Table of contents

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, −σ/x, is also amenable for polynomial solutions.

We obtain the first passage time density for Lévy random processes (LRPs) from a subordination scheme, demonstrating that the first passage time density cannot be inferred uniquely from the probability density function P(x, t) governing the random process. This is due to the fact that P(x, t) does not contain all information on the trajectory of the underlying LRP.

Commuting periodic operators (CPO) depending on the coordinate and the momentum operators are defined. The CPO are functions of the two basic commuting operators and , with a being an arbitrary constant. A periodic Wigner function (PWF) w(x, p) is defined and it is shown that it is applicable in a normal expectation value calculation to the CPO, as done in the original Wigner paper. Moreover, this PWF is non-negative everywhere, and it can therefore be interpreted as an actual probability distribution. The PWF w(x, p) is shown to be given as an expectation value of the periodic Dirac delta function in the phase plane.

We propose a mean field theory for interfaces growing according to the Kardar–Parisi–Zhang (KPZ) equation in 1 + 1 dimensions. The mean field equations are formulated in terms of densities at different heights, taking surface tension and the influence of the nonlinear term in the KPZ equation into account. Although spatial correlations are neglected, the mean field equations still reflect the spatial dimensionality of the system. In the special case of Edwards–Wilkinson growth, our mean field theory correctly reproduces all features. In the presence of a nonlinear term one observes a crossover to a KPZ-like behaviour with the correct dynamical exponent z = 3/2. In particular we compute the skewed interface profile during roughening, and we study the influence of a co-moving reflecting wall, which has been discussed recently in the context of nonequilibrium wetting and synchronization transitions. Also here the mean field approximation reproduces all qualitative features of the full KPZ equation, although with different values of the surface exponents.

The Ullersma model for the damped harmonic oscillator is coupled to the quantized electromagnetic field. All material parameters and interaction strengths are allowed to depend on position. The ensuing Hamiltonian is expressed in terms of canonical fields, and diagonalized by performing a normal-mode expansion. The commutation relations of the diagonalizing operators are in agreement with the canonical commutation relations. For the proof we replace all sums of normal modes by complex integrals with the help of the residue theorem. The same technique helps us to explicitly calculate the quantum evolution of all canonical and electromagnetic fields. We identify the dielectric constant and the Green function of the wave equation for the electric field. Both functions are meromorphic in the complex frequency plane. The solution of the extended Ullersma model is in keeping with well-known phenomenological rules for setting up quantum electrodynamics in an absorptive and spatially inhomogeneous dielectric. To establish this fundamental justification, we subject the reservoir of independent harmonic oscillators to a continuum limit. The resonant frequencies of the reservoir are smeared out over the real axis. Consequently, the poles of both the dielectric constant and the Green function unite to form a branch cut. Performing an analytic continuation beyond this branch cut, we find that the long-time behaviour of the quantized electric field is completely determined by the sources of the reservoir. Through a Riemann–Lebesgue argument we demonstrate that the field itself tends to zero, whereas its quantum fluctuations stay alive. We argue that the last feature may have important consequences for application of entanglement and related processes in quantum devices.

Heat conduction in one-dimensional (1D) scattering models is studied based on numerical simulations and an analytical S-matrix method which is developed in the mesoscopic electronic transport theory. In the models, it is found that the heat conduction is closely related to a spatial correlation of particle motions: if the correlation exists, the heat conduction is abnormal; otherwise (i.e. if the correlation vanishes), the heat conduction is normal. The randomization of scatterers in the models is found to determine the existence of correlation. Our simulations are in agreement with the theoretical expectations. We generalize the result and study the property of heat conduction by directly analysing the correlation in general 1D dynamical systems.

We discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring. The dynamics are based on the Kuramoto model. In the case of identical oscillators, all phase locking solutions and their stability properties are obtained. For nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling. An algorithm to obtain all phase locking solutions is proposed. These solutions can be classified into classes, each with its own stability properties. The stability properties are obtained by means of a novel extension of Gershgorin's theorem. One class of stable solutions has the property that all phase differences between neighbouring cells are contained in . Contrary to intuition, a second class of stable solutions is established with exactly one of the phase differences contained in . The stability results are extended from sinusoidal interconnections to a class of odd functions. To conclude, a connection with the field of active antenna arrays is made, generalizing some results earlier obtained in this field.

Using the Riemann–Hilbert approach, the Ψ-function corresponding to the solution of the first Painlevé equation with the asymptotic behaviour as |x| → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the power-like expansion to the latter are found.

We consider N-soliton solutions of the KP equation, An N-soliton solution is a solution u(x, y, t) which has the same set of N line soliton solutions in both asymptotics y → ∞ and y → −∞. The N-soliton solutions include all possible resonant interactions among those line solitons. We then classify those N-soliton solutions by defining a pair of N numbers (n⁺, n⁻) with n^± = (n^±₁, ..., n^±_N), n^±_j {1, ..., 2N}, which labels N line solitons in the solution. The classification is related to the Schubert decomposition of the Grassmann manifolds Gr(N, 2N), where the solution of the KP equation is defined as a torus orbit. Then the interaction pattern of N-soliton solution can be described by the pair of Young diagrams associated with (n⁺, n⁻). We also show that N-soliton solutions of the KdV equation obtained by the constraint ∂u/∂y = 0 cannot have resonant interaction.

We introduce an S-function formulation for the recently found rth dispersionless modified KP and rth dispersionless Dym hierarchies, giving also a connection of these S-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the S-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic-type systems. In particular, for the one-component and two-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.

The dynamics of the ring dark soliton in a Bose–Einstein condensate (BEC) with thin disc-shaped potential is investigated analytically and numerically. Analytical investigation shows that the ring dark soliton in the radial non-symmetric cylindrical BEC is governed by a cylindrical Kadomtsev–Petviashvili equation, while the ring dark soliton in the radial symmetric cylindrical BEC is governed by a cylindrical Korteweg–de Vries equation. The reduction to the cylindrical KP or KdV equation may be useful to understand the dynamics of a ring dark soliton. The numerical results show that the evolution properties and the snaking of a ring dark soliton are modified significantly by the trapping.

We obtain solutions of the three-dimensional Dirac equation for radial power-law potentials at rest mass energy as an infinite series of square integrable functions. These are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the Dirac operator is tridiagonal. The 'wave equation' results in a three-term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials. These are modified versions of the Meixner–Pollaczek polynomials and of the continuous dual Hahn polynomials. The choice depends on the values of the angular momentum and the power of the potential.

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yield several sequences of upper and lower limits on the critical value, g^(ℓ)_c, of the coupling constant (strength), g, of the potential, V(r) = −gv(r), for which a first ℓ-wave bound state appears, which converges to the exact critical value.

We give a closed-form solution of von Neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in Hilbert spaces of two and three dimensions. We show that the same type of dependence also exists in higher dimensions albeit with other terms. For non-maximal mixing, the results become more involved and generally depend also on the probability of the states. We also outline a method for measuring both the entropy and the phase experimentally using a simple Mach–Zehnder-type interferometer which explains physically why the two concepts are related.

We choose the five-parameter exponential-type potential model as input, and construct five trigonometric-type potentials via point canonical transformations. Their energy spectra and wavefunctions are obtained in a unified manner by using the expressions for the energy spectra and wavefunctions of the five-parameter exponential-type potential model.

A simple model of indirect pre-measurements on an unstable quantum state is presented in this paper. The model is completely solvable and the solutions are used to compare the time evolution of the unstable state with and without the influence of the pre-measurement. We find that by choosing the details of the process of pre-measurement appropriately, it is possible to observe both suppression and enhancement of the rate of decay of the unstable state. When the pre-measurements are assumed to lead on to actual measurements, we see that the quantum Zeno effect, and in some instances the 'anti-Zeno' effect, can be produced by repeating the measurement many times in succession. The Zeno effect can appear in our model either as a real consequence of repeated measurements or sometimes merely as an artefact of the manner in which the observations on the system are performed. The anti-Zeno effect appears almost exclusively as an artefact of the details of the measurement. Numerical investigations are included to delineate the regimes in which the quantum Zeno effect and possibly the anti-Zeno effect can occur.

The author wishes to make some corrections. Please see the pdf for details.