Sharp Estimates in Smoothing Theorems for Schrِdinger Semigroups

The smoothing properties of Schrödinger semigroups, e−tH, H=−12Δ+V, on the scale of Bessel potential spaces Lp, α are studied. We strengthen the (Lp−Lp, α)-smoothing theorem due to M. A. Kon and the author. The new version of this theorem contains a sharp time-estimate for the norm of the semigroup e−tH. We also get an estimate for the constants arising in the form-boundedness inequality for the Kato class potentials.