Localizing the eigenvalues of matrix-valued functions: analysis and applications
The standard eigenvalue problem of finding the zeros of det(<i>zI − A</i>) is ubiquitous, and comes from studying solutions to <i>x′ = Ax</i> as well as myriad other sources. Similarly, if <i>x′</i>(<i>t</i>) = <i>Ax</i>(<i>t</i>) + <i>Bx</i>(<i>t</i> − 1) is some delay-differential equation (arising, say, from modeling the spread of a disease, or from population growth models), then stability is determined by computing the roots of det(<i>zI − A − Be<sup>−z</sup></i>). Models of physical processes where energy slowly leaks away to infinity lead to similar problems. These physical systems are typically modeled in terms of differential equations which are then discretized using e.g. collocation or finite elements. For example, such a discretization gives a correspondence between the quantum scattering resonances associated to a quantum corral and the zeros of det(<i>A − zB + C</i>(<i>z</i>)), where <i>A</i>, <i>B</i>, <i>C</i>(<i>z</i>) ∈ ℂ<sup><i>n</i>×<i>n</i></sup> and the highly nonlinear entries of <i>C</i>(<i>z</i>) involve square roots and ratios of Bessel functions. In each case, we are led to so-called nonlinear eigenvalue problems of the form <i>T</i>(<i>λ</i>)<i>v</i> = 0, <i>v</i> ≠ 0, where <i>T</i> : <i>Ω</i> ⊂ ℂ → ℂ<sup><i>n</i>×<i>n</i></sup> is a matrix-valued function, and <i>λ</i> is called an eigenvalue of <i>T</i>. The first contribution of this thesis is theorems for localizing eigenvalues of general matrix-valued functions, effectively reducing the region in which eigenvalues of <i>T</i> are known to lie from all of <i>Ω</i> down to a smaller space, and deducing eigenvalue counts within regions that meet certain conditions. These theorems are derived by working with the diagonal entries or diagonal blocks of <i>T</i>, such as our generalization of Gershgorin’s theorem, or by considering nonlinear generalizations of pseudospectra. Localization and counting results allow better initial guesses or shifts for iterative algorithms, guide the selection of an appropriate closed contour for contour integral-based algorithms, and facilitate error analysis in cases where eigenvalues can be confined to tiny regions. The second contribution of this thesis is to exploit these results in several contexts. To start with, a variety of strategies for getting the most out of our main localization theorems will be presented and applied to several test problems. Then we foray into the analysis of higher-order and delay differential equations, using our localization results to help bound asymptotic growth of solutions, and using our generalization of the notion of pseudospectra to concretely bound transient growth both above and below; a model for a semiconductor laser with phase-conjugate feedback acts as the central example. The last application we will treat is about the resonances for electrons trapped in circular quantum corrals, microscopic structures built by placing atoms in a circle on a metal surface. We provide a framework for comparing various elastic scattering models, and use it to bound the error between resonances computed from the naïve particle-in-a-box model and resonances computed from a model that takes quantum tunneling into account.
Applied mathematics; delay differential equations; Gershgorin's theorem; nonlinear eigenvalue problems; pseudospectra; scattering resonances; transient dynamics; Mathematics
Bindel, David S.
Vladimirsky, Alexander B.; Van Loan, Charles Francis
PHD of Applied Mathematics
Doctor of Philosophy
dissertation or thesis