"One of the fundamental problems in electronic structure calculations is to determine the electron density associated with the minimum total energy of a molecular or bulk system. The total energy minimization problem is often formulated as a nonlinear eigenvalue problem. This presentation will focus on one of the most successful approaches to it: the SCF-Anderson (Self Consistent Field method accelerated by Anderson Acceleration) method. We will introduce the SCF-Anderson algorithm, talk about properties of an important parameter in it, study a linearly constrained least squares problem embedded in it, and look at the convergence properties."

University of Macau / Faculty of Science and Technology. / Department of Mathematics

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In this thesis, we consider the inverse Toeplitz eigenvalue problem which recover a real symmetric Toeplitz with desired eigenvalues. First some lower dimensional cases are solved by algebraic methods. This gives more insight on the inverse problem. Next, we explore the geometric meaning of real symmetric Toeplitz matrices. For high dimensional cases, numerical are unavoidable. From our numerical experiments, Newton-like methods are very effective for this problem.

In this thesis, we prove 2 theorems. First let £l0 be a minimizing (or maximizing) density function for the first antiperiodic eigenvalue £f1' in E[h,H,M], then £l0=h£q(a,b)+H£q[0,£k]/(a,b) (or £l0=H£q(a,b)+h£q[0,£k]/(a,b)) a.e. Finally, we prove min£f1'=min£g1=min£h1 where £g1 and £h1 are the first Dirichlet and second Neumann eigenvalues, respectively. Furthermore, we determine the jump point X0 of £l0 and the corresponding eigenvalue £f1', assuming that £l0 is symmetric about £k/2 We derive the nonlinear equations for this jump point X0 and £f1',then use Mathematica to solve the equations numerically.

The optimal estimates of the eigenvalue gaps and eigenvalue ratios for the Sturm-Liouville operators have been of fundamental importance. Recently a series of works by Keller [7],Chern-Shen [3], Lavine [8], Huang [4] and Horvath [6] show that the first eigenvalue gap of the Schrodinger operator under Dirichlet boundary condition and the first eigenvalue ratio¡]£f2/£f1¡^of the string equation under Dirichlet boundary condition are dual problems of each other. Furthermore the problems when the potential functions and density functions are restricted to certain classes of functions can all be solved by a variational calculus method (differentiating the whole equation with respect to a parameter t to find £fn'(t)) together with some elementary analysis. In this thesis, we shall give a short survey of these result. In particular, we shall prove $3$ pairs of theorems. First when q is convex (£l is concave), then £f2-£f1¡Ù3 ¡]£f2/£f1¡Ù4¡^.If q is a single well and its transition point is £k/2 (£l is a single barrier and its transition point is £k/2), then £f2-£f1¡Ù3¡]£f2/£f1¡Ù4¡^.All these lower bounds are optimal when q(£l) is constant. Finally when q is bounded (£l is bounded), then £f2-£f1 is minimized by a step function (£f2/£f1 is minimized by a step function), after some additional conditions. We shell give a unified treatment to the above results.

In this work we discover for the first time a strong relationship between Geršgorin theory and the geometric multiplicities of eigenvalues. In fact, if λ is an eigenvalue of an n × n matrix A with geometric multiplicity k, then λ is in at least k Geršgorin discs of A. Moreover, construct the matrix C by replacing, in every row, the (k − 1) smallest off-diagonal entries in absolute value by 0, then λ is in at least k Geršgorin discs of C. We also state and prove many new applications and consequences of these results as well as we update an improve some important existing ones.

The fundamental process that lies at the foundation of batteries, capacitors, and solar cells is the electron transfer process. This takes place at an interface or boundary in each device and is governed by its corresponding chemical reaction. Making these devices more efficient can help decrease our negative impact on the environment. Recent experiments in the field of Electrochemistry demonstrate that sound waves act as a catalyst for these electron transfer reactions. A model is developed using an Euler equation (conservation of momentum), conservation of mass equation, boundary motion equation, and surface tension equation. Chemically, it is clear that the catalytic phenomenon is derived from the sound waves and how they are affected by the top boundary. When combining these four equations we arrive at a boundary condition involving the top boundary only. We place this condition and the other contributing boundary and initial conditions on the wave equation to understand the interaction that occurs between the waves and the cell. We establish a self-adjoint operator and further use its inverse. Overall, using the Variational form and the Galerkin Method an approximation converges to the solution of the wave equation. With the help of MATLAB these eigenfunctions can be articulated as standing waves.

In this thesis, we study the optimal estimate of eigenvalue ratios £f_n/£f_m of the Sturm-Liouville equation with Dirichlet boundary conditions on (0, £k). In 2005, Horvath and Kiss [10] showed that £f_n/£f_m≤(n/m)^2 when the potential function q ≥ 0 and is a single-well function. Also this is an optimal upper estimate, for equality holds if and only if q = 0. Their result gives a positive answer to a problem posed by Ashbaugh and Benguria [2], who earlier showed that £f_n/£f_1≤n^2 when q ≥ 0. Here we first simplify the proof of Horvath and Kiss [10]. We use a modified Prufer substitutiony(x)=r(x)sin(£s£c(x)), y'(x)=r(x)£scos(£s£c(x)), where £s = ¡Ô£f. This modified phase seems to be more effective than the phases £p and £r that Horvath and Kiss [10] used. Furthermore our approach can be generalized to study the one-dimensional p-Laplacian eigenvalue problem. We show that for the Dirichlet problem of the equation -[(y')^(p-1)]'=(p-1)(£f-q)y^(p-1), where p > 1 and f^(p-1)=|f|^(p-1)sgn f =|f|^(p-2)f. The eigenvalue ratios satisfies £f_n/£f_m≤(n/m)^p, assuming that q(x) ≥ 0 and q is a single-well function on the domain (0, £k_p). Again this is an optimal upper estimate.

This thesis concerns the analysis and sensitivity of nonlinear eigenvalue problems for matrices and linear operators. The first part illustrates that lack of normality may result in catastrophic ill-conditioned eigenvalue problem. Linearization of rational eigenvalue problems for both operators over finite and infinite dimensional spaces are considered. The standard approach is to multiply by the least common denominator in the rational term and apply a well known linearization technique to the polynomial eigenvalue problem. However, the symmetry of the original problem is lost, which may result in a more ill-conditioned problem. In this thesis, an alternative linearization method is used and the sensitivity of the two different linearizations are studied. Moreover, this work contains numerically solved rational eigenvalue problems with applications in photonic crystals. For these examples the pseudospectra is used to show how well-conditioned the problems are which indicates whether the solutions are reliable or not.