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
Full-text PDF Free Access
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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 $S=1$.
References
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- J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0034932
- G. A. Kriegsmann, Scattering matrix analysis of a photonic Fabry-Perot resonator, Wave Motion 37 (2003), no. 1, 43–61. MR 1938952, DOI https://doi.org/10.1016/S0165-2125%2802%2900014-8
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962. MR 0176151
- E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, Oxford, 1976.
- Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0230476
- Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004
References
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338 (16,1022b)
- J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0034932 (11,666a)
- G. A. Kriegsmann, Scattering matrix analysis of a photonic Fabry-Perot resonator, Wave Motion 37 (2003), no. 1, 43–61. MR 1938952 (2004a:78017), DOI https://doi.org/10.1016/S0165-2125%2802%2900014-8
- E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151 (31 \#426)
- E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, Oxford University Press, Oxford, 1976.
- Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0230476 (37 \#6038)
- Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004 (53 \#3842)
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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
MR Author ID:
106665
Email:
gregory.a.kriegsmann@njit.edu
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