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URN etd-0704105-132441
Author Yan-Hsiou Cheng
Author's Email Address No Public.
Statistics This thesis had been viewed 5107 times. Download 1747 times.
Department Applied Mathematics
Year 2004
Semester 2
Degree Ph.D.
Type of Document
Language English
Title Inverse Problems for Various Sturm-Liouville Operators
Date of Defense 2005-06-27
Page Count 94
Keyword
  • Sturm-Liouville problem
  • spectral problem
  • nodal problem
  • Hill's equation
  • Abstract In this thesis, we study the inverse nodal problem and inverse
    spectral problem for various Sturm-Liouville operators, in
    particular, Hill's operators.
    We first show that the space of Schr"odinger operators under
    separated boundary conditions characterized by ${H=(q,al, e)in
    L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic
    to the partition set of the space of all admissible
    sequences $X={X_{k}^{(n)}}$ which form sequences that
    converge to $q, al$ and $ e$ individually. The definition of
    $Gamma$, the space of quasinodal sequences, relies on the $L^{1}$
    convergence of the reconstruction formula for $q$ by the exactly
    nodal sequence.
    Then we study the inverse nodal problem for Hill's equation, and
    solve the uniqueness, reconstruction and stability problem. We do
    this by making a translation of Hill's equation and turning it
    into a Dirichlet  Schr"odinger problem. Then the estimates of
    corresponding nodal length and eigenvalues can be deduced.
    Furthermore, the reconstruction formula of the potential function
    and the uniqueness can be shown. We also show the quotient space
    $Lambda/sim$ is homeomorphic to the space $Omega={qin
    L^{1}(0,1) :
    int_{0}^{1}q = 0, q(x)=q(x+1)
    mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection
    of all admissible
    sequences $X={X_{k}^{(n)}}$ which form sequences that
    converge to $q$.
    Finally we show that if the periodic potential function $q$ of
    Hill's equation is single-well on $[0,1]$, then $q$ is constant if
    and only if the first instability interval is absent. The same is
    also valid for convex potentials. Then we show that similar
    statements are valid for single-barrier and concave density
    functions for periodic string equation. Our result extends that of
    M. J. Huang and supplements the works of Borg and Hochstadt.
    Advisory Committee
  • Chao-Liang Shen - chair
  • Tzon-Tzer Lu - co-chair
  • Jenn-Nan Wang - co-chair
  • Chiu-Ya Lan - co-chair
  • Chung-Tsun Shieh - co-chair
  • Wei-Cheng Lian - co-chair
  • Chao-Nien Chen - co-chair
  • Tzy-Wei Hwang - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0704105-132441.pdf
  • indicate accessible in a year
    Date of Submission 2005-07-04

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