Inverse-covariance matrix of linear operators for quantum spectrum scanning Hamse Y. Mussa1, Jonathan Tennyson2, and Robert C. Glen1 1Unilever Centre for Molecular Sciences Informatics Department of Chemistry, University of Cambridge Lens¯eld Road, Cambridge CB2 1EW, UK 2Department of Physics and Astronomy University College London, London WC1E 6BT, UK E-mail:
[email protected] Abstract. It is demonstrated that the SchrÄodinger operator in H^ j Ãk >= Ek j Ãk > can be associated with a covariance matrix whose eigenvalues are the squares of the spectrum σ(H^ + I³) where ³ is an arbitrarily chosen shift. An e±cient method for extracting σ(H^ ) components, in the vicinity of ³, 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 Keywords: Eigensolver, Mixed state, Quantum Control Inverse-covariance matrix of linear operators for quantum spectrum scanning 2 1. Introduction Quantum mechanics provides our understanding of the behaviour of microscopic objects such as atoms, molecules and their constituents. Thus non-relativistic studies of these objects and any process involving them essentially requires solving the appropriate SchrÄodingerequation.