# Computational Tools for Exploring Eigenvector Localization

Jeffrey Ovall

Spring 2024

5-22-2024

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

## Department

Mathematics and Statistics

English

## DOI

10.15760/etd.3749

## Physical Description

1 online resource (x, 157 pages)

## Abstract

We develop computational tools for exploring eigenvector localization for a class of selfadjoint, elliptic eigenvalue problems regardless of the cause for localization. The user inputs a desired region R (not necessarily connected), a tolerance for the amount localization in R, and the desired energy range [a,b]. The tool outputs eigenvectors concentrated within the tolerance inside R and within [a, b]. We develop ample theory that justifies our algorithm, which involves a complex, compact perturbation of the operator L, Ls = L+isχR, for some (small) s > 0. Our central idea can be summarized as follows: if (λ,ψ) is an eigenpair of L with ψ highly localized in R, then there will be an eigenpair (μ,φ) of the shifted operator Ls such that μ is near λ+is and φ is near ψ i.e. the eigenpair (μ,φ) of Ls is close to the eigenpair (λ + is, ψ) of L + is. The algorithm finds eigenpairs (μ, φ) of Ls with I(μ) near s and R(μ) in or near [a, b]. In practice, the algorithm proves robust, immediately eliminating over 90% of unwanted eigenmodes, and many examples are provided. A post processing feature is included, which consists of (a few) inverse iterations that further eliminate unwanted eigenvectors and even identify computationally difficult cases.

Although most of the theory is developed with one kind of operator in mind, we show that it applies directly to the magnetic Laplacian, Hˆ (A) := (−i∇ − A(x))2 as well. We additionally provide a method for a priori prediction of where eigen- i functions may localize for the magnetic Laplacian. Essentially, if an eigenvector achieves a global maximum (in modulus) at x0 ∈ Ω, then A behaves similarly to a conservative vector field in a neighborhood around x0 that depends on the eigenvalue. We provide numerical examples to show the eigenmodes localize where |curlA| is small, at least for low energies, as expected based on our theory.

## Rights

© 2024 Robyn Ashley Markee Reid

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