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We review and further develop the recently introduced numerical approach forscattering calculations based on a so called pseudo-time Schroedinger equation,which is in turn a modification of the damped Chebyshev polynomial expansionscheme.  The method utilizes a special energy-dependent form for the absorbingpotential in the time-independent Schroedinger equation, in which the complexenergy spectrum E_k is mapped to u_k inside the unit disk, where u_k are theeigenvalues of some explicitly known sparse matrix U.  Most importantly for the numerical implementation, all the physicaleigenvalues u_k are extreme eigenvalues of U, which allows one to extract theseeigenvalues very efficiently by harmonic inversion of a pseudo-timeautocorrelation function using the filter diagonalization method. Thecomputation of 2T steps of the autocorrelation function requires only T sparsereal matrix-vector multiplications.  We describe and compare different schemes, effectively corresponding todifferent choices of the energy-dependent absorbing potential, and test themnumerically by calculating resonances of the HCO molecule. Our numerical testssuggest an optimal scheme that provide accurate estimates for most resonancestates.

FURTHER GENERALIZATION AND NUMERICAL IMPLEMENTATION OF PSEUDO-TIME SCHRÖDINGER EQUATIONS FOR QUANTUM SCATTERING CALCULATIONS