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Computing Spectral Properties of Infinite-Dimensional Operators

Author
Horning, Andrew
Abstract
This dissertation introduces a cohesive framework for numerically computing spectral properties related to the discrete and continuous spectrum of infinite-dimensional operators. Approximations to eigenvalues and eigenvectors, spectral measures, and generalized eigenvectors are constructed by sampling the range of the resolvent operator at strategic points in the complex plane. These algorithms are developed and analyzed directly in the abstract infinite-dimensional Hilbert space setting. They require only two essential computational ingredients: (1) solving linear equations with complex shifts and (2) taking inner products in the Hilbert space. Numerical implementations for a broad class of differential and integral operators, leveraging state-of-the-art adaptive spectral methods, are provided in an accompanying MATLAB package called SpecSolve, which is demonstrated through a collection of examples.
Description
169 pages
Date Issued
2021-12Subject
differential eigenvalue problems; FEAST; generalized eigenfunctions; rational filters; spectral density; spectral measures
Committee Chair
Townsend, Alex John
Committee Member
Damle, Anil; Bindel, David S.
Degree Discipline
Applied Mathematics
Degree Name
Ph. D., Applied Mathematics
Degree Level
Doctor of Philosophy
Type
dissertation or thesis