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Journal Article
The Minimum Ratio of Two Eigenvalues
Joseph B. Keller
SIAM Journal on Applied Mathematics
Vol. 31, No. 3 (Nov., 1976), pp. 485491
Published by: Society for Industrial and Applied Mathematics
Stable URL: http://www.jstor.org/stable/2100496
Page Count: 7
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Topics: Eigenvalues, Mathematical minima, Mathematical functions, Mathematical problems, Sine function, Mathematical constants, Mathematical intervals, Open intervals, Piecewise continuous functions, Minimum value
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Abstract
The first two eigenvalues, λ1 and λ2, of the problem $y" + \lambda\phi(x)y = 0, y(\pm\frac {1}{2}) = 0$ are considered. The minimum of their ratio λ2/λ1 is sought for φ(x) ranging over the class of piecewise continuous functions satisfying the inequalities $0 < a \leq \phi(x) \leq A$. It is found that the minimum is an increasing function of a/A, varying from unity at a/A = 0 to four at a/A = 1. A graph of the minimum is given. The minimizing function φ(x) is found to be piecewise constant, taking on the value a for $x_0 < x < x_0$ and the value A elsewhere, and the jump point x0 is found as a function of a/A. The result provides a lower bound on the ratio λ2/λ1 for any φ(x) in the class considered. The method of analysis is applicable to other similar problems with inequality constraints.
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SIAM Journal on Applied Mathematics © 1976 Society for Industrial and Applied Mathematics