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The discreteness of the spectrum of self-adjoint, even order, one-term, differential operators


Author: Roger T. Lewis
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
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Abstract: An open question which was asked by I. M. Glazman is answered. It is well known that the condition

$\displaystyle \mathop {\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 [Enhancements On Off] (What's this?)

  • [1] Calvin D. Ahlbrandt, Equivalent boundary value problems for self-adjoint differential systems, J. Differential Equations 9 (1971), 420-435. MR 44 #1860. MR 0284636 (44:1860)
  • [2] I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators, Fizmatgiz, Moscow, 1963; English transl., Israel Program for Scientific Translations, Jerusalem, 1965 and Davey, New York, 1966. MR 32 #2938; #8210. MR 0190800 (32:8210)
  • [3] Roger T. Lewis, Oscillation and nonoscillation criteria for some self-adjoint, even order, linear differential operators, Pacific J. Math. (to appear). MR 0350112 (50:2605)

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DOI: https://doi.org/10.1090/S0002-9939-1974-0330608-8
Article copyright: © Copyright 1974 American Mathematical Society

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