Quantum chaotic resonances from short periodic orbits

We present an approach to calculating the quantum resonances and resonance wave functions of chaotic scattering systems, based on the construction of states localized on classical periodic orbits and adapted to the dynamics. Typically only a few of such states are necessary for constructing a resonance. Using only short orbits (with periods up to the Ehrenfest time), we obtain approximations to the longest living states, avoiding computation of the background of short living states. This makes our approach considerably more efficient than previous ones. The number of long lived states produced within our formulation is in agreement with the fractal Weyl law conjectured recently in this setting. We confirm the accuracy of the approximations using the open quantum baker map as an example.Comment: 4 pages, 4 figure

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Periodic orbit bifurcations and scattering time delay fluctuations

We study fluctuations of the Wigner time delay for open (scattering) systems which exhibit mixed dynamics in the classical limit. It is shown that in the semiclassical limit the time delay fluctuations have a distribution that differs markedly from those which describe fully chaotic (or strongly disordered) systems: their moments have a power law dependence on a semiclassical parameter, with exponents that are rational fractions. These exponents are obtained from bifurcating periodic orbits trapped in the system. They are universal in situations where sufficiently long orbits contribute. We illustrate the influence of bifurcations on the time delay numerically using an open quantum map.Comment: 9 pages, 3 figures, contribution to QMC200

Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.Comment: 21 pages, 6 figure

Number fields and function fields:Coalescences, contrasts and emerging applications

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained

Attitude determination of the spin-stabilized Project Scanner spacecraft

Attitude determination of spin-stabilized spacecraft using star mapping techniqu

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

Semiclassical structure of chaotic resonance eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the non-unitary quantum propagator, and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as $\hbar\to 0$ on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates $\{\psi(\hbar)\}_{\hbar\to 0}$ is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker map, for which the probability density in position space is observed to have self-similarity properties.Comment: 4 pages, 4 figures; some minor corrections, some changes in presentatio

A different appetite for sovereignty? Independence movements in subnational island jurisdictions

Local autonomy in a subnational jurisdiction is more likely to be gained, secured or enhanced where there are palpable movements or political parties agitating for independence in these smaller territories. A closer look at the fortunes, operations and dynamics of independence parties from subnational island jurisdictions can offer some interesting insights on the appetite for sovereignty and independence, but also the lack thereof, in the twenty-first century.peer-reviewe

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