Return to search

Ambarzumian¡¦s Theorem for the Sturm-Liouville Operator on Graphs

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.

Identiferoai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0706107-111855
Date06 July 2007
CreatorsWu, Mao-ling
ContributorsWei-Cheng Lian, Chun-Kong Law, Chung-Tsun Shieh
PublisherNSYSU
Source SetsNSYSU Electronic Thesis and Dissertation Archive
LanguageEnglish
Detected LanguageEnglish
Typetext
Formatapplication/pdf
Sourcehttp://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0706107-111855
Rightswithheld, Copyright information available at source archive

Page generated in 0.002 seconds