一些與Stieltjes 弦以及Jacobi 矩陣相關的固有值問題之研究

Some eigenvalue problems related to the Stieltjes string and the Jacobi matrix

鄒采蓉(Tsai-Jung Tsou)

指導教授 ： 沈昭亮

國立清華大學/理學院/數學系/碩士(2013年)

https://doi.org/10.6843/NTHU.2013.00197

摘要

藉由Jacobi連分式和Stieltjes連分式所發展出來的方法，我們可以研究有關Jacobi矩陣方程和Stieltjes弦方程的一些固有值的反問題，而在這裡Jacobi矩陣方程和Stieltjes弦方程可分別視為古典的Sturm-Liouville方程理論中位勢方程和弦方程的離散版本。在這其中，我們證明了Stieltjes弦方程的Dirichlet-Neumann同譜定理，找到可把Jacobi矩陣方程轉成Stieltjes弦方程的充要條件並且提供轉換的方法。我們研究了和Sturm-Liouville方程理論中二譜決定的Borg 定理有關的Jacobi matricial couples的理論。我們也考慮一些有關斜對稱Jacobi矩陣和偶Stieltjes弦在給定固有值的條件下的一些固有值的反問題。

關鍵字

Stieltjes弦 ； Jacobi矩陣 ； 反問題 ； 固有值問題 ； 連分式 ； 斜對稱 ； Dirichlet-Neumann同譜定理 ； 離散型

並列摘要

With the help of methods developed for the Jacobi continued fraction and the Stieltjes continued fraction, we investigate some inverse spectral problems related to the
Jacobi matrix equation and the Stieltjes string equation, which may be viewed respectively as the discrete analogues of the potential equation and the string equation studied in the classical theory of Sturm-Liouville equations. We prove, among others, a Dirichlet-Neumann-isospectral theorem for Stieltjes string equations, we find a necessary and sufficient condition for the transformability of a Jacobi matrix equation into a Stieltjes string equation and provide a transformation method. We investigate a theory of Jacobi matricial couples which is related to the two spectra Borg's theorem in the theory of Sturm-Liouville equations. We also consider some inverse spectral problems related to persymmetric Jacobi matrices and even Stieltjes strings with prescribed eigenvalues.

並列關鍵字

Stieltjes string ； Jacobi matrix ； inverse problem ； spectral problem ； continued fraction ； Jacobi matricial couples ； persymmetric Jacobi matrices ； Dirichlet-Neumann-isospectral theorem ； discrete type

參考文獻

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Akhiezer, N. I. (1965) The Classical Moment Problem and Some Related Questions in Analysis}, Hafner Publishing Company, New York.

Atkinson, F. V. (1964) Discrete and Continuous Boundary Problems}, Academic Press.

Borg, G. (1945) Eine umkehrung der Sturm-Liouvillesehen eigenwertaufgabe}, Acta Math., 78, 1--96.

Chihara, T.S. (1978) An Introduction to Orthogonal Polynomials}, Gordon and Breach, New York.