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.

It is a common belief that nonlinearizable PDEs in (1 + 1) dimensions cannot possess two mutually inverse positive-order recursion operators and that the negative hierarchies for such PDEs, unlike the positive ones, contain at most a finite number of local symmetries. We show that the equation , a generalization of the Hunter–Saxton equation considered by Manna and Neveu, provides a counterexample for both of these assertions. Namely, we find two positive-order integro-differential recursion operators for this equation and show that the corresponding positive and negative hierarchies consist solely of local symmetries. The recursion operators in question turn out to be mutually inverse on symmetries of the equation under study.

The Davis–Fulling model (Fulling and Davies 1976 Proc. R. Soc. Lond. A 348 393; Davies and Fulling 1977 Proc. R. Soc. Lond. A 356 237) is studied in the case of a perfect mirror starting from rest, accelerating for a large but finite time T along the trajectory z(t) = −ln cosh(t), and after time T moving with constant velocity. In this situation (a mirror with an asymptotically inertial trajectory), the 'in' and 'out' states are well defined and thus the average number of produced particles can be calculated using the Bogolubov coefficients. In this letter, we compute rigorously the Bogolubov coefficient , and we prove that the black-body spectrum is obtained in the case 1 ∼ ω ≪ ω' ≪ T. The methods used by other authors to obtain the black-body spectrum are also discussed. Finally, we prove that the number of produced particles in the ω mode per unit time is

In general, quantum field theories (QFT) require regularizations and infinite renormalizations due to ultraviolet divergences in their loop calculations. Furthermore, perturbation series in theories like quantum electrodynamics are not convergent series, but are asymptotic series. We apply neutrix calculus, developed in connection with asymptotic series and divergent integrals, to QFT, obtaining finite renormalizations. While none of the physically measurable results in renormalizable QFT is changed, quantum gravity is rendered more manageable in the neutrix framework.

In the past few years, systems with slow dynamics have attracted considerable theoretical and experimental interest. Ageing phenomena are observed during this everlasting non-equilibrium evolution. A simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high-temperature phase to the critical point. The aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case. Particular emphasis is put on the field-theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities. Fluctuation–dissipation relations are discussed and in particular the concept of fluctuation–dissipation ratio (FDR) is reviewed, emphasizing its connection with the definition of a possible effective temperature. The renormalization-group approach to critical dynamics is summarized and the scaling forms of the time-dependent non-equilibrium correlation and response functions of a generic observable are discussed. From them, the universality of the associated FDR follows as an amplitude ratio. It is then possible to provide predictions for ageing quantities in a variety of different models. In particular, the results for models A, B and C dynamics of the O(N) Ginzburg–Landau Hamiltonian, and model A dynamics of the weakly dilute Ising magnet and of the φ³ theory are reviewed and compared with the available numerical results and exact solutions. The effect of a planar surface on the ageing behaviour of model A dynamics is also addressed within the mean-field approximation.

We consider the phase ordering process of a system quenched into a lamellar phase in the presence of a shear flow. By studying the continuum model based on the Brazowskii free energy in a self-consistent approximation, we analyse the effects of a weak anisotropy in the quartic coupling constant, finding that it radically changes the evolution of the system.

With a view to having further insight into the mathematical content of the non-Hermitian Hamiltonian associated with the diffusion–reaction (D–R) equation in one dimension, we investigate (a) the solitary wave solutions of certain types of its nonlinear versions, and (b) the problem of real eigenvalue spectrum associated with its linear version or with this class of non-Hermitian Hamiltonians. For case (a) we use the standard techniques to handle the quadratic and cubic nonlinearities in the D–R equation whereas for case (b) a newly proposed method, based on an extended complex phase space, is employed. For a particular class of solutions, an Ermakov system of equations is also found for the linear case. Further, corresponding to the 'classical' version of the above one-dimensional complex Hamiltonian, an equivalent integrable system of two, two-dimensional, real Hamiltonians is suggested.

Hopf-algebra quantizations of four-dimensional and six-dimensional real classical Drinfel'd doubles are studied by following a direct 'analytic' approach, and the full quantization is explicitly obtained for most of them. Several new four- and six-dimensional quantum algebras are presented and some general features of the method are emphasized.

When computing multicentre integrals over Slater-type orbitals (STOs) by means of the Shavitt and Karplus Gaussian integral transforms (Shavitt and Karplus 1962 J. Chem. Phys.36 550), one usually ends up with a multiple integral of the form (Shavitt and Karplus 1965 J. Chem. Phys.43 398) in which all the integrals are inter-related. The most widely used approach for computing such an integral is to apply a product of Gauss–Legendre quadratures for the integrals over [0, 1] while the semi-infinite term is evaluated by a special procedure. Although numerous approaches have been developed to accurately perform the integration over [0, ∞) efficiently, it is the aim of this work to add a new tool that could be of some benefit in carrying out the hard task of multicentre integrals over STOs. The new approach relies on a special Gauss quadrature referred to as Gauss–Bessel to accurately evaluate the semi-infinite integral of interest. In this work, emphasis is put on accuracy rather than efficiency since its aim is essentially to bring a proof of concept showing that Gauss–Bessel quadrature can successfully be applied in the context of multicentre integrals over STOs. The obtained accuracy is comparable to that obtained with other methods available in the literature.

We review the Johnson–Moser rotation number and the K₀-theoretical gap labelling of Bellissard for one-dimensional Schrödinger operators. We compare them with two further gap labels, one being related to the motion of Dirichlet eigenvalues, the other being a K₁-theoretical gap label. We argue that the latter provides a natural generalization of the Johnson–Moser rotation number to higher dimensions.

We consider the real stationary two-dimensional Schrödinger equation. With the aid of any of its particular solutions, we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schrödinger equation and the imaginary parts are solutions of an associated Schrödinger equation with a potential having the form of a potential obtained after the Darboux transformation. Using Bers' theory of Taylor series for pseudoanalytic functions, we obtain a locally complete system of solutions of the original Schrödinger equation which can be constructed explicitly for an ample class of Schrödinger equations. For example it is possible when the potential is a function of one Cartesian, spherical, parabolic or elliptic variable. We give some examples of application of the proposed procedure for obtaining a locally complete system of solutions of the Schrödinger equation. The procedure is algorithmically simple and can be implemented with the aid of a computer system of symbolic or numerical calculation.

A new class of discrete dynamical systems is introduced via a duality relation for discrete dynamical systems with a number of explicitly known integrals. The dual equation can be defined via the difference of an arbitrary linear combination of integrals and its upshifted version. We give an example of an integrable mapping with two parameters and four integrals leading to a (four-dimensional) dual mapping with four parameters and two integrals. We also consider a more general class of higher-dimensional mappings arising via a travelling-wave reduction from the (integrable) MKdV partial-difference equation. By differencing the trace of the monodromy matrix we obtain a class of novel dual mappings which is shown to be integrable as level-set-dependent versions of the original ones.

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way, all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson–Lie groups are obtained.

A systematic method which is based on the classical Lie group reduction is used to find the novel exact solution of the nonlinear Schrödinger equation (NLS) with distributed dispersion, nonlinearity and gain or loss. We study the transformations between the standard NLS equation and the NLS equations with distributed dispersion, nonlinearity and gain or loss. Appropriate solitary wave solutions can be applied to discuss soliton propagation in optical fibres, and the amplification and compression of pulses in optical fibre amplifiers.

The method for the determination of the position of the pair of complex conjugate branch points suggested in previous studies is generalized here. The method is modified in order to consider cases where the value of the function at the singularity is not real. A method is proposed for the determination of single isolated singularities located either on the real axis or in the complex plane. These methods are applied to three eigenvalue problems, namely the bounded delta-potential atom, the Mathieu equation and the hydrogen atom in a spherically symmetric cavity. We show that the position of the singularities can be obtained very accurately with minimal number of perturbation coefficients. If we take the characteristic polynomial for variational energy levels as an approximate implicit equation, the method can be used for the investigation of the analytic structure of the energy considered as a function of complex coupling constant. In particular, we show that the first singularity appears at the point of intersection of the ground and the first excited states. The second singularity, when the first and second excited states intersect, can be determined either from the expansion at the first singularity or from the expansion of the second excited state at the origin.

The high-order behaviour of the perturbation expansion in the cubic replica field theory of spin glasses in the paramagnetic phase has been investigated. The study starts with the zero-dimensional version of the replica field theory and this is shown to be equivalent to the problem of finding finite-size corrections in a modified spherical spin glass near the critical temperature. We find that the high-order behaviour of the perturbation series is described, to leading order, by coefficients of alternating signs (suggesting that the cubic field theory is well defined) but that there are also subdominant terms with a complicated dependence of their sign on the order. Our results are then extended to the d-dimensional field theory and in particular used to determine the high-order behaviour of the terms in the expansion of the critical exponents in a power series in = 6 − d. We have also corrected errors in the existing expansions at third order.