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On a Non-Analytic Perturbation Problem

Published online by Cambridge University Press:  20 November 2018

C. A. Swanson
C. A. Swanson*
Affiliation:
University of British Columbia
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The purpose here is to study a type of perturbation problem, arising from a differential equation, which is not included in the realm of analytic or asymptotic perturbation theory [4], [6]. Such a problem arises when the domain of the differential operator has been subjected to a variation (rather than the formal operator). We propose to outline one simple problem of this type, concerned with a second order ordinary differential operator. Our purpose is to obtain asymptotic estimates for the characteristic values of a regular Sturm-Liouville problem on a closed interval [a, b] when b is near a singular point of the differential operator. Similar results have been obtained in [3], [8], and [9] by other methods.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 4 , Issue 3 , September 1961 , pp. 243 - 248
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Coddington, E. A. and Levinson, N., Theory of ordinary differential equations, (New York, 1955).Google Scholar
2. Courant, R. and Hilbert, D., Methods of mathematical physics I, (New York, 1953).Google Scholar
3. Hull, T. E. and Julius, R. S., Enclosed quantum mechanical systems, Canad. J. Phys. 34 (1956), 914-919.Google Scholar
4. Kato, Tosio, Quadratic forms in Hilbert spaces and asymptotic perturbation series, Technical report no. 7, Department of Mathematics, University of California, Berkeley (1955).Google Scholar
5. Rellich, Franz, Spectral theory of a second order ordinary differential operator, New York University (1953).Google Scholar
6. Riesz, F. and Sz-Nagy, B., Functional Analysis, (New York, 1956).Google Scholar
7. Schiffer, M. and D. C. Spencer, Functionals of finite Riemann surfaces, (Princeton University Press, 1954).Google Scholar
8. Swanson, C. A., Differential operators with perturbed domains, J. Rat. Mech. Anal. 6, 6 (1957), 823-846.Google Scholar
9. Swanson, C. A., Asymptotic perturbation series for characteristic value problems, Pacific J. Math. 9 (1959), 591-608.Google Scholar
10. Swanson, C. A., Asymptotic estimates for limit circle problems, to appear in Pacific J. Math, in 1962.Google Scholar