| Abstract: | 本研究的目的是想將 Weyl’s function 在二階微分方程譜的反問題的結果推廣到矩陣 子二階微分算子上，並可用這個結果來研究向量型 Sturm-Liouville 方程的反問題以及 Sturm-Liouviile equation on graphs 的反問題的研究。 令 Q(x) C(I ,M (C)) n ! ，A, A, B, B M (C) n ! ，我們想研究的是下列矩陣微分方程的反問題 !" !# $ % + = % + = % + ' = & = ( ) ( ) 0 (0) (0) 0 [ ( )] 0, [0, ] ( ( ) ( Y Y AY BY Y I Q x Y X I n A B A = A = I 的情況，在2006 V. A. Yurko 已有研究成果 [見 (1)]。但在 A, A 不可逆的情 況，則並無成果，而在這的情況下，他的研究成果可以應用到 inverse spectral problem of Sturm-Liouville problem on trees 上，因此我覺得這是一個相當值得研究的問題。 Reference [1] Yurko, V. A., An Inverse problems for the matrix Sturm-Liouville equation on a finite interval, Inverse Problem 22 (2006), 1139-1149.
The purpose of this project is to investigate the Weyl’s function for a matrix-valued differential equation with more flexible boundary conditions. Roughly speaking, we denote Q(x) C(I,M (C)) n ，A, A, B, B M (C) n , we intend to find out the Weyl’s function for the boundary problem (1) A ( ) B ( ) 0 (0) (0) 0 [ ( )] 0, [0, ] Y Y AY BY Y I Q x Y X I n For $n=1,$ we know the weyl’ function can determine the potential uniquely. For n>1 and A = A = I， V. A. Yurko ( see [1]) show the Weyl’s matrix determine the potential function uniquely. So far, there are no results for the cases: neither A nor A is not invertible. We expect to obtain a uniqueness theorem for (1) and this result can be applied to investigate inverse spectral problems of Sturm-Liouville problem on trees and vectorial Sturm-Liouville equations. Reference [1] Yurko, V. A., An Inverse problems for the matrix Sturm-Liouville equ |
