Quarterly of Applied Mathematics Quarterly of Applied Mathematics
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Instability intervals and growth rates for Hill's equation

Author(s): Joseph B. Keller
Journal: Quart. Appl. Math. 66 (2008), 191-195.
MSC (2000): Primary 34B30
Posted: December 5, 2007
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Abstract | References | Similar articles | Additional information

Abstract: Hill's equation is a real linear second-order ordinary differential equation with a periodic coefficient $ f(t)$:

$\displaystyle y^{\prime\prime} (t) +\left[ \lambda+\varepsilon f\left(t\right) \right] y(t) =0.$ (0.1)

It has unbounded solutions for certain intervals of the real parameter $ \lambda$, called instability intervals. Here these intervals, and the growth rate of the unbounded solutions, are determined for $ \varepsilon$ small, and also for $ \lambda$ large. This is done by constructing a fundamental pair of solutions which are power series in $ \varepsilon/\lambda^{1/2}$, with coefficients that are bounded functions of $ \lambda$.

References:

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Magnus, W. and Winkler, S., Hill's Equation, Interscience Publishers, John Wiley, New York, 1966. MR 0197830 (33:5991)

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Hale, J. K., On the behavior of solutions of linear periodic differential equations near resonance points, Contributions to the theory of Nonlinear Oscillations, Vol. 5, 1960, 55-89, Annals of Math. Studies, Princeton Univ. Press, Princeton. MR 0141827 (25:5224)

3.
Levy, D. M. and Keller, J. B., Instability Intervals of Hill's Equation, Comm. Pure Appl. Math. 16 (1963), 469-476. MR 0153914 (27:3875)

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Harrell, E. On the effect of the boundary conditions on the eigenvalues of ordinary differential equations, Amer. J. Math. supplement dedicated to P. Hartman, Johns Hopkins Univ. Press, Baltimore, 1981. MR 648460 (83c:34031)

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Avron, J. and Simon, B., The asymptotics of the gap in the Mathieu equation, Ann. Physics 134 (1981), 76-84. MR 626698 (82h:34030)

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Erdelyi, A., Ueber die freien Schwingungen in Kondensatorkreisen von veränderlichen Kapazitaet, Ann. Physik, Vol. 19, 1934, 585-622.

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Additional Information:

Joseph B. Keller
Affiliation: Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, California 94305-2125
Email: keller@math.stanford.edu

PII: S0033-569X-07-01083-1
Received by editor(s): June 28, 2007
Posted: December 5, 2007
Copyright of article: Copyright 2007, Brown University
The copyright for this article reverts to public domain after 28 years from publication.


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