Domain perturbations, Brownian motion, capacities, and ground states of Dirichlet Schrödinger operators (Q1323423)

Let \(\Omega \subset \mathbb{R}^ d\), \(d \in \mathbb{N}\), be open and connected and denote by \(-\Delta_ \Omega\) the Dirichlet Laplacian in \(L^ 2(\Omega)\). If \(\Lambda \subset \Omega\) is another open connected set, then monotonicity of the spectrum \(\sigma (-\Delta_ \Omega)\) of \(- \Delta_ \Omega\) with respect to \(\Omega\) yields the well known inequality \[ \inf [\sigma (-\Delta_ \Omega)] \leq \inf [\sigma (- \Delta_ \Lambda)]. \tag{*} \] We prove that if \(\inf [\sigma (-\Delta_ \Lambda)]\) is actually an eigenvalue of \(-\Delta_ \Lambda\), then the inequality (*) is strict if and only if the capacity of \(\Omega \backslash \Lambda\) is strictly positive. This result is in fact proven for Dirichlet Schrödinger operators of the type \(H_ \Omega= - \Delta_ \Omega+q\) under very general assumptions on the potential function \(q\) by purely probabilistic techniques. (It is also shown that this result does not extend to higher eigenvalues of \(H_ \Lambda\), \(H_ \Omega\) in general).