Table of contents

We have devised a variational sinc collocation method (VSCM) which can be used to obtain accurate numerical solutions to many strong-coupling problems. Sinc functions with an optimal grid spacing are used to solve the linear and nonlinear Schrödinger equations and a lattice ϕ⁴ model in (1 + 1). Our results indicate that errors decrease exponentially with the number of grid points and that a limited numerical effort is needed to reach high precision.

A canonical form for a matrix product state representation of a general finitely correlated quantum state on a one-dimensional (finite or infinite) lattice is proposed by exploring the gauge symmetry of the matrix product. This representation is unique in the sense that it is the only one which generates canonical forms for all reduced density matrices of the state.

An exact nonsingular solitary wave solution of the Schäfer–Wayne short pulse equation is derived from the breather solution of the sine-Gordon equation by means of a transformation between these two integrable equations.

We provide a mathematical framework for -symmetric quantum theory, which is applicable irrespective of whether a system is defined on or a complex contour, whether symmetry is unbroken, and so on. The linear space in which -symmetric quantum theory is naturally defined is a Krein space constructed by introducing an indefinite metric into a Hilbert space composed of square integrable complex functions in a complex contour. We show that in this Krein space every -symmetric operator is -Hermitian if and only if it has transposition symmetry as well, from which the characteristic properties of the -symmetric Hamiltonians found in the literature follow. Some possible ways to construct physical theories are discussed within the restriction to the class K(H).

Rare regions, i.e., rare large spatial disorder fluctuations, can dramatically change the properties of a phase transition in a quenched disordered system. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. Stronger effects can be observed at zero-temperature quantum phase transitions, at nonequilibrium phase transitions and in systems with correlated disorder. In some cases, rare regions can actually completely destroy the sharp phase transition by smearing. This topical review presents a unifying framework for rare region effects at weakly disordered classical, quantum and nonequilibrium phase transitions based on the effective dimensionality of the rare regions. Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process.

We propose and discuss some toy models of stock markets using the same operatorial approach adopted in quantum mechanics. Our models are suggested by the discrete nature of the number of shares and of the cash which are exchanged in a real market, and by the existence of conserved quantities, like the total number of shares or some linear combination of cash and shares. The same framework as the one used in the description of a gas of interacting bosons is adopted.

The dynamic evolution at zero temperature of a uniform Ising ferromagnet on a square lattice is followed by Monte Carlo computer simulations. The system always eventually reaches a final, absorbing state, which sometimes coincides with a ground state (all spins parallel), and sometimes does not (parallel stripes of spins up and down). We initiate here the numerical study of 'chaotic time dependence' (CTD) by seeing how much information about the final state is predictable from the randomly generated quenched initial state. CTD was originally proposed to explain how nonequilibrium spin glasses could manifest an equilibrium pure state structure, but in simpler systems such as homogeneous ferromagnets it is closely related to long-term predictability and our results suggest that CTD might indeed occur in the infinite volume limit.

In the class of nonlinear one-parameter real maps, we study those with bifurcation that exhibits a period doubling cascade. The fixed points of such maps form a finite discrete real set of dimension 2ⁿm, where m is the (odd) number of 'primary branches' of the map in the non-chaotic region and n is a non-negative integer. A new special representation of these maps is constructed that should give more insight into the physical interpretation and enhance their applications in mathematical physics and nonlinear dynamics. We associate with the map a nonlinear dynamical system whose Hamiltonian matrix is real, tridiagonal and symmetric. The density of states of the system is calculated and shown to have a band structure. The number of density bands is equal to 2ⁿ⁻¹m unless n = 0 in which case the density has m bands. The location of the bands is independent of the initial state. It depends only on the map parameter and whether the ordering of the fixed points in the set is odd or even. Polynomials orthogonal with respect to this density (weight) function are constructed. The logistic map is taken as an illustrative example.

In one-dimensional transport problems, the scattering matrix S is decomposed into a block structure corresponding to reflection and transmission matrices at the two ends. For S a random unitary matrix, the singular value probability distribution function of these blocks is calculated. The same is done when S is constrained to be symmetric, or to be self-dual quaternion real, as is relevant in the presence of particular time reversal symmetries. The latter results are shown to be very similar to those obtained under the (unphysical) assumption that S has real elements, or has real quaternion elements, respectively. Three methods are used: metric forms, a variant of the Ingham–Seigel matrix integral and a theorem specifying the Jacobi random matrix ensemble in terms of Wishart distributed matrices.

Using internal negations acting on Boolean functions, the notion of Boolean–Lie algebra is introduced. The underlying Lie product is the Boolean analogue of the Poisson bracket. The structure of a Boolean–Lie algebra is determined; it turns out to be solvable, but not nilpotent. We prove that the adjoint representation of an element of the Boolean–Lie algebra acts as a derivative operator on the space of Boolean functions. The adjoint representation is related to the previously known concept of the sensitivity function. Using the notion of adjoint representation we give the definition of a temporal derivative applicable to iterative dynamics of Boolean mappings.

Symmetry preserving difference schemes approximating second- and third-order ordinary differential equations are presented. They have the same three- or four-dimensional symmetry groups as the original differential equations. The new difference schemes are tested as numerical methods. The obtained numerical solutions are shown to be much more accurate than those obtained by standard methods without an increase in cost. For an example involving a solution with a singularity in the integration region, the symmetry preserving scheme, contrary to standard ones, provides solutions valid beyond the singular point.

Given a Lie algebroid and a bundle over its base which is endowed with a localizable Poisson structure and a flat connection, we construct an extended bundle whose dual is endowed with an almost-Poisson structure that is a quadratic Poisson structure when a certain compatibility property is satisfied. This new formalism on Lie algebroids describes systems with internal degrees of freedom.

The method of the determination of the principal axes system (PAS) orientation with respect to the initial reference frame for any quadrupolar component of the crystal-field Hamiltonian is presented. The developed method is based on the extreme points of the axial crystal-field parameter map, B₂₀(α, β), where both the partial derivatives ∂B₂₀/∂α and ∂B₂₀/∂β simultaneously vanish, and α, β stand for the two Euler angles with respect to the initial reference frame. In general, there are three such points corresponding to a maximum, minimum and saddle point on the map. These particular points fix the orthogonal PAS in which three equivalent, two-parameter (B₂₀, B₂₂), orthorhombic-like parameterizations coexist according to the three options of the z-axis. Hence, the standardization of parameterizations becomes simplified and lies in the particular choice between the three forms, conventionally that with the maximal |B₂₀|.

The derivation of the Bäcklund transformations (BTs) is a standard problem of the theory of the integrable systems. Here, I discuss the equations describing the BTs for the Ablowitz–Ladik hierarchy (ALH), which have been already obtained by several authors. The main aim of this work is to solve these equations. This can be done in the framework of the so-called functional representation of the ALH, when an infinite number of the evolutionary equations are replaced, using the Miwa's shifts, with a few equations linking tau-functions with different arguments. It is shown that starting from these equations it is possible to obtain explicit solutions of the BT equations. In other words, the main result of this work is a presentation of the discrete BTs as a superposition of an infinite number of evolutionary flows of the hierarchy. These results are used to derive the superposition formulae for the BTs as well as pure soliton solutions.

For non-zero ℓ values, we present an analytical solution of the radial Schrödinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the asymptotic iteration method. The bound state energy eigenvalues and corresponding wavefunctions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov–Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.

A superoscillatory function—that is, a band-limited function f(x) oscillating faster than its fastest Fourier component—is taken to be the initial state of a freely-evolving quantum wavefunction ψ. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to ψ with complex momenta that are exponentially disparate in magnitude; this is established by applying the asymptotics of integrals, supported by numerics. f(x) can alternatively be regarded as the wave generated by a diffraction grating, propagating paraxially and without evanescence as ψ in the space beyond. The persistence of superoscillations is then interpreted as the propagation of sub-wavelength structure farther into the field than the more familiar evanescent waves.

We reinvestigate the question of whether or not the dynamical symmetries of the two-dimensional interacting boson model can be transformed into each other by the quantum deformation of the algebraic structure. The deformation of both the vibrational and the rotational spectra is considered, and the breaking of the classical symmetry is measured quantitatively, while a systematic search for the best fit is performed. In no case can the quantum-deformed spectrum reach the other classical limit; only a modest improvement is observed with the phase deformation of the rotational spectrum.

A two-dimensional Pauli Hamiltonian describing the interaction of a neutral spin-1/2 particle with a magnetic field having axial and second-order symmetries is considered. After separation of variables, the one-dimensional matrix Hamiltonian is analysed from the point of view of supersymmetric quantum mechanics. Attention is paid to the discrete symmetries of the Hamiltonian and also to the Hamiltonian hierarchies generated by intertwining operators. The spectrum is studied by means of the associated matrix shape invariance. The relation between the intertwining operators and the second-order symmetries is established, and the full set of ladder operators that complete the dynamical algebra is constructed.

In this paper, we investigate the relation between the curvature of the physical space and the deformation function of the deformed oscillator algebra using the nonlinear coherent states approach. For this purpose, we study two-dimensional harmonic oscillators on the flat surface and on a sphere by applying the Higgs model. With the use of their algebras, we show that the two-dimensional oscillator algebra on a surface can be considered as a deformed one-dimensional oscillator algebra where the effect of the curvature of the surface appears as a deformation function. We also show that the curvature of the physical space plays the role of deformation parameter. Then we construct the associated coherent states on the flat surface and on a sphere and compare their quantum statistical properties, including quadrature squeezing and antibunching effect.

The mathematical structure of the reflection coefficients for the one-dimensional Fokker–Planck equation is studied. A new formalism using differential operators is introduced and applied to the analysis in the high- and low-energy regions. Formulae for high-energy and low-energy expansions are derived, and expressions for the coefficients of the expansion, as well as the remainder terms, are obtained for general forms of the potential. Conditions for the validity of these expansions are discussed on the basis of the analysis of the remainder terms.

In the present paper we compute the propagator of a quantum mechanical system whose Hamiltonian consists of two commuting terms, the spin–orbit coupling being one of them. We assume that the propagator corresponding to the first part can be cast into a closed form. A detailed treatment is given when such term is set as the simple harmonic oscillator. Some applications are also included.

We construct an Abelian algebra built by maximally entangled pure states. The partial transposition maps this algebra for odd dimensions into a full matrix algebra. ppt-states are in one-to-one correspondence to states with a positive definite Wigner function. Special extremal ppt-states correspond to projections of various dimensions. In particular, we recover the projections corresponding to a complete set of mutually unbiased bases in prime dimensions.

The present paper analyses the relation between the theory of the time-dependent wave operator and the Berry phase concept. It is proved that the wave operator approach is consistent with the non-adiabatic (Aharonov–Anandan) Berry phase, given that the wave operator and the parallel transport commute. It is then demonstrated that the non-Abelian Aharonov–Anandan phase can be calculated by working inside a reduced active space in the framework of wave operator theory. Finally an adiabatic transport formula is derived in the wave operator context and the influence of this effective Hamiltonian theory on the Berry phase is analysed. The theoretical results concerning the non-adiabatic Berry phase are confirmed numerically by considering a photodissociation process in the framework of the generalized Floquet theory.

We proceed with the construction of normalizable Dirac wave packets for treating chiral oscillations in the presence of an external magnetic field. Both chirality and helicity quantum numbers correspond to variables of fundamental importance in the study of chiral interactions, in particular, in the context of neutrino physics. In order to clarify a subtle aspect in the confront of such concepts which, for massive particles, represent different physical quantities, we are specifically interested in quantifying chiral oscillations for a fermionic Dirac-type particle (neutrino) non-minimally coupling with an external magnetic field B by solving the corresponding interacting Hamiltonian (Dirac) equation. The viability of the intermediate wave-packet treatment becomes clear when we assume B orthogonal/parallel to the direction of the propagating particle.

Dirac's operator and Maxwell's equations in vacuum are derived in the algebra of split octonions. The approximations which lead to classical Maxwell–Heaviside equations from full octonionic equations are given. The non-existence of magnetic monopoles in classical electrodynamics is connected with the use of the associativity limit.

In this paper, a supersymmetric extension of the scalar Born–Infeld equation is constructed through a superspace formalism which involves the addition of one independent fermionic Grassmann variable to the existing bosonic spacetime coordinates. The bosonic scalar field is replaced by a fermionic superfield which is composed of two component fields, one bosonic and one fermionic. The resulting equation is invariant under a space supersymmetric transformation and, in its most general form, involves four arbitrary parameters. For a certain specific case, the Lie superalgebra of symmetries was identified, and the one-dimensional subalgebras were systematically classified into splitting and non-splitting conjugate classes. A number of group-invariant solutions were obtained, including polynomial solutions, solutions by radicals and solitary waves (including bumps, kinks and doubly periodic solutions).

Spin and charge currents in systems with Rashba or Dresselhaus spin–orbit couplings are formulated in a unified version of four-dimensional SU(2) × U(1) gauge theory, with U(1) being the Maxwell field and SU(2) being the Yang–Mills field. While the bare spin current is non-conserved, it is compensated by a contribution from the SU(2) gauge field, which gives rise to a spin torque in the spin transport, consistent with the semi-classical theory of Culcer et al. Orbit current is shown to be non-conserved in the presence of electromagnetic fields. Similar to the Maxwell field inducing forces on charge and charge current, we derive forces acting on spin and spin current induced by the Yang–Mills fields such as the Rashba and Dresselhaus fields and the sheer strain field. The spin density and spin current may be considered as a source generating Yang–Mills field in certain condensed matter systems.

Fluctuations of the electromagnetic field produced by quantized matter in an external electric field are investigated. A general expression for the power spectrum of fluctuations is derived within the long-range expansion. It is found that in the whole measured frequency band, the power spectrum of fluctuations exhibits an inverse frequency dependence. A general argument is given showing that for all practically relevant values of the electric field, the power spectrum of induced fluctuations is proportional to the field strength squared. As an illustration, the power spectrum is calculated explicitly using a kinetic model with a relaxation-type collision term. Finally, it is shown that the magnitude of fluctuations produced by a sample generally has a Gaussian distribution around its mean value, and its dependence on the sample geometry is determined. In particular, it is demonstrated that for geometrically similar samples the power spectrum is inversely proportional to the sample volume. Application of the results obtained to the problem of flicker noise is discussed.

The problem of variable separation in the linear stability equations, which govern the disturbance behaviour in viscous incompressible fluid flows, is discussed. The so-called direct approach, in which a form of the 'ansatz' for a solution with separated variables as well as a form of reduced ODEs are postulated from the beginning, is applied. The results of application of the method are the new coordinate systems and the most general forms of basic flows, which permit the postulated form of separation of variables. Then the basic flows are specified by the requirement that they themselves satisfy the Navier–Stokes equations. Calculations are made for the (1+3)-dimensional disturbance equations written in Cartesian and cylindrical coordinates. The fluid dynamics interpretation and stability properties of some classes of the exact solutions of the Navier–Stokes equations, defined as flows for which the stability analysis can be reduced via separation of variables to the eigenvalue problems of ordinary differential equations, are discussed. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. For some classes of perturbations, the eigenvalue problems can be solved analytically. Those unique examples of exact (explicit) solution of the nonparallel unsteady flow stability problems provide a very useful test for numerical methods of solution of eigenvalue problems, and for methods used in the hydrodynamic stability theory, in general.

In this paper, the nonlinear propagation of the dust-acoustic waves in a strongly coupled dusty plasma with two-temperature nonthermal ions and transverse perturbations is governed by a cylindrical Kadomtsev–Petviashvili–Burgers (KP–Burgers) equation. With the help of the variable-coefficient generalized projected Ricatti equation expansion method, the cylindrical KP–Burgers equation is solved and a shock wave solution is obtained. The effects on the amplitude of the shock wave caused by some important parameters such as ion nonthermal parameter a and temperature parameters β₁, β, etc are shown. The effects caused by dissipation and transverse perturbations are also discussed. It also indicates that the dust density hole can form and enlarge as time goes on.

The authors have discovered some errors in the references of their paper. For corrections, please refer to the PDF of this corrigendum.