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arXiv:chao-dyn/9804025 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Semiclassical spectra and diagonal matrix elements by harmonic inversion of cross-correlated periodic orbit sums

Jörg Main, Kirsten Weibert, Vladimir A. Mandelshtam, Günter Wunner

Published Aug 1, 1999• 12 citations
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Abstract

Semiclassical spectra weighted with products of diagonal matrix elements of operators ${A}_{\ensuremath{\alpha}},$ i.e., ${g}_{\ensuremath{\alpha}{\ensuremath{\alpha}}^{\ensuremath{'}}}(E)={\ensuremath{\sum}}_{n}〈n|{A}_{\ensuremath{\alpha}}|n〉〈n|{A}_{{\ensuremath{\alpha}}^{\ensuremath{'}}}|n〉/(E\ensuremath{-}{E}_{n}),$ are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.

DOI
10.1103/physreve.60.1639

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Semiclassical spectra weighted with products of diagonal matrix elements of operators A\ensuremathα,{A}_{\ensuremath{\alpha}},A\ensuremathα​, i.e., g\ensuremathα\ensuremathα\ensuremath′(E)=\ensuremath∑n〈n∣A\ensuremathα∣n〉〈n∣A\ensuremathα\ensuremath′∣n〉/(E\ensuremath−En),{g}_{\ensuremath{\alpha}{\ensuremath{\alpha}}^{\ensuremath{'}}}(E)={\ensuremath{\sum}}_{n}〈n|{A}_{\ensuremath{\alpha}}|n〉〈n|{A}_{{\ensuremath{\alpha}}^{\ensuremath{'}}}|n〉/(E\ensuremath{-}{E}_{n}),g\ensuremathα\ensuremathα\ensuremath′​(E)=\ensuremath∑n​〈n∣A\ensuremathα​∣n〉〈n∣A\ensuremathα\ensuremath′​∣n〉/(E\ensuremath−En​), are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.
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 v...
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 ~ corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbi...
Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.
Semiclassical angular momentum matrices are calculated using a new contour-integral formula for matrix-elements. WKB spherical harmonic functions are found to be exaclty orthonormal with the contour-integral inner-product. Matrix-elements obtained from these wave-functions are accurate to about 2 % and the matrices obey the commutation relations expected of quantum angular momentum operators. The semiclassical wave-functions are related to a superposition of allowed classical orbits and this illustrates the connection to the Feynman pathsummation formulation of quantum mechanics for electrons in a spherically symmetric potential.
We have shown that interesting physical phenomena can be revealed in high resolution quantum recurrence spectra by application of the harmonic inversion technique, thereby circumventing the restrictions imposed by the uncertainty principle of the conventional Fourier transform. The method allows, e.g., to test the validity of semiclassical theories, to identify hidden ghost or bits in the quantum spectra, and to observe the symmetry breaking in the spectra of the hydrogen atom in crossed magnetic and electric fields. The analysis has been demonstrated here on theoretically calculated quantum spectra but can be applied to experimental spectra as well.
We derive semiclassical periodic orbit expansions for a correlation function of the Wigner time delay. We consider the Fourier transform of the two-point correlation function, the form factor K(τ,x,y,M)K(τ,x,y,M)K(τ,x,y,M), that depends on the number of open channels MMM, a non-symmetry breaking parameter xxx, and a symmetry breaking parameter yyy. Several terms in the Taylor expansion about τ=0τ=0τ=0, which depend on all parameters, are shown to be identical to those obtained from Random Matrix Theory.