Scattering by Obstacles of Finite Capacity

For very general generators of diffusion semigroups we show that the essential and absolutely continuous spectra do not change when one adds an extra Dirichlet boundary condition on a "small" set. This is done by proving that the corresponding semigroup differences are Hilbert-Schmidt or trace class, respectively. Our method consists in a factorization argument which is based on calculating the semigroup difference via the Feynman-Kac formula. We also derive trace class estimates for differences of resolvent powers, provided the underlying semigroup has finite local dimension.