Table of contents

A new (algebraic) approximation scheme to find global solutions of two-point boundary value problems of ordinary differential equations (ODEs) is presented. The method is applicable for both linear and nonlinear (coupled) ODEs whose solutions are analytic near one of the boundary points. It is based on replacing the original ODEs by a sequence of auxiliary first-order polynomial ODEs with constant coefficients. The coefficients in the auxiliary ODEs are uniquely determined from the local behaviour of the solution in the neighbourhood of one of the boundary points. The problem of obtaining the parameters of the global (connecting) solutions, analytic at one of the boundary points, reduces to find the appropriate zeros of algebraic equations. The power of the method is illustrated by computing the approximate values of the 'connecting parameters' for a number of nonlinear ODEs arising in various problems in field theory. We treat in particular the static and rotationally symmetric global vortex, the skyrmion, the Abrikosov–Nielsen–Olesen vortex, as well as the 't Hooft–Polyakov magnetic monopole. The total energy of the skyrmion and of the monopole is also computed by the new method. We also consider some ODEs coming from the exact renormalization group. The ground-state energy level of the anharmonic oscillator is also computed for arbitrary coupling strengths with good precision.

Two-photon decay rates in simple atoms such as hydrogen-like systems represent rather interesting fundamental problems in atomic physics. The sum of the energies of the two emitted photons has to fulfil an energy conservation condition, the decay takes place via intermediate virtual states, and the total decay rate is obtained after an integration over the energy of one of the emitted photons. Here, we investigate cases with a virtual state having an intermediate energy between the initial and the final states of the decay process, and we show that due to non-uniform convergence, only a careful treatment of the singularities infinitesimally displaced from the photon integration contour leads to consistent and convergent results.

In this paper, we study the topological entanglement of uniform random polygons in a confined space. We derive the formula for the mean squared linking number of such polygons. For a fixed simple closed curve in the confined space, we rigorously show that the linking probability between this curve and a uniform random polygon of n vertices is at least . Our numerical study also indicates that the linking probability between two uniform random polygons (in a confined space), of m and n vertices respectively, is bounded below by . In particular, the linking probability between two uniform random polygons, both of n vertices, is bounded below by .

This paper is devoted to a non-perturbative renormalization group (NPRG) analysis of Model A, which stands as a paradigm for the study of critical dynamics. The NPRG formalism has appeared as a valuable theoretical tool to investigate non-equilibrium critical phenomena, yet the simplest—and nontrivial—models for critical dynamics have never been studied using NPRG techniques. In this paper we focus on Model A taking this opportunity to provide a pedagogical introduction to NPRG methods for dynamical problems in statistical physics. The dynamical exponent z is computed in d = 3 and d = 2 and is found to be in close agreement with results from other methods.

A powerful procedure is presented for calculating the Casimir attraction between plane parallel multilayers made up of homogeneous regions with arbitrary magnetic and dielectric properties by the use of the Minkowski energy–momentum tensor. The theory is applied to numerous geometries and shown to reproduce a number of results obtained by other authors. Although the various pieces of theory drawn upon are well known, the relative ease with which the Casimir force density in even complex planar structures may be calculated, appears not to be widely appreciated, and no single paper to the author's knowledge renders explicitly the procedure demonstrated herein. Results may be seen as an important building block in the settling of issues of fundamental interest, such as the long-standing dispute over the thermal behaviour of the Casimir force or the question of what is the correct stress tensor to apply, a discussion requickened by the newly suggested alternative theory due to Raabe and Welsch.

The phase diagram for the bond-interacting self-avoiding walk is calculated using transfer matrices on finite strips. The model is shown to have a richer phase diagram than the related Θ-point model. In addition to the standard collapse transition, we find a line of high-order phase transitions, which we conjecture to be in the Berezinskii–Kosterlitz–Thouless (BKT) universality class, terminating in a critical end point.

Consider that the coordinates of N points are randomly generated along the edges of a d-dimensional hypercube (random point problem). The probability P^(d,N)_m,n that an arbitrary point is the mth nearest neighbour to its own nth nearest neighbour (Cox probabilities) plays an important role in spatial statistics. Also, it has been useful in the description of physical processes in disordered media. Here we propose a simpler derivation of Cox probabilities, where we stress the role played by the system dimensionality d. In the limit d → ∞, the distances between pair of points become independent (random link model) and closed analytical forms for the neighbourhood probabilities are obtained both for the thermodynamic limit and finite-size system. Breaking the distance symmetry constraint drives us to the random map model, for which the Cox probabilities are obtained for two cases: whether a point is its own nearest neighbour or not.

We prove that the integrable–nonintegrable discrete nonlinear Schrödinger equation (AL-DNLS) introduced by Cai, Bishop and Gronbech-Jensen (Phys. Rev. Lett.72 591(1994)) is the discrete gauge equivalent to an integrable–nonintegrable discrete Heisenberg model from the geometric point of view. Then we study whether the transmission and bifurcation properties of the AL-DNLS equation are preserved under the action of discrete gauge transformations. Our results reveal that the transmission property of the AL-DNLS equation is completely preserved and the bifurcation property is conditionally preserved to those of the integrable–nonintegrable discrete Heisenberg model.

The Drinfeld double structure underlying the Cartan series A_n, B_n, C_n, D_n of simple Lie algebras is discussed. This structure is determined by two disjoint solvable subalgebras matched by a pairing. For the two nilpotent positive and negative root subalgebras the pairing is natural and in the Cartan subalgebra is defined with the help of a central extension of the algebra. A new completely determined basis is found from the compatibility conditions in the double, and a different perspective for quantization is presented. Other related Drinfeld doubles on are also considered.

Solutions of the q-deformed Schrödinger equation are presented for the following potentials: shifted oscillator, isotropic oscillator, Rosen–Morse II, Eckart II, and Poschl–Teller I and II potentials. Various properties of solutions to such equations are discussed including the limit case q → 1 that corresponds to the non-deformed Schrödinger equation.

In this paper, we discuss ordinary differential equations (ODEs) which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.

Analytical formulae for functional differentiation under simultaneous K-conservation constraints, with K the integral of some function of the functional variable, are derived, making the proper account for the simultaneous conservation of normalization and statistical averages, e.g., possible in functional differentiation in nonvariationally built physical theories, which gets particular relevance for nonequilibrium, time-dependent theories.

A Lagrangian is proposed for the quasi-rigid extended charged particle, which consists of a bare point particle term plus the standard electromagnetic minimal coupling. The quasi-rigid motion is imposed as a constraint. The extension of the particle and the quasi-rigid motion appear inside the current density. The Lorentz contraction of the extended particle makes the interaction term dependent on the acceleration. This dependence produces the additional terms in the equations of motion that are necessary for the proper energy and momentum conservation, and that were previously identified as the inertial effects of stress. The momentum of stress is obtained as an explicit function of the electromagnetic field.

A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on spherical averages |B_k0|_av or |∑_kB_k0|_av, k = 2, 4, 6, of the crystal-(ligand)-field Hamiltonian parametrizations, i.e. on the axial crystal-field (CF) parameters modules averaged over all the reference frame orientations. It is proved that they are equal to and , respectively. While the traditional CF strength measure has been established on the parametrization modules, it means on the second moment of the CF energy levels, the new introduced scale employs rather the first moment of the energy modules and reveals a better resolving power. This complementary measure enables us to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent ones. Using both the compatible CF strength scales one may draw more accurate conclusions about the Stark levels arrays and, in particular, their total splitting.

The problem of persistent currents in loops interrupted by Josephson junctions is considered. We prove that persistent currents are related to local minimizers of the Ginzburg Landau energy functional. Although the order parameter is discontinuous at the junction, we show that the local minimizers are related to the homotopy types of the domain. Therefore, persistent currents will occur in any multiply connected domain if the junction strength is weak enough.

In accordance with the fact that quantum measurements are described in terms of positive operator measures (POMs), we consider certain aspects of a quantization scheme in which a classical variable is associated with a unique positive operator measure (POM) E^f, which is not necessarily projection valued. The motivation for such a scheme comes from the well-known fact that due to the noise in a quantum measurement, the resulting outcome distribution is given by a POM and cannot, in general, be described in terms of a traditional observable, a selfadjoint operator. Accordingly, we note that the noiseless measurements are those which are determined by a selfadjoint operator. The POM E^f in our quantization is defined through its moment operators, which are required to be of the form , with Γ being a fixed map from classical variables to Hilbert space operators. In particular, we consider the quantization of classical questions, that is, functions taking only values 0 and 1. We compare two concrete realizations of the map Γ in view of their ability to produce noiseless measurements: one being the Weyl map, and the other defined by using phase space probability distributions.

Introducing asymmetry into the Weyl representation of operators leads to a variety of phase-space representations and new symbols. Specific generalizations of the Husimi and the Glauber–Sudarshan symbols are explicitly derived.

A simple model of self-pulsation in lasers is considered. The laser is described by the system of two ordinary differential equations for the number of photons in the cavity and the number of excitations in the active medium, leading to the equation for the oscillator Toda with damping. For the case of strong spiking, the damping is considered as perturbation; the estimates in terms of elementary functions are suggested for the period of pulsation, damping rate, amplitude and phase of pulsation, quasi-energy and the output power. These estimates are compared to the numerical solution and to the experimental data.

We study the magnetic properties of an electron in a constant magnetic field and confined by a isotropic two-dimensional harmonic oscillator on a space where the coordinates and momenta operators obey generalized commutation relations leading to the appearance of a minimal length. Using the momentum space representation we determine exactly the energy eigenvalues and eigenfunctions. We prove that the usual degeneracy of Landau levels is removed by the presence of the minimal length in the limits of weak and strong magnetic field.The thermodynamical properties of the system, at high temperature, are also investigated showing a new magnetic behaviour in terms of the minimal length.

The pointed weak energy for an arbitrarily evolving quantum state defines a complex valued phase which is the sum of a dynamic phase and a purely geometric phase—the pointed geometric phase. The pointed geometric phase is concisely expressed as a time integral which depends upon the energy uncertainty, the associated evolving state, and its orthogonal companion state. The real part of the pointed geometric phase is to within a sign the geometric phase for arbitrary evolutions defined by Mukunda and Simon and that of Aharonov and Anandan for cyclic evolutions. The imaginary part of the pointed geometric phase governs the survival probability of the initial state. Several general rate of change relationships associated with the real and imaginary parts of the pointed geometric phase are deduced from this concise expression, and it is used to calculate the pointed geometric phase acquired as a spin- particle precesses under the influence of a uniform magnetic field.

We study a weakly local, but nonlocal model in spacetime dimension d ⩾ 2 and prove that it is maximally nonlocal in a certain specific quantitative sense. Nevertheless, depending on the number of dimensions d, it has string-localized or brane-localized operators which commute at spatial distances. In two spacetime dimensions, the model even comprises a covariant and local subnet of operators localized in bounded subsets of Minkowski space which has a nontrivial scattering matrix. The model thus exemplifies the algebraic construction of local operators from algebras associated with nonlocal fields.

We consider critical curves—conformally invariant curves—that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm–) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.

The HMW effect in noncommutative quantum mechanics is studied. By solving the Dirac equations on noncommutative (NC) space and noncommutative phase space, we obtain topological HMW phase on NC space and NC phase space, respectively, where the additional terms related to the space–space and momentum–momentum noncommutativity are given explicitly.

A simple standard equation for the evolution of the electrons and electromagnetic fields in a gyrotron cavity is studied. A number of mathematical properties are shown: existence and uniqueness of solutions for a limited axial extent and existence and uniqueness for all axial lengths in one case of particular interest. A Poynting theorem is obtained directly from the model and the Hamiltonian character of the electron motion is demonstrated. The start-up and final state in the gyrotron cavity are also examined. The efficiency of energy flux transfer from the electron beam to the wave is estimated.

Leslie, in the book Developments in the Theory of Turbulence (1973 Oxford: Clarendon), offered a very simple and intuitively founded model for the case of homogeneous anisotropic turbulence. Here, we offer a rigorous reformulation of Leslie's model leading to a general form for the velocity correlation tensor that satisfies realizability conditions like symmetry in its tensor indices and the condition of solenoidality arising from the incompressibility condition. The anisotropic part of the correlation tensor involves two non-dimensional constants—one arising from the pure strain term and the other from the term that induces distortion in the wave vector space; the rapid pressure term does not contribute. The estimates for these non-dimensional constants yield encouraging results within this framework.