Table of contents

This is a call for contributions to a Special Issue ofJournal of Physics A: Mathematical and General dedicated to the subject of `Random Matrix Theory'. This Issue should be a venue for high quality original work covering this topic in a broadly interpreted way. We seek contributions both on the analysis of random matrix theories and on their applications to many branches of physics and mathematics. It is therefore expected that researchers in the subdisciplines of Quantum Chaos, Complex Systems, Mesoscopic Physics, Combinatorics, Random Surfaces, Statistical Mechanics, and Lattice Field Theories, as well as branches of Number Theory and Probability, will be well placed to be part of this Special Issue.

The Editorial Board has invited P J Forrester, N C Snaith and J J M Verbaarschot to serve as Guest Editors for this Special Issue. Their criteria for acceptance of contributions are the following:

• The subject of the paper should relate to Random Matrix Theory and its applications in both physics and mathematics.

• Contributions will be refereed and processed according to the usual mechanisms of the journal.

• Papers should contain a sufficient amount of original content (i.e. they should not be simply reviews of authors' own work that is already published elsewhere).

• Review articles may also be considered for inclusion in the Special Issue at the discretion of the Guest Editors. These should be of a sufficiently broad nature to provide an overview of the state of the art in a subfield of the subject (again, not covering only the work of a single author or group). Authors wishing to submit such a Review article should contact one of the Guest Editors at p.forrester@ms.unimelb.edu.au or N.C.Snaith@bristol.ac.uk orverbaarschot@tonic.physics.sunysb.edu in advance of submission. The guidelines for the preparation of contributions are the following:

• DEADLINE for submission of contributions is 31 July 2002. This deadline will allow the Special Issue to appear approximately in February 2003.

• There is a nominal page limit of 15 printed pages per research contribution. The contributions that have been approved by the Guest Editors as Review articles will have a limit of 30 printed pages. Papers exceeding these limits may be accepted at the discretion of the Guest Editors. Further advice on publishing your work in Journal of Physics A: Mathematical and General may be found at www.iop.org/Journals/jphysa.

• Contributions to the Special Issue should if possible be submitted electronically by web upload at www.iop.org/Journals/jphysa or by e-mail at jphysa@iop.org, quoting `JPhysA Special Issue -- Random Matrix Theory'. Submissions should ideally be in standard LaTeX form; we are, however, able to accept most formats including Microsoft Word. Please see the web site for further information on electronic submissions.

• Authors unable to submit electronically may send hard copy contributions to: Publishing Administrators, Journal of Physics A, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK, enclosing the electronic code on floppy disk if available and quoting `JPhysA Special Issue -- Random Matrix Theory'.

• All contributions should be accompanied by a readme file or covering letter giving the postal and e-mail addresses for correspondence. Any subsequent change of address should be notified to the Publishing Office. This Special Issue will be published in the paper and online editions of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue, in addition to the usual 25 free offprints of their article.

P J Forrester, N C Snaith and J J M Verbaarschot

Guest Editors

p.forrester@ms.unimelb.edu.au      www.ms.unimelb.edu.au/~matpjf/matpjf.html

N.C.Snaith@bristol.ac.uk      www.maths.bris.ac.uk/~mancs/

verbaarschot@tonic.physics.sunysb.edu      tonic.physics.sunysb.edu/~verbaarschot/

We present new analytical results concerning the spectral distributions for (2×2) random real symmetric matrices which generalize the Wigner surmise.

We study the relaxation process in a two-dimensional lattice gas model, where the interactions originate from the excluded volume. In this model particles have three arms with an asymmetrical shape, which results in geometrical frustration that inhibits full packing. Relaxation functions are well fitted at long times by a stretched exponential form, with a β exponent decreasing when the density is raised until the percolation transition is reached, and constant for higher densities. The structural arrest of the model seems to happen only at the maximum density of the model, where both the inverse diffusivity and the relaxation times diverge with a power law. The dynamical non-linear susceptibility, defined as the fluctuations of the self-overlap autocorrelation, exhibits a peak at some characteristic time, which also seems to diverge at the maximum density.

A two-lane extension of a recently proposed cellular automaton model for traffic flow is discussed. The analysis focuses on the reproduction of the lane usage inversion and the density dependence of the number of lane changes. It is shown that the single-lane dynamics can be extended to the two-lane case without changing the basic properties of the model, which are known to be in good agreement with empirical single-vehicle data. Therefore it is possible to reproduce various empirically observed two-lane phenomena, like the synchronization of the lanes, without fine tuning of the model parameters.

We consider the Friedel sum rule (FSR) in the context of the scattering theory for the Schrödinger operator -D_x² + V(x) on graphs made of one-dimensional wires connected to external leads. We generalize the Smith formula for graphs. We give several examples of graphs where the state counting method given by the FSR does not work. The reason for the failure of the FSR to count the states is the existence of states localized in the graph and not coupled to the leads, which occurs if the spectrum is degenerate and the number of leads too small.

We consider a possible zero mass limit of the relativistic Thomas–Fermi–von Weizsäcker model of atoms and molecules. We find sharp bounds for the critical atomic number below which there is stability and above which the system collapses.

We address the question of radiation emission from mirrors following prescribed relativistic trajectories. The trajectories considered are asymptotically inertial: the mirror starts from rest and eventually reverts to motion at uniform velocity. This enables us to provide a description in terms of in and out states. We calculate exactly the Bogolubov α and β coefficients for a specific form of the trajectory, and stress the analytic properties of the amplitudes and the constraints imposed by unitarity. A formalism for the description of emission of radiation from a dispersive mirror is presented.

We address the question of radiation emission from a perfect mirror that starts from rest and follows the trajectory z = − ln(cosh t) until t → ∞. We show that a correct derivation of the black body spectrum via the calculation of the Bogolubov amplitudes requires consideration of the whole trajectory and not just of its asymptotic part.

Canonical maps on a two-torus in phase space are quantized under most general conditions. Recent results by Keating et al (1999 Nonlinearity12 579) are thus fully extended in two directions: (a) The translational component of a general canonical map is included in the quantization. (b) All values of Planck's constant, consistent with the toral boundary conditions (BCs), are considered; generically, these values are rational numbers whose numerator must satisfy a number-theoretical condition. Besides the condition on Planck's constant, the quantization is possible only for particular, 'allowed' BCs on the torus. The general equation determining these BCs is derived. Allowed BCs may not exist in some cases; representative examples are the irrational skew translations and Kronecker maps. Exact versions of Egorov's theorem are shown to hold under some conditions. Composition and representation properties of the quantization scheme are studied.

A formula expressing explicitly the derivatives of Jacobi polynomials of any degree and for any order in terms of the Jacobi polynomials themselves is proved. Another explicit formula, which expresses the Jacobi expansion coefficients of a general-order derivative of an infinitely differentiable function in terms of its original Jacobi coefficients, is also given. The results for the special case of ultraspherical polynomials are considered. The results for Chebyshev polynomials of the first and second kinds and for Legendre polynomials are also noted.

An application of how to use Jacobi polynomials for solving ordinary and partial differential equations is described.

We investigate the two-dimensional magnetic Schrödinger operator H_B,β = (−i∇ − A)² − βδ(· − Γ), where Γ is a smooth loop and the vector potential A corresponds to a homogeneous magnetic field B perpendicular to the plane. The asymptotics of negative eigenvalues of H_B,β for β → ∞ is found. It shows, in particular, that for large enough positive β the system exhibits persistent currents.

For each of the exceptional Lie groups, a complete determination is given of those pairs of finite-dimensional irreducible representations whose tensor products (or squares) may be resolved into irreducible representations that are multiplicity free, i.e. such that no irreducible representation occurs in the decomposition of the tensor product more than once. Explicit formulae are presented for the decomposition of all those tensor products that are multiplicity free, many of which exhibit a stability property.

This paper is devoted to the group classification of steady viscous gas dynamics equations in the two-dimensional case (with plane or cylindrical symmetry) with arbitrary state equations. Representations of all invariant solutions are given.

Here we explore the link between the moments of the Laguerre polynomials or Laguerre moments and the generalized functions (as the Dirac delta-function and its derivatives), presenting several interesting relations. A useful application is related to a procedure for calculating mean values in quantum optics that makes use of the so-called quasi-probabilities. One of them, the P-distribution, can be represented by a sum over Laguerre moments when the electromagnetic field is in a photon-number state. Consequently, the P-distribution can be expressed in terms of Dirac delta-function and derivatives. More specifically, we found a direct relation between P-distributions and the Laguerre factorial moments.

We study the positive-operator-valued measures (POVMs) on the projective real line covariant with respect to the projective group. We interpret the projective line as a compactified time axis and we assume that the energy is a positive operator. This formalism may describe a time-of-arrival observable for a free particle covariant with respect to linear canonical transformations. The problem is similar to the more complicated and physically more relevant problem of finding the POVMs on the compactified Minkowski space-time covariant with respect to the conformal group.

Using the Thomas-Fermi (TF) model we describe the electron density distribution that neutralizes two static positive charge distributions, one being planar and the other linear. The first case admits an analytical solution, the second solution is numerical. The various energy terms are calculated and the virial theorem verified. These results are compared with the electron distribution that neutralizes a point-like positive charge, i.e. the familiar TF model for the atom.