Cantor spectrum of CMV matrices, Jacobi matrices and Schrodinger operators with dynamically defined coefficients and potentials

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dc.contributor.advisorDamanik, David
dc.creatorJun, Hyunkyu
dc.date.accessioned2020-04-23T16:24:19Z
dc.date.available2020-04-23T16:24:19Z
dc.date.created2020-05
dc.date.issued2020-04-22
dc.date.submittedMay 2020
dc.date.updated2020-04-23T16:24:19Z
dc.description.abstractIn this thesis, we consider CMV matrices, Jacobi matrices and Schr\"{o}dinger operators while assuming that the coefficients and potentials are generated by dynamical systems. One of the major parts investigates continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map. Another major part concerns the Fibonacci Hamiltonian. In the classical Fibonacci Hamiltonian, a sampling function is a locally constant function of a very special form. In this thesis, we study whether spectral results of the classical Fibonacci Hamiltonian can be extended to more general sampling functions. We provide the trace map description of the spectrum and extend the results for the classical Fibonacci Hamiltonian to arbitrary locally constant sampling functions.
dc.format.mimetypeapplication/pdf
dc.identifier.citationJun, Hyunkyu. "Cantor spectrum of CMV matrices, Jacobi matrices and Schrodinger operators with dynamically defined coefficients and potentials." (2020) Diss., Rice University. <a href="https://hdl.handle.net/1911/108346">https://hdl.handle.net/1911/108346</a>.
dc.identifier.urihttps://hdl.handle.net/1911/108346
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.subjectspectral theory
dc.subjectdynamical systems
dc.subjectergodic theory
dc.subjectmathematical physics
dc.titleCantor spectrum of CMV matrices, Jacobi matrices and Schrodinger operators with dynamically defined coefficients and potentials
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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