Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

Instability intervals and growth rates for Hill's equation


Author: Joseph B. Keller
Journal: Quart. Appl. Math. 66 (2008), 191-195
MSC (2000): Primary 34B30
Published electronically: December 5, 2007
MathSciNet review: 2396657
Full-text PDF

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

  • 1. Wilhelm Magnus and Stanley Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons New York-London-Sydney, 1966. MR 0197830
  • 2. Jack K. Hale, On the behavior of the solutions of linear periodic differential systems near resonance points, Contributions to the theory of nonlinear oscillations, Vol. V, Princeton Univ. Press, Princeton, N.J., 1960, pp. 55–89. MR 0141827
  • 3. Dorothy M. Levy and Joseph B. Keller, Instability intervals of Hill’s equation, Comm. Pure Appl. Math. 16 (1963), 469–476. MR 0153914
  • 4. Evans M. Harrell II, On the effect of the boundary conditions on the eigenvalues of ordinary differential equations, Contributions to analysis and geometry (Baltimore, Md., 1980) Johns Hopkins Univ. Press, Baltimore, Md., 1981, pp. 139–150. MR 648460
  • 5. Joseph Avron and Barry Simon, The asymptotics of the gap in the Mathieu equation, Ann. Physics 134 (1981), no. 1, 76–84. MR 626698, 10.1016/0003-4916(81)90005-1
  • 6. Erdelyi, A., Ueber die freien Schwingungen in Kondensatorkreisen von veränderlichen Kapazitaet, Ann. Physik, Vol. 19, 1934, 585-622.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 34B30

Retrieve articles in all journals with MSC (2000): 34B30


Additional Information

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

DOI: https://doi.org/10.1090/S0033-569X-07-01083-1
Received by editor(s): June 28, 2007
Published electronically: December 5, 2007
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website