Geometric unification of Schroedinger and Yang-Mills equations and Riemannian spectra of vector bundles.
Item
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Title
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Geometric unification of Schroedinger and Yang-Mills equations and Riemannian spectra of vector bundles.
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Identifier
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AAI9605666
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identifier
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9605666
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Creator
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Sowa, Artur.
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Contributor
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Adviser: Jozef Dodziuk
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Date
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1995
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | Physics, Nuclear
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Abstract
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We consider a Riemannian submersion metric on a total space of a principal fiber bundle and its geometry. It turns out that a natural second order PDE, constructed from intrinsic geometric quantities, leads to a weakly coupled system of Yang-Mills and stationary Schrodinger equations.;The system leads to numerical invariants depending on both the topology of a bundle and the Riemannian geometry of the base manifold. In the 'limiting' case of a trivial bundle they become the eigenvalues of the Laplacian of the base-manifold.;We display solutions of the nonlinear eigenvalue problem associated with this system in case of linear bundles over compact two-manifolds and prove their uniqueness.;We further show, that existence of solutions of the nonlinear eigenvalue problem on manifolds of more than four dimensions is equivalent to the existence of pure Yang-Mills fields on those manifolds with conformally deformed metrics.;We consider also the elliptic theory of this system on the Euclidean plane with U(1) as the gauge group, where it reduces to a single scalar equation: {dollar}-\Delta f{dollar} = {dollar}f\sp{lcub}-3{rcub}.{dollar} We prove in particular that there are no finite energy solutions. We further show that there are radially symmetric solutions and we give a complete description of their properties. The results generalize in part to {dollar}-\Delta f{dollar} = {dollar}f\sp{lcub}-p{rcub},{dollar} where {dollar}p >{dollar} 0.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
