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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Local completeness for eigenfunctions of regular maximal ordinary differential operators

Author: Robert Carlson
Journal: Proc. Amer. Math. Soc. 79 (1980), 400-404
MSC: Primary 47E05; Secondary 34B25
MathSciNet review: 567980
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Abstract: Let $ L = \sum\nolimits_{m = 0}^n {{A_m}(x){D^m}} $ be a differential operator on $ { \oplus ^k}{L^2}[0,1]$ with infinitely differentiable $ k \times k$ matrix valued coefficients. Assume that $ \det \,{A_n}(x) \ne 0$ for $ x \in [0,1]$. The domain of L is the set of k-vector valued functions f such that $ f \in {C^{n - 1}}([0,1]),{f^{(n - 1)}}$ is absolutely continuous on [0, 1] and $ Lf \in { \oplus ^k}{L^2}[0,1]$. Let $ {x_0} \in (0,1)$. Then there is a neighborhood $ U({x_0})$ containing $ {x_0}$ such that the restrictions of the eigenfunctions of L to $ U({x_0})$ have dense span in $ { \oplus ^k}{L^2}[U({x_0})]$. The example $ L = {e^{ - ix}}d/dx$ shows that this is the best possible abstract result.

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Article copyright: © Copyright 1980 American Mathematical Society