Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-30T06:36:45.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Green's Functions for Singular Ordinary Differential Operators

Published online by Cambridge University Press:  20 November 2018

Fred Brauer
Fred Brauer*
Affiliation:
University of Wisconsin and Massachusetts Institute of Technology
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There are several ways to approach the eigenfunction expansion problem for ordinary differential operators via the spectral theorem for self-ad joint linear operators in Hilbert space. One can examine the resolvent, which requires a detailed study of the Green's function (4, 5, 7), or one can use the spectral theorem for unbounded operators (2, 3, 9). Since the eigenf unction expansion theorem also requires some multiplicity theory, unless one is prepared to use a rather powerful form of the spectral theorem for unbounded operators, as in (2, 9), the proof requires a good deal of work in addition to the spectral theorem.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 571 - 582
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Akhiezer, N. I. and Glazman, I. M., Theory of linear operators in Hilbert space, vol. 2 (translation from the Russian ; New York, 1963).Google Scholar
2. Brauer, F., Spectral theory for the differential equation Lu = λMu, Can. J. Math., 10 (1958), 431–446.Google Scholar
3. Coddington, E. A., The spectral representation of ordinary self-adjoint differential operators, Ann. of Math., 60 (1954), 192–211.Google Scholar
4. Coddington, E. A., The spectral matrix and Green's function far singular self-adjoint boundary value problems, Can. J. Math., 6 (1954), 169–185.Google Scholar
5. Coddington, E. A., On self-adjoint ordinary differential operators, Math. Scand., 4 (1956), 9–21.Google Scholar
6. Coddington, E. A., On maximal symmetric ordinary differential operators, Math. Scand., 4 (1956), 22–28.Google Scholar
7. Coddington, E. A., Generalized resolutions of the identity for symmetric ordinary differential operators, Ann. of Math., 68 (1958), 378–392.Google Scholar
8. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (New York, 1955).Google Scholar
9. Garding, L., Kvantmekanikens matematiska bakgrund, mimeographed notes (Swedish ; Lund, 1956).Google Scholar
10. Levinson, N., The expansion theorem for singular self-adjoint linear differential operators, Ann. of Math., 59 (1954), 300–315.Google Scholar
11. Levinson, N., Transform and inverse transform expansions for singular self-adjoint differential operators, Illinois J. Math., 2 (1958), 224–235.Google Scholar