2,983 research outputs found
Nodal domain distributions for quantum maps
The statistics of the nodal lines and nodal domains of the eigenfunctions of
quantum billiards have recently been observed to be fingerprints of the
chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett.,
Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88
(2002), 114102). These statistics were shown to be computable from the random
wave model of the eigenfunctions. We here study the analogous problem for
chaotic maps whose phase space is the two-torus. We show that the distributions
of the numbers of nodal points and nodal domains of the eigenvectors of the
corresponding quantum maps can be computed straightforwardly and exactly using
random matrix theory. We compare the predictions with the results of numerical
computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction
Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations
We calculate the negative integer moments of the (regularized) characteristic
polynomials of N x N random matrices taken from the Gaussian Orthogonal
Ensemble (GOE) in the limit as . The results agree nontrivially
with a recent conjecture of Berry & Keating motivated by techniques developed
in the theory of singularity-dominated strong fluctuations. This is the first
example where nontrivial predictions obtained using these techniques have been
proved.Comment: 13 page
On the resonance eigenstates of an open quantum baker map
We study the resonance eigenstates of a particular quantization of the open
baker map. For any admissible value of Planck's constant, the corresponding
quantum map is a subunitary matrix, and the nonzero component of its spectrum
is contained inside an annulus in the complex plane, . We consider semiclassical sequences of eigenstates, such that the
moduli of their eigenvalues converge to a fixed radius . We prove that, if
the moduli converge to , then the sequence of eigenstates
converges to a fixed phase space measure . The same holds for
sequences with eigenvalue moduli converging to , with a different
limit measure . Both these limiting measures are supported on
fractal sets, which are trapped sets of the classical dynamics. For a general
radius , we identify families of eigenstates with
precise self-similar properties.Comment: 32 pages, 2 figure
Spectral statistics for unitary transfer matrices of binary graphs
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits. It
is proposed here to generalise this approach by considering arbitrary, directed
graphs with unitary transfer matrices. An exponentially increasing contribution
to the form factor is identified when performing a diagonal summation over
periodic orbit degeneracy classes. A special class of graphs, so-called binary
graphs, is studied in more detail. For these, the conditions for periodic orbit
pairs to be correlated (including correlations due to the unitarity of the
transfer matrix) can be given explicitly. Using combinatorial techniques it is
possible to perform the summation over correlated periodic orbit pair
contributions to the form factor for some low--dimensional cases. Gradual
convergence towards random matrix results is observed when increasing the
number of vertices of the binary graphs.Comment: 18 pages, 8 figure
Renormalization of Quantum Anosov Maps: Reduction to Fixed Boundary Conditions
A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on
a torus for general boundary conditions (BCs), whose number () is always
finite. It is shown that the quasienergy eigenvalue problem of a QAM for {\em
all} BCs is exactly equivalent to that of the renormalized QAM (with
Planck's constant ) at some {\em fixed} BCs that can
be of four types. The quantum cat maps are, up to time reversal, fixed points
of the renormalization transformation. Several results at fixed BCs, in
particular the existence of a complete basis of ``crystalline'' eigenstates in
a classical limit, can then be derived and understood in a simple and
transparent way in the general-BCs framework.Comment: REVTEX, 12 pages, 1 table. To appear in Physical Review Letter
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
Signatures of homoclinic motion in quantum chaos
Homoclinic motion plays a key role in the organization of classical chaos in
Hamiltonian systems. In this Letter, we show that it also imprints a clear
signature in the corresponding quantum spectra. By numerically studying the
fluctuations of the widths of wavefunctions localized along periodic orbits we
reveal the existence of an oscillatory behavior, that is explained solely in
terms of the primary homoclinic motion. Furthermore, our results indicate that
it survives the semiclassical limit.Comment: 5 pages, 4 figure
Stochastic stabilization of cosmological photons
The stability of photon trajectories in models of the Universe that have
constant spatial curvature is determined by the sign of the curvature: they are
exponentially unstable if the curvature is negative and stable if it is
positive or zero. We demonstrate that random fluctuations in the curvature
provide an additional stabilizing mechanism. This mechanism is analogous to the
one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean
curvature is negative it is capable of stabilizing the photon trajectories;
when the mean curvature is zero or positive it determines the characteristic
frequency with which neighbouring trajectories oscillate about each other. In
constant negative curvature models of the Universe that have compact topology,
exponential instability implies chaos (e.g. mixing) in the photon dynamics. We
discuss some consequences of stochastic stabilization in this context.Comment: 4 pages, 3 postscript figures in color which are also appropriate for
black and white printers; v2 emphasizes relevance to flat as well as
negatively curved cosmologies; to appear in J. Phys.
Two-point correlations of the Gaussian symplectic ensemble from periodic orbits
We determine the asymptotics of the two-point correlation function for
quantum systems with half-integer spin which show chaotic behaviour in the
classical limit using a method introduced by Bogomolny and Keating [Phys. Rev.
Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the
leading terms of the two-point correlation function of the Gaussian symplectic
ensemble. Special attention has to be paid to the role of Kramers' degeneracy.Comment: 7 pages, no figure
On the duality between periodic orbit statistics and quantum level statistics
We discuss consequences of a recent observation that the sequence of periodic
orbits in a chaotic billiard behaves like a poissonian stochastic process on
small scales. This enables the semiclassical form factor to
agree with predictions of random matrix theories for other than infinitesimal
in the semiclassical limit.Comment: 8 pages LaTe
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