Harmonic inversion of time signals and its applications

Abstract

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.

Fractal Weyl law for chaotic microcavities: Fresnel’s laws imply multifractal scattering

“…This time signal is analyzed, in the second step, by a high-resolution harmonic inversion method. The harmonic inversion method for signal processing has been introduced by Wall and Neuhauser [27] and then refined and improved by Mandelshtam and Taylor [28,29]. It has proved to be a powerful tool for both the highresolution analysis of quantum spectra [30] and the semiclassical quantization of non-integrable systems [31,32,33,34].…”

Pseudotime Schrödinger Equation with Absorbing Potential for Quantum Scattering Calculations

“…, 2K) satisfying (19) for the sequence y α (t) computed by (16) and (18). Since by (15) the sequence y α (t) is bounded and the spectral mapping (4) moves the physically relevant eigenvalues u k close to the unit circle, this is an efficiently tractable problem, even in very large dimensions [14,9].…”

“…A similar 3-term-recurrence with another choice of special initial conditions leading to "modified Chebyshev recurrence" was considered in refs. [9,11].) Since D is diagonal and matrix-vector products Hφ are usually cheap to form, φ(0), .…”

“…For j = k, we may achieve (9) by normalizing the eigenvectors, provided that the left hand side of (9) does not vanish. Using (9) and (8) we find (10):…”

“…Thus, we may take difference, multiply by 2u j u k /(u k −u j ), and find (9). For j = k, we may achieve (9) by normalizing the eigenvectors, provided that the left hand side of (9) does not vanish.…”