On the Lower Bounds of Weinstein and Romberg in Quantum Mechanics

Pulak Ranjan Giri

Donald H. Kobe

Walter Kohn

Weinstein's modification of the Ritz principle is used to derive (1) a lower bound for the $n$-th energy-level of a quantum mechanical system if a lower bound for the $(n+1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known; and (2) an upper bound for the $n$-th level if an upper bound for the $(n\ensuremath{-}1)\ensuremath{-}\mathrm{s}\mathrm{t}$ level is known.

Maurice Cohen, Tova Feldmann

The classical procedure of Weinstein has been employed to obtain rigorous upper and lower bounds to the eigenvalues E of a quantum mechanical Hamiltonian operator H. The new bounds represent an improvement over Weinstein's bounds for any reasonable choice of variational trial function. In the case of the lowest eigenvalue E0, for which the Rayleigh–Ritz procedure gives the optimum upper bound, the new lower bound is an improvement over the lower bound formula of Stevenson and Crawford.

Valia Allori, Nino Zanghı̀

M. Cohen, Tova Feldmann

A simple unified procedure, based on a generalization of a method first used by Weinstein in 1934, yields effective lower and upper bounds to energy levels and overlap integrals of quantum-mechanical systems. A further generalization yields bounds to expectation values.