Table of contents

We study a zero-range process with two species of interacting particles. We show that the steady state assumes a simple factorized form, provided the dynamics satisfy certain conditions, which we derive. The steady state exhibits a new mechanism of condensation transition wherein one species induces the condensation of the other. We study this mechanism for a specific choice of dynamics.

In 1941 H Kramers and G Wannier discovered a special symmetry which relates low-temperature and high-temperature phases in the planar Ising model. The corresponding transformation, the Kramers–Wannier transform, is a special non-local substitution in the partition function. The existence of such transformations is a general property of lattice spin systems. Generalization of the KW transform to spin systems with non-Abelian symmetry is essential for many problems in statistical physics and field theory. This problem is very difficult and cannot be carried out by classical methods (like Fourier transform in the commutative case). We present new results which solve this problem for finite non-Abelian groups.

We present an analytical description of the motion in the singular logarithmic potential of the form Φ = ln √x²₁/b² + x²₂, a potential which plays an important role in the modelling of triaxial systems (such as elliptical galaxies) or bars in the centres of galaxy discs. In order to obtain information about the motion near the singularity, we resort to McGehee-type transformations and regularize the vector field. In the axis-symmetric case (b = 1), we offer a complete description of the global dynamics. In the non-axis-symmetric case (b < 1), we prove that all orbits, with the exception of a negligible set, are centrophobic and retrieve numerically partial aspects of the orbital structure.

Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. They also correspond to a class of vertex algebras. An automorphism of a Novikov algebra is a linear isomorphism φ satisfying φ(xy) = φ(x)φ(y) which keeps the algebraic structure. The set of automorphisms of a Novikov algebra is a Lie group whose Lie algebra is just the Novikov algebra's derivation algebra. The theory of automorphisms plays an important role in the study of Novikov algebras. In this paper, we study the automorphisms of Novikov algebras. We get some results on their properties and classification in low dimensions. These results are fundamental in a certain sense, and they will serve as a guide for further development. Moreover, we apply these results to classify Gel'fand–Dorfman bialgebras and Novikov–Poisson algebras. These results also can be used to study certain phase spaces and geometric classical r-matrices.

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature (Morozov A 1992 Mod. Phys. Lett. A 7 3503, Shatashvili S L 1993 Commun. Math. Phys.154 421): we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated bi-orthogonal polynomials. The generating function of these correlations then becomes a determinant involving the recursion coefficients of the bi-orthogonal polynomials.

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

The purpose of this paper is to show the advantages that represent the use of numerical methods that preserve invariant quantities in the study of solitary wave interactions for the regularized long wave equation. It is shown that the so-called conservative methods are more appropriate to study the phenomenon and provide a dynamic point of view that allows us to estimate the changes in the parameters of the solitary waves after the collision.

We show that many integrals containing products of confluent hypergeometric functions follow directly from one single integral that has a very simple formula in terms of Appell's double series F₂. We present some techniques for computing such series. Applications requiring the matrix elements of singular potentials and the perturbed Kratzer potential are presented.

The non-Hermitian matrix Schrödinger equation obtained by the Bäcklund–Darboux transformation (BDT) is treated. The potentials, fundamental solutions and Weyl functions are constructed explicitly. A symmetric reduction of the BDT is introduced and this case is studied in greater detail, including potentials, fundamental solutions, bound states, the reality of the discrete spectrum and spontaneous break of the symmetry, the sign-indefinite scalar product, and examples.

We construct a kind of new multipartite entangled states of continuum variables, which are related to unitary group U(n). Using the technique of integral within an ordered product of operators we prove that such states make up a complete representation in multimode Fock space. The new state can be generated by an optical network whose operation on an incoming photon distributes the photon among the outputs according to the unitary group transform. The potential use of the new state in quantum teleportation is briefly discussed.

Exactly solvable model of the linear singular oscillator in the relativistic configurational space is considered. We have found wavefunctions and energy spectrum for the model under study. It is shown that they have the correct non-relativistic limits.

From the partial differential Calogero's (three-body) and Smorodinsky–Winternitz (superintegrable) Hamiltonians in two variables we separate the respective angular Schrödinger equations and study the possibilities of their 'minimal' symmetric complexification. The simultaneous loss of the Hermiticity and solvability of the respective angular potentials V(φ) is compensated by their replacement by solvable, purely imaginary and piece-wise constant multiple wells V₀(φ). We demonstrate that the spectrum remains real and that it exhibits a rich 'four series' structure in the double-well case.

We discuss Coleman's theorem concerning the energy density of the ground state of the sine-Gordon model proved in Coleman S (1975 Phys. Rev. D 11 2088). According to this theorem the energy density of the ground state of the sine-Gordon model should be unbounded from below for coupling constants β² > 8π. The consequence of this theorem would be the non-existence of the quantum ground state of the sine-Gordon model for β² > 8π. We show that the energy density of the ground state in the sine-Gordon model is bounded from below even for β² > 8π. This result is discussed in relation to Coleman's theorem (Coleman S 1973 Commun. Math. Phys.31 259), particle mass spectra and soliton–soliton scattering in the sine-Gordon model.