Use of Harmonic Inversion Techniques in Semiclassical Quantization and Analysis of Quantum Spectra

Abstract

Harmonic inversion is introduced as a powerful tool for both the analysis of quantum spectra and semiclassical periodic orbit quantization. The method allows to circumvent the uncertainty principle of the conventional Fourier transform and to extract dynamical information from quantum spectra which has been unattainable before, such as bifurcations of orbits, the uncovering of hidden ghost orbits in complex phase space, and the direct observation of symmetry breaking effects. The method also solves the fundamental convergence problems in semiclassical periodic orbit theories - for both the Berry-Tabor formula and Gutzwiller's trace formula - and can therefore be applied as a novel technique for periodic orbit quantization, i.e., to calculate semiclassical eigenenergies from a finite set of classical periodic orbits. The advantage of periodic orbit quantization by harmonic inversion is the universality and wide applicability of the method, which will be demonstrated in this work for various open and bound systems with underlying regular, chaotic, and even mixed classical dynamics. The efficiency of the method is increased, i.e., the number of orbits required for periodic orbit quantization is reduced, when the harmonic inversion technique is generalized to the analysis of cross-correlated periodic orbit sums. The method provides not only the eigenenergies and resonances of systems but also allows the semiclassical calculation of diagonal matrix elements and, e.g., for atoms in external fields, individual non-diagonal transition strengths. Furthermore, it is possible to include higher order terms of the hbar expanded periodic orbit sum to obtain semiclassical spectra beyond the Gutzwiller and Berry-Tabor approximation.

151 pages, 40 figures, LATEX (Elsevier-style), to be published in Physics Reports (1999)

References in corpus (11)

• Bifurcations of Periodic Orbits and Uniform Approximations
• Quantum Mechanics and Semiclassics of Hyperbolic n-Disk Scattering Systems
• Symmetry breaking in crossed magnetic and electric fields
• Harmonic inversion as a general method for periodic orbit quantization
• Semiclassical interference of bifurcations
• Periodic Orbit Quantization of Mixed Regular Chaotic Systems
• On the Accuracy of the Semiclassical Trace Formula
• Semiclassical spectra from periodic-orbit clusters in a mixed phase space
• Semiclassical Calculation of Transition Matrix Elements for Atoms in External Fields
• Uniform semiclassical approximations for umbilic bifurcation catastrophes
• hbar expansion for the periodic orbit quantization by harmonic inversion

Cited by in corpus (38)

• Classical wave experiments on chaotic scattering
• Resonance widths in open microwave cavities studied by harmonic inversion
• Open circular billiards and the Riemann hypothesis
• Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering
• Experimental Observation of the Spectral Gap in Microwave n-Disk Systems
• Decimation and Harmonic Inversion of Periodic Orbit Signals
• Selectively exciting quasi-normal modes in open disordered systems
• Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems
• Lifetime statistics in chaotic dielectric microresonators
• A microwave realization of the chiral orthogonal, unitary, and symplectic ensembles
• Harmonic inversion analysis of exceptional points in resonance spectra
• Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields
• Semiclassical Quantization by Pade Approximant to Periodic Orbit Sums
• Observation of diffractive orbits in the spectrum of excited NO in a magnetic field
• Ghost orbits in the diamagnetic hydrogen spectrum using harmonic inversion
• Semiclassical quantization with bifurcating orbits
• Time propagation of constrained coupled Gaussian wave packets
• Quantization Ambiguity, Ergodicity, and Semiclassics
• Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits
• Fictitious time wave packet dynamics: II. Hydrogen atom in external fields
• Semiclassical Quantization by Harmonic Inversion: Comparison of Algorithms
• Signatures of exciton orbits in quantum mechanical recurrence spectra of Cu${}_2$O
• Periodic orbit quantization of a Hamiltonian map on the sphere
• Semiclassical Accuracy in Phase Space for Regular and Chaotic Dynamics
• $\hslash$ corrections in semi-classical formula for smooth chaotic dynamics
• The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits
• Periodic orbit quantization of chaotic maps by harmonic inversion
• Classical dynamics and semiclassical analysis of excitons in cuprous oxide
• Periodic orbit quantization of chaotic systems with strong pruning
• Verification of exceptional points in the collapse dynamics of Bose-Einstein condensates
• Superiority of semiclassical over quantum mechanical calculations for a three-dimensional system
• Classical, semiclassical, and quantum investigations of the 4-sphere scattering system
• hbar expansions in semiclassical theories for systems with smooth potentials and discrete symmetries
• Semi-classical analysis of real atomic spectra beyond Gutzwiller's approximation
• Exceptional points in the elliptical three-disk scatterer using semiclassical periodic orbit quantization
• Creation Mechanism of Devil's Staircase Surface and Unstable and Stable Periodic Orbit in the Anisotropic Kepler Problem
• Deterministic diffusion in one-dimensional maps - Calculation of diffusion constants by harmonic inversion of periodic orbit sums
• Semiclassical quantization of the diamagnetic hydrogen atom with near action-degenerate periodic-orbit bunches