= Affine-parameter Sturm-Liouville eigenvalue bound
{title2=$\lambda_{\min}(\epsilon)\leq\lambda_{\min}(0)+\epsilon\langle U_0,L_1U_0\rangle$}
For a self-adjoint <Sturm-Liouville operator> $L_\epsilon=L_0+\epsilon L_1$ with common boundary conditions and an $L^2$-normalized lowest <eigenfunction> $U_0$ of $L_0$, evaluating the <Rayleigh quotient> on $U_0$ gives the displayed upper bound. More generally the lowest eigenvalue is the infimum of affine functions of $\epsilon$, so it is concave in that parameter. The trial state must satisfy the boundary requirements of the variational domain; changing the operator without changing the trial function's normalization can still be legitimate when the weight is fixed.
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