325 research outputs found
Exterior-Interior Duality for Discrete Graphs
The Exterior-Interior duality expresses a deep connection between the Laplace
spectrum in bounded and connected domains in , and the scattering
matrices in the exterior of the domains. Here, this link is extended to the
study of the spectrum of the discrete Laplacian on finite graphs. For this
purpose, two methods are devised for associating scattering matrices to the
graphs. The Exterior -Interior duality is derived for both methods.Comment: 15 pages 1 figur
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
An estimate for the average spectral measure of random band matrices
For a class of random band matrices of band width , we prove regularity of
the average spectral measure at scales , and find its
asymptotics at these scales.Comment: 19 pp., revised versio
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
Isospectral domains with mixed boundary conditions
We construct a series of examples of planar isospectral domains with mixed
Dirichlet-Neumann boundary conditions. This is a modification of a classical
problem proposed by M. Kac.Comment: 9 figures. Statement of Theorem 5.1 correcte
Can One Hear the Shape of a Graph?
We show that the spectrum of the Schrodinger operator on a finite, metric
graph determines uniquely the connectivity matrix and the bond lengths,
provided that the lengths are non-commensurate and the connectivity is simple
(no parallel bonds between vertices and no loops connecting a vertex to
itself). That is, one can hear the shape of the graph! We also consider a
related inversion problem: A compact graph can be converted into a scattering
system by attaching to its vertices leads to infinity. We show that the
scattering phase determines uniquely the compact part of the graph, under
similar conditions as above.Comment: 9 pages, 1 figur
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
“Free Will and Affirmation: Assessing Honderich’s Third Way”
In the third and final part of his A Theory of Determinism (TD) Ted Honderich addresses the fundamental question concerning “the consequences of determinism.” The critical question he aims to answer is what follows if determinism is true? This question is, of course, intimately bound up with the problem of free will and, in particular, with the question of whether or not the truth of determinism is compatible or incompatible with the sort of freedom required for moral responsibility. It is Honderich’s aim to provide a solution to “the problem of the consequences of determinism” and a key element of this is his articulation and defence of an alternative response to the implications of determinism that collapses the familiar Compatibilist/Incompatibilist dichotomy. Honderich offers us a third way – the response of “Affirmation” (HFY 125-6). Although his account of Affirmation has application and relevance to issues and features beyond freedom and responsibility, my primary concern in this essay will be to examine Honderich’s theory of “Affirmation” as it concerns the free will problem
The Statistics of the Points Where Nodal Lines Intersect a Reference Curve
We study the intersection points of a fixed planar curve with the
nodal set of a translationally invariant and isotropic Gaussian random field
\Psi(\bi{r}) and the zeros of its normal derivative across the curve. The
intersection points form a discrete random process which is the object of this
study. The field probability distribution function is completely specified by
the correlation G(|\bi{r}-\bi{r}'|) = .
Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point
correlation function of the point process on the line, and derive other
statistical measures (repulsion, rigidity) which characterize the short and
long range correlations of the intersection points. We use these statistical
measures to quantitatively characterize the complex patterns displayed by
various kinds of nodal networks. We apply these statistics in particular to
nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of
special interest is the observation that for monochromatic random waves, the
number variance of the intersections with long straight segments grows like , as opposed to the linear growth predicted by the percolation model,
which was successfully used to predict other long range nodal properties of
that field.Comment: 33 pages, 13 figures, 1 tabl
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