此論文旨在把古典的Ambarzumyan
定理推廣到一般的邊界值條件。在1929年，Ambarzumyan 證明：若
$left{ n^{2}:n=0,1,2,cdots
ight} $ 是 $-y^{^{prime prime
}}+q(x)y=lambda y$ 和 $y^{^{prime }}(0)=y^{^{prime }}(pi )=0$
的譜集合，則 $q$ 在 $left[ 0,pi

ight] $
區間幾乎處處為0。 在1997年，沈昭亮老師和程華淮老師
把Ambarzu-myan定理推廣至向量的型式，他們證明：假設 $left{
k^{2}:k=0,1,2, cdots
ight} $ 是 $-phi ^{^{prime prime
}}+P(x)phi =lambda phi $ 和 $phi ^{^{prime }}(0)=phi
^{^{prime }}(pi )=0$ 的譜集，且每一個特徵值$k^{2}$
的次數皆為$d$，則此連續矩陣$P(x)=0$。

在此，我們先探討，對應於一般邊界值條件的純量Sturm-Liouville問題：
egin{eqnarray*}
-y^{^{prime prime }}+q(x)y&=&lambda y
y(0)cos alpha +y'(0)sin alpha &=&0
y(pi )cos eta +y'(pi )sin eta &=&0.
end{eqnarray*}
如果$q=0$，則 $alpha =eta
eq
frac{pi}{2}$ 若且唯若 ${n^{2}:nin
N}$ 是此方程組的譜集合。接著，在增加 $int_{0}^{pi }q(x)cos
2(x-alpha ), dx=0$ 這一個條件後，我們證明了：當
$alpha=eta
eqfrac{pi}{2}$ 且 ${n^{2}:nin N}$
是此方程組的譜集合，則
$q$幾乎處處為0。此結果包含Dirichlet邊界條件，並且
補充了Poschel-Trubowitz的譜反演理論。


最後，我們也平行地將結果推廣至向量型Sturm-Liouville問題。

We extend the classical Ambarzumyan's theorem for the
Sturm-Liouville equation, which is concerned only with the
Neumann boundary conditions, to the general boundary conditions,
by imposing an additional condition on the potential function. Our
result supplements Poschel-Trubowitz's inverse spectral theory.
We also have parallel results for the vectorial Sturm-Liouville
system.

chapter1:Introduction
chapter2:Scalar case
chapter3:The vectorial case

1.V.A. Ambarzumyan (1929), "{U}ber eine Frage der Eigenwerttheorie,
Z. Phys., 53 690-695.
2.R. Carlson (1999), Large eigenvalues and trace formulas for matrix Sturm-Liouville problems, SIAM J. Math. Anal., 30 949-962.
3.N.K. Chakravarty and S.K. Acharyya (1988), On an extension of the theorem
of V.A. Ambarzumyan, Proc. R. Soc. Edinburgh A, 116A 79-84.
4.H.H. Chern and C.L. Shen (1997), On the $n$-dimensional
Ambarzumyan's theorem, Inverse problems, 13 15-18.
5.C.K. Law and J. Tsay (2000), On the well-posedness of the inverse nodal problem, submitted.
6.V.A. Marchenko (1986), Sturm-Liouville Operators and Applications,
Birkh"{a}user Verlag Basel Boston tuttgart.
7.J. P"{o}schel and E. Trubowitz (1987), Inverse Spectral Theory, Academic Press, Boston.
8.X.F. Yang (1997), A solution of the inverse nodal problem,
Inverse Problems, vol.13, 203-213. Academic Press, Boston.