The Minimum Ratio of Two Eigenvalues

The first two eigenvalues, $\lambda _1 $ and $\lambda _2 $, of the problem $y'' + \lambda \phi ( x )y = 0$, $y( { \pm \frac{1}{2}} ) = 0$ are considered. The minimum of their ratio ${\lambda _2 / \lambda _1 }$ is sought for $\phi ( x )$ ranging over the class of piecewise continuous functions satisfying the inequalities $0 < a \leqslant \phi ( x ) \leqslant A$. 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 $\phi ( x )$ 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 $x_0 $ is found as a function of ${a / A}$. The result provides a lower bound on the ratio ${\lambda _2 / \lambda _1 }$ for any $\phi ( x )$ in the class considered. The method of analysis is applicable to other similar problems with inequality constraints.