15,209 research outputs found
On the Role of Non-Periodic Orbits in The Semiclassical Quantization of the Truncated Hyperbola Billiard
Based on an accurate computation of the first 1851 quantal energy levels of
the truncated hyperbola billiard, we have found an anomalous long-range
modulation in the integrated level density. It is shown that the observed
anomaly can be explained by an additional term in Gutzwiller's trace formula.
This term is given as a sum over families of closed, non-periodic orbits which
are reflected in a point of the billiard boundary where the boundary is
continuously differentiable, but its curvature radius changes discontinuously.Comment: 8 pages, uu-encoded ps-fil
Mode fluctuations as fingerprint of chaotic and non-chaotic systems
The mode-fluctuation distribution is studied for chaotic as well as
for non-chaotic quantum billiards. This statistic is discussed in the broader
framework of the functions being the probability of finding energy
levels in a randomly chosen interval of length , and the distribution of
, where is the number of levels in such an interval, and their
cumulants . It is demonstrated that the cumulants provide a possible
measure for the distinction between chaotic and non-chaotic systems. The
vanishing of the normalized cumulants , , implies a Gaussian
behaviour of , which is realized in the case of chaotic systems, whereas
non-chaotic systems display non-vanishing values for these cumulants leading to
a non-Gaussian behaviour of . For some integrable systems there exist
rigorous proofs of the non-Gaussian behaviour which are also discussed. Our
numerical results and the rigorous results for integrable systems suggest that
a clear fingerprint of chaotic systems is provided by a Gaussian distribution
of the mode-fluctuation distribution .Comment: 44 pages, Postscript. The figures are included in low resolution
only. A full version is available at
http://www.physik.uni-ulm.de/theo/qc/baecker.htm
Linear recursive odometers and beta-expansions
The aim of this paper is to study the connection between different properties
related to -expansions. In particular, the relation between two
conditions, both ensuring pure discrete spectrum of the odometer, is analysed.
The first one is the so-called Hypothesis B for the -odometers and the
second one is denoted by (QM) and it has been introduced in the framework of
tilings associated to Pisot -numerations
- …