近日，在圖上的 Sturm-Liouville 問題 (即量子圖)
研究越來越熱門 。然而在環狀圖上的研究較少。
在本篇論文，我們首次考慮一個特徵方程 (稱 Floquet 方程) 在對於 Schrodinger 算子 H 作用在類似石墨烯圖上的譜分析，此類石墨烯圖由 3 條不同長度的鄰邊所組成的六邊形，而覆蓋成一平面。而算子讓不同邊長對應不同的 potential 函數。
我們應用 Floquet-Bloch 定理來推導出 Floquet 方程，其根即為所有 H 的譜值。
且我們證明此類石墨烯圖的譜是連續的。我們推廣了 Kuchment-Post 和 Korotyaev-Lobanov 的結果，而且我們的方法較簡單直接。
其次，我們研究兩個在圖上的 Ambarzumyan 問題，一個是在石墨烯上，另外一個則是在環狀圖形 (3 個邊和 2 端點) 上。
最後，我們解出一個在同一環狀圖上的 Hochstadt-Lieberman 型式的譜反演問題。

Recently there is a lot of interest in the study of Sturm-Liouville problems on graphs,
called quantum graphs. However the study on cyclic quantum graphs are scarce. In
this thesis, we shall rst consider a characteristic function approach to the spectral
analysis for the Schrodinger operator H acting on graphene-like graphs|in nite periodic
hexagonal graphs with 3 distinct adjacent edges and 3 distinct potentials de ned
on them. We apply the Floquet-Bloch theory to derive a Floquet equation with parameters
theta_1, theta_2, whose roots de ne all the spectral values of H. Then we show that the
spectrum of this operator is continuous. Our results generalize those of Kuchment-Post
and Korotyaev-Lobanov. Our method is also simpler and more direct.
Next we solve two Ambarzumyan problems, one for graphene and another for a cyclic
graph with two vertices and 3 edges. Finally we solve an Hochstadt-Lieberman type
inverse spectral problem for the same cyclic graph with two vertices and 3 edges.
Keywords : quantum graphs, graphene, spectrum, Ambarzumyan problem, inverse
spectral problem.

1 Introduction 1
1.1 Quantum graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Graphene and nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Chapter summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 The spectrum of Schrodinger operator acting on graphene-like graphs 14
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Floquet equation for graphene-like graphs . . . . . . . . . . . . . . . . 19
2.3 Asymptotic behavior for the Floquet equation . . . . . . . . . . . . . . 22
2.4 Floquet equation for nanotube-like graphs . . . . . . . . . . . . . . . . 29
2.5 An Ambarzumyan problem for Dirichlet eigenvalues . . . . . . . . . . . 32
3 An Ambarzumyan problem on a cyclic quantum graph 36
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Proof of Theorem 3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Expansion of characteristic functions . . . . . . . . . . . . . . . . . . . 40
4 An inverse spectral problem on a cyclic quantum graph 50
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Properties of the eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3 Proof of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Appendix 60
5.1 Floquet-Bloch theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.2 Analytically bered operators . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Properties of even potentials . . . . . . . . . . . . . . . . . . . . . . . . 64
6 Bibliography 66

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