Rayleigh quotient with one Neumann endpoint ($\mathcal{R}=k^2-[\varphi {\varphi }^{\prime }{]}_{z_1}^{z_2}/\int {\varphi }^{2}dz$)

For a nonzero real solution of ${\varphi }^{\prime \prime }+(C-k^2)\varphi =0$ on a finite interval, integration by parts gives $\mathcal{R}=\int (C{\varphi }^2-{\varphi }^{\prime 2})dz/\int {\varphi }^{2}dz$ 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 $\varphi (z_2){\varphi }^{\prime }(z_2)=0$. 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.