= Stationary Rayleigh quotient need not be a minimum
For a <self-adjoint operator> with two normalized orthogonal <eigenvectors> $u_0,u_1$ and $E_0<E_1$, the <Rayleigh quotient> of $\psi_t=u_1+tu_0$ is $(E_1+t^2E_0)/(1+t^2)$. It is stationary at $t=0$ but has second derivative $2(E_0-E_1)<0$ there. Thus an exact excited <eigenfunction> can be a maximum within a trial family. A constrained minimum requires exclusion of lower states, as in an <excited-state variational constraint>.
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