Further generalization and numerical implementation of pseudo-time   Schroedinger equations for quantum scattering calculations
  We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schroedinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme.   The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schroedinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U.   Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications.   We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states. 
V.A. Mandelshtam and A. Neumaier