Semiclassical Trace Formulae and Eigenvalue Statistics in Quantum Chaos

A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed.Comment: 48 pages, LaTeX, uses amssym

A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators

We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space L^2(\rz^d)\otimes\kz^n into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of L^2(\rz^d)\otimes\kz^n a classical system on a product phase space \TRd\times\cO, where \cO is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic

On the Rate of Quantum Ergodicity on hyperbolic Surfaces and Billiards

The rate of quantum ergodicity is studied for three strongly chaotic (Anosov) systems. The quantal eigenfunctions on a compact Riemannian surface of genus g=2 and of two triangular billiards on a surface of constant negative curvature are investigated. One of the triangular billiards belongs to the class of arithmetic systems. There are no peculiarities observed in the arithmetic system concerning the rate of quantum ergodicity. This contrasts to the peculiar behaviour with respect to the statistical properties of the quantal levels. It is demonstrated that the rate of quantum ergodicity in the three considered systems fits well with the known upper and lower bounds. Furthermore, Sarnak's conjecture about quantum unique ergodicity for hyperbolic surfaces is confirmed numerically in these three systems.Comment: 19 pages, Latex, This file contains no figures. A postscript file with all figures is available at http://www.physik.uni-ulm.de/theo/qc/ (Delay is expected to 23.7.97 since our Web master is on vacation.

Numerical computation of Maass waveforms and an application to cosmology

We compute numerically eigenvalues and eigenfunctions of the Laplacian in a three-dimensional hyperbolic space. Applying the results to cosmology, we demonstrate that the methods learned in quantum chaos can be used in other fields of research.Comment: A version of the paper with high resolution figures is available at http://www.physik.uni-ulm.de/theo/qc/publications.htm

Semiclassical Transition from an Elliptical to an Oval

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when ¯h is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed. PACS numbers: 03.65.Ge Solutions of wave equations: bound states. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems

Quintessence and the Curvature of the Universe after WMAP

We study quintessence models with a constant (effective) equation of state...

Spectral Statistics for Quantized Skew Translations on the Torus

We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level-spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit