Table of contents

We consider a planar Brownian motion starting from O at time t = 0 and stopped at t = 1 and a set F = {OI_i; i = 1, 2, ..., n} of n semi-infinite straight lines emanating from O. Denoting by g the last time when F is reached by the Brownian motion, we compute the probability law of g. In particular, we show that, for a symmetric F and even n values, this law can be expressed as a sum of arcsin or (arcsin)² functions. The original result of Levy is recovered as the special case n = 2. A relation with the problem of reaction–diffusion of a set of three particles in one dimension is discussed.

A τ function formalism for Sakai's elliptic Painlevé equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta–Ramani–Grammaticos. We also give a simple geometric description of the elliptic Painlevé equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painlevé equation expressed in terms of the elliptic hypergeometric function ₁₀E₉.

In this paper we reexamine the geometric structure of extended irreversible thermodynamics in the context of contact geometry. First, we consider the interplay between the contact manifold (M, ω) with thermodynamic state space B_N as its base, and the cotangent bundle T*B_N equipped with a nondegenerate 2-form Ω = dω. We then show that the Legendre submanifold L of M and the Lagrangian submanifold of T*B_N are intimately related to the entropy surface of the thermodynamic system. Second, we further generalize the symmetry transformations considered in our previous work that preserve the laws of thermodynamics as well as the pseudo-Riemannian metric in L. Finally, we consider some examples on coordinate transformations in M that illustrate the transformation between the entropy surface and the energy surface, and the relationship between Legendre involution and the submanifold of (T*B_N, Ω).

The dynamics of a phase ordering system with non-conserved order parameter under a plain shear flow with rate γ is solved analytically in the large-N limit. A phase transition is observed at a critical temperature T_c(γ). After a quench from a high temperature equilibrium state to a lower temperature T a non-equilibrium stationary state is entered when T > T_c(γ), while aging dynamics characterizes the phases with T ≤ T_c(γ). Two-time quantities are computed and the off-equilibrium generalization of the fluctuation–dissipation theorem is provided.

It is shown explicitly that the fermion particle density ϱ(r) in d dimensions for isotropic harmonic confinement and odd d is determined by the one-dimensional wavefunctions of the highest occupied state. Knowledge of the Dirac density matrices for d = 1 and 2 suffices to completely determine the general d-dimensional matrices.

This paper investigates the occurrence of dangerous situations (DS) within the framework of the Nagel–Schreckenberg model. The conditions of the DS are modified. It is shown that when v_max = 1, there will be no DS in both deterministic and non-deterministic cases. The situation is different for v_max > 1. We show that in the deterministic case, the probability of DS covers a two-dimensional region, which depends on both the density and the initial configuration. As for the non-deterministic case, our results are qualitatively the same as previous ones, but are quantitatively different.

We propose a probabilistic model for a system at a threshold of instability. The distribution of residence times below the threshold that characterizes the properties of the system is studied analytically in various cases.

A simple and practical semiclassical method based on an amplitude-free quasi-correlation function is proposed to quantize the energy spectrum of classically chaotic systems. Since classical trajectories used in this function are required to satisfy only a simple condition, the present method can be readily applied to a relatively large system. Numerical examples, which compare the semiclassical spectra with those of the full quantum mechanics in a two-dimensional system covering the regular and highly chaotic energy regions, show very clearly that the proposed amplitude-free quasi-correlation function can quantize chaos surprisingly well.

We give a computer-aided proof of the non-integrability of an important collinear configuration of the three-body problem in atomic physics. We consider the configuration of helium-like atoms where two electrons are on the same side of the atom. Numerical evidence shows that this configuration for helium atom has a Poincaré section that is hardly distinguishable from an integrable system. We extend the model for several helium-like atoms with different values of Z and also consider the case where a heavier particle takes the place of an electron, such as the muon.

A system of nonlinear equations is presented for the solution of the Cox–Thompson inverse scattering problem (1970 J. Math. Phys.11 805) at fixed energy. From a given finite set of phase shifts for physical angular momenta, the nonlinear equations determine related sets of asymptotic normalization constants and nonphysical (shifted) angular momenta from which all quantities of interest, including the inversion potential itself, can be calculated. As a first application of the method we use input data consisting of a finite set of phase shifts calculated from Woods–Saxon and box potentials representing interactions with diffuse or sharp surfaces, respectively. The results for the inversion potentials, their first moments and asymptotic properties are compared with those provided by the Newton–Sabatier quantum inversion procedure. It is found that in order to achieve inversion potentials of similar quality, the Cox–Thompson method requires a smaller set of phase shifts than the Newton–Sabatier procedure.

We consider the Hirota equation (the discrete analogue of the generalized Toda system) over a finite field. We present the general algebro-geometric method of construction of solutions of the equation. As an example we construct analogues of the multisoliton solutions for which the wavefunctions and the τ-function can be found using rational functions. Within the class of multisoliton solutions we isolate generalized breather-type solutions which have no direct counterparts in the complex field case.

A perturbation theory based on inverse scattering theory is developed for the coupled nonlinear Schrödinger equations. The theory finds useful application to the study of pulse propagation down a birefringent optical fibre and is used to examine features of the radiation field shed by a soliton pulse propagating down the fibre. The radiation field is linked to the scattering data through a transform pair which in the linear limit reduces to the forward and inverse Fourier transform pair. A complementary approach, which is in total agreement to these results, is also discussed.

We show that the supersymmetric radial ladder operators of the three-dimensional isotropic harmonic oscillator are contained in the spherical components of the creation and annihilation operators of the system. Also, we show that the constants of motion of the problem, written in terms of these spherical components, lead us to second-order radial operators. Further, we show that these operators change the orbital angular momentum quantum number by two units and are equal to those obtained by the Infeld–Hull factorization method.

In a recent paper (Buyarov V S, López-Artés P, Martínez-Finkelshtein A and Van Assche W 2000 J. Phys. A: Math. Gen.33 6549–60), an efficient method was provided for evaluating in closed form the information entropy of the Gegenbauer polynomials C^(λ)_n(x) in the case when λ = l ∊ . For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, P(x) and H(x), of degrees 2l − 2 and 2l − 4, respectively. Here it is shown that P(x) is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights w_l(x) = (1 − x²)^l−1/2, and this fact is used to obtain the explicit expression of P(x). From this result, an explicit formula is also given for the polynomial S(x) = lim_n→∞P(1 − x/(2n²)), which is relevant to the study of the asymptotic (n → ∞ with l fixed) behaviour of the entropy.

By reconsidering soliton solutions of the Toda lattice and differential-difference KdV equation in the Casorati determinant form with new entries, we obtain rational and mixed rational–soliton solutions in the Casorati determinant form. All these solutions are verified by direct substitutions into bilinear equations. The method used is general and can apply to other discrete systems.

We study a free quantum motion on periodically structured manifolds composed of elementary two-dimensional 'cells' connected either by linear segments or through points where the two cells touch. The general theory is illustrated with numerous examples in which the elementary components are spherical surfaces arranged into chains in a straight or zigzag way, or two-dimensional square lattice 'carpets'. We show that the spectra of such systems have an infinite number of gaps and that the latter dominate the spectrum at high energies.

A simple classical probabilistic system (a simple card game) classically exemplifies Aharonov and Vaidman's 'three-box 'paradox'' (1991 J. Phys. A: Math. Gen.24 2315), implying that the three-box example is neither quantal nor a paradox and leaving one with less difficulty to busy the interpreters of quantum mechanics. An ambiguity in the usual expression of the retrodiction formula is shown to have misled Albert et al (1985 Phys. Rev. Lett.54 5) to a result not, in fact, 'curious'; the discussion illustrates how to avoid this ambiguity.

A realization of the ladder operators for the infinitely deep well potential is presented. Both the asymmetric and symmetric wells are discussed. It is shown that these operators satisfy the commutation relations for the SU(1, 1) group. Closed analytical expressions for the matrix elements of some relevant functions are obtained. Perelomov coherent states for this system are explicitly constructed. The behaviour of the dispersion relation and the time evolution for such states are examined.

This paper describes supersymmetric quantum mechanical models (D) of D fermions invariant under the adjoint action of a compact simple Lie algebra of dimension D. It determines the spectrum, ground state properties and the full Fock space structure of such interacting models. It also discusses hidden supersymmetries, partner theories and spectrum generating algebras for the models (D).

We introduce a gauge invariant and string independent two-point fermion correlator which is analysed in the context of the Schwinger model (QED₂). We also derive an effective infrared worldline action for this correlator, thus enabling the computation of its infrared behaviour. Finally, we briefly discuss the possible perspectives for the string independent correlator in the QED₃ effective models for the normal state of HTc superconductors.

We investigate the fermionic sector of a given theory, in which massive and charged Dirac fermions interact with an Abelian gauge field, including a non-standard contribution that violates both Lorentz and CPT symmetries. We offer an explicit calculation in which the radiative corrections due to the fermions seem to generate a Chern–Simons-like effective action. Our results are obtained under the general guidance of dimensional regularization, and they show that there is no room for Lorentz and CPT violation in both commutative and noncommutative spacetimes.