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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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