在這篇論文中，我們將討論兩個新的節點反演問題，這兩個問題曾經分別被
X.F Yang、沈昭亮和謝忠村討論過。

考慮古典的 Sturm-Liouville 問題：$$ left{
egin{array}{c}
-phi'+q(x)phi=la phi
phi(0)cosalpha+phi'(0)sinalpha=0
phi(1)coseta+phi'(1)sineta=0
end{array}
ight. ,
$$ 其中 $qin L^{1}(0,1)$，$al$、$ein[0,pi)$
。節點反演問題是討論如何利用已知的特徵函數的零點（節點）來決定以上問題的參數
$(q,al,e)$。1999年，X.F Yang
證明只需要知道某部分的節點就可以得到唯一性了。簡單說來，他證明了只需要
$(0,b) (frac{1}{2}<bleq 1)$ 中所有的節點，就足以決定
$(q,al,e)$。


在這篇論文中，我們將證明，在 $(0,b)$ 區間中的所有節點的一個 twin
和 dense
的子集合，就足以得到相同的結論。除了減弱他的條件外，我們也簡化了他的證明。


在這篇論文的第二部分，我們將討論另一個節點反演問題，也就是向量型的
Sturm-Liouville 問題：
egin{center}
$
left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x)
A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f
y}(1)+B_{2}{f y}'(1)={f 0}
end{array}
ight.
$。
end{center}

假設 ${f y}(x)$ 是一個定義在 $[0,1]$ 區間 $d$
維的連續向量函數。如果 ${f y}(x_{0})={f 0}$，我們稱 $x_{0}$
是 ${f y}(x)$ 的一個節點。如果 ${f y}(x)$
所有分量的零點都是節點，我們稱 ${f y}(x)$ 有(CZ)性質。 我們稱
$P(x)$ 是可以被同時對角化的，若存在一個常數矩陣 $S$
和一個對角矩陣函數 $U(x)$，使得 $P(x)=S^{-1}U(x)S$。

如果 $P(x)$
是可以被同時對角化的，則明顯存在無窮多個有(CZ)性質的特徵函數。
最近，沈昭亮和謝忠村證明了二維時候的反問題也成立：
如果有無窮多個特徵函數都有(CZ)性質，則 $P(x)$
是可以被同時對角化的。


但他們只討論 Dirichlet
的邊界條件。我們將簡化他們的工作，並且將這個問題擴充到一般的邊界條件上討論。

In this thesis, we study two new inverse nodal problems
introduced by Yang, Shen and Shieh respectively.

Consider the classical Sturm-Liouville problem: $$ left{
egin{array}{c}
-phi'+q(x)phi=la phi
phi(0)cosalpha+phi'(0)sinalpha=0
phi(1)coseta+phi'(1)sineta=0
end{array}
ight. ,
$$ where $qin L^1(0,1)$ and $al,ein [0,pi)$. The inverse
nodal problem involves the determination of the parameters
$(q,al,e)$ in the problem by the knowledge of the nodal points
in $(0,1)$. In 1999, X.F. Yang got a uniqueness result which only
requires the knowledge of a certain subset of the nodal set. In
short, he proved that the set of all nodal points just in the
interval $(0,b) (frac{1}{2}<bleq 1)$ is sufficient to determine
$(q,al,e)$ uniquely.

In this thesis, we show that a twin and dense subset of all nodal
points in the interval $(0,b)$ is enough to determine
$(q,al,e)$ uniquely. We improve Yang's theorem by weakening
its conditions, and simplifying the proof.

In the second part of this thesis, we will discuss an inverse
nodal problem for the vectorial Sturm-Liouville problem: $$
left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x)
A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f
y}(1)+B_{2}{f y}'(1)={f 0}
end{array}
ight. .
$$

Let ${f y}(x)$ be a continuous $d$-dimensional vector-valued
function defined on $[0,1]$. A point $x_{0}in [0,1]$ is called a
nodal point of ${f y}(x)$ if ${f y}(x_{0})=0$. ${f y}(x)$
is said to be of type (CZ) if all the zeros of its components are
nodal points. $P(x)$ is called simultaneously diagonalizable if
there is a constant matrix $S$ and a diagonal matrix-valued
function $U(x)$ such that $P(x)=S^{-1}U(x)S.$

If $P(x)$ is simultaneously diagonalizable, then it is easy to
show that there are infinitely many eigenfunctions which are of
type (CZ). In a recent paper, C.L. Shen and C.T. Shieh (cite{SS})
proved the converse when $d=2$: If there are infinitely many
Dirichlet eigenfunctions which are of type (CZ), then $P(x)$ is
simultaneously diagonalizable.

We simplify their work and then extend it to some general
boundary conditions.

1. Introduction
1.1 A Review of Inverse nodal problems
1.2 A New Inverse Nodal problem
1.3 A Vectorial Inverse Nodal Problem
2. The Inverse Nodal Problem due to Yang
2.1 Preliminaries
2.2 Proof of Main Theorem
2.3 Applications
3. A Vectorial Inverse Nodal Problem
3.1 Dirichlet Boundary Conditions
3.2 Other Boundary Conditions

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