published_or_final_version / abstract / Mechanical Engineering / Master / Master of Philosophy

<p>Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t>0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|<\gamma (t), t>0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.</p>

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

The theme of this thesis is combinatorics, complex analysis and algebraic geometry. The thesis consists of six articles divided into four parts. Part A: Spectral properties of the Schrödinger equation This part consists of Papers I-II, where we study a univariate Schrödinger equation with a complex polynomial potential. We prove that the set of polynomial potentials that admit solutions to the Schrödingerequation is connected, under certain boundary conditions. We also study a similar result for even polynomial potentials, where a similar result is obtained. Part B: Graph monomials and sums of squares In this part, consisting of Paper III, we study natural bases for the space of homogeneous, symmetric and translation-invariant polynomials in terms of multigraphs. We find all multigraphs with at most six edges that give rise to non-negative polynomials, and which of these that can be expressed as a sum of squares. Such polynomials appear naturally in connection to expressing certain non-negative polynomials as sums of squares. Part C: Eigenvalue asymptotics of banded Toeplitz matrices This part consists of Papers IV-V. We give a new and generalized proof of a theorem by P. Schmidt and F. Spitzer concerning asymptotics of eigenvalues of Toeplitz matrices. We also generalize the notion of eigenvalues to rectangular matrices, and partially prove the a multivariate analogue of the above. Part D: Stretched Schur polynomials This part consists of Paper VI, where we give a combinatorial proof that certain sequences of skew Schur polynomials satisfy linear recurrences with polynomial coefficients. / <p>At the time of doctoral defence the following papers were unpublished and had a status as follows: Paper 5: Manuscript; Paper 6: Manuscript</p>

Abstract: Considering the Schrödinger equation $\Delta_x u = i\partial{u}/\partial{t}$, we have a solution $u$ on the form $$u(x, t)= (2\pi)^{-n} \int_{\RR} {e^{i x\cdot \xi}e^{it|\xi|^2}\widehat{f}(\xi)}\, d \xi, x \in \RR, t \in \mathbf{R}$$ where $f$ belongs to the Sobolev space. It was shown by Sjögren and Sjölin, that assuming $\gamma : \mathbf{R}_+ \rightarrow \mathbf{R}_+ $ being a strictly increasing function, with $\gamma(0) = 0$ and $u$ and $f$ as above, there exists an $f \in H^{n/2} (\RR)$ such that $u$ is continuous in $\{ (x, t); t&gt;0 \}$ and $$\limsup_{(y,t)\rightarrow (x,0),|y-x|&lt;\gamma (t), t&gt;0} |u(y,t)|= + \infty$$ for all $x \in \RR$. This theorem was proved by choosing $$\widehat{f}(\xi )=\widehat{f_a}(\xi )= | \xi | ^{-n} (\log | \xi |)^{-3/4} \sum_{j=1}^{\infty} \chi _j(\xi)e^{- i( x_{n_j} \cdot \xi + t_j | \xi | ^a)}, \, a=2,$$ where $\chi_j$ is the characteristic function of shells $S_j$ with the inner radius rapidly increasing with respect to $j$. The purpose of this essay is to explain the proof given by Sjögren and Sjölin, by first showing that the theorem is true for $\gamma (t)=t$, and to investigate the result when we use $$S^a f_a (x, t)= (2 \pi)^{-n}\int_{\RR} {e^{i x\cdot \xi}e^{it |\xi|^a}\widehat{f_a}(\xi)}\, d \xi$$ instead of $u$.

The objective of the thesis is to treat the Schrodinger equation in parallel with a standard treatment of the heat equation. In the books of the Rubensteins and Kress, the heat equation initial value problem is converted into a Volterra integral equation of the second kind, and then the Picard algorithm is used to find the exact solution of the integral equation. Similarly, the Schrodinger equation boundary initial value problem can be turned into a Volterra integral equation. We follow the books of the Rubinsteins and Kress to show for the Schrodinger equation similar results to those for the heat equation. The thesis proves that the Schrodinger equation with a source function does indeed have a unique solution. The Poisson integral formula with the Schrodinger kernel is shown to hold in the Abel summable sense. The Green functions are introduced in order to obtain a representation for any function which satisfies the Schrodinger initial-boundary value problem. The Picard method of successive approximations is to be used to construct an approximate solution which should approach the exact Green function as n goes to infinity. To prove convergence, Volterra kernels are introduced in arbitrary Banach spaces, and the Volterra and General Volterra theorems are proved and used in order to show that the Neumann series for the L^1 kernel, the L^infinity kernel, the Hilbert-Schmidt kernel, the unitary kernel, and the WKB kernel converge to the exact Green function. In the WKB case, the solution of the Schrodinger equation is given in terms of classical paths; that is, the multiple scattering expansions are used to construct from, the action S, the quantum Green function. Then the interior Dirichlet problem is converted into a Volterra integral problem, and it is shown that Volterra integral equation with the quantum surface kernel can be solved by the method of successive approximations.

A novel method for integrating the time-dependent Schrödinger equation is presented. The moving boundary truncation (MBT) method is a time-dependent adaptive method that can significantly reduce the number of grid points needed to perform accurate wave packet propagation while maintaining stability. Hydrodynamic quantum trajectories are used to adaptively define the boundaries and boundary conditions of a fixed grid. The result is a significant reduction in the number of grid points needed to perform accurate calculations. A variety of model potential energy surfaces are used to evaluate the method. Excellent agreement with fixed boundary grids was obtained for each example. By moving only the boundary points, stability was increased to the level of the full fixed grid. Variations of the MBT method are developed which allow it to be applied to any potential energy surface and used with any propagation method. A variation of MBT is applied to the collinear H+H₂ reaction (using a LEPS potential) to demonstrate the stability and accuracy. Reaction probabilities are calculated for the three dimensional non-rotating O(³P)+H₂ and O(³P)+HD reactions to demonstrate that the MBT can be used with a variety of numerical propagation techniques. / text

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Some semiconductor devices such as lasers have long had critical dimensions on the nanoscale where quantum effects are critical. Others such as MOSFETs are now being scaled to within this regime. Quantum effects neglected in semiclassical models become increasing important at the nanoscale. Meanwhile, scattering remains important even in MOSFETs of 10 nm and below. Therefore, accurate quantum transport simulators with scattering are needed to explore the essential device physics at the nanoscale. The work of this dissertation is aimed at developing accurate quantum transport simulation tools for deep submicron device modeling, as well as utilizing these simulation tools to study the quantum transport and scattering effects in the nano-scale semiconductor devices. The basic quantum transport method "Schrödinger Equation Monte Carlo" (SEMC) provides a physically rigorous treatment of quantum transport and phasebreaking inelastic scattering (in 3D) via real (actual) scattering processes such as optical and acoustic phonon scattering. The SEMC method has been used previously to simulate carrier transport in nano-scaled devices in order to gauge the potential reliability of semiclassical models, phase-coherent quantum transport, and other limiting models as the transition from classical to quantum transport is approached. In this work, SEMC-1D and SEMC-2D versions with long range polar optical scattering processes have been developed and used to simulate quantum transport in tunnel injection lasers and nanoscaled III-V MOSFETs. Simulation results serve not only to demonstrate the capabilities of the developed quantum transport simulators, but also to illuminate the importance of physically accurate simulation of scattering for the predictive modeling of transport in nano-scaled devices.

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