Topological properties of quantum periodic Hamiltonians

We consider periodic quantum Hamiltonians on the torus phase space (Harper-like Hamiltonians). We calculate the topological Chern index which characterizes each spectral band in the generic case. This calculation is made by a semi-classical approach with use of quasi-modes. As a result, the Chern index is equal to the homotopy of the path of these quasi-modes on phase space as the Floquet parameter (\theta) of the band is varied. It is quite interesting that the Chern indices, defined as topological quantum numbers, can be expressed from simple properties of the classical trajectories.Comment: 27 pages, 14 figure

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces

We study the recurrence and ergodicity for the billiard on noncompact polygonal surfaces with a free, cocompact action of $\Z$ or $\Z^2$. In the $\Z$-periodic case, we establish criteria for recurrence. In the more difficult $\Z^2$-periodic case, we establish some general results. For a particular family of $\Z^2$-periodic polygonal surfaces, known in the physics literature as the wind-tree model, assuming certain restrictions of geometric nature, we obtain the ergodic decomposition of directional billiard dynamics for a dense, countable set of directions. This is a consequence of our results on the ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure

Zitterbewegung and semiclassical observables for the Dirac equation

In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in \hbar by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac-Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity

Linear stability in billiards with potential

A general formula for the linearized Poincar\'e map of a billiard with a potential is derived. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. For the case without potential this gives well known formulas. Four billiards with potentials for which the free motion is integrable are treated as examples: The linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three body problem. The linear stability of periodic orbits with period one and two is analyzed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.Comment: 22 pages, LaTex, 4 Figure

Perturbation analysis of weakly discrete kinks

We present a perturbation theory of kink solutions of discrete Klein-Gordon chains. The unperturbed solutions correspond to the kinks of the adjoint partial differential equation. The perturbation theory is based on a reformulation of the discrete chain problem into a partial differential equation with spatially modulated mass density. The first order corrections to the kink solutions are obtained analytically and are shown to agree with exact numerical results. We discuss the problem of calculating the Peierls-Nabarro barrier.Comment: 13 pages, 6 figures, REVTE

Quantum chaos in a deformable billiard: Applications to quantum dots

We perform a detailed numerical study of energy-level and wavefunction statistics of a deformable quantum billiard focusing on properties relevant to semiconductor quantum dots. We consider the family of Robnik billiards generated by simple conformal maps of the unit disk; the shape of this family of billiards may be varied continuously at fixed area by tuning the parameters of the map. The classical dynamics of these billiards is well-understood and this allows us to study the quantum properties of subfamilies which span the transition from integrability to chaos as well as families at approximately constant degree of chaoticity (Kolmogorov entropy). In the regime of hard chaos we find that the statistical properties of interest are well-described by random-matrix theory and completely insensitive to the particular shape of the dot. However in the nearly-integrable regime non-universal behavior is found. Specifically, the level-width distribution is well-described by the predicted $\chi^2$ distribution both in the presence and absence of magnetic flux when the system is fully chaotic; however it departs substantially from this behavior in the mixed regime. The chaotic behavior corroborates the previously predicted behavior of the peak-height distribution for deformed quantum dots. We also investigate the energy-level correlation functions which are found to agree well with the behavior calculated for quasi-zero-dimensional disordered systems.Comment: 25 pages (revtex 3.0). 16 figures are available by mail or fax upon request at [email protected]

Visual Explorations of Dynamics: the Standard Map

The Macintosh application \textit{StdMap} allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program.Comment: Corrections in a couple of equations, and updated to the latest version of StdMa