From Generalized Fourier Transforms to Coupled Supersymmetry

Williams, Cameron Louis 1990-

From Generalized Fourier Transforms to Coupled Supersymmetry

dc.contributor.advisor	Bodmann, Bernhard G.
dc.contributor.advisor	Kouri, Donald J.
dc.contributor.committeeMember	Kalantar, Mehrdad
dc.contributor.committeeMember	Labate, Demetrio
dc.contributor.committeeMember	Klauder, John R.
dc.creator	Williams, Cameron Louis 1990-
dc.date.accessioned	2019-09-14T16:41:41Z
dc.date.available	2019-09-14T16:41:41Z
dc.date.created	May 2017
dc.date.issued	2017-05
dc.date.submitted	May 2017
dc.date.updated	2019-09-14T16:41:42Z
dc.description.abstract	The Fourier transform, the quantum mechanical harmonic oscillator, and supersymmetric quantum mechanics are well-studied objects in mathematics. The relations between them are also well-understood, though the traditional understanding of each has led to a rigidity in their design. In this thesis, we will develop generalizations of the Fourier transform and explore their relations. Exploring the analytic structure of these integral transforms, particularly their related Hamiltonians, naturally leads to Hamiltonian-like operators which have rich analytic and algebraic structures. The algebraic structure underlying the Hamiltonian-like operators associated to generalized Fourier transforms suggests a much-more-general abstract formulation. To this end, we introduce the coupled supersymmetry (coupled SUSY) algebraic framework which unifies the quantum mechanical harmonic oscillator and supersymmetric quantum mechanics in a more complete way. The coupled SUSY framework subsumes the quantum mechanical harmonic oscillator and provides a broader class of systems which have rich functional analytic and algebraic structures. In this setting, one is able to develop further generalizations of the Fourier transform. A further generalization of coupled SUSY is briefly presented which appears to be a very new insight into supersymmetry.
dc.description.department	Mathematics, Department of
dc.format.digitalOrigin	born digital
dc.format.mimetype	application/pdf
dc.identifier.citation	Portions of this document appear in: Williams, Cameron L., Bernhard G. Bodmann, and Donald J. Kouri. "Fourier and beyond: Invariance properties of a family of integral transforms." Journal of Fourier Analysis and Applications 23, no. 3 (2017): 660-678.
dc.identifier.uri	https://hdl.handle.net/10657/4597
dc.language.iso	eng
dc.rights	The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).
dc.subject	Fourier transforms
dc.subject	Supersymmetry
dc.subject	Supersymmetric quantum mechanics
dc.subject	Generalized Fourier transforms
dc.subject	Coherent states
dc.subject	Superalgebra
dc.subject	Algebra
dc.subject	Harmonic oscillator
dc.subject	Generalized harmonic oscillator
dc.subject	Quantum mechanics
dc.title	From Generalized Fourier Transforms to Coupled Supersymmetry
dc.type.dcmi	Text
dc.type.genre	Thesis
thesis.degree.college	College of Natural Sciences and Mathematics
thesis.degree.department	Mathematics, Department of
thesis.degree.discipline	Mathematics
thesis.degree.grantor	University of Houston
thesis.degree.level	Doctoral
thesis.degree.name	Doctor of Philosophy