992 research outputs found
Beyond the Heisenberg time: Semiclassical treatment of spectral correlations in chaotic systems with spin 1/2
The two-point correlation function of chaotic systems with spin 1/2 is
evaluated using periodic orbits. The spectral form factor for all times thus
becomes accessible. Equivalence with the predictions of random matrix theory
for the Gaussian symplectic ensemble is demonstrated. A duality between the
underlying generating functions of the orthogonal and symplectic symmetry
classes is semiclassically established
Intermediate statistics for a system with symplectic symmetry: the Dirac rose graph
We study the spectral statistics of the Dirac operator on a rose-shaped
graph---a graph with a single vertex and all bonds connected at both ends to
the vertex. We formulate a secular equation that generically determines the
eigenvalues of the Dirac rose graph, which is seen to generalise the secular
equation for a star graph with Neumann boundary conditions. We derive
approximations to the spectral pair correlation function at large and small
values of spectral spacings, in the limit as the number of bonds approaches
infinity, and compare these predictions with results of numerical calculations.
Our results represent the first example of intermediate statistics from the
symplectic symmetry class.Comment: 26 pages, references adde
NGC 2419, M92, and the Age Gradient in the Galactic Halo
The WFPC2 camera on HST has been used to obtain deep main sequence photometry
of the low-metallicity ([Fe/H]=-2.14), outer-halo globular cluster NGC 2419. A
differential fit of the NGC 2419 CMD to that of the similarly metal-poor \
standard cluster M92 shows that they have virtually identical principal
sequences and thus the same age to well within 1 Gyr. Since other
low-metallicity clusters throughout the Milky Way halo have this same age to
within the 1-Gyr precision of the differential age technique, we conclude that
the earliest star (or globular cluster) formation began at essentially the same
time everywhere in the Galactic halo throughout a region now almost 200 kpc in
diameter. Thus for the metal-poorest clusters in the halo there is no
detectable age gradient with Galactocentric distance. To estimate the absolute
age of NGC 2419 and M92, we fit newly computed isochrones transformed through
model-atmosphere calculations to the (M_V,V-I) plane, with assumed distance
scales that represent the range currently debated in the literature.
Unconstrained isochrone fits give M_V(RR) = 0.55 \pm 0.06 and a resulting age
of 14 to 15 Gyr. Incorporating the full effects of helium diffusion would
further reduce this estimate by about 1 Gyr. A distance scale as bright as
M_V(RR) = 0.15 for [Fe/H] = -2, as has recently been reported, would leave
several serious problems which have no obvious solution in the context of
current stellar models.Comment: 32 pages, aastex, 9 postscript figures; accepted for publication in
AJ, September 1997. Also available by e-mail from [email protected]
The Selberg trace formula for Dirac operators
We examine spectra of Dirac operators on compact hyperbolic surfaces.
Particular attention is devoted to symmetry considerations, leading to
non-trivial multiplicities of eigenvalues. The relation to spectra of
Maass-Laplace operators is also exploited. Our main result is a Selberg trace
formula for Dirac operators on hyperbolic surfaces
Level spacings and periodic orbits
Starting from a semiclassical quantization condition based on the trace
formula, we derive a periodic-orbit formula for the distribution of spacings of
eigenvalues with k intermediate levels. Numerical tests verify the validity of
this representation for the nearest-neighbor level spacing (k=0). In a second
part, we present an asymptotic evaluation for large spacings, where consistency
with random matrix theory is achieved for large k. We also discuss the relation
with the method of Bogomolny and Keating [Phys. Rev. Lett. 77 (1996) 1472] for
two-point correlations.Comment: 4 pages, 2 figures; major revisions in the second part, range of
validity of asymptotic evaluation clarifie
Globular Clusters around Galaxies in Groups
We have obtained deep photometry of NGC 1199 (in HCG 22) and NGC 6868 (in the
Telescopium group). Both galaxies are the optically brightest galaxies of their
groups. Our analysis of B and R images taken with the Keck II and the VLT/ESO
telescopes, detected a population of globular clusters around both galaxies,
with total specific frequencies S_N=1.7\pm0.6 for NGC 1199 and S_N = 1.3\pm0.6
for NGC 6868. The color distributions of the globular cluster systems shows
bimodal peaks centered at (B-R)_0 = 1.13\pm0.10 and 1.42\pm0.10 (NGC 1199) and
(B-R)_0=1.12\pm0.10 and 1.42\pm0.10 (NGC 6868).Comment: 3 pages, 1 figure. To appear in the proceedings of IAU Symp. 207,
"Extragalactic Star Clusters", eds. E. Grebel, D. Geisler, D. Minnit
Hubble Space Telescope Observations of the Oldest Star Clusters in the LMC
We present V, V-I color-magnitude diagrams (CMDs) for three old star clusters
in the Large Magellanic Cloud (LMC): NGC 1466, NGC 2257 and Hodge 11. Our data
extend about 3 magnitudes below the main-sequence turnoff, allowing us to
determine accurate relative ages and the blue straggler frequencies. Based on a
differential comparison of the CMDs, any age difference between the three LMC
clusters is less than 1.5 Gyr. Comparing their CMDs to those of M 92 and M 3,
the LMC clusters, unless their published metallicities are significantly in
error, are the same age as the old Galactic globulars. The similar ages to
Galactic globulars are shown to be consistent with hierarchial clustering
models of galaxy formation. The blue straggler frequencies are also similar to
those of Galactic globular clusters. We derive a true distance modulus to the
LMC of (m-M)=18.46 +/- 0.09 (assuming (m-M)=14.61 for M 92) using these three
LMC clusters.Comment: 22 pages; to be published in Ap
On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds
Asymptotic laws for mean multiplicities of lengths of closed geodesics in
arithmetic hyperbolic three-orbifolds are derived. The sharpest results are
obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o)
and some congruence subgroups. Similar results hold for cocompact arithmetic
quaternion groups, if a conjecture on the number of gaps in their length
spectra is true. The results related to the groups above give asymptotic lower
bounds for the mean multiplicities in length spectra of arbitrary arithmetic
hyperbolic three-orbifolds. The investigation of these multiplicities is
motivated by their sensitive effect on the eigenvalue spectrum of the
Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as
the Hamiltonian of a three-dimensional quantum system being strongly chaotic in
the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT
Parabolic maps with spin: Generic spectral statistics with non-mixing classical limit
We investigate quantised maps of the torus whose classical analogues are
ergodic but not mixing. Their quantum spectral statistics shows non-generic
behaviour, i.e.it does not follow random matrix theory (RMT). By coupling the
map to a spin 1/2, which corresponds to changing the quantisation without
altering the classical limit of the dynamics on the torus, we numerically
observe a transition to RMT statistics. The results are interpreted in terms of
semiclassical trace formulae for the maps with and without spin respectively.
We thus have constructed quantum systems with non-mixing classical limit which
show generic (i.e. RMT) spectral statistics. We also discuss the analogous
situation for an almost integrable map, where we compare to Semi-Poissonian
statistics.Comment: 29 pages, 20 figure
Selberg Supertrace Formula for Super Riemann Surfaces III: Bordered Super Riemann Surfaces
This paper is the third in a sequel to develop a super-analogue of the
classical Selberg trace formula, the Selberg supertrace formula. It deals with
bordered super Riemann surfaces. The theory of bordered super Riemann surfaces
is outlined, and the corresponding Selberg supertrace formula is developed. The
analytic properties of the Selberg super zeta-functions on bordered super
Riemann surfaces are discussed, and super-determinants of Dirac-Laplace
operators on bordered super Riemann surfaces are calculated in terms of Selberg
super zeta-functions.Comment: 43 pages, amste
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