Table of contents

This is a call for contributions to a special issue ofJournal of Physics A: Mathematical and General entitled `Trends in Quantum Chaotic Scattering'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions clearly related to quantum chaotic scattering in any of it repercussions, in particular dealing with resonance phenomena, delay times, survival probability, S-matrix statistics, absorption from random media, random lasers, atoms in optical lattices and chaotic cavities as well as scattering and transport in mesoscopic and/or disordered quantum systems. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field.

The Editorial Board has invited Y V Fyodorov, T Kottos and H-J Stöckmann to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are the following:

• The subject of the paper should relate to quantum chaotic scattering in the sense described above.

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

• Papers should be original; reviews of a work published elsewhere will not be accepted.

The guidelines for the preparation of contributions are as follows:

• The DEADLINE for submission of contributions is 1 July 2005. This deadline will allow the special issue to appear in December 2005 or early 2006.

• There is a strict page limit of 16 printed pages (approximately 9600 words) per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. 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 to jphysa@iop.org, quoting `JPhysA Special Issue—Trends in quantum chaotic scattering'. 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—Trends in quantum chaotic scattering'.

• All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.

This special issue will be published in the paper and online version of the journal. The corresponding author of each contribution will receive a complimentary copy of the issue.

The nonequilibrium reweighting technique, which was recently developed by the present authors, is used for the study of nonequilibrium steady states. The renewed formulation of nonequlibrium reweighting enables us to use the very efficient multi-spin coding. We apply the nonequilibrium reweighting technique to the driven diffusive lattice gas model. Combining with the dynamical finite-size scaling theory, we estimate the critical temperature T_c and the dynamical exponent z. We also argue that this technique has an interesting feature that enables explicit calculation of derivatives of thermodynamic quantities without resorting to numerical differences.

We give an explicit example of an exactly solvable -symmetric Hamiltonian with unbroken symmetry which has one eigenfunction with zero norm. The set of its eigenfunctions is not complete in the corresponding Hilbert space and it is non-diagonalizable. In the case of a regular Sturm–Liouville problem any diagonalizable-symmetric Hamiltonian with unbroken symmetry has a complete set of positive -normalizable eigenfunctions. For non-diagonalizable Hamiltonians, a complete set of -normalizable functions is possible but the functions belonging to the root subspace corresponding to multiple zeros of the characteristic determinant are no longer eigenfunctions of the Hamiltonian.

We show that a new integrable two-component system of KdV type studied by Karasu (Kalkanlı) et al (2004 Acta Appl. Math.83 85–94) is bi-Hamiltonian, and its recursion operator, which has a highly unusual structure of nonlocal terms, can be written as a ratio of two compatible Hamiltonian operators found by us. Using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution, and the evolution systems corresponding to these symmetries are bi-Hamiltonian as well. We also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form, with the integration operator D⁻¹_x appearing in each term at most once.

The close similarity between the hierarchies of multiple-point correlation functions for the diffusion-limited coalescence and annihilation processes has caused some recent confusion, raising doubts as to whether such hierarchies uniquely determine an infinite particle system. We elucidate the precise relations between the two processes, arriving at the conclusion that the hierarchy of correlation functions does provide a complete representation of a particle system on the line. We also introduce a new hierarchy of probability density functions for finding particles at specified locations and none in between. This hierarchy is computable for coalescence, through the method of empty intervals, and is naturally suited for questions concerning the ordering of particles on the line.

In this paper the percolation of monomers on a square lattice is studied as the particles interact with either repulsive or attractive energies. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattice are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in terms of simple considerations related to the interactions present in the problem. The influence of the phase transitions occurring in the system is reflected by the phase diagram. In addition, a scaling treatment maintaining constant the surface coverage and varying the temperature of the system is performed. In all the considered cases, the universality class of the model is found to be the same as for the random percolation model.

We show that lattice Boltzmann simulations can be used to model the radiation force on an object in a standing wave acoustic field and comparisons are made to theoretical predictions. We show how viscous effects change the radiation force and predict the motion of a particle placed near a boundary where viscous effects are important.

We present a new calculation of the radiation force on a cylinder in a standing wave acoustic field. We use the formula to calculate the force on a cylinder which is free to move in the field and one which is fixed in space.

We compute the Bray and Moore (BM) TAP complexity for the Sherrington–Kirkpatrick model through the cavity method, showing that some essential modifications are needed with respect to the standard formulation of the method. This allows us to understand various features recently discovered and to unveil at last the physical meaning of the parameters of the BM theory. We also reconsider the supersymmetric (SUSY) formulation of the problem finding that the BM solution satisfies some proper SUSY Ward identities that are different from the standard ones. The SUSY relationships encode the physical meaning of the parameters obtained through the cavity method. The problem of the vanishing prefactor is addressed, showing how it can be avoided.

We find a solution of the dark soliton lying on a cnoidal wave background in a defocusing medium. We use the method of Darboux transformation, which is applied to the cnoidal wave solution of the defocusing nonlinear Schrödinger equation. Interesting characteristics of the dark soliton, i.e., the velocity and greyness, are calculated and compared with those of the dark soliton lying on a continuous wave background. We also calculate the shift of the crest of the cnoidal wave along the soliton.

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth Painlevé equation.

We consider the continuum limit of the relativistic Toda lattice. In particular, we propose a method in order to 'integrate' this system of nonlinear partial differential equations for some particular initial data and boundary conditions, before possible shocks. First, we recall the relation between the finite relativistic Toda lattice and the theory of discrete Laurent orthogonal polynomials. Our analysis is then based on some results for the asymptotic theory of discrete Laurent orthogonal polynomials with varying recurrence coefficients and the connection with a constrained and weighted extremal problem for logarithmic potentials.

This paper concerns processes described by a nonlinear partial differential equation that is an extension of the Fisher and KPP equations including density-dependent diffusion and nonlinear convection. The set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized. There is a minimal wave speed. For this wave speed there is a unique wavefront which can be found explicitly. It displays a sharp propagation front. For all greater wave speeds there is a unique wavefront which does not possess this property. For such waves, the asymptotic behaviour as the equilibrium states are approached is determined.

A Lie-algebraic approach successfully used to describe one-dimensional Bloch oscillations in a tight-binding approximation is extended to two dimensions. This extension has the same algebraic structure as the one-dimensional case while the dynamics shows a much richer behaviour. The Bloch oscillations are discussed using analytical expressions for expectation values and widths of the operators of the algebra. It is shown under which conditions the oscillations survive in two dimensions and the centre of mass of a wavepacket shows a Lissajous-like motion. In contrast to the one-dimensional case, a wavepacket shows systematic dispersion that depends on the direction of the field and the dispersion relation of the field-free system.

It is well known that integrable hierarchies in (1+1) dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms. We resolve this apparent discrepancy by providing simple and universal sufficient conditions for a (nonlocal) recursion operator in (1+1) dimensions to generate a hierarchy of local symmetries. These conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work. We also give explicit formulae for the nonlocal parts of higher recursion, Hamiltonian and symplectic operators of integrable systems in (1+1) dimensions. Using these two results we prove, under some natural assumptions, the Maltsev–Novikov conjecture stating that higher Hamiltonian, symplectic and recursion operators of integrable systems in (1+1) dimensions are weakly nonlocal, i.e., the coefficients of these operators are local and these operators contain at most one integration operator in each term.

We obtain L²-series solutions of the three-dimensional Schrödinger wave equation for a large class of non-central potentials that includes, as special cases, the Aharonov–Bohm, Hartmann and magnetic monopole potentials. It also includes contributions of the potential term, cos θ/r² (in spherical coordinates). The solutions obtained are for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The L² bases of the solution space are chosen such that the matrix representation of the wave operator is tridiagonal. The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations resulting from the matrix wave equation.

We explore the reflection–transmission quantum Yang–Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow us to establish, for the first time, a direct relationship with the different previous works on the subject and make evident the advantages of the reflection–transmission algebra as a universal approach to integrable systems with impurities.

We study the bound-state solutions of vanishing angular momentum in a quaternionic spherical square-well potential of finite depth. As in standard quantum mechanics, such solutions occur for discrete values of energy. At first glance, it seems that the continuity conditions impose a very restrictive constraint on the energy eigenvalues and, consequently, no bound states were expected for energy values below the pure quaternionic potential. Nevertheless, a careful analysis shows that pure quaternionic potentials do not remove bound states. It is also interesting to compare these new solutions with the bound state solutions of the trial-complex potential. The study presented in this paper represents a preliminary step towards a full understanding of the role that quaternionic potentials could play in quantum mechanics. Of particular interest for the authors is the analysis of confined wave packets and tunnelling times in this new formulation of quantum theory.

We consider the weak localization correction to the conductance of a ring connected to a network. We analyse the harmonics content of the Al'tshuler–Aronov–Spivak (AAS) oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by AAS. The physical origin of this behaviour is the anomalous diffusion of Brownian trajectories around the ring, due to the diffusion in the wires. We show that this problem is related to the anomalous diffusion along the skeleton of a comb. We study in detail the winding properties of Brownian curves around a ring connected to an arbitrary network. Our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales. A general expression of this length is derived for arbitrary networks. More specifically we consider the case of a ring connected to wires, to a square network and to a Bethe lattice.

There is an error in the proof of lemma 3.4 which can be remedied by using (3.22) and (3.13) directly. The proof of lemma 3.2 can also be simplified. Please see PDF for details.