Table of contents

A sequence of entanglement swapping of continuous variables is considered. It is classified into one-way entanglement swapping and two-way entanglement swapping, where the former (the latter) uses one-way (two-way) classical communication. When resources of quantum entanglement are bipartite Gaussian states, it is shown that the one-way entanglement swapping is superior to the two-way entanglement swapping. This means that although the entanglement swapping is performed the same number of times, there is a case that the one-way entanglement swapping can yield an entangled state while the two-way entanglement swapping cannot.

Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep.339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

We derive a novel multiple integral representation for a generating function of the σ^z–σ^z correlation functions of the spin-XXZ chain at finite temperature and finite, longitudinal magnetic field. Our work combines algebraic Bethe ansatz techniques for the calculation of matrix elements with the quantum transfer matrix approach to thermodynamics.

We study extremely diluted spin models of neural networks in which the connectivity evolves in time, although adiabatically slowly compared to the neurons, according to stochastic equations which on average aim to reduce frustration. The (fast) neurons and (slow) connectivity variables equilibrate separately, but at different temperatures. Our model is exactly solvable in equilibrium. We obtain phase diagrams upon making the condensed ansatz (i.e. recall of one pattern). These show that, as the connectivity temperature is lowered, the volume of the retrieval phase diverges and the fraction of mis-aligned spins is reduced. Still one always retains a region in the retrieval phase where recall states other than the one corresponding to the 'condensed' pattern are locally stable, so the associative memory character of our model is preserved.

We study the spectral properties of a charged particle confined to a two-dimensional plane and submitted to homogeneous magnetic and electric fields and an impurity potential V. We use the method of complex translations to prove that the lifetimes of resonances induced by the presence of electric field are at least Gaussian as long as the electric field tends to zero.

These notes explore the consequences of simple representation theory of for the expressions for Padé approximants and discuss the role of a generalized Hirota derivative therein.

We study operator pencils on generators of the Lie algebras sl₂ and the oscillator algebra. These pencils are linear in a spectral parameter λ. The corresponding generalized eigenvalue problem gives rise to some sets of orthogonal polynomials and Laurent biorthogonal polynomials (LBP) expressed in terms of the Gauss ₂F₁ and degenerate ₁F₁ hypergeometric functions. For special choices of the parameters of the pencils, we identify the resulting polynomials with the Hendriksen–van Rossum LBP which are widely believed to be the biorthogonal analogues of the classical orthogonal polynomials. This places these examples under the umbrella of the generalized bispectral problem which is considered here. Other (non-bispectral) cases give rise to some 'nonclassical' orthogonal polynomials including Tricomi–Carlitz and random-walk polynomials. An application to solutions of relativistic Toda chain is considered.

We discuss the algebraic and analytic structure of rational Lax operators. With algebraic reductions of Lax equations we associate a reduction group—a group of automorphisms of the corresponding infinite-dimensional Lie algebra. We present a complete study of dihedral reductions for Lax operators with simple poles and corresponding integrable equations. In the last section we give three examples of dihedral reductions for Lax operators.

We calculate the first integrals of the Kepler problem by the method of Jacobi's last multiplier using the symmetries for the equations of motion. Also we provide another example which shows that Jacobi's last multiplier together with Lie symmetries unveils many first integrals neither necessarily algebraic nor rational whereas other published methods may yield just one.

Using special quasigraded Lie algebras we obtain new hierarchies of integrable nonlinear vector equations admitting zero-curvature representations. Among them the most interesting is an extension of the generalized Landau–Lifshitz hierarchy called the 'doubled' generalized Landau–Lifshitz hierarchy. This hierarchy can also be interpreted as an anisotropic vector generalization of 'modified' sine–Gordon hierarchy or as a very special vector generalization of so(3) anisotropic chiral field hierarchy.

We modify the J-matrix method for scattering to improve its convergence and reduce the computational cost. Our method applies to the oscillator basis J-matrix method. We distinguish three regions in the space of wavefunction coefficients. In the asymptotic region the free-space boundary conditions hold. In the far interaction region, semi-classical approximations to the matrix elements reduce the Schrödinger equation to an inhomogeneous three-term recurrence relation, and in the near-interaction region one has the full Schrödinger matrix equation. We apply the modified J-matrix method to scattering off a Yukawa potential. The examples show that the number of matrix elements that need to be calculated is significantly smaller than that for the J-matrix method.

An energy eigenproblem for a relativistic N-electron system confined to the interior of a finite volume is considered. The confinement is modelled by imposing a local impedance boundary condition at a hypersurface enclosing the hypervolume in the configuration space. It is shown that energy eigenvalues are non-increasing functions of the hypersurface impedance. Variational principles for energy eigenvalues, admitting the use of trial functions which do not obey the boundary condition imposed on exact eigenfunctions, are constructed in a systematic manner. The Dirac–Hartree–Fock method is applied to derive integro-differential equations and local boundary conditions satisfied by one-electron spin orbitals from which the best determinantal approximations to exact eigenfunctions are built. It is proved that the Dirac–Hartree–Fock estimates of exact energy eigenvalues are also non-increasing functions of the hypersurface impedance.

The evolution of both quantum and classical ensembles may be described via the probability density P on configuration space, its canonical conjugate S, and an ensemble Hamiltonian . For quantum ensembles this evolution is, of course, equivalent to the Schrödinger equation for the wavefunction, which is linear. However, quite simple constraints on the canonical fields P and S correspond to nonlinear constraints on the wavefunction. Such constraints act to prevent certain superpositions of wavefunctions from being realized, leading to superselection-type rules. Examples leading to superselection for energy, spin direction and 'classicality' are given. The canonical formulation of the equations of motion, in terms of a probability density and its conjugate, provides a universal language for describing classical and quantum ensembles on both continuous and discrete configuration spaces, and is briefly reviewed in an appendix.

Two damped coupled oscillators have been used to demonstrate the occurrence of exceptional points in a purely classical system. The implementation was achieved with electronic circuits in the kHz-range. The experimental results perfectly match the mathematical predictions at the exceptional point. A discussion about the universal occurrence of exceptional points—connecting dissipation with spatial orientation—concludes this paper.

The authors would like to correct two errors. Please see pdf for details.