= Rayleigh quotient with one Neumann endpoint
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{title2=$\mathcal R=k^2-[\phi\phi']_{z_1}^{z_2}/\int\phi^2dz$}
For a nonzero real solution of $\phi''+(C-k^2)\phi=0$ on a finite interval, <integration by parts> gives $\mathcal R=\int(C\phi^2-\phi'^2)dz/\int\phi^2dz$ in the displayed form. A <Neumann boundary condition> only at the lower endpoint does not remove the upper boundary term. Equality with $k^2$ additionally requires $\phi(z_2)\phi'(z_2)=0$. With the lower condition, nonconstant oscillatory solutions satisfy this when their phase advance is an integer multiple of $\pi/2$; affine solutions must be constant, while nonzero hyperbolic-cosine solutions fail the condition. Infinite intervals additionally require convergence of both integrals.
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