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dc.contributor.advisorGesztesy, Fritz, 1953-eng
dc.contributor.authorBatchenko, Vladimir, 1978-eng
dc.date.issued2005eng
dc.date.submitted2005 Springeng
dc.descriptionThe entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file.eng
dc.descriptionTitle from title screen of research.pdf file viewed on (May 23, 2006)eng
dc.descriptionIncludes bibliographical references.eng
dc.descriptionVita.eng
dc.descriptionThesis (Ph. D.) University of Missouri-Columbia 2005.eng
dc.descriptionDissertations, Academic -- University of Missouri--Columbia -- Mathematics.eng
dc.description.abstractIn this thesis we characterize the spectrum of one-dimensional Schrödinger operators. H = -d2/dx2+V in L2(R; dx) with quasi-periodic complex-valued algebro geometric, potentials V (i.e., potentials V which satisfy one (and hence infinitely many) equation(s) of the stationary Korteweg-de Vries (KdV) hierarchy) associated with nonsingular hyperelliptic curves. The spectrum of H coincides with the conditional stability set of H and can explicitly be described in terms of the mean value of the inverse of the diagonal Green's function of H. As a result, the spectrum of H consists of finitely many simple analytic arcs and one semi-infinite simple analytic arc in the complex plane. Crossings as well as confluences of spectral arcs are possible and discussed as well. These results extend to the Lp(R; dx)-setting for p 2 [1,1). In addition, we apply these techniques to the discrete case and characterize the spectrum of one-dimensional Jacobi operators H = aS+ + a-S- b in 2(Z) assuming a, b are complex-valued quasi-periodic algebro-geometric coefficients. In analogy to the case of Schrödinger operators, we prove that the spectrum of H coincides with the conditional stability set of H and can also explicitly be described in terms of the mean value of the Green's function of H. The qualitative behavior of the spectrum of H in the complex plane is similar to the Schrödinger case: the spectrum consists of finitely many bounded simple analytic arcs in the complex plane which may exhibit crossings as well as confluences.eng
dc.identifier.merlinb55434423eng
dc.identifier.urihttp://hdl.handle.net/10355/4136
dc.languageEnglisheng
dc.publisherUniversity of Missouri--Columbiaeng
dc.relation.ispartofcollectionUniversity of Missouri--Columbia. Graduate School. Theses and Dissertationseng
dc.subject.lcshSpectral theory (Mathematics)eng
dc.subject.lcshSchrödinger operatoreng
dc.subject.lcshJacobi operatorseng
dc.titleOn the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficientseng
dc.typeThesiseng
thesis.degree.disciplineMathematics (MU)eng
thesis.degree.grantorUniversity of Missouri--Columbiaeng
thesis.degree.levelDoctoraleng
thesis.degree.namePh. D.eng


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