Table of contents

We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is given by formulae (11), (14), (25)–(27) and is a rational determinantal expression in which both the numerator and the denominator have a simple structure when expressed in terms of Schur polynomials.

We use boundary quantum group symmetry to obtain recursion formulae which determine nondiagonal solutions of the boundary Yang–Baxter equation (reflection equation) of the XXZ type for any spin j.

We discuss the spin glass states of the deterministic spin model with long range anti-ferromagnetic interactions. To apply the replica method, the partial statistical summation with fixed overlap parameters is introduced. By this formulation, we find that the replica theory for this model reduces to that of the anti-Hebbian model in the long range limit, suggesting the existence of dynamical phase transition and glassy states. Results of simulated annealing are also presented to confirm these results.

In 1959, March and Young (Nucl. Phys.12 237) rewrote the equation of motion for the Dirac density matrix γ(x, x₀) in terms of sum and difference variables. Here, γ(, ₀) for the d-dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables ∣ + ₀∣/2 and ∣ − ₀∣/2, a generalized partial differential equation embracing the March–Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix γ(, ₀) and the diagonal density ϱ(r) = γ(, ₀)∣_{0 =} , this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for d = 1 through 3. Some comments are finally made on the d-dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrödinger equations (the so-called Slater–Kohn–Sham equations).

We investigate the resonance mechanisms for discrete breathers in finite-size Klein–Gordon lattices, when some harmonic of the breather frequency enters the linear phonon band. For soft on-site potentials, the second-harmonic resonances typically result in the appearance of solutions with non-zero tails, phonobreathers. However, these tails may be very weak, and for small systems where the phonon frequencies are sparsely distributed, we identify 'phantom breathers' as being practically localized solutions, existing with frequencies in-between the phonon frequencies. For particular parameter values the tails completely vanish, and the phantom breathers decay exponentially over the whole system. We also describe briefly a first-harmonic resonance with a constant-amplitude wave and the generation of phonobreathers for hard potentials.

In this paper we establish a multipolar Hamiltonian in QED for an arbitrary system of moving charges. The particular cases of 2 and 3 charges are considered together with the Breit–Fermi Hamiltonian for two fermions. A scaling analysis shows the perturbative character of the dipole approximation of the multipolar Hamiltonian. The electric-dipole and magnetic-dipole terms of the dilated Hamiltonian are of order 1 and 2, respectively, with respect to the fine structure constant.

Starting from a differential realization of the generators of the so(2, 2) algebra we connect the eigenvalue equation of the Casimir invariant either with the hypergeometric equation, or the Schrödinger equation. In the latter case we consider problems for which so(2, 2) appears as a potential algebra, connecting states with the same energy in different potentials. We analyse the role of the two so(2, 1) subalgebras and point out their importance for -symmetric problems, where the doubling of bound states is known to occur. We present two mechanisms for this and illustrate them with the example of the Scarf and the Pöschl–Teller II potentials. We also analyse scattering states, transmission and reflection coefficients for these potentials.

The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that cannot be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of geometric symmetry of a linearly singular differential equation is introduced as a transformation that preserves the geometric data that define the problem. It is proved that such symmetries are essentially equivalent to dynamic symmetries, that is, transformations mapping solutions into solutions. Similar results are given for infinitesimal symmetries. To study the invariance of several objects under the flows of vector fields, a careful study of infinitesimal variations is performed, with a special emphasis on infinitesimal vector bundle automorphisms.

In this paper, by means of the Lax representations, we demonstrate the existence of infinitely many conservation laws for the general Toda-type lattice equation, the relativistic Volterra lattice equation, the Suris lattice equation and some other lattice equations. The conserved density and the associated flux are given formulaically. We also give an integrable discretization for a lattice equation with n dependent coefficients.

In this paper we would like to suggest a matrix form of the string field for any configuration of N D-instantons in bosonic string field theory.

A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to holonomic quantum computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated with a non-Abelian gauge connection. When applied to the optical holonomic computer, these conditions determine that the holonomy group of the two-qubit interaction model contains SU(2) × SU(2). In particular, a universal two-qubit logic gate is attainable for this model.

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang–Baxter equation. As is well known, the latter can be generically constructed from quantum affine superalgebras U_q(ĝ). These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter q when periodic boundary conditions are imposed. If q is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-Abelian subalgebras which are translation invariant and commute with the transfer matrix and therefore with all charges of the model. In the line of argument, we introduce the restricted quantum superalgebra U^res_q(ĝ) and investigate its root of unity limit. We prove several new formulae involving supercommutators of arbitrary powers of the Chevalley–Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.

The Helmholtz decomposition theorem for an anisotropic medium is explicitly stated in terms of two symmetric and positive definite dyadics, which carry the directional characteristics of the medium. It is shown that this general decomposition is not unique, and that once the scalar invariant with respect to one of the dyadics and the vector invariant with respect to the other dyadic are given the initial field can be completely reconstructed.