Spectral Analysis of One-Dimensional Operators

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dc.contributor.advisorDamanik, David T
dc.contributor.committeeMemberHardt, Robert M
dc.contributor.committeeMemberCox, Steven J
dc.creatorFillman, Jacob D
dc.date.accessioned2016-01-15T21:52:41Z
dc.date.available2016-01-15T21:52:41Z
dc.date.created2015-05
dc.date.issued2015-02-25
dc.date.submittedMay 2015
dc.date.updated2016-01-15T21:52:42Z
dc.description.abstractWe study the spectral analysis of one-dimensional operators, motivated by a desire to understand three phenomena: dynamical characteristics of quantum walks, the interplay between inverse and direct spectral problems for limit-periodic operators, and the fractal structure of the spectrum of the Thue-Morse Hamiltonian. Our first group of results comprises several general lower bounds on the spreading rates of wave packets defined by the iteration of a unitary operator on a separable Hilbert space. By using tools within the class of CMV matrices, we apply these general lower bounds to deduce quantitative lower bounds for the spreading of the time-homogeneous Fibonacci quantum walk. Second, we construct several classes of limit-periodic operators with homogeneous Cantor spectrum, which connects problems from inverse and direct spectral analysis for such operators. Lastly, we precisely characterize the gap structure of the canonical periodic approximants to the Thue-Morse Hamiltonian, which constitutes a first step towards understanding the fractal structure of its spectrum. This thesis contains joint work with David Damanik, Milivoje Lukic, Paul Munger, and Robert Vance.
dc.format.mimetypeapplication/pdf
dc.identifier.citationFillman, Jacob D. "Spectral Analysis of One-Dimensional Operators." (2015) Diss., Rice University. <a href="https://hdl.handle.net/1911/87874">https://hdl.handle.net/1911/87874</a>.
dc.identifier.urihttps://hdl.handle.net/1911/87874
dc.language.isoeng
dc.rightsCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.
dc.subjectSchodinger operators
dc.subjectCMV matrices
dc.subjectJacobi matrices
dc.subjectspectral theory
dc.subjectfunctional analysis
dc.titleSpectral Analysis of One-Dimensional Operators
dc.typeThesis
dc.type.materialText
thesis.degree.departmentMathematics
thesis.degree.disciplineNatural Sciences
thesis.degree.grantorRice University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy
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