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.References
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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