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dc.contributor.advisorWehr, Janen_US
dc.contributor.authorBishop, Michael Anthony
dc.creatorBishop, Michael Anthonyen_US
dc.date.accessioned2013-06-04T22:34:56Z
dc.date.available2013-06-04T22:34:56Z
dc.date.issued2013
dc.identifier.urihttp://hdl.handle.net/10150/293431
dc.description.abstractQuantum mechanics is a theory developed to explain both particle and wave-like properties of small matter such as light and electrons. The consequences of the theory can be counter-intuitive but lead to mathematical and physical theory rich in fascinating phenomena and challenging questions. This dissertation investigates the nature of quantum systems in Bernoulli distributed random potentials for systems on the one dimensional lattice {0, 1, ..., L, L+1} ⊂ Z in the large system limit L → ∞. For single particle systems, the behavior of the low energy states is shown to be approximated by systems where positive potential is replaced by infinite potential. The approximate shape of these states is described, the asymptotics of their eigenvalues are calculated in the large system limit L → ∞, and a Lifschitz tail estimate on the sparsity of low energy states is proven. For interacting multi-particle systems, a Lieb-Liniger model with Bernoulli distributed potential is studied in the Gross-Pitaevskii approximation. First, to investigate localization in these settings, a general inequality is proven to bound from below the support of the mean-field. The bound depends on the per particle energy, number of particles, and interaction strength. Then, the ground state for the one-dimensional lattice with Bernoulli potential is studied in the large system limit. Specifically, the case where the product of interaction strength and particle density is near zero is considered to investigate whether localization can be recovered.
dc.language.isoenen_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.subjectDisorderen_US
dc.subjectGross-Pitaevskiien_US
dc.subjectGround Stateen_US
dc.subjectInteractionen_US
dc.subjectMean fielden_US
dc.subjectMathematicsen_US
dc.subjectBernoullien_US
dc.titleQuantum Systems in Bernoulli Potentialsen_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
thesis.degree.grantorUniversity of Arizonaen_US
thesis.degree.leveldoctoralen_US
dc.contributor.committeememberFriedlander, Leoniden_US
dc.contributor.committeememberMcLaughlin, Kennethen_US
dc.contributor.committeememberSims, Roberten_US
dc.contributor.committeememberWehr, Janen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.namePh.D.en_US
refterms.dateFOA2018-08-30T06:28:59Z
html.description.abstractQuantum mechanics is a theory developed to explain both particle and wave-like properties of small matter such as light and electrons. The consequences of the theory can be counter-intuitive but lead to mathematical and physical theory rich in fascinating phenomena and challenging questions. This dissertation investigates the nature of quantum systems in Bernoulli distributed random potentials for systems on the one dimensional lattice {0, 1, ..., L, L+1} ⊂ Z in the large system limit L → ∞. For single particle systems, the behavior of the low energy states is shown to be approximated by systems where positive potential is replaced by infinite potential. The approximate shape of these states is described, the asymptotics of their eigenvalues are calculated in the large system limit L → ∞, and a Lifschitz tail estimate on the sparsity of low energy states is proven. For interacting multi-particle systems, a Lieb-Liniger model with Bernoulli distributed potential is studied in the Gross-Pitaevskii approximation. First, to investigate localization in these settings, a general inequality is proven to bound from below the support of the mean-field. The bound depends on the per particle energy, number of particles, and interaction strength. Then, the ground state for the one-dimensional lattice with Bernoulli potential is studied in the large system limit. Specifically, the case where the product of interaction strength and particle density is near zero is considered to investigate whether localization can be recovered.


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