Resolvent Differences for General Obstacles

Abstract

In case of potential perturbations the second resolvent equation transforms the resolvent difference into a product of operators. For obstacle perturbations this behavior is maintained due to Dynkin’s formula. In the present article we study generalized obstacle perturbations, e.g., perturbations by measures with infinite weight. It turns out that the resolvent difference equals a product of two operators, one factor is the free resolvent, the second factor contains all the interaction. This result is applicable to differential operators of arbitrary order and to a wide class of perturbations.