Table of contents

It is shown that applicability of standard high-temperature expansions in multifractal thermodynamics is restricted by complex-temperature singularities. An analytic continuation method (finite-temperature expansions) has been developed to improve the analytic expansions approach. The quantum intermittency is then used as an example of applicability of the finite-temperature expansion.

A stochastic simulation algorithm for the computation of multitime correlation functions which is based on the quantum-state diffusion model of open systems is developed. The crucial point of the proposed scheme is a suitable extension of the quantum master equation to a doubled Hilbert space which is then unravelled by a stochastic differential equation.

The solution of the logistic map at the generic nth iteration is given in terms of the initial datum and the parameter as the characteristic polynomial of a simple matrix.

We determine the transition amplitude for multimagnon scattering induced through an inhomogeneous distribution of the coupling constant in the ferromagnetic XXX-model. The two- and three-particle amplitudes are explicitly calculated at small momenta. This suggests a rather plausible conjecture also for a formula of the general n-particle amplitude.

An exact calculation of the phase diagram for a loop-gas model on the brickwork lattice is presented. The model includes a bending energy. In the dense limit, where all the lattice sites are occupied, a phase transition occurring at an asymmetric Lifshitz tricritical point is observed as the temperature associated with the bending energy is varied. Various critical exponents are calculated. At lower densities, two lines of transitions (in the Ising universality class) are observed, terminated by a tricritical point, where there is a change in the modulation of the correlation function. To each tricritical point an associated disorder line is found.

We analyse the problem of constructing global complex action-angle variables for a class of Hamiltonians rational in q and quadratic in p.

We give a complete description of the monodromy of the action variables in terms of subgroups of unimodular transformations associated to the singular energy points.

We also consider a peculiar subclass of Hamiltonian systems which are reducible to algebraically complete integrable systems and show that the reduction preserves the symplectic structure. As a consequence the monodromy of the action variable is induced by the one of the reduced system.

Finally, we consider the problem of defining global angle variables. We propose to define a set of angle variables which correspond to a group of Poisson structures associated to .

Isotropic diffusion processes on cosets and and their zero-curvature limit are studied from a unified viewpoint. As indicated in our previous works the projection of the Fokker-Planck equation onto the maximal commutative subgroup of these cosets can be described by using the radial part of the Laplace-Beltrami operator. By taking the zero-curvature limit, an integral which is a natural extension of the Itzykson-Zuber integral to the rectangular matrices is explicitly evaluated. The probability density function obtained from the diffusion on is studied in detail, which can be applied to the quantum transport problem. The explicit expressions for the probability density function in the metallic and insulating regimes are obtained. For the metallic regime the integral representation for the hypergeometric function is used and the results are exact. Furthermore, by using the orthogonal polynomial method n-point correlation functions are obtained exactly for arbitrary n.

A new spectral parameter independent R-matrix (that depends on all of the dynamical variables) is proposed for the elliptic Calogero-Moser models. The necessary and sufficient conditions for the existence this R-matrix reduces to a determinantal equality involving elliptic functions. The required identity appears new and is still unproven in full generality; we present it as conjecture.

The vacuum energies corresponding to massive Dirac fields with the boundary conditions of the MIT bag model are obtained. The calculations are carried out with the fields occupying the regions inside and outside the bag, separately. The renormalization procedure for each of the situations is studied in detail, in particular the differences occurring with respect to the case when the field extends over the whole space. The final result contains several constants that undergo renormalization and can be determined experimentally only. The non-trivial finite parts which appear in the massive case are found exactly, providing a precise determination of the complete, renormalized zero-point energy in the fermionic case. The vacuum energy behaves as an inverse power of the mass, for large mass of the field.

We investigate the propagation of cw (continuous wave) circularly symmetric Gaussian beams in a nonlinear saturable medium using a modified variational approach. We find the equations that describe the characteristics of the beam, solving them analytically in various regimes. We also determine the conditions under which these solutions may be stable in two transverse dimensions. Finally, we solve these equations numerically in the case of loss, comparing them with the lossless analytical solutions in the limit of small losses.

q-Legendre polynomials can be treated as some special `functions in the quantum double cosets '. They form a family (depending on a parameter q) of polynomials in one variable. We get their further generalization by introducing a two-parameter family of polynomials. If the former family arises from an algebra which is in a sense `q-commutative', the latter one is related to its non-commutative counterpart. We also introduce a two-parameter deformation of the invariant integral on a quantum sphere.

The analytic properties of the lattice Green function

where t lies in a complex plane which is cut along the real axis from -2 to +2, are investigated. In particular, it is proved that tG(t) can be written in the product form

where and denotes a hypergeometric function. This result and the analytic continuation formulae for the function are then used to obtain various exact closed-form expressions for the related functions

where . It is also shown that G(t) is a solution of a third-order Fuchsian differential equation.

We present new expressions of form factors of the XXZ model which satisfy Smirnov's three axioms. These new form factors are obtained by applying the affine quantum group to the known ones obtained in our previous works. We also find the relations among all the new and known form factors, i.e. all other form factors are obtained by applying to a singlet form factor.

Quantum state diffusion (QSD) provides a natural unravelling of a mixed-state open quantum system into component pure states. We investigate the semiclassical limit of QSD in a phase-space approach using the Wigner function. As , QSD exhibits two very different dynamical regimes, depending on the volume of phase space covered by the quantum state. For large volumes there is a localization regime represented by classical nonlinear and nonlocal diffusion processes. For small volumes, comparable in size with a Planck cell, there is a wavepacket regime. Here, the centroid of the wavepacket follows a classical Langevin equation, obtained through the adiabatic elimination of the dynamics of the second-order moments of the wavepacket. The corresponding Fokker-Planck equation is identical to the one obtained from the classical limit of the original mixed-state dynamics. In the companion paper we present an axiomatic approach to a classical theory of quantum localization without using the underlying QSD theory.

Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.

The maximal Abelian subalgebras (MASAs) of the Euclidean and pseudo-euclidean Lie algebras are classified into conjugacy classes under the action of the corresponding Lie groups and , and also under the conformal groups and , respectively. The results are presented in terms of decomposition theorems. For orthogonally indecomposable MASAs exist only for p = 1 and p = 2. For , on the other hand, orthogonally indecomposable MASAs exist for all values of p. The results are used to construct new coordinate systems in which wave equations and Hamilton-Jacobi equations allow the separation of variables.

The supersymmetry of the electron in both the nonstationary magnetic and electric fields in a two-dimensional case is studied. The supercharges which are the integrals of motion and their algebra are established. Using the obtained algebra the solutions of nonstationary Pauli equation are generated.

We studied the dynamics of surface and wake charges induced by a charged particle traversing a dielectric slab. It is shown that, after crossing the first boundary of the slab, the charge induced on the slab surface (image charge) is transformed into the wake charge, which `overflows' to the second boundary when the particle crosses that. It is also shown, that the polarization of the slab is of an oscillatory nature, and the net induced charge in a slab remains zero at all stages of the motion.