Harmonic Inversion Techniques Research Articles

Semiclassical Vibrational Spectroscopy of Real Molecular Systems by Means of Cross-Correlation Filter Diagonalization.

We applied the harmonic inversion technique to extract vibrational eigenvalues from the semiclassical initial value representation (SC-IVR) propagator of molecular systems described by explicit potential surfaces. The cross-correlation filter-diagonalization (CCFD) method is used for the inversion problem instead of the Fourier transformation, which allows much shorter propagation time and is thus capable of avoiding numerical divergence issues while getting rid of approximations like the separable one to the pre-exponential factor. We also used the "Divide-and-Conquer" technique to control the total dimensions under consideration, which helps to further enhance the numerical behavior of SC-IVR calculations and the stability of harmonic inversion methods. The technique is tested on small molecules and water trimer to justify its applicability and reliability. Results show that the CCFD method can effectively extract the vibrational eigenvalues from short trajectories and reproduce the original spectra conventionally obtained from long-time ones, with no loss on accuracy while the numerical behavior is much better. This work demonstrates the possibility to apply the combined method of CCFD and SC-IVR to real molecular potential surfaces, which might be a new way to overcome the numerical instabilities caused by the increase of dimensions.

Exact analytical representations for broadband transmission properties of quarter-wave multilayers

The formalism of the scattering matrix is applied to describe the transmission properties of multilayered structures with deep variations of the refractive index and arbitrary arrangements of the layers. We show that there is an exact analytical formula for the transmission spectrum, which is valid for the full spectral range and which contains only a limited number of parameters for structures satisfying the quarter-wave condition. These parameters are related to the poles of the scattering matrix, and we present an efficient algorithm to find them, which is based on considering the ray propagation inside the structure and subsequent application of the harmonic inversion technique. These results are significant for analyzing the reshaping of ultrashort pulses in multilayered structures.

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

We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [Lu, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.

Semiclassical quantization by harmonic inversion: comparison of algorithms.

Harmonic inversion techniques have been shown to be a powerful tool for the semiclassical quantization and analysis of quantum spectra of both classically integrable and chaotic dynamical systems. Various computational procedures have been proposed for this purpose. Our aim is to find out which method is numerically most efficient. To this end, we summarize and discuss the different techniques and compare their accuracies by way of two example systems.

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: Firstly, the periodic orbit quantization will be extended to include higher order hbar corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.

Rayleigh wave phase velocities in the Pacific with implications for azimuthal anisotropy and lateral heterogeneities

The lateral distribution of fundamental-mode Rayleigh wave phase velocities in the Pacific has been calculated in order to determine the variation in velocity as a function of the age of the oceanic plate, the importance of azimuthal anisotropy, and the presence of secondary lateral heterogeneities. The data set used in this study consists of phase velocity measurements in the period range 20–125 s for 178 paths traversing the Pacific. Both the pure-path and spherical harmonic inversion techniques are used in this investigation with an emphasis placed on determining the effectiveness and resolving power of these methods in calculating the velocity distribution in an oceanic regime. The first-order velocity-age relation which dominantly characterizes surface wave dispersion in an oceanic environment can be adequately modelled by either the pure-path or spherical harmonic techniques. Azimuthal anisotropy is the dominant second-order effect and is best developed in regions less than 80 Myr in age. In this age zone, the faster direction of wave propagation is either in the direction of fossil seafloor spreading or in the direction of present-day absolute plate motion. In the western Pacific, there is a correlation with paleo-relative motion (fossil seafloor spreading) for periods less than 50 s. Lateral velocity heterogeneities which are not related to the age of the oceanic plate are found by pure-path inversions and are correlated with regions associated with shallow residual depth anomalies. This effect (slow velocities with respect to the velocity-age model) monotonically decreases as a function of period and is indicative of a shallow origin. Age independent anomalies were also modelled by the application of the pure-path and spherical harmonic methods in a sequential inversion.