Table of contents

We consider the specific models of Zhu–Kroemer and BenDaniel–Duke in a sech²-mass background and point out interesting correspondences with the stationary 1-soliton and 2-soliton solutions of the KdV equation in a supersymmetric framework.

For a maximally entangled eigenstate of a system of two non-interacting identical one-dimensional harmonic oscillators, at the semiclassical level, it is not obviously true that a nonlinear interaction with one of the subsystems leaves the reduced semiclassical Wigner function of the other subsystem unaffected. Once stated, we advance various speculations regarding this puzzle.

Analytic expressions for the probability density distribution of the linear entropy and the purity are derived for bipartite pure random quantum states. The explicit distributions for a state belonging to a product of Hilbert spaces of dimensions p and q are given for p = 3 and any q ⩾ 3, as well as for p = q = 4.

It is commonly thought that a state-dependent quantity, after being averaged over a classical ensemble of random Hamiltonians, will always become independent of the state. We point out that this is in general incorrect: if the ensemble of Hamiltonians is time-reversal invariant, and the quantity involves the state in higher than bilinear order, then we show that the quantity is only a constant over the orbits of the invariance group on the Hilbert space. Examples include fidelity and decoherence in appropriate models.

We investigate solutions to the Bethe equations for the isotropic S = 1/2 Heisenberg chain involving complex, string-like rapidity configurations of arbitrary length. Going beyond the traditional string hypothesis of undeformed strings, we describe a general procedure to construct eigenstates including strings with generic deformations, discuss general features of these solutions and provide a number of explicit examples including complete solutions for all wavefunctions of short chains. We finally investigate some singular cases and show from simple symmetry arguments that their contribution to zero-temperature correlation functions vanishes.

In this paper, we study the asymptotic distributions of the heterogeneous random coagulation-fragmentation processes (HCFP) which model the coagulation, fragmentation and diffusion of clusters of particles on a lattice. Based on a closed form of the stationary distribution for the HCFP, we prove that the mutually dependent clusters with a finite size (finite particles) in the sub-critical stage will become independent in the critical and super-critical stages, and the asymptotic distributions of the number of clusters may converge to Gaussian or Poisson distribution according to its size, to be small or large in the critical and super-critical stages.

We present an algorithm for enumerating exactly the number of Hamiltonian chains on regular lattices in low dimensions. By definition, these are sets of k disjoint paths whose union visits each lattice vertex exactly once. The well-known Hamiltonian circuits and walks appear as the special cases k = 0 and k = 1, respectively. In two dimensions, we enumerate chains on L × L square lattices up to L = 12, walks up to L = 17 and circuits up to L = 20. Some results for three dimensions are also given. Using our data we extract several quantities of physical interest.

This paper proposes a new algorithm for identifying backbones in the application of percolation theory. This algorithm is based on the current-carrying definition of backbone and is carried out on the predetermined spanning cluster. It is fairly easy to implement and further parallelize. The efficiency is enhanced by the fact that the conductivity of a percolating system can be obtained in the same processing of backbone identification. The critical exponents of backbone mass, red bonds (sites) and conductivity obtained by this algorithm are in very good agreement with the existing results.

Linear stability characteristics of an anomalous sub-diffusive activator–inhibitor system are investigated through a mesoscopic model, where anomaly ensues by a memory integro-differential operator with an equal anomaly exponent for all species. It is shown that monotonic instability, known from normal diffusion, persists in the anomalous system, thereby allowing Turing pattern occurrence. Presence of anomaly stabilizes the system by diminution of the range of diffusion coefficients' ratio involving instability. In addition, the maximal growth rate is diminished, proving the existence of an absolutely stable anomalous system.

We construct lattice parafermions—local products of order and disorder operators—in nearest-neighbor Z_N models on regular isotropic planar lattices and show that they are discretely holomorphic, that is they satisfy discrete Cauchy–Riemann equations, precisely at the critical Fateev–Zamolodchikov (FZ) integrable points. We generalize our analysis to models with anisotropic interactions, showing that, as long as the lattice is correctly embedded in the plane, such discretely holomorphic parafermions exist for particular values of the couplings which we identify as the anisotropic FZ points. These results extend to more general inhomogeneous lattice models as long as the covering lattice admits a rhombic embedding in the plane.

Based on the heat bath system approach where the bath is nonlinearly modulated by an external Gaussian random force, we propose a new microscopic model to study directed motion in the overdamped limit for a nonequilibrium open system. Making use of the coupling between the heat bath and the external modulation as a small perturbation, we construct a Langevin equation with multiplicative noise- and space-dependent dissipation and the corresponding Fokker–Planck–Smoluchowski equation in the overdamped limit. We examine the thermodynamic consistency condition and explore the possibility of observing a phase-induced current as a consequence of state-dependent diffusion and, necessarily, nonlinear driving of the heat bath by the external noise.

The dynamic-mechanical response of a polymer network has been calculated using a stress-based Rouse model formalism. In contrast to the previous work, this improved formulation incorporates appropriate boundary conditions and provides a smooth crossover from the classical equilibrium result of rubber elasticity to the short time-scale relaxation. We develop a consistent implementation of the classical tube model, which is merged with the Rouse dynamics to take into account the entanglement effects. In a polymer network, crosslinks prevent the global reptation and constraint release. Entanglements thus acquire a different topological meaning and have a much stronger effect on the resulting mechanical response. We construct a dynamic stress tensor for a polymer network, which naturally covers the whole frequency/time range. Using this stress tensor, we first examine the equilibrium response to small shear and uniaxial deformations, and then investigate the linear dynamic response of a network for all the cases where the stress–tensor computations are analytically tractable.

We study centred second order in time and space discretizations of the inviscid Burgers equation. Although this equation in its continuum formulation supports non-smooth shock wave solutions, the discrete equation generically supports smooth solitary wave solutions. Using backward error analysis we derive the modified equation associated with the numerical scheme. We identify three different equations, the Korteweg–de Vries (KdV) equation, the Camassa–Holm (CH) equation and the b = 0 member of the b-family. Solutions of the first two equations are solitary waves and do not converge to the shock solutions of the Burgers equation. The third equation however supports solutions which strongly approximate weak solutions of the Burgers equation. We corroborate our analytical results with numerical simulations.

In this paper we apply the machinery of the previous paper of this series to results reported in our first paper devoted to these topics: in particular, we extend and prove certain Diophantine findings due to Christophe Smet reported there.

We analyse under which conditions the missing label problem associated with a reduction chain of (simple) Lie algebras can be completely solved by means of an Inönü–Wigner contraction naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labelling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semi-simple algebras.

The multipole moments of a homogeneously polarized ellipsoid and a homogeneously charged ellipsoidal disc are calculated. The resulting hypergeometric functions are expressed as finite polynomials of the semi-axes a, b and c of the ellipsoid. The polynomial form exists for any order of the multipole moments. It is shown that the solution also applies to a two-domain ellipsoid with antiparallel polarized domains and to a system with radially changing polarization density. The results allow us to calculate the potential as well as interaction energies within the framework of multipole expansion.

In the frame of a vectorial Pauli algebraic approach it is shown that the product of two exponentials of any two-by-two linear operators can be put in a single-exponential closed form. As a first application, a compact vectorial expression for the characteristics (angle and axis) of the product of two R³ rotations is established. The same mathematics can be used in a large diversity of problems of the whole class of two-states physical systems. An exemplification in the field of polarization optics is given.

We show that the ground state of a Bose–Einstein condensate of atoms with hyperfine spin f = 2 can be either spin aligned, condensed into pairs of atoms coupled to F = 0 or condensed into triplets of atoms coupled to F = 0. The complete phase diagram is constructed for f = 2 and the generic properties of the phase diagram are obtained for f > 2.

We consider the Hamiltonian of a system of three quantum mechanical particles (two identical fermions and a boson) on the three-dimensional lattice interacting by means of zero-range attractive potentials. We describe the location and structure of the essential spectrum of the three-particle discrete Schrödinger operator H_γ(K), K being the total quasi-momentum belonging to the three-dimensional torus and γ > 0 the ratio of the mass of fermion to boson. We choose for γ > 0 the interaction μ(γ) in such a way that the system consisting of one fermion and one boson has a zero-energy resonance. For all nonzero values of the quasi-momentum , we prove the finiteness of the number N(K, γ; τ_γ(K)) of eigenvalues of H_γ(K) below the bottom τ_γ(K) of the essential spectrum and we give for N(K, γ; 0) an asymptotics as K → 0. Moreover, we prove the existence of infinitely many eigenvalues of the operator H_γ(0) and give for the number N(0, γ; z) of eigenvalues lying below z < 0 an asymptotics as z → 0.

We obtain an analytic solution of the time-independent Schrödinger equation in two dimensions for a charged particle moving in the field of an electric quadrupole. The solution is written as a series in terms of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. This solution is for all energies, bound as well as scattering states. The expansion coefficients of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. The charged particle could become bound to the quadrupole only if its moment exceeds a certain critical value.

It is shown that the spectral points (bound states and resonances) generated by a central potential of a single-channel problem, can be found using rational parametrization of the S-matrix. To achieve this, one only needs values of the S-matrix along the real positive energy axis. No calculations of the S-matrix at complex energies or a complex rotation are necessary. The proposed method is therefore universal in that it is applicable to any potential (local, non-local, discontinuous, etc) provided that there is a way of obtaining the S-matrix (or scattering phase shifts) at real collision energies. Besides this, combined with any method that extracts the phase shifts from the scattering data, the proposed rational parametrization technique would be able to do the spectral analysis using the experimental data.

We study entanglement in the scattering processes by the spin-fixed impurity and the Kondo impurity, respectively. The spin-fixed impurity is employed as a filter to concentrate entanglement. One Kondo impurity can create entanglement between two noninteracting particles and one scattered particle can do so between two noninteracting Kondo impurities.

We study some quantum systems described by noncanonical commutation relations formally expressed as , where is the associated (harmonic-oscillator-like) Hamiltonian of the system and is a Hermitian (constant) operator, i.e. . In passing, we also consider a simple ( canonical) model, in the framework of a relativistic Klein–Gordon-like wave equation.

In our recent paper (Szmytkowski 2006 J. Phys. A: Math. Gen.39 15147; 2007 J. Phys. A: Math. Theor.40 7819 (corrigendum)), we have investigated the derivative of the Legendre function of the first kind, P_ν(z), with respect to its degree. In this addendum, we derive some further representations of [∂P_ν(z)/∂ν]_ν=n with . The results are used to obtain in an elementary way two formulae for the Legendre function of the second kind of integer degree, Q_n(z).