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 $H = H_1 + H_2$ (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 $H^q = H^q_1 + H^q_2$ whose spacing distribution agrees with the Wigner law and we show that in general the spacing distribution of $H^q$ is not the Poisson law, even if it has often the same qualitative behaviour. We have found that the spacings of $H^q$ 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 $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. 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