Table of contents

We calculate the vacuum polarization-induced ellipticity acquired by a linearly polarized laser beam of angular frequency on traversing a region containing a transverse magnetic field rotating with a small angular velocity Ω around the beam axis. The transmitted beam contains the fundamental frequency and weak sidebands of frequency , but no other sidebands. To first order in small quantities, the ellipticity acquired by the transmitted beam is independent of Ω, and is the same as would be calculated in the approximation of regarding the magnetic field as fixed at its instantaneous angular orientation, using the standard vacuum birefringence formulae for a static magnetic field. Also to first order, there is no rotation of the polarization plane of the transmitted beam. Analogous statements hold when the magnetic field strength is slowly varying in time.

The Korteweg-de Vries equation u_t + uu_x + u_xxx = 0 is symmetric (invariant under spacetime reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the symmetry. The result is the family of complex nonlinear wave equations u_t − iu(iu_x) + u_xxx = 0, where is real. The features of these equations are discussed. Special attention is given to the = 3 equation, for which conservation laws are derived and solitary waves are investigated.

We investigate random, discrete Schrödinger operators which arise naturally in the theory of random matrices, and depend parametrically on Dyson's Coulomb gas inverse temperature β. They are similar to the class of 'critical' random Schrödinger operators with random potentials which diminish as . We show that as a function of β they undergo a transition from a regime of (power-law) localized eigenstates with a pure point spectrum for β < 2 to a regime of extended states with a singular continuous spectrum for β ⩾ 2.

A new class of semi-analytically solvable MHD α²-dynamos is found based on a global diagonalization of the matrix part of the dynamo differential operator. Close parallels to SUSY QM are used to relate these models to the Dirac equation and to extract non-numerical information about the dynamo spectrum.

For large clause-to-variable ratios, typical K-SAT instances drawn from the uniform distribution have no solution. We argue, based on statistical mechanics calculations using the replica and cavity methods, that rare satisfiable instances from the uniform distribution are very similar to typical instances drawn from the so-called planted distribution, where instances are chosen uniformly between the ones that admit a given solution. It then follows, from a recent article by Feige, Mossel and Vilenchik (2006 Complete convergence of message passing algorithms for some satisfiability problems Proc. Random 2006 pp 339–50), that these rare instances can be easily recognized (in O(log N) time and with probability close to 1) by a simple message-passing algorithm.

We study a probabilistic cellular automaton to describe two population biology problems: the threshold of species coexistence in a predator–prey system and the spreading of an epidemic in a population. By carrying out mean-field approximations and numerical simulations we obtain the phase boundaries (thresholds) related to the transition between an active state, where prey and predators present a stable coexistence, and a prey absorbing state. The numerical estimates for the critical exponents show that the transition belongs to the directed percolation universality class. In the limit where the cellular automaton maps into a model for the spreading of an epidemic with immunization we observe a crossover from directed percolation class to the dynamic percolation class. Patterns of growing clusters related to species coexistence and spreading of epidemic are shown and discussed.

Using an environmentally friendly renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, y = f(x), that exhibits all desired asymptotic and analyticity properties in the three limits x → 0, x → ∞ and x → −1. The only necessary inputs are the Wilson functions γ_λ, γ_φ and , associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson–Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N = 1, we derive a one-loop equation of state for 2 < d < 4 naturally parameterized by a ratio of nonlinear scaling fields. For d = 3 we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.

We study the problem of a Brownian particle diffusing in finite dimensions in a potential given by ψ = ϕ²/2 where ϕ is Gaussian random field. Exact results for the diffusion constant in the high temperature phase are given in one and two dimensions and it is shown to vanish in a power-law fashion at the dynamical transition temperature. Our results are confronted with numerical simulations where the Gaussian field is constructed, in a standard way, as a sum over random Fourier modes. We show that when the number of Fourier modes is finite the low temperature diffusion constant becomes non-zero and has an Arrhenius form. Thus we have a simple model with a fully understood finite size scaling theory for the dynamical transition. In addition we analyse the nature of the anomalous diffusion in the low temperature regime and show that the anomalous exponent agrees with that predicted by a trap model.

We consider quantum systems with a chaotic classical limit that depends on an external parameter, and study correlations between the spectra at different parameter values. In particular, we consider the parametric spectral form factor K(τ, x) which depends on a scaled parameter difference x. For parameter variations that do not change the symmetry of the system we show by using semiclassical periodic orbit expansions that the small τ expansion of the form factor agrees with random matrix theory for systems with and without time reversal symmetry.

The existence and properties of coherent pattern in the multisoliton solutions of the dKP equation over a finite field are investigated. To that end, starting with an algebro-geometric construction over a finite field, we derive a 'travelling wave' formula for N-soliton solutions in a finite field. However, despite it having a form similar to its analogue in the complex field case, the finite-field solutions produce patterns essentially different from those of classical interacting solitons.

The Darboux-dressing transformations are applied to the Lax pair associated with systems of coupled nonlinear wave equations in the case of boundary values which are appropriate to both 'bright' and 'dark' soliton solutions. The general formalism is set up and the relevant equations are explicitly solved. Several instances of multicomponent wave equations of applicative interest, such as vector nonlinear Schrödinger-type equations and three resonant wave equations, are considered.

This paper reports on a study of the motion of moving space curves induced from integrable equations having variable spectral parameters. We explain the geometric structure of the curves using the associated linear equations of integrable equations. A generalized form of the Hasimoto transformation is introduced. This form relates the curvature and torsion of the curves to the variables of integrable equations. Some explicit curve motions, including the solitonic and Kelvin-type curves, are calculated using the Sym–Tafel formula.

The Green's function for the Helmholtz differential operator ∇² + λ(λ + N − 1) on the N-dimensional (with N ⩾ 1) hyperspherical surface of unit radius is investigated. Its closed form is shown to be where S_N is the area of , C^(α)_λ(x) is the Gegenbauer function of the first kind, while n and n' are radius vectors, with respect to the centre of , of the observation and source points, respectively. The Green's function G^(N)(λ; n, n') fails to exist whenever λ is such that it holds that λ(λ + N − 1) = L(L + N − 1), with . For these exceptional cases, the generalized (known also as 'modified' or 'reduced') Green's function is considered. It is shown that may be expressed compactly in terms of the Gegenbauer polynomial C^((N−1)/2)_L(n⋅n') and the derivative [∂C^((N−1)/2)_λ(−n⋅n')/∂λ]_λ=L. Explicit expressions for the derivatives [∂C⁽ⁿ⁾_λ(x)/∂λ]_λ=L and [∂C^(n+1/2)_λ(x)/∂λ]_λ=L, with , are found and used to transform the functions and to potentially more useful forms.

For a single particle of mass m experiencing the potential −α/|x|, the 1D Klein–Gordon equation is mathematically underdefined even when α ≪ 1: unique solutions require some physically motivated prescription for handling the singularity at the origin. The procedure appropriate in most cases is to soften the singularity by means of a cutoff. Here we study the bound states of spin-zero particles in the potential −α/(|x| + R), extending the nonrelativistic results of Loudon (1959 Am. J. Phys.27 649) to allow for relativistic effects, which become appreciable and eventually dominant for small enough mR: they are totally different from conclusions based hitherto on mathematically simple-seeming matching conditions on the wavefunction at x = 0. For realizable R, all relativistic effects remain very small; but with mR decreasing to order α² the ground-state energy E decreases through zero, and soon after that mR reaches a finite critical value below which E becomes complex, signalling a breakdown of the single-particle theory. At this critical point of the curve E(mR) the Klein–Gordon norm changes sign: the curve has a lower branch describing a bound antiparticle state, with positive energy −E, which exists for mR between the critical and some higher value where E reaches −m. Though apparently unanticipated in this context, similar scenarios are in fact familiar for strong short-range potentials (1D or 3D), and also for strong 3D Coulomb potentials with α of order unity.

This paper deals with different ways to extract the effective two-dimensional lower level dynamics of a lambda system excited by off-resonant laser beams. We present a commonly used procedure for elimination of the upper level, and we show that it may lead to ambiguous results. To overcome this problem and better understand the applicability conditions of this scheme, we review two rigorous methods which allow us both to derive an unambiguous effective two-level Hamiltonian of the system and to quantify the accuracy of the approximation achieved: the first relies on the exact solution of the Schrödinger equation, while the second resorts to the Green's function formalism and the Feshbach projection operator technique.

We present an analysis of the two-dimensional Schrödinger equation for two electrons interacting via Coulombic force and confined in an anisotropic harmonic potential. The separable case of ω_y:ω_x = 2 is studied particularly carefully. The closed-form expressions for bound-state energies and the corresponding eigenfunctions are found at some particular values of ω_x. For highly accurate determination of energy levels at other values of ω_x, we apply an efficient scheme based on the Fröbenius expansion.

The analytic expression of the time evolution wavefunction of the two-dimensional harmonic oscillator with time-dependent mass and frequency in a static magnetic field is obtained using an operator-algebraic method slightly different from the usual Lie algebraic technique. The evolution operator of the one-dimensional harmonic oscillator with time-dependent mass and frequency is established first by forming an operator differential equation with the su(1, 1) Lie algebra, which is deduced from the time-dependent linear unitary transformation for boson operators (a, a†), and then by comparing this operator equation with the time evolution equation of the one-dimensional oscillator.

An algebraic approach based on the multimode two-photon Lie algebra and its corresponding Lie group is followed to derive a formal solution to the time-dependent Schrödinger equation. This solution is written as an expansion series whose leading term corresponds to the thawed Gaussian approximation (TGA). Our scheme provides the most general expression reported so far for this approximation. By using the coherent state representation of the formal solution, the correction term to the TGA is analysed in the zero ℏ asymptotic limit. The error is generally found not to vanish in this semiclassical limit. The same approach is followed to study the remainder to the TGA initial value representation (IVR) of the quantum propagator. This correction is found not to vanish either in the zero ℏ limit. Hence, the TGA IVR would not be the correct semiclassical asymptotic form of the quantum propagator. The origin of this behaviour is shown to be in the existence of contributions from unphysical saddle points in the semiclassical limit. These would unveil an incorrect analytic structure of the TGA IVR propagator in that limit.

We calculate the first-order energy shifts for the N-dimensional hydrogen atom exposed to a static electric field. The results are compared with numerical diagonalization of the Hamiltonian in a finite basis. Using simple scaling relations, we show how corrections to arbitrarily high order may be obtained from known results for the three-dimensional Coulomb problem.

We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. As a new example we find that while the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in four spacetime dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations.

We construct Lagrangians for non-relativistic massive fields with arbitrary spin. We use a Bargmann–Wigner construction, together with a Galilean covariant approach based on the reduction from an extended (4, 1) Minkowski manifold to the Galilean (3, 1) spacetime. Fierz identities are developed within this framework. By using symmetric spinor fields of rank 2 and rank 3, we can avoid the difficulty arising from the introduction of the minimal electromagnetic interaction in the Bargmann–Wigner wave equations. For fields with spin S, the minimal electromagnetic coupling thereby leads to the gyromagnetic ratio g_S = 1/S.

We present the perturbative Yangian symmetry at next-to-leading order in the sector of planar SYM. Just like the ordinary symmetry generators, the bi-local Yangian charges receive corrections acting on several neighbouring sites. We confirm that the bi-local Yangian charges satisfy the necessary conditions: they transform in the adjoint of , they commute with the dilatation generator and they satisfy the Serre relations. This proves that the sector is integrable at two loops.

This paper develops an efficient method for simulation of breakdown in a gas, which explicitly makes use of the real underlying physics, in a case where standard numerical schemes are likely to fail. We develop a 'time-dependent capacitor model' (TDCM) for 2D or 3D, which ensures that the ionization rate is consistent with energy conservation and which disallows almost all numerical diffusion (and hence allows larger (Δx, Δt)). To avoid spurious ionization in the TDCM, density is only added in a cell when the density and electric field are high enough so that the density could physically grow to the expected final density within the cell. Numerical diffusion is negligible in the TDCM, in part because we only inject density into cells/capacitors when enough time has elapsed for density to be physically present. The direction of injection is controlled, so if, for example, density from cell [i, j] in reality moves to cell [i ± 1, j ± 1], it goes there directly, giving a physically correct direction of propagation. A simple scheme for accelerating convergence, exploiting the very different time scales which arise, is also discussed.

Using molecular dynamics simulation, we calculate the mean square displacement, the pair correlation function, the velocity autocorrelation function and the spectrum function of a two-dimensional (2D) charged dust system. Results show that the critical coupling constant Γ* for the phase transition from a liquid-like state to a solid-like state decreases with increasing charge and mass of particles in the one-component system. A scaling relation between Γ* and μ (Γ* ∝ μ^{−1.72±0.07}) is given, where μ represents the quantities of charge and mass. In the system with two species of particles, the coagulation process is found to be delayed by the presence of one species, and Γ* is closer to that of the 'smaller' particles species, which has the smaller charge and mass.