The Titchmarsh-Weyl Eigenfunction Expansion Theorem for Sturm-Liouville Differential Equations

Christer Bennewitz () and W. Norrie Everitt ()
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Christer Bennewitz: University of Lund, Department of Mathematics
W. Norrie Everitt: University of Birmingham Edgbaston, School of Mathematics and Statistics

A chapter in Sturm-Liouville Theory, 2005, pp 137-171 from Springer

Abstract: Abstract This paper involves a revisit to the original works of Hermann Weyl in 1910 and of Edward Charles Titchmarsh in 1941, concerning Sturm-Liouville theory and the corresponding eigenfunction expansions. For this account the essential results of Weyl concern the regular, limit-circle and limit-point classifications of Sturm-Liouville differential equations; the eigenfunction expansion theory from Titchmarsh is based on classical function theory methods, in particular complex function theory. The eigenfunction expansion theory presented in this paper is based on the Titchmarsh-Weyl m-coefficient; the proofs are essentially in classical function theory but are related to operator theoretic methods in Hilbert space. One important innovation here is that for the Sturm-Liouville problem considered on the open interval (a,b), the endpoint a can be classified as either regular or limit-circle, whilst the endpoint b can be regular, limit-circle or limit-point; nevertheless it is shown that these conditions lead to the definition of a single Titchmarsh-Weyl m-coefficient. From this coefficient the complex function theory methods of Titchmarsh provide a guide to a new proof of the general eigenfunction expansion theorem.

Keywords: Sturm-Liouville theory; Titchmarsh-Weyl m-coefficient; eigenfunction expansion (search for similar items in EconPapers)
Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-7359-7_7

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DOI: 10.1007/3-7643-7359-8_7

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