Hamse Y. Mussa, Jonathan Tennyson, and Robert C. Glen
Inverse-covariance matrix of linear operators for quantum spectrum scanning
(340K, pdf)

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: Filter-Diagonalization, Shift-and-invert 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 near-dissociation vibrational 
spectrum of the HOCl molecule is presented.