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We consider a one--particle bound quantum mechanical system governed by aSchr\"odinger operator $\mathscr{H} = -\Delta + v\,f(r)$, where $f(r)$ is anattractive central potential, and $v>0$ is a coupling parameter. If $\phi \in\mathcal{D}(\mathscr{H})$ is a `trial function', the local energy theorem tellsus that the discrete energies of $\mathscr{H}$ are bounded by the extremevalues 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 monotoneincreasing with definite convexity and $h(r)$ is a potential for which theeigenvalues $H_n(u)$ of the operator $\mathcal{H}=-\Delta + u\, h(r)$, forappropriate $u >0$, are known. It is shown that the eigenfunctions of$\mathcal{H}$ provide local-energy trial functions $\phi$ which necessarilylead 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 forthe excited-state energies when the trial function is chosen to be aneigenfunction of such an operator $\mathcal{H}$. Moreover, we prove that thelocal-energy approximations obtained are identical to `envelope bounds', whichcan be obtained directly from the spectral data $H_n(u)$ without explicitreference to the trial wave functions.

Potential envelope theory and the local energy theorem