IJPAM: Volume 71, No. 2 (2011)

EIGENVALUES OF STURM-LIOUVILLE PROBLEMS AND
THE ZEROS OF ENTIRE FUNCTIONS OF SINE TYPE

Mihaela-Cristina Drignei
Division of Physical and Computational Sciences
University of Pittsburgh at Bradford
Bradford, PA 16701, USA


Abstract. The main contribution of this paper is a result about the eigenvalues of two classes of Sturm-Liouville problems on a finite interval: if $\{\lambda_n\}_{n\geq 1}$ is the sequence of Dirichlet, respectively of Dirichlet-Robin eigenvalues of the canonical Sturm-Liouville operator with coefficient function $q\in L^2_{\mathbb{R}}(0,a)$ (and boundary parameter $\alpha\in\mathbb{R}$ - for the Dirichlet-Robin case), then $\{\pm\sqrt{\lambda_n}\}_{n\geq 1}\cup\{0\}$, and respectively $\{\pm\sqrt{\lambda_n}\}_{n\geq 1}$ are the zeros of some entire functions of sine type $a$.

Received: March 30, 2011

AMS Subject Classification: 42C99, 34A55, 34B24, 34L20

Key Words and Phrases: eigenvalues, eigenfunctions, characteristic functions of Sturm-Liouville operators, function of sine type, Riesz basis, Gelfand-Levitan kernel, Goursat problem

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 71
Issue: 2