, Volume 4, Issue 2, pp 179-243
Date: 14 Jul 2009

Sturm-Liouville Operators with Singularities and Generalized Nevanlinna Functions

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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 \({\dfrac{q_0}{x^2}+\dfrac{q_1}{x},\,-\dfrac 14\le q_0 < \infty}\) .

Communicated by Daniel Alpay.
The work of C. Fulton was partially supported by National Science Foundation Grant DMS-0109022 to Florida Institute of Technology.