= Two-sine variational bound for a Dirichlet Mathieu equation
{title2=$\lambda_0\leq[5\pi^2-\sqrt{9\pi^4+1}]/2$}
For $-\psi''+\cos(\pi x)\psi=\lambda\psi$ on $(0,1)$ with zero <Dirichlet boundary conditions>, use $\psi=\sin(\pi x)+\alpha\sin(2\pi x)$. Its <Rayleigh quotient> is $(\pi^2+\alpha+4\pi^2\alpha^2)/(1+\alpha^2)$. In the normalized sine basis the restricted operator has <matrix> $\begin{pmatrix}\pi^2&1/2\\1/2&4\pi^2\end{pmatrix}$. Its smaller <eigenvalue> gives the displayed upper bound, attained within this trial space at $\alpha=3\pi^2-\sqrt{9\pi^4+1}$. The other stationary parameter gives the larger restricted <eigenvalue>. This is a finite-dimensional <Rayleigh-Ritz method> for a regular <Sturm-Liouville problem> with a sign-changing potential.
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