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 ${\psi }_t=u_1+t{u}_0$ is $(E_1+t^2{E}_0)/(1+t^2)$. 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.