L^p-uniqueness for elliptic operators with unbounded coefficients in R^N

Let $\cal A$ be an elliptic operator with unbounded and sufficiently smooth coefficients and let $\mu$ be a (sub)-invariant measure of the operator $\cal A$. In this paper we give sufficient conditions guaranteeing that the closure of the operator $(\cal A, C^\infty_c(\R^N))$ generates a sub-Markovian strongly continuous semigroup of contractions in $L^p(\R^N,\mu)$. Applications are given in the case when $\cal A$ is a generalized Schrodinger operator.