Ontology type: schema:Chapter
2002
AUTHORS ABSTRACTThe spectral problem for the Laguerre equation on (0, ∞) with real parameter a in the case 0 <|α|< 1 is closely related to the Nevanlinna \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Q}_{\alpha }}(z) = - \pi \Gamma ( - z)/(\sin \pi \alpha )\Gamma ( - z - \alpha ). $$\end{document} function If |α| and |α|≠ 2, 3,… this function belongs to the generalized Nevanlinna class Nm, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ m = [\tfrac{{|\alpha | + 1}} {2}]. $$\end{document} A natural question appears: to what spectral problem does this function correspond? For α< -1, α ≠-2, -3,…, an answer was given by Derkach [D]. He obtained an operator representation for the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m_\alpha }(z) = - {Q_\alpha }( - Z)/{\mathcal{T}^2}(1 + \alpha ) $$\end{document} in terms of a self-adjoint operator in a Pontryagin space and an interpretation of mα,(z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That an indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for α> 1, α ≠ 2, 3,… we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Qa(z) and compare them with the models for -a. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Qa(z) and -Qa(z)-1 Finally, we describe the connection between the self-adjoint linear relations in the representations of Qa(z) and Q-α (z +α) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions. More... »
PAGES141-181
Operator Methods in Ordinary and Partial Differential Equations
ISBN
978-3-0348-9479-1
978-3-0348-8219-4
http://scigraph.springernature.com/pub.10.1007/978-3-0348-8219-4_13
DOIhttp://dx.doi.org/10.1007/978-3-0348-8219-4_13
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