Improvement of Weyl’s Inequality

Abstract

Consider the construction of an operator from the sum of two component operators. Weyl’s inequality gives a lower bound to an eigenvalue of the constructed operator using a single eigenvalue from each of the component operators. Using such minimal information gives a poor bound, however, and when the eigenvectors that correspond to the said eigenvalues of the component operators are known, Weyl’s inequality can be significantly improved by considering the overlap of the two eigenvectors. This improvement can sometimes be further improved when several eigenvectors of each component operator are known so that the overlap of sub-eigenspaces are considered instead. The improvement is best when there is minimal overlap and Weyl’s inequality returns when the overlap is complete. An example with the hydrogen molecular ion is presented which illustrates the superiority over Weyl’s inequality when eigenvector or sub-eigenspace information is utilized.