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論文名稱 Title |
半無窮區間上的節點反演問題 An inverse nodal problem on semi-infinite intervals |
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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 |
35 |
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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 |
2006-06-15 |
繳交日期 Date of Submission |
2006-07-07 |
關鍵字 Keywords |
節點反演問題、重構、奇異的Sturm-Liouvulle 算子、Parseval方程、半無窮區間 inverse nodal problem, semi-infinite interval, reconstruction, Parseval's equation, singular Sturm-Liouville operator |
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統計 Statistics |
本論文已被瀏覽 5794 次,被下載 1739 次 The thesis/dissertation has been browsed 5794 times, has been downloaded 1739 times. |
中文摘要 |
節點反演問題是討論如何從節點(特徵方程的零點)來了解古典Sturm-Liouville算子之勢函數的問題。這個問題首先由McLaughlin定義。到目前為止,在有限區間上的問題已經研究的相當徹底。唯一性、重構式和穩定性問題都已經解決了。 在這篇論文中,我觀察在半無窮區間的節點反演問題如下:考慮古典的 Sturm-Liouville 問題: 其中 q(x) 是在半無窮區間的連續實變函數並且 q(x) 趨近無窮,當x趨於無窮。我們會有下列性質。L是屬於limit-point狀況 在式中微分算子的頻譜函數是一個階躍函數且 有不連續點。和所對應的解(特徵方程) 恰有k個零點在半無窮區間。再者,特徵方程 表現出正交行為。最後,我們也討論節點的稠密性和重構公式在半無窮區間上。 |
Abstract |
The inverse nodal problem is the problem of understanding the potential function of the Sturm-Liouville operator from the set of the nodal data ( zeros of eigenfunction ). This problem was first defined by McLaughlin[12]. Up till now, the problem on finite intervals has been studied rather thoroughly. Uniqueness, reconstruction and stability problems are all solved. In this thesis, I investigate the inverse nodal problem on semi-infinite intervals q(x) is real and continuous on [0,1) and q(x)!1, as x!1. we have the following proposition. L is in the limit-point case. The spectral function of the differential operator in (1) is a step function which has discontinuities at { k} , k = 0, 1, 2, .... And the corresponding solutions (eigenfunction) k(x) = (x, k) has exactly k zeros on [0,1). Furthermore { k} forms an orthogonal set. Finally we also discuss that density of nodal points and a reconstruction formula on semiinfinite intervals. |
目次 Table of Contents |
Contents 1 Introduction 1 2 Analysis of the Singular Sturm-Liouville Operator 10 3 Density of nodal points and reconstruction formula 21 |
參考文獻 References |
Bibliography [1] M. Bartuˇsek, Z. Doˇsl´a and J. R. Graef, The Nolinear Limit-point/Limit- Circle Problem, Birkh¨auser, Boston, Basel, Berlin, (2004) . [2] G. Birkhoff and G.C. Rota, Ordinary Differential Equations, Fourth Edition, Wiley, (1989) . [3] Y.H. Cheng and C.K. Law, On the quasinodal map for the Strum-Liouville problem, to appear in Proc. Royal Soc. Edinburgh, 136A, 71-86 (2006) . [4] Y.H. Cheng, The inverse nodal problem for Hill’s equations, Unpublished Ph.D. Thesis, National Sun Yat-sen University, Kaohsiung, (2005) . [5] Y.H. Cheng, C.K. Law and J. Tsay, Remarks on a new inverse nodal problem, J. Math. Anal. Appl. 248 (2000), no. 1, 145-155. [6] E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, New York : McGraw-Hill, (1955) . [7] G.B. Folland, Fourier analysis and its applications, Pacific Grove, California, (1992) . [8] C.K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problem 17 (2001), 1493-1512. [9] C.K. Law, C.L. Shen and C.F. Yang, The inverse nodal problem on the smoothness of the potential function, Inverse Problems 15 (1999), 253-263; Errata, 17 (2001), 361-364. [10] C.K. Law and C.F. Yang, Reconstructing the potential fuction and its derivatives using nodal data, Inverse Problem 14 (1998), 299-312. [11] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Dordrecht, Boston, London, (1991) . [12] J.R. McLaughlin, Inverse spectral theory using nodal points as data - a uniqueness result, J. Diff. Eq. 73, (1988), 354-362. [13] E.C. Titchmarsh, Eigenfunction Expansions-Associated with Second-Order Differential Equations, Part I, Oxford University Press, 2nd edition, (1962). [14] A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, second edition of Russian original, Moscow, (1968) . [15] C.J. Wu, Reconstruction formulas for periodic potential functions of Hill’s equation using nodal data, Unpublished Master Thesis, National Sun Yatsen University, Kaohsiung, (2005) . [16] X.F. Yang, A new inverse nodal problem, J. Differential Equations 169 (2001), no. 2, 633-653. [17] X.F. Yang, A solution of the inverse nodel problem, Inverse Problems 13, 203-213, (1997) . |
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