Complex Analysis and Operator Theory

, Volume 4, Issue 2, pp 179–243

Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions

Article

DOI: 10.1007/s11785-009-0026-0

Cite this article as:
Fulton, C. & Langer, H. Complex Anal. Oper. Theory (2010) 4: 179. doi:10.1007/s11785-009-0026-0

Abstract

The Titchmarsh–Weyl function, which was introduced in Fulton (Math Nachr 281(10):1418–1475, 2008) for the Sturm-Liouville equation with a hydrogen-like potential on (0, ∞), is shown to belong to a generalized Nevanlinna class \({\bf N_\kappa}\). As a consequence, also in the case of two singular endpoints for the Fourier transformation defined by means of Frobenius solutions there exists a scalar spectral function. This spectral function is given explicitly for potentials of the form \({\dfrac{q_0}{x^2}+\dfrac{q_1}{x},\,-\dfrac 14\le q_0 < \infty}\).

Keywords

Sturm-Liouville operatorTitchmarsh–Weyl functionSpectral functionSingular potentialGeneralized Nevanlinna functionWhittaker functionBessel function

Mathematics Subject Classification (2000)

Primary 34B2047E05Secondary 34B2434B3034B4034L1034L4033C1581Q10

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria