Table of contents

Systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are, in the semiclassical limit, a combination of Poisson and random-matrix, with relative weightings determined by the corresponding measures of regular and chaotic orbits. We here identify an additional component in long-range spectral statistics, associated with periodic orbit bifurcations, which can be semiclassically large. This is illustrated for a family of perturbed cat maps.

For the isotropic O(3) model we prove that percolation in the Wolff representation is a necessary and sufficient condition for positivity of the spontaneous magnetization.

We discuss (1 + 1)-dimensional sigma models and Heisenberg models in which the target space has the topology of a cylinder. In these integrable systems, the solutions are classified by a winding number.

We constructed a factorized representation of the Felder's elliptic quantum group by using the intertwining vectors, and further derived its explicit form which is, in a special case, identical with what Bo-yu Hou et al gave, i.e. a cyclic representation of the Sklyanin algebra up to a factor.

To find critical points of O(n) models on the triangular lattice we apply two methods. First we investigate the Yang-Baxter equations on the triangular lattice. We find only solvable points directly related to those for the square lattice. Second we construct intersections with the Potts model. This yields eight branches of critical points, parametrized by n. We establish the equivalence of these branches with the known critical points of the O(n) model on the square lattice. Transfer-matrix calculations are performed to obtain the conformal anomaly and the thermal exponent of these branches. These results include a numerical analysis of a q = 3 Potts tricritical point. We find analytic relations between Potts and O(n) models, as well as among O(n) models with different values of n, and among Potts models with different values of q.

An artificial neural network can be used to generate a series of numbers. A Boolean perceptron generates bit sequences with a periodic structure. The corresponding spectrum of cycle lengths is investigated analytically and numerically; it has similarities with properties of rational numbers.

We derive the exact Green functions for segmented potentials, i.e. for potentials of the form , where each has compact support. The expressions depend explicitly on the wavefunctions' quantum amplitudes (e.g. transmission and reflection coefficients) for the individual 's. We also show that the exact Green functions are in fact generalizations of the usual semiclassical formula. Finally, the possibility of applying the method to a general potential is briefly discussed.

The interaction between a trapped ion and a resonant laser standing wave is studied under the analytical approximation. Periodic orbits next to the heteroclinic one and their stability conditions are derived from the Rayleigh perturbation method. Theoretical analysis reveals the stable periodic orbits to be embedded in the Melnikov chaotic attractor. The corresponding numerical results show that fitting control parameters into the stability conditions can control chaos in the system.

The conventional approach to the modelling of semiconductor devices operation is based on a numerical solution of transport equations. This paper exposes a complete analytical treatment of transport equations for a two-valley semiconductor. We consider the case when the geometry of the analysed structure permits us to assume that the transport in one of the dimensions of the structure is dominant, and when the electric field is homogeneous and stationary. The obtained solution for the case of generally set initial conditions is reduced to the quadratures. We used the example of a p-i-n diode to demonstrate the superiority of the analytical treatment as compared with the numerical one. The proposed approach offers the possibility to determine exactly and quickly the spatial-temporal distribution of the electron concentration within the structure. Considering the accuracy of the analytical procedure, compared with that of the numerical one, the obtained solution could also be used as a kind of standard for the analysis of particular approximate solutions of the considered type of partial differential equation systems.

The consistent analytical theory of multiphoton ionization in a superstrong laser field is developed. The corresponding initial problem for the Schrödinger equation is reduced to the initial problem for the pair of integro-differential equations. The appropriate asymptotic technique is derived and the leading asymptotic term of the solution is obtained. It is shown, that the atomic stabilization is present under fairly general conditions. The possibility of population revivals is also shown.

We derive the equations that describe the evolution of the modulation of a short localized optical pulse in a bulk medium presenting a nonzero second-order nonlinear susceptibility, far from the the phase-matching. We give the equations in the general case, and for the symmetry classes of crystals and 3m. A wave interaction between the wave and some DC field, due to optical rectification and the electro-optic effect, strongly modifies the obtained system with regard to the three-dimensional NLS equation. An exhaustive list of the cases where the derived model can be reduced to the one-dimensional NLS equation is given. This reduction is possible for only a few special choices of the polarization and of the modulation direction, in a way corresponding to the symmetry of the crystal. The reduction of the same system to two coupled one-dimensional NLS equations is also investigated. With regard to the integrable systems of this type, an additional nonlinear term arises in all but a few particular cases.

This paper concerns the total internal reflection of an elastic wavefield at the boundary between a compressible fluid and an adjacent elastic solid. The system is forced by a prescribed high-frequency wave incident upon the interface from the fluid and emphasis is placed on the construction of the local acoustic response near the points where the shear or longitudinal components of the total elastic field propagate tangentially to the boundary.

A methodology for this construction, which is based on multiple-scales techniques, is presented and detailed calculations are carried out for both flat and curved interfaces. These analyses not only yield local amplitude balances and governing solutions but also furnish certain diffraction coefficients which might be of importance in other global scattering problems involving coupled fluid-solid configurations.

The Hilbert space , to which the wavefunction of the three-dimensional Schrödinger equation belongs, has been replaced by , where is a bounded region. The energy spectrum of the usual unbounded system is then determined by showing that the Dirichlet and Neumann problems in generate upper and lower bounds, respectively, to the eigenvalues required. Highly accurate numerical results for the quartic and sextic oscillators are presented for a wide range of the coupling constants.

The definitions of δ and γ given in this comment are, in fact, the definitions for δ^-1 and γ^-1, respectively. The author wishes to thank Dr B Sahu (Department of Physics, Revenshaw College, Cuttak, India) for pointing this out.

In reference [3], please read Zhakhariev instead of Zhakhirev.

On page 7021, line 15, reference [18] should be replaced by reference [12].

In equation (2.21), `Exp' should be replaced by `Ext'.

On page 7034, the final sentence of the first paragraph of section 6 should read as follows. `In this sense, there is no real difference between equilibrium and metastable states, since equilibrium is a collective property.'

In equation (11), `+2A' should be replaced by `-2A' at three occurrences.