URN 
etd0706107111855 
Author 
Maoling Wu 
Author's Email Address 
m942040027@student.nsysu.edu.tw 
Statistics 
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Department 
Applied Mathematics 
Year 
2006 
Semester 
2 
Degree 
Master 
Type of Document 

Language 
English 
Title 
Ambarzumianâ€™s Theorem for the SturmLiouville Operator on Graphs 
Date of Defense 
20070629 
Page Count 
29 
Keyword 
Ambarzumyan Theorem
graphs
SturmLiouville operators

Abstract 
The Ambarzumyan Theorem states that for the classical SturmLiouville 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 SturmLiouville operators on starshaped 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 RaleighRitz inequality to prove the inverse problem. Following Pivovarchik's work on starshaped 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 
WeiCheng Lian  chair
ChungTsun Shieh  cochair
ChunKong Law  advisor

Files 
indicate accessible in a year 
Date of Submission 
20070706 