Some results on the spectral theory of one-dimensional operators and associated problems

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dc.contributor.advisorLukic, Milivoje
dc.creatorYoung, Giorgio F
dc.date.accessioned2022-09-26T15:16:26Z
dc.date.available2022-09-26T15:16:26Z
dc.date.created2022-05
dc.date.issued2022-04-21
dc.date.submittedMay 2022
dc.date.updated2022-09-26T15:16:26Z
dc.description.abstractThis thesis discusses results in the area of spectral theory of Schr\"odinger operators, and their discrete analogs, Jacobi matrices, as well as some closely associated problems. The first result we present relates to the quantum dynamics generated by a particular class of almost periodic Schr\"odinger operators. We show that the dynamics generated by Schr\"odinger operators whose potentials are approximated exponentially quickly by a periodic sequence exhibit a strong form of ballistic transport. The second result exploits the connection between the KdV hierarchy and one-dimensional Schr\"odinger operators to prove a uniqueness result for the KdV hierarchy with reflectionless initial data via inverse spectral theoretic techniques. The third and fourth results concern orthogonal and Chebyshev rational functions with poles on the extended real line. In the process of extending some of the existing theory for polynomials and exploring some of the new phenomena that arise, we present a proof of a conjecture of Barry Simon's. This thesis contains joint work with Benjamin Eichinger and Milivoje Luki\'c.
dc.format.mimetypeapplication/pdf
dc.identifier.citationYoung, Giorgio F. "Some results on the spectral theory of one-dimensional operators and associated problems." (2022) Diss., Rice University. <a href="https://hdl.handle.net/1911/113355">https://hdl.handle.net/1911/113355</a>.
dc.identifier.urihttps://hdl.handle.net/1911/113355
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.subjectSchr\"odinger operators
dc.subjectKdV hierarchy
dc.subjectJacobi matrices
dc.subjectChebyshev Extremal problems
dc.subjectorthogonal rational functions
dc.titleSome results on the spectral theory of one-dimensional operators and associated problems
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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