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研究生:翁鵬絜
研究生(外文):Wong, Peng-Jie
論文名稱:關於Jacobi矩陣與Green矩陣的一些固有值問題與反問題之研究
論文名稱(外文):Some inverse problems with mixed spectral data for the Jacobi matrix and the Green's matrix
指導教授:沈昭亮沈昭亮引用關係
指導教授(外文):Shen, Chao-Liang
學位類別:碩士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2012
畢業學年度:100
論文頁數:32
中文關鍵詞:反問題固有值問題Jacobi矩陣Green矩陣Stieltjes弦
外文關鍵詞:inverse problemeigenvalue problemJacobi matrixGreen's matrixStieltjes string
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我們藉由某些特殊形式的矩陣函數的分解法去研究關於Jacobi矩陣與Stieltjes弦的反問題。關於Stieltjes弦,我們也試圖估計Stieltjes弦的第一固有值。我們證明,當弦之密度數列之上下界與總質量固定時,我們可以決定使Stieltjes弦的第一固有值為極大或極小的密度數列為何。除此之外,我們也證明幾個關於Stieltjes弦的第一、第二固有值之比值的比較定理。
Applying the factorization of some related matrix functions, we investigate some inverse problems with mixed spectral data for Jacobi matrices and Stieltjes strings. Besides, we prove a discrete analogue of Borg's theorem for the Green's matrix. We also study the first eigenvalue, and the ratio of the first two eigenvalues of the Stieltjes string equation. With certain restrictions on the class of density sequences $p$, we determine the shapes of the extremal density sequence for the first eigenvalue, and the minimum for the ratio of the first two eigenvalues.
1. Introduction 1
2. Preliminaries 4
3. Decompositions of the Jacobi continued fraction and the Stieltses continued
fraction 6
4. Some inverse spectral problems with mixed spectral data 10
5. The extrema of the rst eigenvalue 17
6. The bounds for the ratio of the rst and second eigenvalues 23
References 30
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