Table of contents

The Riemann zeta function is an important object of number theory. We argue that it is related to the Heisenberg spin-1/2 anti-ferromagnet. In the XXX spin chain we study the probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in the thermodynamics limit. We prove that for short strings the probability can be expressed in terms of the Riemann zeta function with odd arguments.

Recent numerical developments in the study of glassy systems have shown that it is possible to give a purely geometric interpretation of the dynamic glass transition by considering the properties of unstable saddle points of the energy. Here we further develop this approach in the context of a mean-field model, by analytically studying the properties of the closest saddle point to an equilibrium configuration of the system. We prove that when the glass transition is approached the energy of the closest saddle goes to the threshold energy, defined as the energy level below which the degree of instability of the typical stationary points vanishes. Moreover, we show that the distance between a typical equilibrium configuration and the closest saddle is always very small and that, surprisingly, it is almost independent of the temperature.

We extend a Gaussian model for the internal electrical potential of a two-dimensional Coulomb gas by a non-Gaussian measure term, which singles out the physically relevant configurations of the potential. The resulting Hamiltonian, expressed as a functional of the internal potential, has a surprising large-scale limit: the additional term simply counts the number of maxima and minima of the potential. The model allows for a transparent derivation of the divergence of the correlation length upon lowering the temperature to the Kosterlitz-Thouless transition point.

The finite XXZ Heisenberg spin chain with twisted boundary conditions is considered. For the case of an even number of sites N, anisotropy parameter -1/2 and twisting angle 2π/3 the Hamiltonian of the system possesses an eigenvalue -3N /2. The explicit form of the corresponding eigenvector was found for N ⩽12. Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, we conjecture that the squared norm of the ground state vector coincides with the number of half-turn symmetric alternating sign N×N matrices.

The Moller operators and the asociated Lippman-Schwinger equations obtained from different partitionings of the Hamiltonian for a step-like potential barrier are worked out, compared and related.

This paper explores potential symmetries of the nonlinear wave equation u_tt = (uu_x)_x, as well as related new similarity reductions and exact solutions of this equation. New approximate solutions of the perturbed nonlinear equations stemming from the exact solutions of the equation are obtained by applying a new approach to the use of the Lie group technique for differential equations dependent on a small parameter. In addition, some nonlinear wave equations exactly reducible to the equation u_tt = (uu_x)_x are constructed using this approach.

Generalized intelligent states (coherent and squeezed states) are derived for an arbitrary quantum system by using the minimization of the so-called Robertson-Schrödinger uncertainty relation. The Fock-Bargmann representation is also considered. As a direct illustration of our construction, the Pöschl-Teller potentials of trigonometric type will be chosen. We will show the advantage of the Fock-Bargmann representation in obtaining the generalized intelligent states in an analytical way. Many properties of these states are studied.

We study a pair of commuting difference operators arising from the elliptic solution of the dynamical Yang-Baxter equation of type C₂. The operators act on the space of meromorphic functions on the weight space of (4,). We show that these operators can be identified with the system by van Diejen and by Komori-Hikami with special parameters. It turns out that our case can be related to the difference Lamé operator (two-body Ruijsenaars operator) and thereby we diagonalize the system on the finite-dimensional space spanned by the level-one characters of the C₂⁽¹⁾-affine Lie algebra.

We discuss the explicit construction of the Schrödinger equations admitting representation through some family of general nonorthogonal polynomials. The specific choice of the third-order polynomial coefficient functions, that lead to quasi-solvable families of Schrödinger potentials, is considered in detail.

It is shown that a magnetic field satisfying the mean-field magnetohydrodynamics equation with zero mean velocity and without energy input from the outside possesses a Lyapunov function, which is a combination of the magnetic energy and the helicity. As a consequence, if the mean magnetic field remains uniformly bounded for all time, the field tends asymptotically in time to an attractor formed by force-free states.

A general expression is derived for the (paramagnetic) shielding factor for a nuclear spin embedded in a d-dimensional noninteracting electron gas and a parabolic quantum dot. We find that for d = 2 the Knight shift has no intrinsic magnetic field dependence and that for the quantum dot the shift is negligible unless the nuclear spin is near the centre.

We complete our study of non-Abelian gauge theories in the framework of the Epstein-Glaser approach to renormalization theory including in the model an arbitrary number of Dirac fermions. We consider the consistency of the model up to the third order of the perturbation theory. In the second order we obtain pure group theoretical relations expressing a representation property of the numerical coefficients appearing in the left- and right-handed components of the interaction, Lagrangian. In the third order of the perturbation theory we obtain the the condition of cancellation of the axial anomaly.

Let M ⁿ be a space-like submanifold in a de Sitter space M_p^{n + p}(c) with flat normal bundle. This paper gives some intrinsic conditions for M ⁿ to be totally umbilical.