Table of contents

We propose a generalization of the formula that gives an integrable quantum 1D spin Hamiltonian with nearest-neighbour interactions as a logarithmic derivative of a vertex model transfer matrix in order to include in this scheme more realistic integrable models. We compute exactly this generalized formula using the R matrix of the XXX model, obtaining the Majumdar-Ghosh Hamiltonian plus a charge-like interaction term. We diagonalize this Hamiltonian using the quantum inverse scattering method and present the Bethe ansatz equations of the model.

In this Letter we announce rigorous results that elucidate the relation between metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerably greater precision than has so far been available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable state and sharp estimates of the convergence of the probability distribution of the metastable transition times to the exponential distribution.

We discuss the analytic properties of the Callan-Symanzik β-function β(g) associated with the zero-momentum four-point coupling g in the two-dimensional ϕ⁴ model with O(N) symmetry. Using renormalization-group arguments, we derive the asymptotic behaviour of β(g) at the fixed point g*. We argue that β'(g) = β'(g*) + O(|g-g*|^1/7) for N = 1 and β'(g) = β'(g*) + O(1/log |g-g*|) for N⩾3. Our claim is supported by an explicit calculation in the Ising lattice model and by a 1/N calculation for the two-dimensional ϕ⁴ theory. We discuss how these non-analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g.

We compute the renormalized four-point coupling in the 2D Ising model using transfer-matrix techniques. We greatly reduce the systematic uncertainties which usually affect this type of calculation by using the exact knowledge of several terms in the scaling function of the free energy. Our final result is g₄ = 14.697 35(3).

We study the KPZ equation (in D = 2, 3 and 4 spatial dimensions) by using a restricted solid-on-solid discretization of the surface. We measure the critical exponents very precisely, and we show that the rational guess is not appropriate, and that D = 4 is not the upper critical dimension. We are also able to determine very precisely the exponent of the sub-leading scaling corrections, that turns out to be close to unity in all cases. We introduce and use a multi-surface coding technique, that allows a gain of the order of 30-fold over usual numerical simulations.

Conditions are established under which a system of hydrodynamic-type equations with time-dependent coefficients allows an infinite-dimensional group of hydrodynamic-type symmetries. When this group exists it is used to linearize and solve the original system.

Starting from the classical r matrix of the non-standard (or Jordanian) quantum deformation of the sl(2,) algebra, new triangular quantum deformations for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously constructed by using a graded contraction scheme; these are realized as deformations of conformal algebras of (1+1)-dimensional spacetimes. Time- and space-type quantum algebras are considered according to the generator that remains primitive after deformation: either the time or the space translation, respectively. Furthermore, by introducing differential-difference conformal realizations, these families of quantum algebras are shown to be the symmetry algebras of either a time or a space discretization of (1+1)-dimensional (wave and Laplace) equations on uniform lattices; the relationship with the known Lie symmetry approach to these discrete equations is established by means of twist maps.

In this paper we prove an upper bound for the Lyapunov exponent γ(E) and a two-sided bound for the integrated density of states N(E) at an arbitrary energy E>0 of random Schrödinger operators in one dimension. These Schrödinger operators are given by potentials of identical shape centred at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show, in particular, that both γ(E) and N(E)-(E)^1/2/π decay at infinity at least like 1/(E)^1/2. As an example we consider the random Kronig-Penney model.

We consider the heap formation of granular materials contained in a cylindrical vertically vibrating bed under slight vibration. Using the surface profile of the heap as the dynamical variable, the heap equation is generalized to the three-dimensional case. The steady state heap profiles are calculated. Our results indicate a change of downward to upward heaps as the vibration strength is increased, similar to those observed in two-dimensional vibrating bed experiments. The effective current in this model is also calculated, which can describe the convection phenomena. We also discuss the relationship between the heap formation times and the vibration strength and system size.

We suggest a procedure for the construction of Baxter Q-operators for the Toda chain. Apart from the one-parametric family of Q-operators, considered in our recent paper (Pronko 2000 Commun. Math. Phys. to appear) we also give the method of construction of two basic Q-operators and the derivation of the functional relations for these operators. Also we have found the relation of the basic Q-operators with Bloch solutions of the quantum linear problem.

The full set of polynomial solutions of the nested Bethe ansatz is constructed for the case of the A₂ rational spin chain. The structure and properties of these associated solutions are more various than the case of the usual XXX (A₁) spin chain but their role is similar.

Belavin's _n-symmetric elliptic model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We find non-diagonal K-matrices for n>2 that satisfy the reflection equation (boundary Yang-Baxter equation), and also find non-diagonal Boltzmann weights for the A⁽¹⁾_n-1-face model even for n⩾2. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the boundary model. The boundary spontaneous polarization is obtained by solving the simplest difference equations in the case of the free boundary condition. The resulting quantity is the square of the spontaneous polarization for the bulk _n-symmetric model, up to a phase factor.

We propose quantum mechanics of smeared particles that account for the delocalization of a particle defined via its Compton wavelength. The Hilbert space representation theory of such quantum mechanics is presented and its invariance under spatial translations and rotations is examined. The quantum mechanics of smeared particles is then applied to two paradigm examples, namely, the smeared harmonic oscillator and the Yukawa potential. In the second example, we theoretically predict the phenomenological coupling constant of the ω meson, which mediates the short range and repulsive nucleon force, as well as the repulsive core radius.

We consider the three n-dimensional nonlinear wave equations

We also consider a special class of point transformations. Motivated by the results on the corresponding one-dimensional equations we present a class of discrete symmetries for these equations. In some cases these discrete symmetries form cyclic groups of finite order. Furthermore, point transformations exist that relate different equations but of the same class. The equivalence point transformations for each of the above general equations are presented.