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Journal of Physics A Mathematical and General

A projection method for generating bounds to overlap

Anthea Grubb

Published Feb 1, 1976• 3 citations
Publisher

Abstract

A simple method is presented which unifies and generalizes a variety of upper and lower bounds for overlaps in a Hilbert space context.
DOI
10.1088/0305-4470/9/2/004

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A general formalism is presented which furnishes upper and lower variational bounds on squared overlaps mod ( theta , chi ) mod 2 , where chi and theta are vectors in a Hilbert space. The bounding functionals involve a self-adjoint operator A and a second operator B, such that A theta =0 and B chi =b chi . Most of the non-variational bounds derived by other workers on quantum-mechanical overlaps emerge from the formalism, together with optimal variational improvements in every case. In addition many new bounds are obtained.
The accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ | ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.