Table of contents

A category which generalizes to higher dimensions many of the features of the Temperley–Lieb category is introduced.

Continuous models for anomalous diffusion have previously been tested in the subdiffusive case by making comparisons to diffusion on a Sierpinski gasket. This paper extends this discussion to the superdiffusive case by comparing performance to diffusion on a tree model. Although there is reasonable agreement within limited regimes for all four models, one model, due to Compte and Jou, stands out as being consistently sound over all regimes studied.

Stochastic fractals, generated from combinations of deterministic fractals, have the advantage of being tractable to some extent, but also being closer to real materials, since they are partially disordered. In the present work, we focus our attention on the remarkable nonlinear mixing behavior exhibited by fractals generated as random combinations of two different Sierpinski carpet generators. When patterns with different anomalous diffusion exponents and the same or different fractal dimensions are combined together, the effective diffusion exponent cannot in general be expressed as a linear weighted average of the diffusion exponents of the constituents. The effective exponent may show a maximum or minimum for certain compositions. An explanation of this interesting phenomenon is offered on the basis of details of the carpet generator, particularly on the number and position of 'connection points', which determine the connectivity of the 'fractal composite'.

Scaling properties of Chirikov's standard map are investigated by studying the average value of I², where I is the action variable, for initial conditions in (a) the stability island and (b) the chaotic component. Scaling behavior appears in three regimes, defined by the value of the control parameter K: (i) the integrable to non-integrable transition (K ≈ 0) and K < K_c (K_c ≈ 0.9716); (ii) the transition from limited to unlimited growth of I², K ≳ K_c; (iii) the regime of strong nonlinearity, K ≫ K_c. Our scaling results are also applicable to Pustylnikov's bouncer model, since it is globally equivalent to the standard map. We also describe the scaling properties of a stochastic version of the standard map, which exhibits unlimited growth of I² even for small values of K.

A new efficient method for calculating the photoionization of a hydrogen atom in a strong magnetic field is developed based on the Kantorovich approach to the parametric boundary problems in spherical coordinates using the orthogonal basis set of angular oblate spheroidal functions. The progress as compared with our previous paper (Dimova M G, Kaschiev M S and Vinitsky S I 2005 J. Phys. B: At. Mol. Opt. Phys.38 2337–52) consists of the development of the Kantorovich method for calculating the wavefunctions of a continuous spectrum, including the quasi-stationary states imbedded in the continuum. Resonance transmission and total reflection effects for scattering processes of electrons on protons in a homogenous magnetic field are manifested. The photoionization cross sections found for the ground and excited states are in good agreement with the calculations by other authors and demonstrate correct threshold behavior. The estimates using the calculated photoionization cross section show that due to the quasi-stationary states the laser-stimulated recombination may be enhanced by choosing the optimal laser frequency.

For complex two-dimensional Riemannian spaces every classical or quantum second-order superintegrable system can be obtained from a single generic 3-parameter potential on the complex 2-sphere by delicate limit operations and through Stäckel transforms between manifolds. Here we derive families of finite- and infinite-dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2-sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system. The algebra is exactly the one that describes the Wilson and Racah polynomials in their full generality.

The new arguments based on the Majorana fermions model indicating that non-completely positive maps can describe open quantum evolution are presented.

We introduce, in the spirit of Witten's gauging of exterior differential, a deformed Lie derivative that allows a geometrical interpretation of λ- and μ-symmetries, in complete analogy with standard symmetries. The case of variational symmetries (both for ODEs and for PDEs) is also considered in this approach, leading to λ- and μ-conservation laws.

We consider two simple models of higher derivative and nonlocal quantum systems. It is shown that, contrary to some claims found in literature, they can be made unitary.

The decoherence and entanglement dynamics of two interacting qubits coupled through Heisenberg XY interactions to a spin bath in thermal equilibrium are studied. The exact form of the reduced density matrix is derived for finite and infinite numbers of environmental spins. It is shown that decoherence can be minimized at low bath temperatures and strong coupling between the qubits. Some initial product states evolve into entangled ones, initially entangled states lose completely or partially their entanglement. The relation between the fidelity and the concurrence is also investigated.

Linear chains of quantum scatterers are studied in the process of lengthening, which is treated and analysed as a discrete dynamical system defined over the manifold of scattering matrices. Elementary properties of such dynamics relate the transport through the chain to the spectral properties of individual scatterers. For a single-scattering channel case, some new light is shed on known transport properties of disordered and noisy chains, whereas the translationally invariant case can be studied analytically in terms of a simple deterministic dynamical map. The many-channel case was studied numerically by examining the statistical properties of scatterers that correspond to a certain type of transport of the chain, i.e. ballistic or (partially) localized.

A class of transparent potentials (i.e., potentials with trivial scattering operator S = 1) for the three-dimensional Schrödinger equations is studied. We find the underlying group explaining the transparency phenomenon for these Hamiltonians.

We have studied the stability of two entangled spins dressed by electrons by calculating the scattering phase shifts. The interaction between electrons is interpreted by fully relativistic QED and the screening effect is described phenomenologically in the Debye exponential form e^−αr. Our results show that if the (Einstein–Podolsky–Rosen-) EPR-type states are kept stable under the interaction of QED, the spatial wavefunction will be constrained to be parity dependent. The spin-singlet state s = 0 and the polarized state along the z-axis give rise to two different kinds of phase shifts. Interestingly, the interaction between electrons in the spin-singlet pair is found to be attractive. Using this attraction we propose a mechanism to produce entangled pairs of massive spin-1/2 particles.

This paper describes an electromagnetic field analogue of the classical Brayton–Moser formulation. It is shown that Maxwell's curl equations constitute a pseudo-gradient system with respect to a single electromagnetic mixed-potential functional and a metric defined by the constitutive relations of the fields. Besides its use for the generation of power-based Lyapunov functionals for stability analysis and Poynting-like flow balances, the electromagnetic mixed-potential formulation suggests a family of alternative variational principles. This yields a novel Lagrangian boundary control system formulation admitting nonzero energy flow through the boundary. The corresponding symplectic Hamiltonian system is still associated with the total electromagnetic field energy.

We derive rigorously explicit formulae of the Casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by a zeta regularization method. We study both the low and high temperature expansions of the free energy. In each case, we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms. We show that the free energy is always a decreasing function of temperature. In the cases of massless scalar field with the Dirichlet boundary condition and electromagnetic field, the zero temperature Casimir free energy might be positive. In each of these cases, there is a unique transition temperature (as a function of the side lengths of the cavity) where the Casimir energy changes from positive to negative. When the space dimension is equal to two and three, we show graphically the dependence of this transition temperature on the side lengths of the cavity. Finally we also show that we can obtain the results for a non-closed rectangular cavity by letting the size of some directions of a closed cavity go to infinity, and we find that these results agree with the usual integration prescription adopted by other authors.

On behalf of the authors, Dr M Andrecut would like to apologise sincerely to the authors of the following two papers for the reproduction of a number of phrases throughout this work without citation:

Duarte M, Wakin M and Baraniuk R 2005 Fast reconstruction of piecewise smooth signals from random projections Proc. SPARS Workshop Preprint

Tropp J and Gilbert A 2005 Signal recovery from partial information via orthogonal matching pursuit Preprint