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博碩士論文 etd-0707106-122050 詳細資訊
Title page for etd-0707106-122050
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
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
35
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
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
統計
Statistics
本論文已被瀏覽 5794 次,被下載 1739
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
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Circle Problem, Birkh¨auser, Boston, Basel, Berlin, (2004) .
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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) .
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J. Math. Anal. Appl. 248 (2000), no. 1, 145-155.
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New York : McGraw-Hill, (1955) .
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Inverse Problem 17 (2001), 1493-1512.
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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
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[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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