752 research outputs found
Missing levels in correlated spectra
Complete spectroscopy (measurements of a complete sequence of consecutive
levels) is often considered as a prerequisite to extract fluctuation properties
of spectra. It is shown how this goal can be achieved even if only a fraction
of levels are observed. The case of levels behaving as eigenvalues of random
matrices, of current interest in nuclear physics, is worked out in detail.Comment: 14 pages and two figure
Distance matrices and isometric embeddings
We review the relations between distance matrices and isometric embeddings
and give simple proofs that distance matrices defined on euclidean and
spherical spaces have all eigenvalues except one non-negative. Several
generalizations are discussed.Comment: 17 page
RESONANCES GEANTES DANS LES NOYAUX
Pas de Résumé disponibl
On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates
We consider a single particle spectrum as given by the eigenvalues of the
Wigner-Dyson ensembles of random matrices, and fill consecutive single particle
levels with n fermions. Assuming that the fermions are non-interacting, we show
that the distribution of the total energy is Gaussian and its variance grows as
n^2 log n in the large-n limit. Next to leading order corrections are computed.
Some related quantities are discussed, in particular the nearest neighbor
spacing autocorrelation function. Canonical and gran canonical approaches are
considered and compared in detail. A semiclassical formula describing, as a
function of n, a non-universal behavior of the variance of the total energy
starting at a critical number of particles is also obtained. It is illustrated
with the particular case of single particle energies given by the imaginary
part of the zeros of the Riemann zeta function on the critical line.Comment: 28 pages in Latex format, 5 figures, submitted for publication to
Physica
Spectral spacing correlations for chaotic and disordered systems
New aspects of spectral fluctuations of (quantum) chaotic and diffusive
systems are considered, namely autocorrelations of the spacing between
consecutive levels or spacing autocovariances. They can be viewed as a
discretized two point correlation function. Their behavior results from two
different contributions. One corresponds to (universal) random matrix
eigenvalue fluctuations, the other to diffusive or chaotic characteristics of
the corresponding classical motion. A closed formula expressing spacing
autocovariances in terms of classical dynamical zeta functions, including the
Perron-Frobenius operator, is derived. It leads to a simple interpretation in
terms of classical resonances. The theory is applied to zeros of the Riemann
zeta function. A striking correspondence between the associated classical
dynamical zeta functions and the Riemann zeta itself is found. This induces a
resurgence phenomenon where the lowest Riemann zeros appear replicated an
infinite number of times as resonances and sub-resonances in the spacing
autocovariances. The theoretical results are confirmed by existing ``data''.
The present work further extends the already well known semiclassical
interpretation of properties of Riemann zeros.Comment: 28 pages, 6 figures, 1 table, To appear in the Gutzwiller
Festschrift, a special Issue of Foundations of Physic
The Riemannium
The properties of a fictitious, fermionic, many-body system based on the
complex zeros of the Riemann zeta function are studied. The imaginary part of
the zeros are interpreted as mean-field single-particle energies, and one fills
them up to a Fermi energy . The distribution of the total energy is shown
to be non-Gaussian, asymmetric, and independent of in the limit
. The moments of the limit distribution are computed
analytically. The autocorrelation function, the finite energy corrections, and
a comparison with random matrix theory are also discussed.Comment: 10 pages, 2 figures, 1 tabl
Connection between low energy effective Hamiltonians and energy level statistics
We study the level statistics of a non-integrable one dimensional interacting
fermionic system characterized by the GOE distribution. We calculate
numerically on a finite size system the level spacing distribution and
the Dyson-Mehta correlation. We observe that its low energy spectrum
follows rather the Poisson distribution, characteristic of an integrable
system, consistent with the fact that the low energy excitations of this system
are described by the Luttinger model. We propose this Random Matrix Theory
analysis as a probe for the existence and integrability of low energy effective
Hamiltonians for strongly correlated systems.Comment: REVTEX, 5 postscript figures at the end of the fil
Random-Matrix Approach to RPA equations. I
We study the RPA equations in their most general form by taking the matrix
elements appearing in the RPA equations as random. This yields either a
unitarily or an orthogonally invariant random-matrix model which is not of the
Cartan type. The average spectrum of the model is studied with the help of a
generalized Pastur equation. Two independent parameters govern the behaviour of
the system: The strength of the coupling between positive- and
negative-energy states and the distance between the origin and the centers of
the two semicircles that describe the average spectrum for , the
latter measured in units of the equal radii of the two semicircles. With
increasing , positive- and negative-energy states become mixed and
ever more of the spectral strength of the positive-energy states is transferred
to those at negative energy, and vice versa. The two semicircles are deformed
and pulled toward each other. As they begin to overlap, the RPA equations yield
non--real eigenvalues: The system becomes unstable. We determine analytically
the critical value of the strength for the instability to occur. Several
features of the model are illustrated numerically.Comment: 29 pages, 6 figure
Structure of trajectories of complex matrix eigenvalues in the Hermitian-non-Hermitian transition
The statistical properties of trajectories of eigenvalues of Gaussian complex
matrices whose Hermitian condition is progressively broken are investigated. It
is shown how the ordering on the real axis of the real eigenvalues is reflected
in the structure of the trajectories and also in the final distribution of the
eigenvalues in the complex plane.Comment: 12 pages, 3 figure
Family of generalized random matrix ensembles
Using the Generalized Maximium Entropy Principle based on the nonextensive q
entropy a new family of random matrix ensembles is generated. This family
unifies previous extensions of Random Matrix Theory and gives rise to an
orthogonal invariant stable Levy ensemble with new statistical properties. Some
of them are analytically derived.Comment: 13 pages and 2 figure
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