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Quantum chaos

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Interference Effects in Semiclassical Orbits truecm

We investigate the relationship between the power spectrum of a wave field and that of a spatially uncorrelated source exciting it. Correlation functions are also examined. It is found that interference effects can significantly alter the structure of the power spectrum, leading to oscillations even when the power spectrum of the source is smooth. We derive a semiclassical approximation in which these oscillations are related to orbits of the geometrical limit of the wave system. We also derive a trace formula that approximates a spatial average of the wave power spectrum as a sum over periodic orbits. These calculations explain the structure of a measured power spectrum of the fluctuations in the height of a fluid surface generated by the circular hydraulic jump, which provided the motivation for the study. truecm

(S.C. Creagh) and P. Dimon truecm

Periodic Orbits and Spectral Correlations of Pseudointegrable Systems truecm

Properties of periodic orbit families in pseudointegrable billiards are studied and it is found that they obey the sum rule where and refer respectively to the length and area occupied by a primitive periodic orbit family. The proliferation law follows from this. Numerical investigations reveal that the average area increases initially before saturating. This is perhaps the first demonstration that asymptotically even for systems that are not almost-integrable.

It is further found that off-diagonal terms in the spectral form factor do contribute unlike the case of intergable dynamics. Finally, it is shown that periodic orbits give good estimates of two-point correlations in the semiclassical quantum spectrum. Diffraction corrections are thus more important near the ground state. truecm

D. Biswas truecm

The Classical Spectrum of Rational Polygons and Their Quantization Using Arbitrary Trajectories truecm

The trace of the classical evolution operator, for the flow in rational polygonal billiards is studied.

By restricting to the 2-dimensional invariant surface, it is shown that the trace, where N refers to the number of billiard copies in the invariant surface, is the area occupied by a primitive periodic orbit family of length and l is the length traversed by the particle in time t with a velocity v. Using semi-classical methods is then re-expressed as where are the quantum energy eigenvalues and is a Bessel function. This result is also derived directly for integrable polygons.

Any function constructed using the phase-space variable and averaged over the constant energy surface thus has oscillatory contributions from quantal eigenenergies. This is used to demonstrate that arbitrary non-periodic trajectories can be used as an effective quantization tool in such systems. truecm

D. Biswas truecm

Topological Chaos in a Non-ergodic System truecm

An exponential proliferation of periodic orbits in a dynamical system is associated with metric chaos or the fact that trajectories that are initially close separate exponentially with time. The proliferation rate is in fact linked to a basic sum rule obeyed by periodic orbits arising from the fact that in a bounded system the particle never escapes.

A situation in which trajectories suffer mode conversion and splitting is studied as a small wave-length limit of the Navier equation which describes elasto-mechanical displacements. While most geometries lead to ergodicity, it is found that for a square, the dynamics is non-ergodic and hence non-chaotic. It is argued however that due to an exponential decay in intensity, the basic conservation law should lead to an exponential proliferation of periodic orbits. This is demonstrated numerically and this novel pheomenon is referred to as topological chaos in a non-ergodic situation. truecm

D. Biswas truecm

Intermittency and Periodic Orbit Zeta Functions truecm

The understanding of intermittency, i.e. the coexistence of (almost) regular and chaotic behaviour in connected ergodic components of a dynamical system is a major challenge in describing and characterising dynamical properties of a generic system. In Hamiltonian flows, intermittency is generally introduced through stable islands or marginal stable fixed points. The influence of intermittency on periodic orbit product formulas, which are suitable representations of the Fredholm determinant or (in semiclassical approximation) of the quantum spectral determinant, was studied. A regularisation of divergences originating from the regular part of the dynamics good be given and could be applied successfully to carry out the first purely semiclassical quantisation of the resonance spectrum of Hydrogen in a strong magnetic field. truecm

G. Tanner and (K.T. Hansen) truecm

Entire Spectral Determinants in Semiclassical Quantization, truecm

We investigate spectral determinants in semiclassical quantization that are entire functions in the complex plane since they are given by a multiplicative quasi classical evolution operator. The spectra are being compared to the usual Gutzwiller-Voros results aswell as to the exact quantum mechanics. truecm

G. Tanner, P. Cvitanovic, P. Rosenqvist, (G. Vattay) and (A. Wirzba) truecm

Fredholm Theory and the Semiclassical Approximation truecm

The semiclassical approximation for chaotic systems usually produces a divergent sum over classical trajectories. The Fredholm method is a way of combining Bogomolny's T-operator with the old Fredholm theory for solving integral equation, to obtain a general scheme of resummation valid for various quantities, not necessarily the density of states. We are applying it to individual wavefunctions, to get a resummed formula permitting to predict the occurence of scars (enhancement of wavefunctions around classical periodic orbits). We also plan to apply it to scattering systems such as the three-disks system or mesoscopic systems. truecm

B. Georgeot, (S. Fishman and R.E. Prange) truecm

Ray-Splitting and Quantum Chaos truecm

Ray-splitting is a phenomenon whereby a ray incident on a boundary splits into more than one ray travelling away from the boundary. This corresponds to the short-wavelength limit of wave systems where at some place (discontinuity of the potential, boundary...) a wave is decomposed into two other (for example, one transmitted and one reflected). We extend results and techniques from quantum chaos to this kind of systems, and test these extensions on a simple model. This should enable to treat afterwards various kind of problems, in particular in accoustics. truecm

B. Georgeot, (T.M. Antonsen, Jr., R. Bluemel, E. Ott and R.E. Prange) truecm

Diffraction Effects in the Gutzwiller Trace Formula truecm

By using the geometrical theory of diffraction introduced by Keller, we have incorporated diffractive or creeping periodic orbits in the Gutzwiller trace formula and in the related spectral determinant. The numerical calculations has been performed on the 2- and 3-disk scatering systems and on the confocal hyperbolae. Also point scatterers has been considered. A new type of system with 3 disks on a line is now under investigation. In this system the semiclassical degeneracy of the scattering resonances is lifted purely due to inclusion of creeping periodic orbits. truecm

P. Rosenqvist, (G. Vattay), (A. Wirzba) and (N. Whelan) truecm

Differential Equations to Compute Corrections to the Gutzwiller Trace Formula truecm

By considering local Schrödinger problems in the neighborhood of classical periodic orbits it is possible to construct an expansion of the local eigenvalues. The corrections can be found successively by using ordinary differential equations. By defining the local spectral determinants we can get an corrected expression for the global spectral determinant by multiplying the corrected local determinants together. The method has been specialized to billard systems, and a general fortran code has been develloped that can calculate the first correction to any two-dimensional billard system. Numerical calculations has been performed on the 2- and 3-disk scattering systems. truecm

(G. Vattay) and P. Rosenqvist.



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