Table of contents

A long-term aim of density functional theory is to obtain a differential equation for the ground-state electron density ρ(r) in a closed-shell atom, the simplest example being He. Since He remains intractable analytically, artificial atoms with harmonic repulsive potential energy u(r₁₂) have therefore been studied. Here we exploit recent work on ρ(r) for such a two-electron system, with u(r₁₂) = λ/r²₁₂, to construct a second-order linear differential equation for ρ(r). This is compared and contrasted with available results for different choices of u(r₁₂).

We prove that for a Hamiltonian system on a cotangent bundle that is Liouville-integrable and has monodromy the vector of Maslov indices is an eigenvector of the monodromy matrix with eigenvalue 1. As a corollary, the resulting restrictions on the monodromy matrix are derived.

The ideal anti-Zeno effect means that a perpetual observation leads to an immediate disappearance of the unstable system. We present a straightforward way to derive sufficient conditions under which such a situation occurs expressed in terms of the decaying states and spectral properties of the Hamiltonian. They show, in particular, that the gap between Zeno and anti-Zeno effects is in fact very narrow.

A simple one-dimensional model is introduced describing a two particle 'atom' approaching a point at which the interaction between the particles is lost. The wavefunction is obtained analytically and analysed to display the entangled nature of the subsequent state.

In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange–Poincaré (Hamilton–Poincaré) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincaré (Hamilton–Poincaré) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.

In this paper, we present a model describing the time evolution of two-dimensional surface waves in gravity and infinite depth. The model of six interacting modes derives from the normal form of the system describing the dynamics of surface waves and is governed by a Hamiltonian system of equations of cubic order in the amplitudes of the waves. We derive a Hamiltonian system with two degrees of freedom from this Hamiltonian using conserved quantities. The interactions are those of two coupled Benjamin–Feir resonances. The temporal evolution of the amplitude of the different modes is described according to the parameters of the system. In particular, we study the energy exchange produced by the modulations of the amplitudes of the modes. The evolution of the modes reveals a chaotic dynamics.

The decay behaviour of radial distribution functions for large distances r is investigated for classical Coulomb fluids where the ions interact with an r⁻⁶ potential (e.g. a dispersion interaction) in addition to the Coulombic and the short-range repulsive potentials (e.g. a hard core). The pair distributions and the density–density (NN), charge–density (QN) and charge–charge (QQ) correlation functions are investigated analytically and by Monte Carlo simulations. It is found that the NN correlation function ultimately decays like r⁻⁶ for large r, just as it does for fluids of electroneutral particles interacting with an r⁻⁶ potential. The prefactor is proportional to the squared compressibility in both cases. The QN correlations decay in general like r⁻⁸ and the QQ correlations like r⁻¹⁰ in the ionic fluid. The average charge density around an ion decays generally like r⁻⁸ and the average electrostatic potential like r⁻⁶. This behaviour is in stark contrast to the decay behaviour for classical Coulomb fluids in the absence of the r⁻⁶ potential, where all these functions decay exponentially for large r. The power-law decays are, however, the same as for quantum Coulomb fluids. This indicates that the inclusion of the dispersion interaction as an effective r⁻⁶ interaction potential in classical systems yields the same decay behaviour for the pair correlations as in quantum ionic systems. An exceptional case is the completely symmetric binary electrolyte for which only the NN correlation has a power-law decay but not the QQ correlations. These features are shown by an analysis of the bridge function.

In a recent work, N = 2 supersymmetry has been proposed as a tool for the analysis of itinerant, correlated fermions on a lattice. In this paper, we extend these considerations to the case of lattice fermions with spin 1/2. We introduce a model for correlated spin-1/2 fermions with a manifest N = 4 supersymmetry, and analyse its properties. The supersymmetric ground states that we find represent holes in an anti-ferromagnetic background.

Coarsening and persistence of Ising spins on a ladder is examined under voter dynamics. The density of domain walls decreases algebraically with time as t^−1/2 for sequential as well as parallel dynamics. The persistence probability decreases as under sequential dynamics and as under parallel dynamics where θ_p = 2θ_s ≈ 0.88. Numerical values of the exponents are explained. The results are compared with the voter model on one- and two-dimensional lattices, as well as Ising model on a ladder under zero-temperature Glauber dynamics.

A well-known ansatz ('trace method') for soliton solutions turns the equations of the (non-commutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the non-commutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the non-commutative KP hierarchy. Relations with Rota–Baxter algebras are established.

Ladder operators and a triangular relation are used to derive the five-dimensional surface harmonics with definite angular momentum, as used in studies of the dynamics of a quadrupole shape in the nuclear collective model. A new basis is used which leads to solutions in terms of associated Legendre functions. The role of the Octahedral symmetry group and the limit of large quantum numbers are discussed.

The paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and Hermitian matrices due to an arbitrary complex perturbation near a diabolic point. General asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived. As a physical application, singularities of the surfaces of refractive indices in crystal optics are studied.

Symmetry properties of the Green function in magnetic multilayers with non-collinear magnetization of the layers are investigated on the basis of the transfer-matrix method. The Green function symmetric with respect to permutation of its arguments is constructed. It is shown how the boundary conditions can be imposed on this Green function.

For a system of multiphoton Jaynes–Cummings model, which represents the interaction of a two-level atom with the radiation field, we study the connection between the atomic inversion, in particular the occurrence of revival–collapse phenomenon (RCP), and the higher-order fluctuation factors. We assume that the atom and the field are initially prepared in the excited and the k-photon coherent states, respectively. We show that there is a class of states for which the higher-order fluctuation factor can provide RCP similar to that involved in the corresponding atomic inversion. Moreover, for initial coherent light we prove that the higher-order fluctuation factor of the three-photon transition can provide RCP similar to that occurring in the atomic inversion of the one-photon transition. As an example we discuss the fourth-order fluctuation in detail.

By considering uncommon factors as spacetime events that influence the spin orientations in the EPRB thought experiment, it is intended to show that one can still introduce the correlation functions. These uncommon factors are positioned inside the common lightcone of two particles. Then, Bell inequalities are proved with the preassumptions of local realism and spin conservation law in the context of a new scenario of hidden variables.

We propose an algorithm which proves a given bipartite quantum state to be separable in a finite number of steps. Our approach is based on the search for a decomposition via a countable subset of product states, which is dense within all product states. Performing our algorithm simultaneously with the algorithm by Doherty, Parrilo and Spedalieri (which proves a quantum state to be entangled in a finite number of steps) leads to a two-way algorithm that terminates for any input state. Only for a set of arbitrary small measure near the border between separable and entangled states is the result inconclusive.

How can we relate the constraint structure and constraint dynamics of the general gauge theory in the Hamiltonian formulation to specific features of the theory in the Lagrangian formulation, especially relate the constraint structure to the gauge transformation structure of the Lagrangian action? How can we construct the general expression for the gauge charge if the constraint structure in the Hamiltonian formulation is known? Whether we can identify the physical functions defined as commuting with first-class constraints in the Hamiltonian formulation and the physical functions defined as gauge invariant functions in the Lagrangian formulation? The aim of the present paper is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions. To fulfil such a programme, we demonstrate the existence of the so-called superspecial phase-space variables in terms of which the quadratic Hamiltonian action takes a simple canonical form. On the basis of such a representation, we analyse a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analysing a symmetry equation. We then use these results to identify the two definitions of physical functions and thus prove the Dirac conjecture.