Higher-order <Emphasis Type="Bold">?</Emphasis> corrections in the semiclassical quantization of chaotic billiards

In the periodic orbit quantization of physical systems, usually only the leading-order ? contribution to the density of states is considered. Therefore, by construction, the eigenvalues following from semiclassical trace formulae generally agree with the exact quantum ones only to lowest order of ?. In different theoretical work the trace formulae have been extended to higher orders of ?. The problem remains, however, how to actually calculate eigenvalues from the extended trace formulae since, even with ? corrections included, the periodic orbit sums still do not converge in the physical domain. For lowest-order semiclassical trace formulae the convergence problem can be elegantly, and universally, circumvented by application of the technique of harmonic inversion. In this paper we show how, for general scaling chaotic systems, also higher-order ? corrections to the Gutzwiller formula can be included in the harmonic inversion scheme, and demonstrate that corrected semiclassical eigenvalues can be calculated despite the convergence problem. The method is applied to the open three-disk scattering system, as a prototype of a chaotic system. Received 10 September 2001 and Received in final form 3 January 2002     

Quantum mechanics of classically non-integrable systems

In the semiclassical limit, quantum mechanics shows differences between classically integrable abd chaotic systems. Here we review recent developments in this field. Topics dealt with include formal integrability of quantum mechanics, semiclassical quantization, statistical properties of eigenvalues, semiclassical eigenfunctions, effects on the time-evolution and localization due to classical diffusion. A large bibliography supplements the text.     

Discrete Hamiltonian symmetries and semiclassical quantization

Most quantum Hamiltonian systems exhibit discrete symmetries. Allowing for these is crucial when properly calculating the fluctuation properties of the quantal spectrum. These properties are then employed to distinguish between classically chaotic or non-chaotic quantum systems. In general, semiclassical quantization procedures do not take into account irreducible representations of the Hamiltonian. A procedure is presented to take these into account in semiclassical quantization schemes and calculate some of the energy eigenvalues belonging to a specific irreducible representation.     

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Padé approximant, or signal diagonalization. The second method is based upon the direct application of the Padé approximant to the periodic orbit sum. The Padé approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.     

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 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. Received 24 February 2000 and Received in final form 22 May 2000     

Semiclassical mechanics of bound chaotic potentials

Semiclassical methods for determining quantum eigenvalues in chaotic systems are discussed. A recent calculation for an open scattering system with Axiom-A properties serves as a starting point for the discussion. How deviation from Axiom-A properties, such as intermittency and occurrence of small stability islands, normally arise in bound Hamiltonian systems, and how these deviations complicate the calculation of semiclassical eigenvalues are demonstrated. It is also stressed that since such deviations are typical of bound Hamiltonian systems, they might be of crucial importance for the statistical properties of the energy levels.     

Microscopic theory for the quantum to classical crossover in chaotic transport

We present a semiclassical theory for the scattering matrix S of a chaotic ballistic cavity at finite Ehrenfest time. Using a phase-space representation coupled with a multibounce expansion, we show how the Liouville conservation of phase-space volume decomposes S as S=S(cl) plus sign in circle S(qm). The short-time, classical contribution S(cl) generates deterministic transmission eigenvalues T=0 or 1, while quantum ergodicity is recovered within the subspace corresponding to the long-time, stochastic contribution S(qm). This provides a microscopic foundation for the two-phase fluid model, in which the cavity acts like a classical and a quantum cavity in parallel, and explains recent numerical data showing the breakdown of universality in quantum chaotic transport in the deep semiclassical limit. We show that the Fano factor of the shot-noise power vanishes in this limit, while weak localization remains universal.     

Periodic orbit quantization of chaotic maps by harmonic inversion

A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.     

The semiclassical regime of the chaotic quantum-classical transition

An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.     

Accuracy of semiclassical dynamics in the presence of chaos

We review some of the issues facing semiclassical methods in classically chaotic systems, then demonstrate the long-time accuracy of semiclassical propagation of a nonstationary wave packet using the quantum baker's map of Balazs and Voros. We show why some of the standard arguments against the efficacy of semiclassical dynamics for long-time chaotic motion are incorrect.     

On semi-classical bounds for eigenvalues of Schrödinger operators

Our principal results is that if the semiclassical estimate is a bound for some moment of the negative eigenvalues (as is known in some cases in one-dimension), then the semiclassical estimates are also bounds for all higher moments.     

Periodic orbit quantization of the anisotropic Kepler problem

The periodic orbit quantization on the anisotropic Kepler problem is tested. By computing the stability and action of some 2000 of the shortest periodic orbits, the eigenvalue spectrum of the anisotropic Kepler problem is calculated. The aim is to test the following claims for calculating the quantum spectrum of classically chaotic systems: (1) Curvature expansions of quantum mechanical zeta functions offer the best semiclassical estimates; (2) the real part of the cycle expansions of quantum mechanical zeta functions cut at appropriate cycle length offer the best estimates; (3) cycle expansions are superfluous; and (4) only a small subset of cycles (irreducible cycles) suffices for good estimates for the eigenvalues. No evidence is found to support any of the four claims.     

Time domain approach to semiclassical dynamics: Breaking the log time barrier

The presence of chaos in the classical dynamics is not necessarily the destructive element it was thought to be for semiclassical approximations in the time domain. The method of calculating the semiclassical propagation of initial states and correlation functions for nonlinear and chaotic dynamics is shown, and the excellent accuracy is noted for rather long times. The breakdown timescale is much longer than the infamous "log time" for the cases investigated here.     

Eigenstates ignoring regular and chaotic phase-space structures

We report the failure of the semiclassical eigenfunction hypothesis if regular classical transport coexists with chaotic dynamics. All eigenstates, instead of being restricted to either a regular island or the chaotic sea, ignore these classical phase-space structures. We argue that this is true even in the semiclassical limit for extended systems with transporting regular islands such as the standard map with accelerator modes.     

Harmonic Inversion of Recurrence Spectra of Nonhydrogen Atom in an Electric Field

An extended harmonic inversion method is analytically continued to approach bifurcation region of the dosed orbits thus to obtain highly resolved spectra of lithium atom in external field. The suitable band-limited signal is generated by a semielassieal uniform approximation. By decimating the selected signal window and solving the algebraic set of nonlinear equations the quantum eigenvalues are properly fitted, which reveal the fine resonance structure hidden in low resolution spectrum. The study is made at the sealed energy ε= -2.7, relevant bifurcation effects and corescattered impacts have to be taken into account. It is demonstrated that the present method is a useful technique for the semiclassieal quantization of system with mixed regular-chaotic classical dynamics.     

Quantum-chaotic scattering effects in semiconductor microstructures

We show that classical chaotic scattering has experimentally measurable consequences for the quantum conductance of semiconductor microstructures. These include the existence of conductance fluctuations-a sensitivity of the conductance to either Fermi energy or magnetic field-and weak-localization-a change in the average conductance upon applying a magnetic field. We develop a semiclassical theory and present numerical results for these two effects in which we model the microstructures by billiards attached to leads. We find that the difference between chaotic and regular classical scattering produces a qualitative difference in the fluctuation spectrum and weak-localization lineshape of chaotic and nonchaotic structures. While the semiclassical theory within the diagonal approximation accounts well for the weak-localization lineshape and for the spectrum of the fluctuations, we uncover a surprising failure of the semiclassical diagonal-approximation theory in describing the magnitude of these quantum transport effects.     

Quantization rules for low dimensional quantum dots

This paper applies the analytical transfer matrix method （ATMM） to calculate energy eigenvalues of a particle in low dimensional sharp confining potential for the first time, and deduces the quantization rules of this system. It presents three cases in which the applied method works very well. In the first quantum dot, the energy eigenvalues and eigenfunction are obtained, and compared with those acquired from the exact numerical analysis and the WKB （Wentzel, Kramers and Brillouin） method; in the second or the third case, we get the energy eigenvalues by the ATMM, and compare them with the EBK （Einstein, Brillouin and Keller） results or the wavefunction outcomes. From the comparisons, we find that the semiclassical method （WKB, EBK or wavefunction） is inexact in such systems.     

Semiclassical matrix model for quantum chaotic transport with time-reversal symmetry

We show that the semiclassical approach to chaotic quantum transport in the presence of time-reversal symmetry can be described by a matrix model. In other words, we construct a matrix integral whose perturbative expansion satisfies the semiclassical diagrammatic rules for the calculation of transport statistics. One of the virtues of this approach is that it leads very naturally to the semiclassical derivation of universal predictions from random matrix theory.     

The Van Vleck formula,Maslov theory,and phase space geometry

The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslov's theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.