Measures not charging polar sets and Schroedinger equations in L ( p )

We study the Schroedinger equation (q - )u + ku = f, where is the generator of a Borel right process and k is a signed measure on the state space. We prove the existence and uniqueness results in L ( p ), 1 p < !. Since we consider measures k charging no polar set, we have to use new tools: the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation (subordination) operators (in the sense of G. Mokobodzki) induced by the regular strongly supermedian kernels. We extend the results on the Schroedinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on L ( p ). If the measure k is positive then the perturbed process solves the martingale problem for - k and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having k as Revuz measure.