特徵函數是一個定義實數上的函數，其零點恰好是一個量子圖的特徵值。我們將給出一個遞迴公式，來幫忙建造出一些複雜的量子樹的特徵函數。在勢能函數Q=0的情況下，我們定義一個修飾特徵函數，對於複雜的量子樹，可列出一個直接的公式。我們將利用以上兩種方法來說明存在兩個具有一樣的特徵函數的量子圖。換句話說，它們是同構譜的量子圖。另外，我們也舉出其它量子圖與它們的特徵函數。本論文的理論部分主要依據[3,6]。

The characteristic function is a function on ℝ where zeros are exactly the eigenvalues of a quantum graph. We shall give a recursive formula which helps to build up the characteristic function of complicated quantum trees. In the case when the potential Q = 0, there is also a modified characteristic function which can have a direct formula for complicated quantum trees. We shall use the above two methods to show that there are two distinct quantum graphs having the same set of eigenvalues. In other words, they are isospectral quantum graphs. Many other examples of quantum graphs and their modified characteristic functions will also be given. The theoretical part of this thesis follow from the papers [3, 6].

1 Overview 1
1.1 Characteristic function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Recursive formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Isospectral quantum graphs . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Modified characteristic functions 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Formulas for modified characteristic functions . . . . . . . . . . . . . . 16
2.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Spectra of some quantum graphs 23
3.1 Two isospectral quantum trees . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Some variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Appendix 31
A Multiple angle formula for tangent . . . . . . . . . . . . . . . . . . . . 31
B Figures and Mathematica programs . . . . . . . . . . . . . . . . . . . . 34
C Nevanlinna function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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