Finite Capacities in Spectral Theory

Abstract

The notion of capacity comprises a natural link between the spectrum of selfadjoint Feller operators in L ² (Σ), Σ $\subseteq$ ℝ^d, with Dirichlet boundary conditions on ∂Σ and geometric properties of the region Σ. Here we describe two complementary results. On changing the boundary of Σ, the lowest eigenvalue (ground state) turns out to be shifted if and only if the capacity of the difference set is positive. On the other hand the absolutely continuous spectra of Feller operators with Dirichlet conditions are not affected by changing the perturbations arbitrarily on sets of finite capacities, because the corresponding scattering systems turn out to be complete.