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URN etd-0704105-182611
Author YU-HAO LEE
Author's Email Address m922040010@student.nsysu.edu.tw
Statistics This thesis had been viewed 5059 times. Download 1446 times.
Department Applied Mathematics
Year 2004
Semester 2
Degree Master
Type of Document
Language English
Title The theory of transformation operators and its application in inverse spectral problems
Date of Defense 2005-06-03
Page Count 63
Keyword
  • eigenvalue
  • Transformation operator
  • spectral problem
  • eigenvector
  • Abstract The inverse spectral problem is the problem of
    understanding the potential function of the Sturm-Liouville
    operator from the set of eigenvalues plus some additional
    spectral data. The theory of transformation operators, first
    introduced by Marchenko, and then reinforced by Gelfand and
    Levitan, is a powerful method to deal with the different stages
    of the inverse spectral problem: uniqueness, reconstruction,
    stability and existence. In this thesis, we shall give a survey
    on the theory of transformation operators. In essence, the theory
    says that the transformation operator $X$ mapping the solution of
    a Sturm-Liouville operator $varphi$ to the solution of a
    Sturm-Liouville operator, can be written as
    $$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$ where the
    kernel $K$ satisfies the Goursat problem
    $$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0$$ plus some initial boundary
    conditions. Furthermore, $K$ is related by a function $F$ defined
    by the spectral data ${(lambda_{n},alpha_{n})}$ where
    $alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2})^{frac{1}{2}}$
    through the famous Gelfand-Levitan equation
    $$K(x,y)+F(x,y)+int_{o}^{x}K(x,t)F(t,y)dt=0.$$ Furthermore, all
    the above relations are bilateral, that is $$qLeftrightarrow
    KLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.$$
    hspace*{0.25in}We shall give a concise account of the above
    theory, which involves Riesz basis and order of entire functions.
    Then, we also report on some recent applications on the
    uniqueness result of the inverse spectral problem.
    Advisory Committee
  • none - chair
  • none - co-chair
  • C.K.Low - advisor
  • Files
  • etd-0704105-182611.pdf
  • indicate access worldwide
    Date of Submission 2005-07-04

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