Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-06T10:12:05.871Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Theory for the Differential Equation Lu= λ Mu

Published online by Cambridge University Press:  20 November 2018

Fred Brauer
Fred Brauer*
Affiliation:
University of Chicago
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.

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958 , pp. 431 - 446
Copyright
Copyright © Canadian Mathematical Society 1958

References

1.Brauer, F., Singular self-adjoint boundary value problems for the differential equation Lx = λMx (to appear in Trans. Amer. Math. Soc).Google Scholar
2.Browder, F. E., Eigenfunction expansions for formally self-adjoint partial differential operators, Proc. Nat. Acad. Sci., 42 (1956), 769-771, 870-872.Google Scholar
3. Coddington, E. A., The spectral representation of ordinary self-adjoint differential operators. Ann. Math., 60 (1954), 192-211.Google Scholar
4. Coddington, E. A. & Levinson, N., Theory of ordinary differential equations (New York, 1955).Google Scholar
5. Friedrichs, K. O., Spektraltheorie halbbeschrdnkter Operatoren, Math. Annalen, 109 (1934), 465-487, 685-713.Google Scholar
6. Garding, L., Application of the theory of direct integrals of Hilbert spaces to some integral and differential operators, Institute for Fluid Dynamics and Applied Mathematics (Univ. of Maryland, 1954).Google Scholar
7. Garding, L., Kvantmekanikens matematiska bakgrund, Mimeographed notes (Swedish) (Lund, 1956).Google Scholar
8. Garding, L., Eigenfunction expansions, Mimeographed notes, Seminar in Applied Mathematics (Boulder, Colo., 1957).Google Scholar
9. Gelfand, I. M. & Kostyucenko, A. G., Expansions in eigenfunctions of differential and other operators, Dokl. Akad. Nauk. (Russian), 103 (1955), 349-352.Google Scholar
10. John, F., Plane waves and spherical means applied to partial differential equations (New York, 1955).Google Scholar
11. Kodaira, K., On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Amer. J. Math., 72 (1950), 502-544.Google Scholar
12. von Neumann, J., On rings of operators, reduction theory. Ann. Math., 50 (1949), 401-485.Google Scholar
13.Riesz, F. & Nagy, B. Sz., Functional analysis (New York, 1955).Google Scholar
14. Titchmarsh, E. C., Eigenfunction expansions associated with second order differential equations (Oxford, 1946).Google Scholar