  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.
