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On the oscillation of almost-periodic Sturm-Liouville operators with an arbitrary coupling constant


Authors: S. G. Halvorsen and A. B. Mingarelli
Journal: Proc. Amer. Math. Soc. 97 (1986), 269-272
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1986-0835878-3
MathSciNet review: 835878
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Abstract: In this paper we characterize those (Bohr) almost periodic functions $ V$ on $ {\mathbf{R}}$ for which the Sturm-Liouville equations

$\displaystyle - y'' + \lambda V(x)y = 0,\quad x \in \mathbf{R},$

are oscillatory at $ \pm \infty $ for every real $ \lambda \ne 0$, or, equivalently, for which there exists a real $ \lambda \ne 0$ such that the equation has a positive solution on $ {\mathbf{R}}$.

References [Enhancements On Off] (What's this?)

  • [1] A. Besicovitch, Almost periodic functions, Dover, New York, 1954, xiii, 180 pp. MR 0068029 (16:817a)
  • [2] H. Bohr, Almost periodic functions, Chelsea, New York, 1951, 114 pp. MR 0020163 (8:512a)
  • [3] S. G. Halvorsen and A. B. Mingarelli, The large-scale structure of the domains of non-oscillation of second order differential equations with two parameters, preprint.
  • [4] L. Markus and R. A. Moore, Oscillation and disconjugacy for linear differential equations with almost-periodic coefficients. Acta Math. 96 (1956), 79-123. MR 0080813 (18:306d)
  • [5] R. A. Moore, The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955), 125-145. MR 0068690 (16:925b)
  • [6] S. Staněk, A note on the oscillation of the differential equation $ y'' = \lambda qy$ with a periodic coefficient, Czechoslovak Math. J. 29 (1979), 318-323. MR 529519 (80c:34032)
  • [7] A. Winter, On the non-existence of conjugate points, Amer. J. Math. 73 (1951), 368-380. MR 0042005 (13:37d)

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DOI: https://doi.org/10.1090/S0002-9939-1986-0835878-3
Article copyright: © Copyright 1986 American Mathematical Society

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