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The Spectral Matrix and Green's Function for Singular Self-Adjoint Boundary Value Problems

Published online by Cambridge University Press:  20 November 2018

E. A. Coddington
E. A. Coddington*
Affiliation:
University of California, Los Angeles
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Let L denote the formal ordinary differential operator

,

where we assume the p k are complex-valued functions with n-k continuous derivativeson an open real interval a < x < b (a = — ∞, b = + ∞, or both may occur), p 0(x) ≠ 0 on a < x < b, and L coincides with its Lagrange adjoint L + given by

.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954, pp. 169 - 185
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Coddington, E. A., On the spectral representation of ordinary self-adjoint differential operators, Proc. Nat. Acad. Sci., 38 (1952), 732–737.Google Scholar
2. Kodaira, K., Eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S'matrices, Amer. J. Math., 71 (1949), 921–945.Google Scholar
3. Kodaira, K., On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Amer. J. Math., 72 (1950), 502–544.Google Scholar
4. Levinson, N., A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J., 18 (1951), 57–71.Google Scholar
5. Levinson, N., Addendum to above, Duke Math. J., 18 (1951), 719–722.Google Scholar
6. Levinson, N., The expansion theorem for singular self-adjoint linear differential operators, to appear in Annals of Math.Google Scholar
7. Levitan, B. M., Proof of a theorem on eigenfunction expansions for self-adjoint differential equations, Doklady Akad. Nauk. USSR, 73 (1950), 651–654.Google Scholar
8. Stone, M. H., Linear transformations in Hilbert space, Amer. Math. Soc. Coll. Publication (1932).Google Scholar
9. Titchmarsh, E. C, Eigenfunction expansions associated with second order differential equations (Oxford, 1946).Google Scholar
10. Titchmarsh, E. C, Eigenfunction expansions associated with partial differential equations, Proc. London Math. Soc, 1 (1951), 1–27.Google Scholar
11. Weyl, H., Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Annalen, 68 (1910), 220–269.Google Scholar
12. Yosida, K., On Titchmarsh-Kodaira's formula concerning Weyl-Stone's eigenfunction expansion, Nagoya Math. J., 1 (1950) 49–58.Google Scholar