| Recently joined a new MITACS research group whose main focus is to improve semiconductor manufacturing through scientific modeling. The formal title is: Mathematical and Computational Modeling of Semi-Conductor Manufacturing Processes. |
| Invited Talk: Mathematical Modelling of Crystal Growth. Presented at the Third Annual MITACS AGM, University of British Columbia, Vancouver. |
| Some photos from the MITACS AGM lectures can be found here. |
| Statement of Research Interests DATE: December 1999 |
1 Research Accomplishments
1.1 Wigner transform and Wigner-Poisson system This work which appears in [8] discusses the derivation of the quantum Liouville equation and the Wigner-Poisson system (or quantum Vlasov equation) from elementary quantum mechanical principles and the application of the Wigner transformation. The point of this work was to illustrate the physical content of the Wigner function as well as to derive the quantum Liouville equation. Although the work is semi-formal, it can be made rigorous as in [16] with minimal effort. It replaces the somewhat heuristic arguments of [13] and [15].
1.2 Newtonian limit of axisymmetric spacetimes In collaboration with G.P. Perry [17], we illustrated how Ehlers' formal mathematical definition of the Newtonian limit of a particular spacetime [11] requires additional information to successfully determine the Newtonian limit. It was shown that this information can be obtained through the physical arguments of presented by Cooperstock [9]. Furthermore, in the case of axisymmetric spacetimes we showed that the method used by Ehlers' and method used by Cooperstock are mathematically equivalent.
1.3 Schrödinger-Poisson and Wigner-Poisson systems: Galerkin approximations In [7] a study of the Galerkin approximation to the to the periodic Schrödinger-Poisson problem in the unit cube [0,1]3 was presented. Two of the authors had previously shown that there exists a solution to the periodic system with sufficient regularity. This work established convergence rates for a Galerkin approximation to this solution. In addition the error estimates were transformed into corresponding L\infty-error estimates for the Wigner distribution function.
1.4 Dirac-Maxwell solitons In addition to my work with the Schrödinger-Poisson and Wigner-Poisson systems, I have completed some research with the coupled Dirac-Maxwell equations in collaboration with F.I. Cooperstock (to appear in Physical Review A). This work is a result of the attempt to model elementary particles as self-interacting soliton structures of a nonlinear field theory. Originally suggested in [10] and initially developed in [5], this work arose from a recent revival of interest in this field and in particular, the issue of gravitational coupling in the Dirac-Maxwell system has been considered [12]. However, there was the misconception that gravitation was a necessary ingredient for the creation of the soliton. We illustrated that this was not the case by performing a detailed analysis of the coupled Dirac-Maxwell equations and presenting the structure of the solutions. Numerical solutions of the field equations in the case of spherical symmetry with negligible gravitational self-interaction revealed the existence of families of solitons with electric field dominance that are completely determined by the observed charge and mass of the underlying particles. A soliton was found which has the charge and mass of the electron as well as a charge radius of 10-23 m. This is well within the present experimentally determined upper limit of 10-18 m. 2 Research Accomplishments
2.1 Schrödinger-Poisson and Wigner-Poisson systems: External coulomb fileds This was the main topic of my doctoral thesis [6]. Currently I am in the process of writing three separate papers related to this topic. The first paper is concerned with the existence, uniqueness and regularity properties of the solution to this system and the second paper focuses on the asymptotic in time behaviour of the solutions. In addition, conditions are found that ensure that the solutions decay in time even when the Coulomb field is attractive. The third paper, in collaboration with H. Teismann, deals with the linear Schrödinger equation with an external Coulomb field. Through various Sobolev embeddings we determine a range of function classes that the initial wave function must belong to in order to guarantee the existence of a C0 group of operators. This work is important in that it simplifies the analysis of the nonlinear Schrödinger equation. The reason for this is that some sort of regularization procedure is required if the singular potential is treated as a perturbation rather than as part of the free propagator.
2.2 Connections with industry Over the last few years I have been involved with the Pacific Institute of Mathematical Sciences (PIMS) and in particular the Industrial Mathematics workshops. These workshops are typically one week in duration and are designed to foster ties between the academic and industrial communities. Some of the problems that I have recently worked on have included a diverse range of topics including the fields of combinatorics [3], variational equations [1], materials science [2] and two-phase flow [4].
2.3 Consulting As a direct result of my work with industry, I have recently begun a consulting role with a local hydroelectric power company, Columbia Power Corporation, to develop a mathematical model of one of their sites. By carefully modelling the fluid flow through the site, they hope to determine the theoretical properties that an additional turbine should have so as to maximize the power produced. 3 Future Research Plans I have a wide range of research interests which cover both the theoretical and applied aspects of many-body quantum mechanical systems. This necessarily includes partial differential equations, scientific computing and the mathematical investigation of multibody systems. In the near future, I will continue my current research of the Schrödinger-Poisson and the Wigner-Poisson systems in the presence of external potentials. In particular, the questions of existence and uniqueness of the initial value problem as well as the investigation of conditions that ensure solutions decay (or don't decay) in time. My goals are to develop effective mathematical models for the intrinsically nonlinear problems that arise in many-body quantum mechanical systems as well as in other areas. This includes the establishment of rigorous analytical and numerical theories for these models. Tentative new areas of my future research include: the extension of the current analysis to other singular potentials; the transition from quantum to classical behaviour; the inclusion of spin effects. My research on all of these subjects will necessarily involve modeling, computing, and analysis in collaboration with other mathematicians, scientists, and engineers. References
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