Potential envelope theory and the local energy theorem

We consider a one--particle bound quantum mechanical system governed by a Schr\"odinger operator $\mathscr{H} = -\Delta + v\,f(r)$, where $f(r)$ is an attractive central potential, and $v>0$ is a coupling parameter. If $\phi \in \mathcal{D}(\mathscr{H})$ is a `trial function', the local energy theorem tells us that the discrete energies of $\mathscr{H}$ are bounded by the extreme values of $(\mathscr{H}\phi)/\phi,$ as a function of $r$. We suppose that $f(r)$ is a smooth transformation of the form $f = g(h)$, where $g$ is monotone increasing with definite convexity and $h(r)$ is a potential for which the eigenvalues $H_n(u)$ of the operator $\mathcal{H}=-\Delta + u\, h(r)$, for appropriate $u >0$, are known. It is shown that the eigenfunctions of $\mathcal{H}$ provide local-energy trial functions $\phi$ which necessarily lead to finite eigenvalue approximations that are either lower or upper bounds. This is used to extend the local energy theorem to the case of upper bounds for the excited-state energies when the trial function is chosen to be an eigenfunction of such an operator $\mathcal{H}$. Moreover, we prove that the local-energy approximations obtained are identical to `envelope bounds', which can be obtained directly from the spectral data $H_n(u)$ without explicit reference to the trial wave functions.