Kac regular sets and Sobolev spaces in geometry, probability and quantum physics

Let $\Omega\subset M$ be an open subset of a Riemannian manifold $M$ and let $V:M\to \IR$ be a Kato decomposable potential. With $W^{1,2}_{0}(M;V)$ the natural form domain of the Schr\"odinger operator $-\Delta+V$ in $L^2(M)$, in this paper we study systematically the following question: Under which assumption on $\Omega$ is the statement $$\text{ for all $f\in W^{1,2}_{0}(M;V)$ with $f=0$ a.e. in $M\setminus \Omega$ one has $f|_\Omega\in W^{1,2}_{0}(\Omega;V)$}$$ true for every such $V$? We prove that without any further assumptions on $V$, the above property is satisfied, if $\Omega$ is Kac regular, a probabilistic property which means that the first exit time of Brownian motion on $M$ from $\Omega$ is equal to its first penetration time to $M\setminus \Omega$. In fact, we treat more general covariant Schr\"odinger operators acting on sections in metric vector bundles, allowing new results concerning the harmonicity of Dirac spinors on singular subsets. Finally, we prove that locally Lipschitz regular $\Omega$'s are Kac regular.