W.W. Zachary

A new class of Banach spaces

In this paper, we construct a new class of separable Banach spaces , for 1 ≤ p ≤ ∞, each of which contains all of the standard Lp spaces, as well as the space of finitely additive measures, as compact dense embeddings. Equally important is the fact that these spaces contain all Henstock–Kurzweil integrable functions and, in particular, the Feynman kernel and the Dirac measure, as norm bounded elements. As a first application, we construct the elementary path integral in the manner originally intended by Feynman. We then suggest that is a more appropriate Hilbert space for quantum theory, in that it satisfies the requirements for the Feynman, Heisenberg and Schrodinger representations, while the conventional choice only satisfies the requirements for the Heisenberg and Schrodinger representations. As a second application, we show that the mixed topology on the space of bounded continuous functions, , used to define the weak generator for a semigroup T(t), is stronger than the norm topology on . (This means that, when extended to is strongly continuous, so that the weak generator on becomes a strong generator on .)

Analytic representation of the Dirac equation

In this paper, we construct an analytical separation (diagonalization) of the full (minimal coupling) Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wavefunctions, as is done by the Foldy–Wouthuysen method, and reveals the nonlocal time behaviour of the particle–antiparticle relationship. We then show explicitly that the Pauli equation is not completely valid for the study of the Dirac hydrogen atom problem in s-states (hyperfine splitting). We conclude that there are some open mathematical problems with any attempt to explicitly show that the Dirac equation is insufficient to explain the full hydrogen spectrum. If the perturbation method can be justified, our analysis suggests that the use of cut-offs in QED is already justified by the eigenvalue analysis that supports it. Using a new method, we are able to effect separation of variables for full coupling, solve the radial equation and provide graphs of the probability density function for the 2p- and 2s-states, and compare them with those of the Dirac–Coulomb case.

Nonlinear pulse propagation in elliptically birefringent optical fibers

Abstract In the framework of a variational approach, we analyze nonlinear pulse propagation in a two-component nonlinear Schrodinger model of an elliptically birefringent optical fiber with reference to soliton switching. Through our approach, we also discuss stability of solitons with different polarizations and obtain a threshold condition for the soliton amplitude as a function of birefringence to bypass instability corresponding to a phase mismatch in soliton positions.

Dimensionality of invariant sets for nonautonomous processes

The existence of global attractors and estimates of their dimensions have been investigated by various authors for a number of dissipative nonlinear partial differential equations which are either autonomous or are subject to time-periodic forcing. In the presence of more general forcing (e.g., almost periodic but not periodic), the usual estimates of the dimensionality of global attractors in terms of uniform (or global) Lyapunov exponents are not valid. This article investigates the estimation of Hausdorff and fractal dimensions of invariant sets corresponding to differential equations of the above type, subject to time-dependent forcing of a quite general class. Working in the framework of skew-product semiflows associated with these equations, the authors consider invariant sets defined in terms of global attractors of semigroups determined by these semiflows. In autonomous situations these invariant sets coincide with the usual global attractors. Upper bounds for the Hausdorff and fractal dimensions ...

Chirped solitary pulses in low birefringent optical fibres: a theoretical model

Non-linear coupling of polarized solitons with initial linear frequency chirp in a low birefringent optical fibre is analysed on the basis of a theoretical model of coupled non-linear Schrödinger equations. A threshold condition is obtained for the bound state of two polarizations corresponding to a phase mismatch in solition positions.

Letter polarization dynamics of stable vector solitons in a birefringent single-mode optical fibre

Polarization dynamics of the vector solitons in a birefringent single-mode optical fibre in the framework of a system of coupled nonlinear Schrodinger equations with oscillating terms has been studied, and a condition for their stability has been deduced. This study should evoke considerable interest because of its potential applications in optical logic devices.