Variational projection onto an involutive symmetry (${\mathcal{E}}_{\pm }=(a\pm c)/(1\pm b)$)

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 $\varphi$, define $a=⟨H{⟩}_{\varphi }$, $b=⟨I{⟩}_{\varphi }$, $c=⟨HI{⟩}_{\varphi }$. The Rayleigh quotient on $\varphi +tI\varphi$ 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 $\varphi \pm I\varphi$, 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 $H$.