32 research outputs found
Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms
We exhibit scarring for certain nonlinear ergodic toral automorphisms. There
are perturbed quantized hyperbolic toral automorphisms preserving certain
co-isotropic submanifolds. The classical dynamics is ergodic, hence in the
semiclassical limit almost all eigenstates converge to the volume measure of
the torus. Nevertheless, we show that for each of the invariant submanifolds,
there are also eigenstates which localize and converge to the volume measure of
the corresponding submanifold.Comment: 17 page
Intermediate statistics in quantum maps
We present a one-parameter family of quantum maps whose spectral statistics
are of the same intermediate type as observed in polygonal quantum billiards.
Our central result is the evaluation of the spectral two-point correlation form
factor at small argument, which in turn yields the asymptotic level
compressibility for macroscopic correlation lengths
Delocalization of slowly damped eigenmodes on Anosov manifolds
We look at the properties of high frequency eigenmodes for the damped wave
equation on a compact manifold with an Anosov geodesic flow. We study
eigenmodes with spectral parameters which are asymptotically close enough to
the real axis. We prove that such modes cannot be completely localized on
subsets satisfying a condition of negative topological pressure. As an
application, one can deduce the existence of a "strip" of logarithmic size
without eigenvalues below the real axis under this dynamical assumption on the
set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two
reference
Semi-classical study of the Quantum Hall conductivity
The semi-classical study of the integer Quantum Hall conductivity is
investigated for electrons in a bi-periodic potential .
The Hall conductivity is due to the tunnelling effect and we concentrate our
study to potentials having three wells in a periodic cell. A non-zero
topological conductivity requires special conditions for the positions, and
shapes of the wells. The results are derived analytically and well confirmed by
numerical calculations.Comment: 23 pages, 13 figure
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
Eigenfunction Statistics on Quantum Graphs
We investigate the spatial statistics of the energy eigenfunctions on large
quantum graphs. It has previously been conjectured that these should be
described by a Gaussian Random Wave Model, by analogy with quantum chaotic
systems, for which such a model was proposed by Berry in 1977. The
autocorrelation functions we calculate for an individual quantum graph exhibit
a universal component, which completely determines a Gaussian Random Wave
Model, and a system-dependent deviation. This deviation depends on the graph
only through its underlying classical dynamics. Classical criteria for quantum
universality to be met asymptotically in the large graph limit (i.e. for the
non-universal deviation to vanish) are then extracted. We use an exact field
theoretic expression in terms of a variant of a supersymmetric sigma model. A
saddle-point analysis of this expression leads to the estimates. In particular,
intensity correlations are used to discuss the possible equidistribution of the
energy eigenfunctions in the large graph limit. When equidistribution is
asymptotically realized, our theory predicts a rate of convergence that is a
significant refinement of previous estimates. The universal and
system-dependent components of intensity correlation functions are recovered by
means of an exact trace formula which we analyse in the diagonal approximation,
drawing in this way a parallel between the field theory and semiclassics. Our
results provide the first instance where an asymptotic Gaussian Random Wave
Model has been established microscopically for eigenfunctions in a system with
no disorder.Comment: 59 pages, 3 figure
Quantum cat maps with spin 1/2
We derive a semiclassical trace formula for quantized chaotic transformations
of the torus coupled to a two-spinor precessing in a magnetic field. The trace
formula is applied to semiclassical correlation densities of the quantum map,
which, according to the conjecture of Bohigas, Giannoni and Schmit, are
expected to converge to those of the circular symplectic ensemble (CSE) of
random matrices. In particular, we show that the diagonal approximation of the
spectral form factor for small arguments agrees with the CSE prediction. The
results are confirmed by numerical investigations.Comment: 26 pages, 3 figure
Weyl's law and quantum ergodicity for maps with divided phase space
For a general class of unitary quantum maps, whose underlying classical phase
space is divided into several invariant domains of positive measure, we
establish analogues of Weyl's law for the distribution of eigenphases. If the
map has one ergodic component, and is periodic on the remaining domains, we
prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the
equidistribution of eigenfunctions with respect to the ergodic component of the
classical map (quantum ergodicity). We apply our main theorems to quantised
linked twist maps on the torus. In the Appendix, S. Zelditch connects these
studies to some earlier results on `pimpled spheres' in the setting of
Riemannian manifolds. The common feature is a divided phase space with a
periodic component.Comment: Colour figures. Black & white figures available at
http://www2.maths.bris.ac.uk/~majm. Appendix by Steve Zelditc
Distribution of resonances for open quantum maps
We analyze simple models of classical chaotic open systems and of their
quantizations (open quantum maps on the torus). Our models are similar to
models recently studied in atomic and mesoscopic physics. They provide a
numerical confirmation of the fractal Weyl law for the density of quantum
resonances of such systems. The exponent in that law is related to the
dimension of the classical repeller (or trapped set) of the system. In a
simplified model, a rigorous argument gives the full resonance spectrum, which
satisfies the fractal Weyl law. For this model, we can also compute a quantity
characterizing the fluctuations of conductance through the system, namely the
shot noise power: the value we obtain is close to the prediction of random
matrix theory.Comment: 60 pages, no figures (numerical results are shown in other
references
Quantum ergodicity for graphs related to interval maps
We prove quantum ergodicity for a family of graphs that are obtained from
ergodic one-dimensional maps of an interval using a procedure introduced by
Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take
the L^2 functions on the interval. The proof is based on the periodic orbit
expansion of a majorant of the quantum variance. Specifically, given a
one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an
increasingly refined sequence of partitions of the interval. To this sequence
we associate a sequence of graphs, whose directed edges correspond to elements
of the partitions and on which the classical dynamics approximates the
Perron-Frobenius operator corresponding to the map. We show that, except
possibly for subsequences of density 0, the eigenstates of the quantum graphs
equidistribute in the limit of large graphs. For a smaller class of observables
we also show that the Egorov property, a correspondence between classical and
quantum evolution in the semiclassical limit, holds for the quantum graphs in
question.Comment: 20 pages, 1 figur