= Variational projection onto an involutive symmetry
{title2=$\mathcal E_\pm=(a\pm c)/(1\pm b)$}
Let $H$ be a lower-bounded self-adjoint <Hamiltonian operator> and $I$ a commuting self-adjoint operator with $I^2=1$. For a normalized trial vector $\phi$, define $a=\langle H\rangle_\phi$, $b=\langle I\rangle_\phi$, $c=\langle HI\rangle_\phi$. The <Rayleigh quotient> on $\phi+tI\phi$ is $[(1+t^2)a+2tc]/[1+t^2+2tb]$, with derivative $2(c-ab)(1-t^2)/(1+t^2+2tb)^2$. Unless $c=ab$, its extrema are the normalized nonzero vectors $\phi\pm I\phi$, with energies $(a\pm c)/(1\pm b)$. Their difference is $2(c-ab)/(1-b^2)$, identifying the maximum and minimum.
The orthogonal projections $(1\pm I)/2$ split any trial energy into a weighted mean of the two sector energies, so one nonzero sector projection never raises the trial energy. If $|b|=1$, one projected vector is zero and its quotient is undefined; the remaining family has constant energy. Exchange symmetry makes these sectors bosonic and fermionic spatial wavefunctions. Stationarity within this two-dimensional trial space does not generally make a vector an exact eigenvector of $H$.
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