Spectral Analysis of One-Dimensional Operators

dc.contributor.advisorDamanik, David Ten_US
dc.contributor.committeeMemberHardt, Robert Men_US
dc.contributor.committeeMemberCox, Steven Jen_US
dc.creatorFillman, Jacob Den_US
dc.date.accessioned2016-01-15T21:52:41Zen_US
dc.date.available2016-01-15T21:52:41Zen_US
dc.date.created2015-05en_US
dc.date.issued2015-02-25en_US
dc.date.submittedMay 2015en_US
dc.date.updated2016-01-15T21:52:42Zen_US
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.en_US
dc.format.mimetypeapplication/pdfen_US
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>.en_US
dc.identifier.urihttps://hdl.handle.net/1911/87874en_US
dc.language.isoengen_US
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.en_US
dc.subjectSchodinger operatorsen_US
dc.subjectCMV matricesen_US
dc.subjectJacobi matricesen_US
dc.subjectspectral theoryen_US
dc.subjectfunctional analysisen_US
dc.titleSpectral Analysis of One-Dimensional Operatorsen_US
dc.typeThesisen_US
dc.type.materialTexten_US
thesis.degree.departmentMathematicsen_US
thesis.degree.disciplineNatural Sciencesen_US
thesis.degree.grantorRice Universityen_US
thesis.degree.levelDoctoralen_US
thesis.degree.nameDoctor of Philosophyen_US
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