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.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0706107-111855 |
| Date | 06 July 2007 |
| Creators | Wu, Mao-ling |
| Contributors | Wei-Cheng Lian, Chun-Kong Law, Chung-Tsun Shieh |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0706107-111855 |
| Rights | withheld, Copyright information available at source archive |
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