Table of contents

We discuss how the semiclassical approximation arises within the (classically improved) Langevin quantization. We derive a new representation of the semiclassical propagator (at imaginary time) in the form of a white noise average, which is valid over an arbitrary (imaginary) time interval. Although our result is ultimately equivalent to the standard representation of the semiclassical propagator, it nevertheless turns out to be more advantageous in dealing with certain problems that involve tunnelling and metastability.

We present a succinct way of obtaining all possible higher-dimensional generalizations of the quantum Yang-Baxter equation (QYBE). Using the scheme, we could generate the two popular three-simplex equations, namely the Zamolodchikov tetrahedron equation and the Frenkel and Moore equation.

In this paper we discuss the equivalence between two integrable and isotropic Osp(2/2) chains via the braid-monoid approach. We use the analytical Bethe ansatz approach in order to solve the spectrum of these systems. The ground state is parametrized by a special structure of the Bethe ansatz roots. Their interpretation in terms of correlated fermionic models is briefly discussed.

We present calculations of the complex-temperature zeros of the partition functions for 2D Ising models on the square lattice with spin s=1, 3/2 and 2. These give an insight into complex-temperature phase diagrams of these models in the thermodynamic limit. Support is adduced for a conjecture that all divergences of the magnetization occur at endpoints of arcs of zeros protruding into the FM phase. We conjecture that there are 4(s²)-2 such arcs for s>or=1, where (x) denotes the integral part of x.

A wide variety of complex physical systems described by unitary matrices have been shown numerically to satisfy level statistics predicted by Dyson's circular ensemble. We argue that the impact of localization in such systems is to impose certain restrictions on the eigenvalues. We consider a solvable model which takes into account such restrictions qualitatively and find that within the model a gap is created in the spectrum, and there is a transition from the universal Wigner distribution towards a Poisson distribution with increasing localization.

Certain problems of quantum physics (for example, the Aharonov-Bohm effect) lead to the eigenvalue problem for a Schrodinger operator with wave multifunctions. For a multiply-connected configuration space with a simplest topology (for example, for a n-dimensional torus) this problem was considered by several authors. In the present paper, by using rigorous mathematical methods we investigate this problem on an arbitrary multi-dimensional smooth manifold (possibly, with a boundary). We carefully define the concept of multifunctions, then we introduce spaces of these objects similar to L₂ and H₀¹. Finally, we present a spectral theorem on the existence of a self-adjoint extension of a Schrodinger operator in the introduced spaces which implies the completeness of the system of eigenfunctions of this operator in the considered functional spaces.

The quantum dynamics of a mechanical subsystem interacting with a thermal bath is studied. A new approach to dissipated quantum mechanics is proposed which is based on the thermodynamic treatment of the energy balance. Simple application of the theory to ideal gases is considered and equations for the evolution of the main average quantities are derived.

The quantal behaviour of a particle in a one-dimensional triangular potential well, driven by a monochromatic electric field, is studied. A classical high-frequency expansion together with semiclassical uniform methods are used to obtain an explicit form of the Floquet evolution operator in the unperturbed basis. A local exact solution is found for the eigenvalue equation of this operator under certain conditions. The local solution provides a tool for the quantitative investigation of the eigenstates. It predicts the appearance of quasi-resonances, or photonic states, and gives their location, shape and width as a function of parameters. It also predicts a local crossover from a decaying region to a more extended region as a function of n, with a point of crossover n_c between them. The results concerning the local structures are used to justify and extend a previously suggested method for the investigation of the asymptotic properties of the eigenstates. These are found to decay with a power depending on the field parameters (first proposed by Benvenuto et al., 1991). The specific system studied here is suggested as a prototype model for a class of driven one-dimensional bound systems. Whose main characteristic is an increasing density of states as a function of energy.

We study the effective diffusivity tensor for particles in a random gradient flow that, statistically, lacks rotational symmetry. The effective diffusivity tensor is computed up to two-loop order in perturbation theory and the 'Ward identity' that relates this tensor to the effective coupling is verified to the same order. We re-examine the renormalization group calculation that produced a very accurate numerical value for the effective diffusivity in the rotationally symmetric case and formulate two versions based on distinct divisions of the random potential field that gives rise to the flow. Both types of renormalization group calculation give good results when compared with numerical simulations. However, at two-loop order in perturbation theory the two methods differ in detail from each other and from the exact perturbation calculation. This is in contrast to the corresponding results in the isotropic case.

Transport properties of the multicomponent quantum many-body systems obeying Haldane's fractional exclusion statistics are studied in one dimension. By computing the finite-size spectrum under twisted boundary conditions, we explicitly express the conductivity and conductance in terms of statistical interactions. Through this analysis, the effective charge and effective mass for collective excitations are determined. We apply the results for l/r² quantum systems as well as for correlated electron systems.

It has been recently suggested that a totally asymmetric exclusion process with two species on an open chain could exhibit spontaneous symmetry breaking in some range of the parameters defining its dynamics. The symmetry breaking is manifested by the existence of a phase in which the densities of the two species are not equal. In order to provide a more rigorous basis to these observations we consider the limit of the process when the rate at which particles leave the system goes to zero. In this limit the process reduces to a biased random walk in the positive quarter plane, with specific boundary conditions. The stationary probability measure of the position of the walker in the plane is shown to be concentrated around two symmetrically located points, one on each axis, corresponding to the fact that the system is typically in one of the two states of broken symmetry in the exclusion process. We compute me average time for the walker to traverse the quarter plane from one axis to the other, which corresponds to the average time separating two flips between states of broken symmetry in the exclusion process. This time is shown to diverge exponentially with the size of the chain.

Monte Carlo simulations of the critical behaviour of disordered Ising systems have led to concentration-dependent critical exponents which seem to violate universality. We apply the renormalization group to investigate the crossover effects which cause the observed behaviour. We improve the three-loop expansion of the Callan-Symanzik and Wilson functions by a Pade-Borel approximation and solve the flow equations for various initial points. The exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point.

A Green function approach to the boundary effects of a free-electron gas in a magnetic field is presented. It is shown that truncation of the multiple-reflection expansion for the Green function of a confined system is not justified if a magnetic field is present, even for a system with flat boundaries. Using the full expansion we calculate the profiles for the particle density, the current density and the components of the pressure tensor near the boundary.

An Abelian sandpile model is considered on the Husimi lattice of square plaquettes. Exact expressions for the distribution of height probabilities in the self-organized critical state are derived. The two-point correlation function for the sites deep inside the Husimi lattice is calculated exactly.

We model a self-interacting branched polymer by a nearest-neighbour contact model of lattice animals with either site or bond counting (k- and k'-models, respectively). 1/d-expansions for the reduced limiting free energy are derived through order l/d⁵ in which the coefficients are temperature dependent. By evaluating the coefficients at certain special temperatures, we obtain 1/d-expansions for the growth constants of various types of lattice animal, again through order 1/d^d. The 1/d-expansions are used to obtain numerical estimates of the growth constants and to study the temperature dependence of the reduced limiting free energies on d-dimensional simple hypercubic lattices for values of d up to the upper critical dimension. The results are compared with ones obtained by more conventional series methods. Unexpected results are obtained for a range of values of the temperature variable.

In supervised learning, learning from queries rather than from random examples can improve generalization performance significantly. We study the performance of query learning for unrealizable tasks, where the student cannot learn from the perfectly. As a simple model scenario of this kind, we consider a linear perceptron student learning a general nonlinear perceptron teacher. Two kinds of queries for maximum information gain, i.e. minimum entropy, are investigated: minimum student space entropy (MSSE) queries, which are appropriate if the teacher space is unknown, and minimum teacher space entropy (MTSE) queries, which can be used if the teacher space is assumed to be known, but a student of a simpler form has deliberately been chosen. We find that for MSSE queries, the structure of the student space determines the efficacy of query learning. MTSE queries, on the other hand, which we investigate for the extreme case of a binary perceptron teacher, lead to a higher generalization error than random examples, due to a lack of feedback about the progress of the student in the way queries are selected.

We introduce a Brownian motion model for the parametric evolution of eigenstates of a complex quantum system, modelled by a random matrix. The model is analogous to Dyson's model for the evolution of the eigenvalues. We use this approach to analyse correlation functions describing the parameter dependence of diagonal and off-diagonal matrix elements of a generic operator. In the case of diagonal matrix elements, we compare our results with a semiclassical approach, which relates sums of matrix elements to periodic classical orbits. For systems with a chaotic classical limit, the semiclassical correlation function agrees exactly with the random matrix theory.

We have studied the phase transition of the Ising model on a family of fractals called Nice trees whose spectral dimension d_s can take values greater than 2. The phase transition is shown to be the trivial zero-temperature one through exactly solving the free energy and the spontaneous magnetization of the system, and is different from that on Cayley trees. The result is independent of d_s of the structure and hence provides an example of trivial phase transition at d_s>or=2, which does not agree with the argument of Yu and Gong (1994). It suggests that the role of d_s in determining the phase transition may be complex.

By using the finite-time thermodynamic method, the optimal performances of a Carnot engine under the influence of thermal resistance, heat loss and other irreversibilities are studied and the fundamental optimal relation for the engine is derived. Thus, a new theory for irreversible Carnot cycles is established, together with some significant discussions. The results obtained here are more general and useful than those in the relevant literature.

A monomer-dimer-monomer (A-B₂-D) lattice surface reaction model is proposed, in which two adsorption parameters X_A and X_D are involved. The phase diagram for this model is determined by the partial simulation method. For the case of no desorption being considered, six phases are found, which are the two-species saturated phases S_AD, S_BD, S_BE and S_AE (E denotes the intermediate product AB), the one-species saturated phase S_B and the steady reactive phase R. The R and S_B phases are restricted to the line X_A=X_D. Phase transitions between two-species phases or between R and S_B are all second order, while that between R and S_AD is first order. It appears that there exists a pentacritical point at X_A=X_D=0.023+or-0.001. If D* desorption is taken into account, the reactive phase R extends to a region of finite area.

We construct GL_h(3) from GL_q(3) by a singular map. We show that there exist two singular maps which map GL_q(3) to new quantum groups. We also construct their R-matrices and will show that, although the maps are singular, their R-matrices are not. Then we generalize these singular maps to the case GL(N) and for C_n series.

We consider the nonlinear complex d'Alembert equation Square Operator Psi =F( mod Psi mod ) Psi with Psi represented in terms of amplitude and phase, in (1+n)-dimensional Minkowski space. We exploit a compatible d'Alembert-Hamilton system to construct new types of exact solutions for some nonlinearities.

The type-I quantum superalgebras are known to admit non-trivial one-parameter families of inequivalent finite dimensional irreps, even for generic q. We apply the recently developed technique to construct new solutions to the quantum Yang-Baxter equation associated with the one-parameter family of irreps of U_q(gl(m/n)), thus obtaining R-matrices that depend not only on a spectral parameter but in addition on further continuous parameters. These extra parameters enter the Yang-Baxter equation in a similar way to the spectral parameter but in a non-additive form.

The analytic Bethe ansatz is executed for a wide class of finite-dimensional U_q(B_r⁽¹⁾) modules. They are labelled by skew-Young diagrams which, in general, contain a fragment corresponding to the spin representation. For the transfer matrix spectra of the relevant vertex models, we establish a number of formulae, which are U_q(B_r⁽¹⁾) analogues of the classical ones due to Jacobi-Trudi and Giambelli on Schur functions. They yield a full solution to the previously proposed functional relation (T-system), which is a Toda equation on discrete spacetime.

A quantum system is described by a density matrix which determines the joint probabilities for simultaneous eigenstates of its observables. Since there do not exist any simultaneous eigenstates of non-commuting observables, the density matrix cannot provide any information about the joint probability of their eigenstates. The quasiprobability distributions (QPDs) constructed by using the density matrix provide a means of determining these joint probabilities. Here the QPDs for a system of N spin-1/2 are constructed and the joint probabilities for their components determined.

The influence functional is introduced as a kernel in an integral equation that gives the probability density at time t and position q in terms of the initial probability density. This functional is applied to tunnelling through a square barrier to determine the influence, at different times, of various regions of the incident packet on the transmitted peak.

A new technique for solving the perturbed Kepler problem is presented. As its approach is essentially algebraic, it can be easily computerized and carried out to any perturbation order. It also eliminates the notorious difficulty of small divisors, and is well suited to deal with commensurable orbital periods, as demonstrated by its explanation of Kirkwood gaps.

We suggest an effective method for reducing the Yang-Mills equations to systems of ordinary differential equations. With the use of this method we construct extensive families of new exact solutions of the Yang-Mills equations. Analysis of the solutions thus obtained shows that they correspond to the conditional (non-classical) symmetry of the equations under study.

We propose a new version of the multipoint Pade technique to solve Schrodinger equations in their first-order Riccati form. Our compact product representation of wavefunctions interpolates or 'matches' available approximants at several points. The mechanism of 'matching' determines the energies and eliminates spurious solutions. On V(x)= mu ²x²+ lambda x²/(1+gx²), its efficiency is illustrated numerically.