關於定義在區間的非對稱形Sturm-Liouville 微分方程式的反問題研究及學習，Yurko ( [24] , 2006)利用Weyl矩陣，提出了矩陣邊界值問題的反問題有唯一性的定理。 在本篇論文，首先；對於Sturm-Liouville矩陣微分方程式含有一般的邊界條件的反問題，我們將証明ㄧ般的h1 , H1，亦可得到Q(x)有唯一性。利用矩陣型式邊界值反問題的唯一性，我們主要工作是在向量微分方程式邊界值反問題上，探求向量頻譜(spectral sets)與位階函數Q(x)唯一性的關係。 對於h1 = H1 = In ,我們找出某些個頻譜就可以決定Q(x)了。而若為一對稱矩陣或對角化矩陣，則個別僅需某些頻譜集合即可。 對於一般的h1 , H1，我們也獲得了一些相關的結果。

Inverse spectral problems are studied for the non-self-adjoint matrix Sturm-Liouville differential equation on a finite interval. Using Weyl function, Yurko([24],2006) solved the inverse spectral problem for the matrix Sturm-Liouville operator on a finite interval with the boundary value problem L(Q(x), h, H ).                                    
At first, in this thesis, we try to solve the uniqueness theorem of the matrix-valued boundary value problem for arbitrary matrices h1 , h0 , H1 , H0  with the general boundary conditions. By the uniqueness theorem of  L(Q(x),h1 , h0 , H1 , H0) described as above, our main work is to find those relations between spectra and potential Q(x) for the vectorial Sturm-Liouville differential equation.
For h1 = H1 = In ,  we will give some characteristic functions corresponding to spectra to determine the Weyl matrix and to prove the uniqueness theorem. Furthermore, we also prove the uniqueness theorems for the vectorial Sturm-Liouville operators with real symmetric potential or real diagonal potential by given some spectra, respectively. We also obtain some results for arbitrary matrices h1 and H1.

Chapter 1.  Introduction
1.1  Sturm-Liouville operators on a finite interval 
                               ..........  3
1.2  Vectorial Sturm-Liouville equation on a finite 
      interval                 ...........6              
1.3  The Weyl matrix .............. 9
1.4  Sturm-Liouville equation on a graph ...... 11

Chapter 2. Uniqueness of the potential function for vectorial Sturm-Liouville equation on a finite interval
2.1  Preliminaries  ............19
2.2  Main results ..............21
 
Chapter 3. Uniqueness theorem for the vectorial Sturm-Liouville equation with general boundary conditions
3.1  Introduction  ....................  31
3.2  Preliminaries    ...............    32 
3.3  Main results   .................... 38

References  .............................50

[1] Andersson E., On the m-function and Borg-Marchenko theorems for vector-valued Sturm- Liouville equations. Journal of Mathematical Physics Vol. 44(2003), Issue 12, pp. 6077-6100.
[2] Brog G., Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe Bestimmung der Differentialgleichung durch die Eigenwerte. Acta Math. 78(1946), 1-96.
[3] Carlson R., An inverse problem for the matrix Schr dinger equation. Journal of Mathematical Analysis and Applications, 267(2002) , pp. 564-575.
[4] Chern H-H and Shen C-L., On the n-dimensional Ambarzumyan’s theorem. Inverse Problems, 13(1997) No 1, 15-18.
[5] Clark S.; Gesztesy F.; Holden H.and Levitan B. M., Borg-Type Theorems for Matrix-Valued Schr dinger Operators. Journal of Differential Equations Vol.(2000),167 (2000), No. 1, pp. 181-210.
[6] Gesztesy F. and Simon B., On the determination of a potential from three spectra, Differential Operators and Spectral Theory. Amer. Math. Soc. Transl. Ser 2, vol 189, American Mathematical Society, Providence, R1 (1999), 85-92. 
[7] Gesztesy F.; Kiselev A. and Makarov K. A., Uniqueness Results for matrix-valued schr dinger, Jacobi, and Dirac-Type Operators. Math. Nachr. 239-240(2002), Issue1, pp. 103-145.
[8] Hochstadt H., The inverse Sturm-Liouville problem. Comm. Pure Appl. Math. 26 (1973), 715-729.
[9] Hochstadt H and Lieberman B., An inverse Sturm-Liouville problem with mixed given data. SIAM J. Appl. Math. 34(1978), 676-680.  

[10] Jodeit M. and Levitan B. M., The Isospectrality Problem for the Classical Sturm-Liouville Equation. Advances in Differential Equations Vol.2(1997), 297-318.
[11] Jodeit M. and Levitan B. M., Isospectral Vector-Valued Sturm-Liouville Problems. Letters in Mathematical Physics, 43(1998), pp. 117-122.
[12] Krein M. G., Solution of the inverse Sturm-Liouville Problem. Dokl. Akad. Nauk. SSSR 76 (1951), 21-24.
[13] Levinson N., The inverse Sturm-Liouville Problem. Mat. Tidsskr. B., (1949), 25-30.
[14] Levitan B. M., Inverse Sturm-Liouville Problems. Utrecht: VNU, 1987.
[15] Levitan B. M. and Gasymov M. G., Determination of a differential equation by two of its spectra. Russ. Math. Surv. 19(1964), 1-63.
[16] Levitan B. M. and Sargsjan I. S., Introduction to Spectral Theory: Selfadjoint ordinary Differential Operators. Transl. Math. Monographs vol 39 (1975).
[17] Marchenko V. A ., Sturm- Liouville Operators and Application. Basel: Birkhauser, 1986. 
[18] Poschel J. and Trubowitz E., Inverse Spectral Theory. New York: Academic, 1987.
[19] Shen C-L., Some eigenvalue problems for the vectorial Hill’s equation. Inverse Problems 16 (2000), No 3 749-783.
[20] Shen C-L ., Some eigenvalue problems for vectorial Sturm- Liouville equations. Inverse Problems 17(2001), No 5, 1253-1294.
[21] Shieh C-T., Isospectral sets and inverse problems for vector-valued Sturm-Liouville Equations. Inverse Problems 23(2007), No 6, 2457-2468.
[22] Yurko V. A., Method of Spectral Mappings in the Inverse Problem Theory. Inverse And Ill-Posed Problems Series, VSP, Utrecht, 2002.
[23] Yurko V. A., Inverse spectral Problems for Sturm- Liouville operators on graphs. Inverse Problems 21(2005), 1075-1086.
[24] Yurko V. A., Inverse Problems for the matrix Sturm- Liouville equation on a finite interval. Inverse Problems 22(2006), 1139-1149.