 07253 Hamse Y. Mussa, Jonathan Tennyson, and Robert C. Glen
 Inversecovariance matrix of linear operators for quantum spectrum scanning
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Oct 27, 07

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Abstract. It is demonstrated that the Schr\"odinger operator in ${\bf\hat{H}}\mid\psi_{k}> = E_{k}\mid\psi_{k}>$
can be associated with a covariance matrix whose eigenvalues are the squares of the spectrum
$\sigma({\bf \hat{H}} +\bf{I}\zeta )$ where $\zeta$ is an arbitrarily chosen shift. An efficient method
for extracting $\sigma({\bf\hat{H}})$ components, in the vicinity of $\zeta $, from a few specially selected
eigenvectors of the inverse of the covariance matrix is derived. The method encapsulates (and improves on) the
three most successful quantum spectrum scanning schemes: FilterDiagonalization, Shiftandinvert Lanczos and
Folded Spectrum Method. It gives physical insight into the scanning process. The new method can also be employed
to probe the nature of underlying potential energy surfaces. A sample application to the neardissociation vibrational
spectrum of the HOCl molecule is presented.
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