Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



A lower bound and estimate of the lowest eigenvalue of a second order Floquet equation

Author: G. A. Kriegsmann
Journal: Quart. Appl. Math. 73 (2015), 599-605
MSC (2010): Primary 34L10, 31A25, 30A99, 30B10, 30E10
DOI: https://doi.org/10.1090/qam/1389
Published electronically: September 15, 2015
MathSciNet review: 3432273
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Abstract | References | Similar Articles | Additional Information

Abstract: A bound on the lowest eigenvalue of a second order Floquet problem is derived by applying the Cauchy Integral Theorem. Specifically, we chose a special function which depends on an arbitrary positive parameter $ S\ge 1$. We use the residue theorem and show that its residue at the origin determines an infinite sum composed of reciprocals of the eigenvalues raised to the $ 2S$ power. A simple bound gives us our result. We show that the residue depends explicitly on the power series expansions, in the eigen-parameter, of the original equation's fundamental solutions. The coefficients of these power series are computed recursively.

Three typical examples are presented, and it is shown for these cases that the lower bound, derived in this paper, actually affords a good approximation to the first eigenvalue. We show this for the case only of $ S=1$.

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Additional Information

G. A. Kriegsmann
Affiliation: Department of Mathematical Sciences, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, University Heights, Newark, New Jersey 07102
Email: gregory.a.kriegsmann@njit.edu

DOI: https://doi.org/10.1090/qam/1389
Received by editor(s): August 21, 2013
Received by editor(s) in revised form: November 18, 2013
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 Brown University

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