The discreteness of the spectrum of self-adjoint, even order, one-term, differential operators
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- by Roger T. Lewis
- Proc. Amer. Math. Soc. 42 (1974), 480-482
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330608-8
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Abstract:
An open question which was asked by I. M. Glazman is answered. It is well known that the condition \[ \lim \limits _{x \to \infty } {x^{2n - 1}}\int _x^\infty {{r^{ - 1}} = 0} \] is sufficient for the discreteness and boundedness from below of the spectrum of selfadjoint extensions of ${( - 1)^n}{(r{y^{(n)}})^{(n)}}$. This paper shows that the condition is also necessary.References
- Calvin D. Ahlbrandt, Equivalent boundary value problems for self-adjoint differential systems, J. Differential Equations 9 (1971), 420–435. MR 284636, DOI 10.1016/0022-0396(71)90015-5
- I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Daniel Davey & Co., Inc., New York, 1966. Translated from the Russian by the IPST staff; Israel Program for Scientific Translations, Jerusalem, 1965. MR 0190800
- Roger T. Lewis, Oscillation and nonoscillation criteria for some self-adjoint even order linear differential operators, Pacific J. Math. 51 (1974), 221–234. MR 350112
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 480-482
- MSC: Primary 34B25
- DOI: https://doi.org/10.1090/S0002-9939-1974-0330608-8
- MathSciNet review: 0330608