New methods of high resolution spectral analysis of short time signals are presented. These methods utilize the filter-diagonalization approach of Wall and Neuhauser [J. Chem. Phys. 102, 8011 (1995)] that extracts the complex frequencies ωk and amplitudes dk from a signal C(t)=∑kdke−itωk in a small frequency interval by recasting the harmonic inversion problem as the one of a small matrix diagonalization. The present methods are rigorously adapted to the conventional case of the signal available on a sparse equidistant time grid and use a more efficient boxlike filter. Various applications are discussed, such as iterative diagonalization of large Hamiltonian matrices for calculating bound and resonance states, scattering calculations in the presence of narrow resonances, etc. For the scattering problem the harmonic inversion is directly applied to the signal cn=(χf,Tn(Ĥ)χi), generated by the dynamical system governed by a modified Chebyshev recursion, avoiding the usual recasting the problem to the time domain. Some challenging numerical examples are presented. The general filter-diagonalization method is shown to be stable and efficient for the extraction of thousands of complex frequencies ωk and amplitudes dk from a signal. When the model signal is “spoiled” by a moderate amount of an additive Gaussian noise the obtained spectral estimate is still superior to the conventional Fourier spectrum.
Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. A low-storage filter diagonalization method for quantum eigenenergy calculation or for spectral analysis of time
The new recently introduced [J. Chem. Phys 102, 7390 (1995)] empirical recursion formula for the scattering solution is here proved to yield an exact polynomial expansion of the operator [E−(Ĥ+Γ̂)]−1, Γ̂ being a simple complex optical potential. The expansion is energy separable and converges uniformly in the real energy domain. The scaling of the Hamiltonian is trivial and does not involve complex analysis. Formal use of the energy-to-time Fourier transform of the ABC (absorbing boundary conditions) Green’s function leads to a recursion polynomial expansion of the ABC time evolution operator that is global in time. Results at any energy and any time can be accumulated simultaneously from a single iterative procedure; no actual Fourier transform is needed since the expansion coefficients are known analytically. The approach can be also used to obtain a perturbation series for the Green’s function. The new iterative methods should be of a great use in the area of the reactive scattering calculations and other related fields.
A new method of implementing scattering calculations is presented. For the S-matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal ℒ2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−Ĥ)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.
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