Kac regular sets and Sobolev spaces in geometry, probability and quantum physics
Abstract
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 Schrödinger 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 Schrödinger 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.05542
 arXiv:
 arXiv:1708.05542
 Bibcode:
 2017arXiv170805542B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Differential Geometry