Table of contents

We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue - Morse quasicrystal, i.e. of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction.

Periodic orbits that participate in a bifurcation contribute collectively to the periodic orbit sum for the quantum density of states. The contributions of multiple windings of isolated orbits are easily obtained from powers of the stability matrix, but it is generally hard to compose the actions that determine the contributions of higher windings of a bifurcation. Here we derive an approximate relation between the amplitude of the contributions of different windings for the saddle-centre bifurcation and the period-doubling bifurcation.

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents and in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent . The samples reduced critical temperature has a power law distribution and we show that the difference between the values of the critical exponents in the pure and in the random system is just . Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.

We analyse the violations of linear fluctuation-dissipation theorem (FDT) in the coarsening dynamics of the antiferromagnetic Ising model on percolation clusters in two dimensions. The equilibrium magnetic response is shown to be nonlinear for magnetic fields of the order of the inverse square root of the number of sites. Two extreme regimes can be identified in the thermoremanent magnetization: (i) linear response and out-of-equilibrium relaxation for small waiting times (ii) nonlinear response and equilibrium relaxation for large waiting times. The function X(C) characterizing the deviations from linear FDT crossovers from unity at short times to a finite positive value for longer times, with the same qualitative behaviour whatever the waiting time. We show that the coarsening dynamics on percolation clusters exhibits stronger long-term memory than usual Euclidean coarsening.

We present a one-mode quantized version of the semiclassical Su-Schrieffer-Heeger model, relevant for the study of conjugated polymers. The model is soluble. We give its equilibrium states at all temperatures and prove the existence of kink-antikink solutions. We impose stretching constraints in order to make the model boundary condition invariant. A uniform stretching of the lattice is considered. A formula for a temperature-dependent sound velocity is derived. The influence on the critical temperature and on the dimerization is rigorously derived.

A new two-parameter integrable model with quantum superalgebra symmetry is proposed, which is an eight-state fermions model with correlated single-particle and pair hoppings as well as uncorrelated triple-particle hopping. The model is solved and the Bethe ansatz equations are obtained.

The Hamiltonian with magnetic impurities coupled to the strongly correlated electron system is constructed from the t-J model. It is diagonalized exactly by using the Bethe ansatz method. Our boundary matrices depend on the spins of the electrons. The Kondo problem in this system is discussed in detail. The integral equations are derived with complex rapidities which describe the bound states in the system. The finite-size corrections for the ground-state energies are obtained.

A new approach is applied to the one-dimensional Anderson model by using a two-dimensional Hamiltonian map. For a weak disorder this approach allows for a simple derivation of correct expressions for the localization length both at the centre and at the edge of the energy band, where standard perturbation theory fails. Approximate analytical expressions for strong disorder are also obtained.

Based on a previously postulated entropy, that now becomes a particular case, we show that there exists an infinite set of entropies, with similar properties, that reduce in a common limit to the Boltzmann-Shannon form. The probabilities for the microcanonical ensemble and for the canonical ensemble are obtained. The method used to construct the set is quite simple and quite general and can be applied to generalizations of physical quantities and to other generalized entropies. The existence of an infinite set of `entropies' with, in principle, similar properties, could be a serious drawback for the actual utility of any of them and points to their utter uselessness unless some reason can be given for a special choice of one of them.

We revisit the model of the two-component plasma in a gravitational field, which mimics charged colloidal suspensions. We concentrate on the computation of the grand potential of the system. Also, a special sum rule for this model is presented.

A generalization of the circular and hyperbolic functions is proposed, based on the Tsallis statistics and on a consistent generalization of the Euler formula. Some properties of the presently proposed q-trigonometry are then investigated. The generalized functions are exact solutions of a nonlinear oscillator. Original circular and hyperbolic relations are recovered as the limiting case.

Given M copies of a q-deformed Weyl or Clifford algebra in the defining representation of a quantum group , we determine a prescription to embed them into a unique, inclusive -covariant algebra. The different copies are `coupled' to each other and are naturally ordered into a `chain'. In the case a modified prescription yields an inclusive algebra which is even explicitly -covariant, where is a symmetry relating the different copies. By the introduction of these inclusive algebras we significantly enlarge the class of -covariant deformed Weyl/Clifford algebras available for physical applications.

An improved method for solving time-dependent problems in quantum mechanics, in the customary cases of constant or harmonic perturbation, is applied to the calculation of the self-energy of electrons interacting with phonons in solids. The mixing of unperturbed Bloch states, resulting from the actual coupling, is self-consistently taken into account, and the related quantum probability amplitudes are determined through direct integration over the quasiparticle spectrum. Laplace transform and elementary mathematics are used, thereby enhancing the physical transparency, and bringing out approximations in every stage. Explicit illustrative results are worked out in the simple case of slowly varying self-energy parameters. The method is critically compared with the standard Green function approach, and further encourages more detailed applications.

A Schrödinger operator with a matrix-valued rational potential U(z) is said to have trivial monodromy if all the solutions of the corresponding Schrödinger equations are single-valued in the complex plane for any . A local criterion of this property in terms of the Laurent coefficients of the potential U near its singularities, which are assumed to be regular, is found. It is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the Darboux transformation from the Schrödinger operator . This generalizes the well known Duistermaat-Grünbaum result to the matrix case and gives the explicit description of the Schrödinger operators with trivial monodromy in this case.

The most general possible central extensions of two whole families of Lie algebras, which can be obtained by contracting the special pseudo-unitary algebras of the Cartan series and the pseudo-unitary algebras , are completely determined and classified for arbitrary p and q. In addition to the and algebras, whose second cohomology group is well known to be trivial, each family includes many non-semisimple algebras; their central extensions, which are explicitly given, can be classified into three types as far as their properties under contraction are involved. A closed expression for the dimension of the second cohomology group of any member of these families of algebras is given.

In this paper, we propose an elliptic algebra which is based on the relations , where R and are the dynamical R-matrices of -type face model with the elliptic moduli shifted by the centre of the algebra. From the Ding-Frenkel correspondence, we find that its corresponding (Drinfeld) current algebra at level one is the algebra of screening currents for q-deformed Virasoro algebra. We realize the elliptic algebra at level one by Miki's construction from the bosonization for the type I and type II vertex operators. We also show that the algebra is related to the algebra by dynamically twisting.

The three-body angular basis has been used to produce two infinite series of identities for the associated Legendre polynomials which are mostly known as two-body objects. The coefficients that are involved in the new sum rules are given in terms of the Clebsh-Gordan coefficients.

The probability density function for the determinant of a random Hermitian matrix taken from the Gaussian unitary ensemble is calculated. It is found to be a Meijer G-function or a linear combination of two Meijer G-functions, depending on the parity of n. The integer moments of this probability density are also given.

By introducing appropriate weights in the energy method we propose a new technique for studying the nonlinear stability of Maxwellian states for discrete velocity models (McKean and Broadwell) of the extended Boltzmann equation.

We study Heisenberg spins on an infinite plane. In the continuum limit the Hamiltonian of the system is given by the nonlinear model. Following an approach developed by Mikeska and Affleck, we find that the angular momentum associated with the order parameter presents a classical spin part, associated with the gauge freedom of a trihedra. We show that this gauge field may induce a non-trivial topological term, the Hopf term (or Chern-Simons term), as initially suggested by Dzyaloshinski, Polyakov and Wiegmann.