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On the Uniqueness of the Green's Function Associated with a Second-order Differential Equation

Published online by Cambridge University Press:  20 November 2018

E. C. Titchmarsh
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The Green's function G(x, ξ, λ) associated with the differential equation is of importance in the theory of the expansion of an arbitrary function in terms of the solutions of the differential equation. It is proved that this function is unique if q(x) ≧ — Ax2— B, where A and B are positive constants or zero. A similar theorem is proved for the Green's function G(x, y, ξ, η, λ) associated with the partial differential equation

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 2 , 01 April 1949 , pp. 191 - 198
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 See Hilbert, D., Grundzûge einer allgemeinen Théorie der linear en Integralgleichungen, Zweiter Abschnitt (Berlin, 1924);Google Scholar Courant, R.and Hilbert, D., Methoden der math. Physik I, (Berlin, 1931), V, § 14;Google Scholar E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Differential Equations (Oxford, 1946), p. 29; Hartman, P. and Wintner, A., “A Criterion for the Non-degeneracy of the Wave Equation,” Amer. J. Math., vol. 71 (1949), 206-213.Google Scholar

2 Different constant factors are attached to the Green's function by different writers. The definition adopted here is that of Courant and Hilbert. The function in my book had the opposite sign.

3 See chap. II of my book.

4 See a forthcoming paper in the Proc. London Math. Soc.

5 See e.g. E. Goursat, Cours d'Analyse mathématique, vol. 3 (Paris, 1927), § 506.

6 It can be verified at once by integration by parts.