System No. U0002-0406201219461400 在有限區間向量型Sturm-Liouville方程式的唯一性定理 Uniqueness of the potential function of the vectorial Sturm- Liouville equations with general boundary conditions 淡江大學 數學學系博士班 Department of Mathematics 100 2 101 張淙華 Tsorng-Hwa Chang 892150052 博士 English 2012-05-17 51page advisor - Chung-Tsun Shieh co-chair - 沈昭亮 co-chair - 羅春光 co-chair - 朱啟平 co-chair - 錢傳仁 co-chair - 陳功宇 co-chair - 楊定揮 頻譜 Weyl matrix Sturm-Liouville equation Potential `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.``` Within Campus： On-campus access to my hard copy thesis/dissertation is open immediately Agree to authorize disclosure on campus Release immediately Outside the Campus： I grant the authorization for the public to view/print my electronic full text with royalty fee and I donate the fee to my school library as a development fund.Release immediately