145 research outputs found
On the properties of level spacings for decomposable systems
In this paper we show that the quantum theory of chaos, based on the
statistical theory of energy spectra, presents inconsistencies difficult to
overcome. In classical mechanics a system described by an hamiltonian (decomposable) cannot be ergodic, because there are always two dependent
integrals of motion besides the constant of energy. In quantum mechanics we
prove the existence of decomposable systems \linebreak
whose spacing distribution agrees with the Wigner law and we show that in
general the spacing distribution of is not the Poisson law, even if it
has often the same qualitative behaviour. We have found that the spacings of
are among the solutions of a well defined class of homogeneous linear
systems. We have obtained an explicit formula for the bases of the kernels of
these systems, and a chain of inequalities which the coefficients of a generic
linear combination of the basis vectors must satisfy so that the elements of a
particular solution will be all positive, i.e. can be considered a set of
spacings.Comment: LateX, 13 page
Random matrix theory and the zeros of zeta'(s)
We study the density of the roots of the derivative of the characteristic
polynomial Z(U,z) of an N x N random unitary matrix with distribution given by
Haar measure on the unitary group. Based on previous random matrix theory
models of the Riemann zeta function zeta(s), this is expected to be an accurate
description for the horizontal distribution of the zeros of zeta'(s) to the
right of the critical line. We show that as N --> infinity the fraction of
roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit
function. We derive asymptotic expressions for this function in the limits x
--> infinity and x --> 0 and compare them with numerical experiments.Comment: 18 pages, 5 figures. Revised version: section 4 expanded; minor
correction
Free fermions and the classical compact groups
There is a close connection between the ground state of non-interacting
fermions in a box with classical (absorbing, reflecting, and periodic) boundary
conditions and the eigenvalue statistics of the classical compact groups. The
associated determinantal point processes can be extended in two natural
directions: i) we consider the full family of admissible quantum boundary
conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded
interval, and the corresponding projection correlation kernels; ii) we
construct the grand canonical extensions at finite temperature of the
projection kernels, interpolating from Poisson to random matrix eigenvalue
statistics. The scaling limits in the bulk and at the edges are studied in a
unified framework, and the question of universality is addressed. Whether the
finite temperature determinantal processes correspond to the eigenvalue
statistics of some matrix models is, a priori, not obvious. We complete the
picture by constructing a finite temperature extension of the Haar measure on
the classical compact groups. The eigenvalue statistics of the resulting grand
canonical matrix models (of random size) corresponds exactly to the grand
canonical measure of non-interacting free fermions with classical boundary
conditions.Comment: 35 pages, 5 figures. Final versio
On the Lagged Diffusivity Method for the solution of nonlinear finite difference systems
In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of the method itself. Moreover, we also analyze the behavior of the method in case of problems presenting boundary layers or blow-up solutions
Density and spacings for the energy levels of quadratic Fermi operators
The work presents a proof of convergence of the density of energy levels to a
Gaussian distribution for a wide class of quadratic forms of Fermi operators.
This general result applies also to quadratic operators with disorder, e.g.,
containing random coefficients. The spacing distribution of the unfolded
spectrum is investigated numerically. For generic systems the level spacings
behave as the spacings in a Poisson process. Level clustering persists in
presence of disorder.Comment: 19 pages, 2 figures. v3: Typos fixe
Moments of the eigenvalue densities and of the secular coefficients of β-ensembles
This is an Open Access Article. It is published by IOP under the Creative Commons Attribution 4.0 Unported Licence (CC BY). Full details of this licence are available at: http://creativecommons.org/licenses/by/4.0/© 2017 IOP Publishing Ltd & London Mathematical Society.We compute explicit formulae for the moments of the densities of the eigenvalues of the classical β-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters
On the moments of characteristic polynomials
We examine the asymptotics of the moments of characteristic polynomials of
matrices drawn from the Hermitian ensembles of Random Matrix
Theory, in the limit as . We focus in particular on the Gaussian
Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a
novel approach to calculate asymptotic formulae for the moments, enabling us to
uncover subtle structure not apparent in previous approaches.Comment: 26 page
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