Estimate of the difference between the Kac operator and the Schrödinger semigroup

An operator norm estimate of the difference between the Kac operator and the Schrödinger semigroup is proved and used to give a variant of the Trotter product formula for Schrödinger operators in theL ^p operator norm. This extends Helffer’s result in theL ² operator norm to the case in theL ^p operator norm for more general scalar potentials and with vector potentials. The method of the proof is probabilistic based on the Feynman—Kac and Feynman—Kac—Itô formula.