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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Local completeness for eigenfunctions of regular maximal ordinary differential operators
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by Robert Carlson PDF
Proc. Amer. Math. Soc. 79 (1980), 400-404 Request permission

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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Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 79 (1980), 400-404
  • MSC: Primary 47E05; Secondary 34B25
  • DOI: https://doi.org/10.1090/S0002-9939-1980-0567980-2
  • MathSciNet review: 567980