Titchmarsh-Weyl theory for Schr\"odinger operators on unbounded domains

Abstract

In this note it is proved that the complete spectral data of selfadjoint Schr\"odinger operators on unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. Furthermore, a sufficient criterion for the absence of singular continuous spectrum is provided. The results are natural multidimensional analogues of classical facts from singular Sturm-Liouville theory.