Lp-Uniqueness of Schrِdinger Operators and the Capacitary Positive Improving Property

We prove several Lp-uniqueness results for Schrödinger operators −L+V by means of the Feynman–Kac formula. Using the (m, p)-capacity theory for general Markov semigroups, we show that the associated Feynman–Kac semigroup is positive improving in the sense of (m, p)-capacity, improving the well known one in the sense of measure. Using that capacitary positive improving property and two new inequalities for generalized Ornstein–Uhlenbeck generators, we show the essential self-adjointness of the ground state diffusion generator Lφ=L+2Γ(φ, ·)/φ associated with two dimensional Euclidean quantum fields.