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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 |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
63 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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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 |
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統計 Statistics |
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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) . 6. H. Hochstadt and B. Liebermann, An inverse Sturm-Liouville problem with mixed given data, SIAM J. Appl. Math. $mathbf{34},$ No.4, 676-680 (1978) . 7. H. Hald, The inverse Sturm-Liouville problems with symmetric potentials, Acta Mathematica, (1978) . 8. E. L. Isaacson, E. Trubowitz, The inverse Sturm-Liouville problem. I., Comm. Pure Appl. Math. 36, no. 6, 767-783, (1983) . 9. M. Jodeit, B. M. Levitan, The isospectrality problems 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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