Table of contents

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett.85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full 'sign problem'. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.

We evaluate the six-fold integral representation for the second-order exchange contribution to the self-energy of a dense three-dimensional electron gas on the Fermi surface.

The phonon distribution function in a weakly anharmonic one-dimensional chain in a thermal gradient is determined in the high temperature or, equivalently, classical regime. Thermal relaxation is dominated by 4-phonon scattering processes in this limit and these are treated using Peierls' Boltzmann equation approach. An analytical analysis of the Boltzmann equation for small wave vector k shows that the distribution is singular and that this singularity at k = 0 is of the form of an infinite sum of power law branch points with powers that are rational and dense on the real axis above some critical value. The value of the smallest discrete power implies that the thermal conductivity is infinite in the thermodynamic limit. For finite systems, whenever 4-phonon scattering is the dominant mechanism inhibiting energy transport, the thermal conductivity will scale as (system length)^2/5/(temperature)^6/5.

We consider a two-dimensional Fermi–Pasta–Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a reduction to a cubic nonlinear Schrödinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher order analysis yielding a generalized NLS, which includes known stabilizing terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary direction, an ellipticity criterion for the wavenumbers of the carrier wave, asymptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximized for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed. Several of the results obtained are similar to those obtained for the square FPU lattice (Butt and Wattis 2006 J. Phys. A: Math. Gen.39 4955), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalized NLS equation.

We review the construction of a q-analogue of the Gaussian measure. We apply that construction to obtain a q-analogue of Feynman integrals and to compute explicitly an example of such integrals.

In the scheme of a quantum nondemolition (QND) measurement, an observable is measured without perturbing its evolution. In the context of studies of decoherence in quantum computing, we examine the 'open' quantum system of a two-level atom or, equivalently, a spin-1/2 system, in interaction with quantum reservoirs of either oscillators or spins, under the QND condition of the Hamiltonian of the system commuting with the system–reservoir interaction. For completeness, we also examine the well-known non-QND spin-Bose problem. For all these many-body systems, we use the methods of functional integration to work out the propagators. The propagators for the QND Hamiltonians are shown to be analogous to the squeezing and rotation operators, respectively, for the two kinds of baths considered. Squeezing and rotation being both phase-space area-preserving canonical transformations, this brings out an interesting connection between the energy-preserving QND Hamiltonians and the homogeneous linear canonical transformations.

Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.

The intrinsic unsharpness of a quantum observable is studied by introducing the notion of resolution width. This quantification of accuracy is shown to be closely connected with the possibility of making approximately repeatable measurements. As a case study, the intrinsic unsharpness and approximate repeatability of position and momentum measurements are examined in detail.

We consider quasinormal modes with complex energies from the point of view of the theory of quasi-exactly solvable (QES) models. We demonstrate that it is possible to find new potentials which admit exactly solvable or QES quasinormal modes by suitable complexification of parameters defining the QES potentials. Particularly, we obtain one QES and four exactly solvable potentials out of the five one-dimensional QES systems based on the sl(2) algebra.

Within a well-known decay model describing a particle confined initially within a spherical δ potential shell, we consider the situation when the initial state has an unusual energy distribution decaying slowly as k → ∞; the simplest example corresponds to a wavefunction constant within the shell. We show that the non-decay probability as a function of time then behaves in a highly irregular, most likely fractal way.

We consider the general form of linear transformation for point rotation coordinate frames. The frames have the rotation axis at every point. In the transformation, the frequency of one frame relative to another is not equivalent to the reverse frequency. Using symmetry of the direct and reverse transformation as well as symmetry of the frame coordinates, we show that three different types of transformation are possible. The first type is a generalization of the Lorentz transformation. This case cannot be checked in optical measurements. In contrast to this, the second unusual type allows us to observe consequences of the transformation in an optical experiment even though the characteristic constant inherent in the transformation is about 'nuclear time' of the order of 10⁻²³ s. We describe the schematic of the experiment. The third type is a supplement to the second type. This type is not applicable to the electromagnetic field and cannot be verified optically.

The problem of a relativistic spinning particle interacting with pseudoscalar potentials in (3 + 1) dimensions is formulated in the framework of covariant supersymmetric path integrals. The relative Green's function is expressed through a functional integral over bosonic trajectories that describe the external motion and fermionic variables that describe the spin degrees of freedom. As an application, we have considered the case of the plane wave, where the pseudoscalar potential is an arbitrary function of the variable (k_μx^μ). The (3 + 1)-dimensional problem is reduced to a (1 + 1)-dimensional one by using an identity. For the case of k² = 0, the relative propagator is exactly calculated and the wavefunctions are extracted.

It is widely recognized that concurrence can be regarded as a measure of two-qubit entanglement. This paper studies the geometry of concurrence for a two-qubit system to show that the concurrence serves as a coordinate of the factor space , where is the space of normalized two-qubit states, and where G = U(1) × SU(2) × SU(2). Any monotonically increasing function of the concurrence can serve as a measure for entanglement. From the viewpoint of Riemannian geometry, a state with concurrence r is shown to be distant from the separable states by , where r ranges over 0 ⩽ r ⩽ 1. In addition, measures of entanglement for n-qubits are discussed on the basis of a bipartite decomposition with ℓ + m = n. They are invariant under the local unitary transformation group U(2^ℓ) × U(2^m).

The temporal evolution of quantum statistical properties of an interacting atom–radiation field system in the presence of a classical homogeneous gravitational field is investigated within the framework of the Jaynes–Cummings model. To analyse the dynamical evolution of the atom–radiation system a quantum treatment of the internal and external dynamics of the atom is presented based on an alternative su(2) dynamical algebraic structure. By solving the Schrödinger equation in the interaction picture, the evolving state of the system is found by which the influence of the gravitational field on the dynamical behaviour of the atom–radiation system is explored. Assuming that initially the radiation field is prepared in a coherent state and the two-level atom is in a coherent superposition of the excited and ground states, the influence of gravity on the collapses and revivals of the atomic population inversion, atomic dipole squeezing, atomic momentum diffusion, photon counting statistics and quadrature squeezing of the radiation field is studied.

The Villain transform plays a key role in spin-wave theory, a bosonization of elementary excitations in a system of extensively many Heisenberg spins. Intuitively, it is a representation of the spin operators in terms of an angle and its canonically conjugate angular momentum operator and, as such, has a few nasty boundary-condition twists. We construct an isometric phase representation of spin operators that conveys a precise mathematical meaning to the Villain transform and is related to both classical mechanics and the Pegg–Barnett–Bialynicki-Birula boson (photon) phase operators by means of suitable limits. In contrast to the photon case, unitary extensions are inadequate because they describe the wrong physics. We also discuss in some detail the application to spin-wave theory, pointing out some examples in which the isometric representation is indispensable.

The electromagnetic field generated by a charged particle moving along a helical orbit inside a dielectric cylinder immersed into a homogeneous medium is investigated. Expressions are derived for the electromagnetic potentials, electric and magnetic fields in the region inside the cylinder. The parts corresponding to the radiation field are separated. The radiation intensity on the lowest azimuthal mode is studied.

We have rigorously derived the perimeter-generating function for the mean-squared radius of gyration of staircase polygons. This function was first obtained by Jensen. His nonrigorous result is based on the analysis of the long series expansions.