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    On the spectra of Schrödinger and Jacobi operators with complex-valued quasi-periodic algebro-geometric coefficients

    Batchenko, Vladimir, 1978-
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    Date
    2005
    Format
    Thesis
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    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.
    URI
    https://hdl.handle.net/10355/4136
    https://doi.org/10.32469/10355/4136
    Degree
    Ph. D.
    Thesis Department
    Mathematics (MU)
    Rights
    This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Copyright held by author.
    Collections
    • 2005 MU dissertations - Freely available online
    • Mathematics electronic theses and dissertations (MU)

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