Inverse-covariance matrix of linear operators for quantum spectrum scanning
نویسندگان
چکیده
It is demonstrated that the Schrödinger operator in Ĥ | ψk >= Ek | ψk > can be associated with a covariance matrix whose eigenvalues are the squares of the spectrum σ(Ĥ + Iζ) where ζ is an arbitrarily chosen shift. An efficient method for extracting σ(Ĥ) components, in the vicinity of ζ, 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. PACS numbers: 03.65.-w,02.70.-c,02.10.10.Yn,02.30.Tb,02.50.Sk,02.30.Zz,02.30Yy
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