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Journal of Spectral Theory

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Volume 6, Issue 1, 2016, pp. 67–87
DOI: 10.4171/JST/118

Published online: 2016-04-04

Titchmarsh–Weyl theory for Schrödinger operators on unbounded domains

Jussi Behrndt[1] and Jonathan Rohleder[2]

(1) TU Graz, Austria
(2) TU Graz, Austria

In this paper it is proved that the complete spectral data of selfadjoint Schrödinger 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 analogs of classical facts from singular Sturm-Liouville theory.

Keywords: Dirichlet-to-Neumann map, Schrödinger operator, spectrum, Titchmarsh–Weyl function

Behrndt Jussi, Rohleder Jonathan: Titchmarsh–Weyl theory for Schrödinger operators on unbounded domains. J. Spectr. Theory 6 (2016), 67-87. doi: 10.4171/JST/118