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On the convergence of Hill's method

Authors: Christopher W. Curtis and Bernard Deconinck
Journal: Math. Comp. 79 (2010), 169-187
MSC (2000): Primary 34L16, 65L07
Published electronically: July 6, 2009
MathSciNet review: 2552222
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Abstract: Hill's method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill's method. We show the method does not produce any spurious approximations, and that for self-adjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eigenvalue, we prove convergence of the associated eigenfunction approximation in the $ L^2$-norm. These results are not restricted to selfadjoint operators. Finally, for certain selfadjoint operators, we prove that the rate of convergence of Hill's method to the least eigenvalue is faster than any polynomial power.

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Additional Information

Christopher W. Curtis
Affiliation: Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3420

Bernard Deconinck
Affiliation: Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3420

Received by editor(s): October 7, 2008
Received by editor(s) in revised form: February 13, 2009
Published electronically: July 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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