- 學位論文
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
摘要
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。
關鍵字
頻譜並列摘要
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.