Exact smoothing properties of Schrödinger semigroups

We study Schrödinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L^p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrödinger operators are generically smoother by exactly two derivatives (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 involve Leibniz and chain rules for fractional derivatives which are of independent interest, as well as a new characterization of the Kato class.