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

Testing the left-handedness of the b \to c transition

We analyse the spin structure of inclusive semileptonic b \to c transitions and the effects of non-standard model couplings on the rates and the spectra. The calculation includes the {\cal O} (\alpha_s) corrections as well as the leading non-perturbative ones.Comment: 15 pages, 3 figure

Instantons and Meson Correlators in QCD

Various QCD correlators are calculated in the instanton liquid model in zeromode approximation and $1/N_c$ expansion. Previous works are extended by including dynamical quark loops. In contrast to the original "perturbative" $1/N_c$ expansion not all quark loops are suppressed. In the flavor singlet meson correlators a chain of quark bubbles survives the $N_c\to\infty$ limit causing a massive $\eta^\prime$ in the pseudoscalar correlator while keeping massless pions in the triplet correlator. The correlators are plotted and meson masses and couplings are obtained from a spectral fit. They are compared to the values obtained from numerical studies of the instanton liquid and to experimental results.Comment: 43 latex pages, 7 1/2 figures included by epsf.tex-macr

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

Habitat width along a latitudinal gradient

We use the Chowdhury ecosystem model, one of the most complex agent-based ecological models, to test the latitude-niche breadth hypothesis, with regard to habitat width, i.e., whether tropical species generally have narrower habitats than high latitude ones. Application of the model has given realistic results in previous studies on latitudinal gradients in species diversity and Rapoport's rule. Here we show that tropical species with sufficient vagility and time to spread into adjacent habitats, tend to have wider habitats than high latitude ones, contradicting the latitude-niche breadth hypothesis.Comment: 13 pages including all figures, draft for a biology journa