Table of contents

It is shown, through an elementary quantum mechanical calculation, that two particles interacting via a short range repulsive force in an external periodic potential can form a bound state. The two-particle wavefunction is labelled by a continuous centre-of-mass momentum. It is bounded and spatially localized in the centre-of-mass system; thus, the spatial wavefunction in the relative distance is square integrable and corresponds to a discrete energy. For instance, a combination of short-range (i.e. screened) binary Coulomb interactions and the periodic potential provided by the stationary ions, can create a two-electron bound state in a crystalline solid (Slater et al 1953 Phys. Rev.91 1323 and Hubbard 1963 Proc. R. Soc. A 276 238). However, the phenomenon delineated here is universal in the sense that, under appropriate conditions, bound states are possible independent of the nature of the particles and/or the mechanism by which the external periodic potential is engineered. Our general wave mechanical result may explain experimental results presenting evidence of such bound pair states in solids (Gross et al 1971 JETP Lett.13) and photonic lattices (Winkler et al 2006 Nature441 853). It has many other potentially interesting consequences even for classical interacting wave systems (e.g. solitons) propagating in a periodic background. This result of wave mechanics and interference is remarkable in that two repulsively interacting particles cannot form a bound state when moving in vacuum. Two non-interacting particles moving in a periodic external potential can only ever form uncorrelated two-particle Bloch states and yet when both physical conditions are present they can move as a 'bound pair'.

We introduce a real-space renormalization group procedure for driven diffusive systems which predicts both steady state and dynamic properties. We apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram, as well as the crossovers in the relaxation dynamics for each phase.

The Maslov index is a topological property of periodic orbits of finite-dimensional Hamiltonian systems that is widely used in semiclassical quantization, quantum chaology, stability of waves and classical mechanics. The Maslov index is determined from the analysis of a linear Hamiltonian system with periodic coefficients. In this paper, a numerical scheme is devised to compute the Maslov index for hyperbolic linear systems when the phase space has a low dimension. The idea is to compute on the exterior algebra of the ambient vector space, where the Lagrangian subspace representing the unstable subspace is reduced to a line. When the exterior algebra is projectified the Lagrangian subspace always forms a closed loop. The idea is illustrated by application to Hamiltonian systems on a phase space of dimension 4. The theory is used to compute the Maslov index for the spectral problem associated with periodic solutions of the fifth-order Korteweg de Vries equation.

It is known that many physical systems which do not exhibit deterministic chaos when treated classically may exhibit such behaviour if treated from the quantum mechanics point of view. In this paper, we will show that an annihilation operator of the unforced quantum harmonic oscillator exhibits distributional chaos as introduced in B Schweizer and J Smítal (1994 Trans. Am. Math. Soc.344 737–54). Our approach strengthens previous results on chaos in this model and provides a very powerful tool to measure chaos in other (quantum or classical) models.

We derive completeness criteria for sequences of functions of the form f(xλ_n), where λ_n is the nth zero of a suitably chosen entire function. Using these criteria, we construct complete nonorthogonal systems of Fourier–Bessel functions and their q-analogues, as well as other complete sets of q-special functions. We discuss connections with uncertainty principles over q-linear grids and the completeness of certain sets of q-Bessel functions is used to prove that, if a function f and its q-Hankel transform both vanish at the points {q⁻ⁿ}^∞_n=1, 0 < q < 1, then f must vanish on the whole q-linear grid {qⁿ}^∞_n=−∞.

I adapt a recently introduced method for integrating over the unitary group (Aubert S and Lam C S 2003 J. Math. Phys.44 6112–31) to the orthogonal group. I derive explicit formulae for a number of one, two and three-vector integrals, as well as recursion formulae for more complicated cases.

The Riemann–Hilbert problem proposed in [2] for the integrable stimulated Raman scattering (SRS) model was shown to be solvable under an additional condition: the boundary data have to be chosen in such a way that a corresponding spectral problem has no spectral singularities. In the general case, it can be shown that a spectral singularity occurs at k = 0. On the other hand, the initial boundary value (IBV) problem for the SRS equations is known to be well posed: using PDE techniques, this has been established in [3]. Therefore, it seems natural to try to find a new RH problem that is solvable in the presence of arbitrary spectral singularities. The formulation of such a RH problem is the main aim of the paper. Then the solution of the nonlinear initial boundary value problem for the SRS equations is expressed in terms of the solution of a linear problem which is the Riemann–Hilbert problem for a sectionally analytic matrix function.

In this paper we define a new kind of quantized enveloping algebra of a generalized Kac–Moody algebra . We denote this algebra by . It is a noncommutative and noncocommutative weak Hopf algebra. It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of .

Two non-isospectral modified KdV equations with self-consistent sources are derived, which correspond to the time-dependent spectral parameter λ satisfying λ_t = λ and λ_t = λ³, respectively. Gauge transformation between the first non-isospectral equation (corresponding to λ_t = λ) and its isospectral counterpart is given, from which exact solutions and conservation laws for the non-isospectral one are easily listed. Besides, solutions to the two non-isospectral modified KdV equations with self-consistent sources are derived by means of the Hirota method and the Wronskian technique, respectively. Non-isospectral dynamics and source effects, including one-soliton characteristics in non-uniform media, two-solitons scattering and special behaviours related to sources (for example, the 'ghost' solitons in the degenerate two-soliton case), are investigated analytically.

We demonstrate a method to characterize the general Heisenberg Hamiltonian with non-uniform couplings by mapping the entanglement it generates as a function of time. Identification of the Hamiltonian in this way is possible as the coefficients of each operator control the oscillation frequencies of the entanglement function. The number of measurements required to achieve a given precision in the Hamiltonian parameters is determined and an efficient measurement strategy designed. We derive the relationship between the number of measurements, the resulting precision and the ultimate discrete error probability generated by a systematic mis-characterization. This has important implications when implementing two-qubit gates for fault-tolerant quantum computation.

A new quantum model with rational functions for the potential and effective mass is proposed in a stretchable region outside which both are constant. Starting from a generalized effective mass kinetic energy operator the matching and boundary conditions for the envelope wavefunctions are derived. It is shown that in a mapping to an auxiliary constant-mass Schrödinger picture one obtains one-period 'associated Lamé' well bounded by two δ-wells or δ-barriers depending on the values of the ordering parameter β. The results for bound states of this new solvable model are provided for a wide variation of the parameters.

For one-dimensional non-relativistic quantum mechanical problems, we investigate the conditions for all the position dependence of the propagator to be in its phase, that is, for the semi-classical approximation to be exact. For velocity-independent potentials we find that: (i) the potential must be quadratic in space, but can have arbitrary time dependence, (ii) the phase may be made proportional to the classical action, and the magnitude ('fluctuation factor') can also be found from the classical solution and (iii) for the driven harmonic oscillator the fluctuation factor is independent of the driving term.

A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, such as for example Airy's equation. In this paper, we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross–Pitaevskii equation describing the macroscopic wavefunction of a Bose–Einstein condensate. The nonlinear Schrödinger equation is mapped to the second Painlevé equation (P_II), which is one of the best-known differential equations with a cubic nonlinearity. A quantization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.

The dynamical equations of an electromagnetic field coupled with a conducting material are studied. The properties of the interaction are described by a classical field theory with tensorial material laws in spacetime geometry. We show that the main features of superconducting response emerge in a natural way within the covariance, gauge invariance and variational formulation requirements. In particular, the Ginzburg–Landau theory follows straightforwardly from the London equations when fundamental symmetry properties are considered. Unconventional properties, such as the interaction of superconductors with electrostatic fields are naturally introduced in the geometric theory, at a phenomenological level. The BCS background is also suggested by macroscopic fingerprints of the internal symmetries. It is also shown that dissipative conducting behaviour may be approximately treated in a variational framework after breaking covariance for adiabatic processes. Thus, nonconservative laws of interaction are formulated by a purely spatial variational principle, in a quasi-stationary time discretized evolution. This theory justifies a class of nonfunctional phenomenological principles, introduced for dealing with exotic conduction properties of matter (Badía and López 2001 Phys. Rev. Lett.87 127004).

In this paper, we study the stability of the space of asymptotic fermion states in (2+1)D, when long range interparticle interactions are present. This is done in the framework of bosonization, where the fermion propagator can be represented in terms of a vortex correlator. In particular, we discuss possible instabilities in the large distance behaviour of the induced action for the vortex worldline.

Five anticommuting property coordinates can accommodate all the known fundamental particles in their three generations plus more. We describe the points of difference between this scheme and the standard model and show how flavour mixing arises through a set of expectation values carried by a single Higgs superfield.

The boundary theory for the c = −2 triplet model is investigated in detail. In particular, we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their OPE coefficients explicitly. We also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization.

We define the dual of a set of generators of the fundamental group of an oriented 2-surface S_g,n of genus g with n punctures and the associated surface S_g,n ∖ D with a disc D removed. This dual is another set of generators related to the original generators via an involution and has the properties of a dual graph. In particular, it provides an algebraic prescription for determining the intersection points of a curve representing a general element of the fundamental group π₁(S_g,n ∖ D) with the representatives of the generators and the order in which these intersection points occur on the generators. We apply this dual to the moduli space of flat connections on S_g,n and show that when expressed in terms of both, the holonomies along a set of generators and their duals, the Poisson structure on the moduli space takes a particularly simple form. Using this description of the Poisson structure, we derive explicit expressions for the Poisson brackets of general Wilson loop observables associated with closed, embedded curves on the surface and determine the associated flows on phase space. We demonstrate that the observables constructed from the pairing in the Chern–Simons action generate infinitesimal Dehn twists and show that the mapping class group acts by Poisson isomorphisms.

Recently, a general proof was given (Tautz et al 2006 J. Phys. A: Math. Gen.39 13831) that for an asymmetric relativistic particle phase-space distribution function, and in the absence of a homogeneous background magnetic field, any unstable linear Weibel modes are isolated, i.e. restricted to discrete wavenumbers. In this paper, for a specific distribution function consisting of mono-energetic counterstreaming electron and positron beams, growth rates and associated wavenumbers for the isolated modes are calculated, proving the existence of discrete values for unstable wavenumbers. Furthermore, electrostatic and electromagnetic Weibel modes are investigated for monoenergetic counterstreaming plasmas, yielding constraints to the momentum components that have to be fulfilled in order to have unstable wave modes.