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An elementary proof of Weyl's limit-classification

Part of: Boundary value problems

Published online by Cambridge University Press:  09 April 2009

J. Das
J. Das
Affiliation:
Department of Pure Mathematics, Calcutta University, 35, Ballygunge Circular Road Calcutta - 700 019, India
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Abstract

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It is known [Herman Weyl, 1910] that every linear second-order differential expression L (with real coefficients) is such that Ly = λy (im λ ≠ 0) has at least one solution belonging to the class L2 = L2[0, ∞) of functions, the squares of whose moduli are Lebesgue-integrable on [0, ∞). This celebrated result was later proved by E. C. Titchmarsh (1940–1944), using sophisticated analysis of bilinear transformation. The aim of the present note is to prove the same result once again, but using only elementary analysis and school geometry. The power of this method will be appreciated further when one realises the amount of simplifications that can be acheived by this expressions. This part of the note course will be taken up in a subsequent paper.

MSC classification

Secondary: 34B20: Weyl theory and its generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 2 , April 1989 , pp. 171 - 176
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Everitt, W. N., ‘Fourth-order singular differential equations’, Math. Ann. 149 (1963), 320–40.CrossRefGoogle Scholar
[2]Titchmarsh, E. C., Eigenfunction expansions associated with second order differential equations’, Part I (2nd ed., Clarendon Press, 1962).CrossRefGoogle Scholar
[3]Titchmarsh, E. C., ‘An extension of the Sturm-Liouville expansion’, Quart. J. Math. Oxford Ser. 15 (1944), 40–48.CrossRefGoogle Scholar
[4]Weyl, H., ‘Über gewöhnliche Differentialgleichungen mit Singularitäten und zugehörigen Entwicklungen willkürlicher Funktionen’, Math. Ann. 68 (1910), 222–269.CrossRefGoogle Scholar