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博碩士論文 etd-0704105-182611 詳細資訊
Title page for etd-0704105-182611
論文名稱
Title
變換算子理論及其在譜反演問題上的應用
The theory of transformation operators and its application in inverse spectral problems
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
63
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2005-06-03
繳交日期
Date of Submission
2005-07-04
關鍵字
Keywords
變換算子、特徵值、譜問題、特徵向量
eigenvalue, Transformation operator, spectral problem, eigenvector
統計
Statistics
本論文已被瀏覽 5701 次,被下載 1616
The thesis/dissertation has been browsed 5701 times, has been downloaded 1616 times.
中文摘要
譜反演問題是透過特徵值,再加上一些關於譜的數據,去了解位勢函數的問題。
而變換算子理論最早是由Marchenko所提出,
後來被Gelfand和Levitan所強化,這對於譜反演的四部曲:唯一性、重建性、平衡性、存在性
是一個強而有力的方法。在這篇論文裡,我們會研究一系列變換算子的理論。感覺上來說,
變換算子$X$是將一個Sturm-Liouville算子的解,映射到另一個Sturm-Liouville算子的解。我們可以把它寫成$$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$
其中,核函數$K$必須滿足這個Goursat問題$$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0,$$再加上一些初始
值的邊界條件。除此之外,透過一個有名的Gelfand-Levitan方程式$$K(x,y)+F(x,y)+int_{0}^{x}K(x,t)F(t,y)dt=0,$$
$K$和$F$就被連結在一起;而$F$可由${(lambda_{n},alpha_{n})}$定義,
其中,$alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2}dt)^{frac{1}{2}}$。此外我們把上述的關係連成以下的式子
hspace*{2.3in}$q$ $Longleftrightarrow K$
hspace*{2.3in}$Downarrow$ hspace*{0.3in} $Updownarrow $
hspace*{2in}${(alpha_{n},lambda_{n})}Rightarrow F$
hspace*{0.25in}我們會利用Riesz basis和整函數的階數,來對這個理論做精要的說明,然後,我們也會報告一些最近有關於譜反演問題的唯一性的應用。
Abstract
The inverse spectral problem is the problem of
understanding the potential function of the Sturm-Liouville
operator from the set of eigenvalues plus some additional
spectral data. The theory of transformation operators, first
introduced by Marchenko, and then reinforced by Gelfand and
Levitan, is a powerful method to deal with the different stages
of the inverse spectral problem: uniqueness, reconstruction,
stability and existence. In this thesis, we shall give a survey
on the theory of transformation operators. In essence, the theory
says that the transformation operator $X$ mapping the solution of
a Sturm-Liouville operator $varphi$ to the solution of a
Sturm-Liouville operator, can be written as
$$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$ where the
kernel $K$ satisfies the Goursat problem
$$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0$$ plus some initial boundary
conditions. Furthermore, $K$ is related by a function $F$ defined
by the spectral data ${(lambda_{n},alpha_{n})}$ where
$alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2})^{frac{1}{2}}$
through the famous Gelfand-Levitan equation
$$K(x,y)+F(x,y)+int_{o}^{x}K(x,t)F(t,y)dt=0.$$ Furthermore, all
the above relations are bilateral, that is $$qLeftrightarrow
KLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.$$
hspace*{0.25in}We shall give a concise account of the above
theory, which involves Riesz basis and order of entire functions.
Then, we also report on some recent applications on the
uniqueness result of the inverse spectral problem.
目次 Table of Contents
Chapter1 Introduction
chapter2 Preliminaries
2.1 Volterra integral equations
2.2 Order of an entire function
2.3 Riesz basis
Chapter3 The theory of transformation operators
3.1 Transformation operators and Goursat problems
3.2 Gelfand-Levitan equation
Chapter4 Uniqueness theorems
4.1 Some preliminary lemmas
4.2 Hochstadt-Libermann's uniqueness theorem
4.3 Mochizuki-Trooshin's uniqueness theorem
Chapter5 Appendix
5.1 Some lemmas for the theory of transformation operators
5.2 Stability and existence theorems
參考文獻 References
1. G. Borg, Eine Umkehrung der Sturm-Liouvilleschen
Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte. Acta. Math. 78, 1-96, (1946) .
2. K. Chadan, D. Colton, L. P$ddot{a}$iv$ddot{a}$rinta and W.Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM., Philadelphia (1997) .
3. G. Freiling and V. Yurko, Inverse Sturm-Liouville
Problems And Their Applications, Nova Science Publishers,New York, (2001) .
4. I. M. Gelfand, B. M. Levitan, On the determination of
a differential equation from its spectra function, Izv. Akad.Nauk SSSR, Ser. Mat. 15, 309-360, (1951); English transl. in Amer. Math. Soc. Transl.(2)1, (1955) .
5. H. Hochstadt, The inverse Sturm-Liouville problem,
Comm. Pure Appl. Math. no. 26, 715-729, (1973) .
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data, SIAM J. Appl. Math. $mathbf{34},$ No.4, 676-680 (1978) .
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problem. I., Comm. Pure Appl. Math. 36, no. 6, 767-783, (1983) .
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for the classical Sturm-Liouville equation, Adv. Differential
Equations $mathbf{2},$ no. 2, 297-318, (1997) .
10. B. Ja. Levin, Distribution of Zeros of Entire
Functions, American Mathematical Society, Providence, (1964) .
11. N. Levison, The inverse Sturm-Liouville
problem, Math. Tidsskr. 13, 25-30, (1949) .
12. B. M. Levitan, M. G. Gasymov, Determination of a
differential equation by two spectra, (Russian) Uspehi Mat. Nauk 19, no. 2(116), 3-63, (1964) .
13. B. M. Levitan, Inverse Sturm-Liouville problems, Nauka
Moscow, (1984); English transl., VNU Sci. Press, Utrecht, (1987) .
14. B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac
Operators, Kluwer Academic Publishers, Dordrecht, (1991) .
15. V. A. Marchenko, Some problems in the theory of a
second order differential operator, Dokl. Akad. Nauk SSSR, 72, 457-460, (1950) .
16. V. A. Marchenko, Sturm-Liouville Operators and
Applications, Birkh$ddot{a}$user, Basel, (1986) .
17. K. Mochizuki and I. Trooshin, Inverse problem for
interior spectal data of the Sturm-Liouville operator. {sl J.Inv. Ill-Posed Problems}, Vol. 9, No. 4, 425-433 (2001) .
18. J. P$ddot{o}$schel, E. Trubowitz, Inverse Spectral
Theory, Academic Press, London, (1987) .
19. L. Sakhnovich, Half-inverse problems on the finite
interval, Inverse Problems $mathbf{17},$ 527-532, (2001) .
20. C. T. Shieh, Transformation operators and its
applications, unpublished lecture notes .
21. R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, (1980) .
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