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dc.contributor.authorStorms, Rebecah Helenen
dc.date.accessioned2020-06-26T08:00:34Z
dc.date.available2020-06-26T08:00:34Z
dc.date.issued2020-06-25
dc.identifier.othervt_gsexam:26691en
dc.identifier.urihttp://hdl.handle.net/10919/99147
dc.description.abstractWe consider the spectrum of the Schrödinger operator on an octagonal lattice using the Floquet-Bloch transform of the Laplacian. We will first consider the spectrum of the Laplacian in detail and prove various properties thereof, including spectral-band limits and locations of singularities. In addition, we will prove that Schrödinger operators with 1-1 periodic potentials can open at most two gaps in the spectrum precisely at energies $pm1$, and that a third gap can open at 0 for 2-2 periodic potentials. We describe in detail the structure of these operators for higher periods, and motivate our expectations of their spectra.en
dc.format.mediumETDen
dc.publisherVirginia Techen
dc.rightsThis item is protected by copyright and/or related rights. Some uses of this item may be deemed fair and permitted by law even without permission from the rights holder(s), or the rights holder(s) may have licensed the work for use under certain conditions. For other uses you need to obtain permission from the rights holder(s).en
dc.subjectSpectral Theoryen
dc.subjectSchrödinger Operatoren
dc.subjectPeriodic Graphsen
dc.titleSpectra of Periodic Schrödinger Operators on the Octagonal Latticeen
dc.typeThesisen
dc.contributor.departmentMathematicsen
dc.description.degreeMaster of Scienceen
thesis.degree.nameMaster of Scienceen
thesis.degree.levelmastersen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.disciplineMathematicsen
dc.contributor.committeechairEmbree, Mark Particken
dc.contributor.committeememberRossi, John F.en
dc.contributor.committeememberElgart, Alexanderen
dc.contributor.committeememberFillman, Jacoben
dc.description.abstractgeneralIn quantum physics, we would like the capability to model environments, such as magnetic fields, that interact with electrons or other quantum entities. The fields of graph theory and functional analysis within mathematics provide tools which relate well-understood mathematical concepts to these physical interactions. In this work, we use these tools to describe these environments using previously employed techniques in new ways.en


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