dc.contributor.advisor | Gesztesy, Fritz, 1953- | eng |
dc.contributor.author | Batchenko, Vladimir, 1978- | eng |
dc.date.issued | 2005 | eng |
dc.date.submitted | 2005 Spring | eng |
dc.description | The 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.description | Title from title screen of research.pdf file viewed on (May 23, 2006) | eng |
dc.description | Includes bibliographical references. | eng |
dc.description | Vita. | eng |
dc.description | Thesis (Ph. D.) University of Missouri-Columbia 2005. | eng |
dc.description | Dissertations, Academic -- University of Missouri--Columbia -- Mathematics. | eng |
dc.description.abstract | In 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.merlin | b55434423 | eng |
dc.identifier.uri | http://hdl.handle.net/10355/4136 | |
dc.language | English | eng |
dc.publisher | University of Missouri--Columbia | eng |
dc.relation.ispartofcollection | University of Missouri--Columbia. Graduate School. Theses and Dissertations | eng |
dc.subject.lcsh | Spectral theory (Mathematics) | eng |
dc.subject.lcsh | Schrödinger operator | eng |
dc.subject.lcsh | Jacobi operators | eng |
dc.title | On the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients | eng |
dc.type | Thesis | eng |
thesis.degree.discipline | Mathematics (MU) | eng |
thesis.degree.grantor | University of Missouri--Columbia | eng |
thesis.degree.level | Doctoral | eng |
thesis.degree.name | Ph. D. | eng |