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The spectral theory of second order two-point differential operators: I. A priori estimates for the eigenvalues and completeness

Published online by Cambridge University Press:  14 November 2011

John Locker
John Locker
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, U.S.A.
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Synopsis

This paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L2[0, 1] determined by a second order formal differential operator l = −D2 + pD + q and by independent boundary values B1, B2. The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B1, B2. For Cases 1–4 it is shown that if λ = ρ2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L2[0, 1].

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 3-4 , 1992 , pp. 279 - 301
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Birkhoff, G. D.. On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Amer. Math. Soc. 9 (1908), 219–231.CrossRefGoogle Scholar
2Birkhoff, G. D.. Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9 (1908), 373–395.CrossRefGoogle Scholar
3Coddington, E. A. and Levinson, N.. Theory of Ordinary Differential Equations (New York: McGraw-Hill, 1955).Google Scholar
4Dunford, N. and Schwartz, J. T.. Linear Operators, I, II, III (New York: Wiley-Interscience, 1958, 1963, 1971).Google Scholar
5Hoffman, S.. Second-order linear differential operators defined by irregular boundary conditions (Dissertation, Yale University, 1957).Google Scholar
6Lang, P. and Locker, J.. Spectral decomposition of a Hilbert space by a Fredholm operator. J. Fund. Anal. 79 (1988), 9–17.CrossRefGoogle Scholar
7Lang, P. and Locker, J.. Spectral representation of the resolvent of a discrete operator. J. Fund. Anal. 79(1988), 18–31.CrossRefGoogle Scholar
8Lang, P. and Locker, J.. Denseness of the generalized eigenvectors of an H-S discrete operator. J. Fund. Anal. 82 (1989), 316–329.CrossRefGoogle Scholar
9Lang, P. and Locker, J.. Spectral theory for a differential operator: Characteristic determinant and Green's function. J. Math. Anal. Appl. 141 (1989), 405–423.CrossRefGoogle Scholar
10Lang, P. and Locker, J.. Spectral theory of two-point differential operators determined by –D2. I. Spectral properties. J. Math. Anal. Appl. 141 (1989), 538–558.CrossRefGoogle Scholar
11Lang, P. and Locker, J.. Spectral theory of two-point differential operators determined by –D2. II. Analysis of cases. J. Math. Anal. Appl. 146 (1990), 148–191.CrossRefGoogle Scholar
12Locker, J.. Functional Analysis and Two-Point Differential Operators, Pitman Research Notes in Mathematics 144 (Harlow: Longman, 1986).Google Scholar
13Locker, J.. The nonspectral Birkhoff-regular differential operators determined by –D2. J. Math. Anal. Appl. 154 (1991), 243–254.CrossRefGoogle Scholar
14Locker, J.. The spectral theory of second order two-point differential operators. II. Asymptotic expansions and the characteristic determinant (to appear).Google Scholar
15Locker, J.. The spectral theory of second order two-point differential operators. III. The eigenvalues and their asymptotic formulae (to appear).Google Scholar
16Locker, J.. The spectral theory of second order two-point differential operators. IV. The associated projections and the subspace S∞) (to appear).Google Scholar
17Naimark, M. A.. Linear Differential Operators, I (Moscow: GITTL, 1954; English transl: New York: Ungar, 1967).Google Scholar
18Schwartz, J.. Perturbations of spectral operators, and applications. I. Bounded perturbations. Pacific J. Math. 4 (1954), 415–458.CrossRefGoogle Scholar
19Stone, M. H.. A comparison of the series of Fourier and Birkhoff. Trans. Amer. Math. Soc. 28 (1926), 695–761.CrossRefGoogle Scholar
20Stone, M. H.. Irregular differential systems of order two and the related expansion problems. Trans. Amer. Math. Soc. 29 (1927), 23–53.CrossRefGoogle Scholar
21Tamarkin, J.. Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions. Math. Z. 27 (1927), 1–54.CrossRefGoogle Scholar
22Walker, P. W.. A nonspectral Birkhoff-regular differential operator. Proc. Amer. Math. Soc. 66 (1977), 187–188.Google Scholar