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URN etd-0717100-110512
Author Yan-Hsiou Cheng
Author's Email Address jengyh@ibm7.math.nsysu.edu.tw
Statistics This thesis had been viewed 5068 times. Download 1718 times.
Department Applied Mathematics
Year 1999
Semester 2
Degree Master
Type of Document
Language English
Title On Some New Inverse nodal problems
Date of Defense 2000-06-02
Page Count 39
Keyword
  • Sturm-Liouville
  • nodal set
  • inverse nodal problem
  • Abstract In this thesis, we study two new inverse nodal problems
    introduced by Yang, Shen and Shieh respectively.
    Consider the classical Sturm-Liouville problem: $$ left{
    egin{array}{c}
    -phi'+q(x)phi=la phi
    phi(0)cosalpha+phi'(0)sinalpha=0 
    phi(1)coseta+phi'(1)sineta=0
    end{array}
    ight. ,
    $$ where $qin L^1(0,1)$ and $al,ein [0,pi)$. The inverse
    nodal problem involves the determination of the parameters
    $(q,al,e)$ in the problem by the knowledge of the nodal points
    in $(0,1)$. In 1999, X.F. Yang got a uniqueness result which only
    requires the knowledge of a certain subset of the nodal set. In
    short, he proved that the set of all nodal points just in the
    interval $(0,b) (frac{1}{2}<bleq 1)$ is sufficient to determine
    $(q,al,e)$ uniquely.
    In this thesis, we show that a twin and dense subset of all nodal
    points in the interval $(0,b)$ is enough to determine
    $(q,al,e)$ uniquely. We improve Yang's theorem by weakening
    its conditions, and simplifying the proof.
    In the second part of this thesis, we will discuss an inverse
    nodal problem for the vectorial Sturm-Liouville problem: $$
    left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x) 
    A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f
    y}(1)+B_{2}{f y}'(1)={f 0}
    end{array}
    ight. .
    $$
    Let ${f y}(x)$ be a continuous $d$-dimensional vector-valued
    function defined on $[0,1]$. A point $x_{0}in [0,1]$ is called a
    nodal point of ${f y}(x)$ if ${f y}(x_{0})=0$. ${f y}(x)$
    is said to be of type (CZ) if all the zeros of its components are
    nodal points. $P(x)$ is called simultaneously diagonalizable if
    there is a constant matrix $S$ and a diagonal matrix-valued
    function $U(x)$ such that $P(x)=S^{-1}U(x)S.$
    If $P(x)$ is simultaneously diagonalizable, then it is easy to
    show that there are infinitely many eigenfunctions which are of
    type (CZ). In a recent paper, C.L. Shen and C.T. Shieh (cite{SS})
    proved the converse when $d=2$: If there are infinitely many
    Dirichlet eigenfunctions which are of type (CZ), then $P(x)$ is
    simultaneously diagonalizable.
    We simplify their work and then extend it to some general
    boundary conditions.
    Advisory Committee
  • Tzon-Tzer Lu - co-chair
  • Hua-Huai Chern - co-chair
  • Jhishen Tsay - co-chair
  • Tzy-Wei Hwang - co-chair
  • Chun-Kong Law - advisor
  • Files
  • graduate.pdf
  • indicate in-campus access immediately and off_campus access in a year
    Date of Submission 2000-07-17

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