285 research outputs found
Edges of the Barvinok-Novik orbitope
Here we study the k^th symmetric trigonometric moment curve and its convex
hull, the Barvinok-Novik orbitope. In 2008, Barvinok and Novik introduce these
objects and show that there is some threshold so that for two points on S^1
with arclength below this threshold, the line segment between their lifts on
the curve form an edge on the Barvinok-Novik orbitope and for points with
arclenth above this threshold, their lifts do not form an edge. They also give
a lower bound for this threshold and conjecture that this bound is tight.
Results of Smilansky prove tightness for k=2. Here we prove this conjecture for
all k.Comment: 10 pages, 3 figures, corrected Lemma 4 and other minor revision
Sum of Two Squares - Pair Correlation and Distribution in Short Intervals
In this work we show that based on a conjecture for the pair correlation of
integers representable as sums of two squares, which was first suggested by
Connors and Keating and reformulated here, the second moment of the
distribution of the number of representable integers in short intervals is
consistent with a Poissonian distribution, where "short" means of length
comparable to the mean spacing between sums of two squares. In addition we
present a method for producing such conjectures through calculations in prime
power residue rings and describe how these conjectures, as well as the above
stated result, may by generalized to other binary quadratic forms. While
producing these pair correlation conjectures we arrive at a surprising result
regarding Mertens' formula for primes in arithmetic progressions, and in order
to test the validity of the conjectures, we present numericalz computations
which support our approach.Comment: 3 figure
On the semiclassical theory for universal transmission fluctuations in chaotic systems: the importance of unitarity
The standard semiclassical calculation of transmission correlation functions
for chaotic systems is severely influenced by unitarity problems. We show that
unitarity alone imposes a set of relationships between cross sections
correlation functions which go beyond the diagonal approximation. When these
relationships are properly used to supplement the semiclassical scheme we
obtain transmission correlation functions in full agreement with the exact
statistical theory and the experiment. Our approach also provides a novel
prediction for the transmission correlations in the case where time reversal
symmetry is present
On the classical-quantum correspondence for the scattering dwell time
Using results from the theory of dynamical systems, we derive a general
expression for the classical average scattering dwell time, tau_av. Remarkably,
tau_av depends only on a ratio of phase space volumes. We further show that,
for a wide class of systems, the average classical dwell time is not in
correspondence with the energy average of the quantum Wigner time delay.Comment: 5 pages, 1 figur
Nodal domains statistics - a criterion for quantum chaos
We consider the distribution of the (properly normalized) numbers of nodal
domains of wave functions in 2- quantum billiards. We show that these
distributions distinguish clearly between systems with integrable (separable)
or chaotic underlying classical dynamics, and for each case the limiting
distribution is universal (system independent). Thus, a new criterion for
quantum chaos is provided by the statistics of the wave functions, which
complements the well established criterion based on spectral statistics.Comment: 4 pages, 5 figures, revte
Counting nodal domains on surfaces of revolution
We consider eigenfunctions of the Laplace-Beltrami operator on special
surfaces of revolution. For this separable system, the nodal domains of the
(real) eigenfunctions form a checker-board pattern, and their number is
proportional to the product of the angular and the "surface" quantum numbers.
Arranging the wave functions by increasing values of the Laplace-Beltrami
spectrum, we obtain the nodal sequence, whose statistical properties we study.
In particular we investigate the distribution of the normalized counts
for sequences of eigenfunctions with where . We show that the distribution approaches
a limit as (the classical limit), and study the leading
corrections in the semi-classical limit. With this information, we derive the
central result of this work: the nodal sequence of a mirror-symmetric surface
is sufficient to uniquely determine its shape (modulo scaling).Comment: 36 pages, 8 figure
Characterization of Quantum Chaos by the Autocorrelation Function of Spectral Determinants
The autocorrelation function of spectral determinants is proposed as a
convenient tool for the characterization of spectral statistics in general, and
for the study of the intimate link between quantum chaos and random matrix
theory, in particular. For this purpose, the correlation functions of spectral
determinants are evaluated for various random matrix ensembles, and are
compared with the corresponding semiclassical expressions. The method is
demonstrated by applying it to the spectra of the quantized Sinai billiards in
two and three dimensions.Comment: LaTeX, 32 pages, 6 figure
Effect of phase relaxation on quantum superpositions in complex collisions
We study the effect of phase relaxation on coherent superpositions of
rotating clockwise and anticlockwise wave packets in the regime of strongly
overlapping resonances of the intermediate complex. Such highly excited
deformed complexes may be created in binary collisions of heavy ions, molecules
and atomic clusters. It is shown that phase relaxation leads to a reduction of
the interference fringes, thus mimicking the effect of decoherence. This
reduction is crucial for the determination of the phase--relaxation width from
the data on the excitation function oscillations in heavy--ion collisions and
bimolecular chemical reactions. The difference between the effects of phase
relaxation and decoherence is discussed.Comment: Extended revised version; 9 pages and 3 colour ps figure
Quantal Consequences of Perturbations Which Destroy Structurally Unstable Orbits in Chaotic Billiards
Non-generic contributions to the quantal level-density from parallel segments
in billiards are investigated. These contributions are due to the existence of
marginally stable families of periodic orbits, which are structurally unstable,
in the sense that small perturbations, such as a slight tilt of one of the
segments, destroy them completely. We investigate the effects of such
perturbation on the corresponding quantum spectra, and demonstrate them for the
stadium billiard
Families of spherical caps: spectra and ray limit
We consider a family of surfaces of revolution ranging between a disc and a
hemisphere, that is spherical caps. For this family, we study the spectral
density in the ray limit and arrive at a trace formula with geodesic polygons
describing the spectral fluctuations. When the caps approach the hemisphere the
spectrum becomes equally spaced and highly degenerate whereas the derived trace
formula breaks down. We discuss its divergence and also derive a different
trace formula for this hemispherical case. We next turn to perturbative
corrections in the wave number where the work in the literature is done for
either flat domains or curved without boundaries. In the present case, we
calculate the leading correction explicitly and incorporate it into the
semiclassical expression for the fluctuating part of the spectral density. To
the best of our knowledge, this is the first calculation of such perturbative
corrections in the case of curvature and boundary.Comment: 28 pages, 7 figure
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