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

TL;DR

This paper extends Titchmarsh-Weyl theory to selfadjoint Schrödinger operators on unbounded domains, linking spectral data to the Dirichlet-to-Neumann map and characterizing various spectral components.

Contribution

It provides a multidimensional analogue of classical Sturm-Liouville results, describing spectral data via the Dirichlet-to-Neumann map for unbounded domains.

Findings

Spectral data characterized by Dirichlet-to-Neumann map

Criteria for absence of singular continuous spectrum

Description of eigenvalues and spectrum types

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.