odinger operators are generically smoother by exactly two derviatives in given Sobolev spaces than their potentials We give applications to the relation between the potential s smoothness and particle kinetic energy in the context of quantum mechanics and characterize kinetic energies in Coulomb systems The techniques of proof invove Leibniz and chain rules for fractional derivatives which are of independent interest as well as a new characterization of the Kato class