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URN etd-0706107-111855
Author Mao-ling Wu
Author's Email Address m942040027@student.nsysu.edu.tw
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Department Applied Mathematics
Year 2006
Semester 2
Degree Master
Type of Document
Language English
Title Ambarzumian’s Theorem for the Sturm-Liouville Operator on Graphs
Date of Defense 2007-06-29
Page Count 29
Keyword
  • Ambarzumyan Theorem
  • graphs
  • Sturm-Liouville operators
  • Abstract The Ambarzumyan Theorem states that for the
    classical Sturm-Liouville problem on $[0,1]$, if the set of Neumann
    eigenvalue $sigma_N={(npi)^2: nin { f N}cup { 0}}$, then
    the potential function $q=0$. In this thesis, we study the analogues
    of Ambarzumyan Theorem for the Sturm-Liouville operators on
    star-shaped graphs with 3 edges of different lengths. We first
    solve the direct problem: to find out the set of eigenvalues when
    $q=0$. Then we use the theory of transformation operators and
    Raleigh-Ritz inequality to prove the inverse problem. Following
    Pivovarchik's work on star-shaped graphs of uniform lengths, we
    analyze the Kirchoff condition in detail to prove our theorems. In
    particular, we study the cases when the lengths of the 3 edges
    satisfy $a_1=a_2=frac{1}{2}a_3$ or
    $a_1=frac{1}{2}a_2=frac{1}{3}a_3$. Furthermore, we work on Neumann
    boundary conditions as well as Dirichlet boundary conditions. In
    the latter case, some assumptions about $q$ have to be made.
    Advisory Committee
  • Wei-Cheng Lian - chair
  • Chung-Tsun Shieh - co-chair
  • Chun-Kong Law - advisor
  • Files
  • etd-0706107-111855.pdf
  • indicate accessible in a year
    Date of Submission 2007-07-06

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