Ambarzumyan定理說明在[0,1]區間的古典Sturm-Liouville問題，若Neumann特徵值形成的集合為$sigma_N={npi^2:nin{ f N}cup{0}}$，則勢函數$q$恆為$0$。在本論文中，我們對在三邊不等長星狀圖
上的Sturm-Liouville算子研究類似Ambarzumyan定理的結果。我們先解決正問題：找出當勢函
數$q=0$時，特徵值形成的集合，然後利用算子變換理論和Raleigh-Ritz不等式去證明反問題。根
據Pivovarchik在等長星狀圖上的方法，我們詳細地分析Kirchoff條件去證明我們的定理。我們特別去研究當三邊長滿足$a_1=a_2=frac{1}{2}a_3$或$a_1=frac{1}{2}a_2=frac{1}{3}a_3$我們是研究Neumann邊界值條件和Dirichlet邊界值條件的，其中後者的勢函數$q$必須再滿足一些假設，才能進行分析。

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.

1 Introduction 5
2 Direct Problems 11
3 Inverse Problems 17

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