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論文名稱 Title |
週期勢函數的節點重構公式 Reconstruction formulas for periodic potential functions of Hill's equation using nodal data |
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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 |
25 |
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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 |
2005-06-30 |
關鍵字 Keywords |
反節點問題、希爾方程、重構、節點、週期勢函數 Hill's equation, inverse nodal problems, periodic potential function, Reconstruction formula, nodal point |
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統計 Statistics |
本論文已被瀏覽 5722 次,被下載 1738 次 The thesis/dissertation has been browsed 5722 times, has been downloaded 1738 times. |
中文摘要 |
希爾方程也就是水丁格方程: -y'+qy=la y在勢函數q具有週期性質的條件下的邊值問題。 不失一般性,我們可以假設邊界條件為週期: y(0)=y(1),y'(0)=y'(1) 或半週期:y(0)=-y(1),y'(0)=-y'(1)。 我們有興趣研究希爾方程的反節點問題,特別是關於勢函數q的重構問題。 我們想利用節點(特徵方程的零點)來重構勢函數q。 本論文中,我們給出了兩個勢函數的重構。 分別是: (1)在周期邊界條件下 (2)在半周期邊界條件下 另外,我們也證明了上式數列的收斂性:若勢函數為連續函數則逐點收斂;若為 L^1 函數則是處處逐點收斂,同時亦是 L^1 收斂。我們利用了平移轉換到狄利克雷問題解決這些問題,這個想法來自寇斯肯和哈雷斯。 |
Abstract |
The Hill's equation is the Schrodinger equation $$-y'+qy=la y$$ with a periodic one-dimensional potential function $q$ and coupled with periodic boundary conditions $y(0)=y(1)$, $y'(0)=y'(1)$ or anti-periodic boundary conditions $y(0)=-y(1)$, $y'(0)=-y'(1)$. We study the inverse nodal problem for Hill's equation, in particular the reconstruction problem. Namely, we want to reconstruct the potential function using only nodal data ( zeros of eigenfunctions ). In this thesis, we give a reconstruction formula for $q$ using the periodic nodal data or using anti-periodic nodal data We show that the convergence is pointwise for all $x in (0,1)$ where $q$ is continuous; and pointwise for $a.e.$ $x in (0,1)$ as well as $L^1$ convergence when $qin L^1(0,1)$. We do this by making a translation so that the problem becomes a Dirichlet problem. The idea comes from the work of Coskun and Harris. |
目次 Table of Contents |
1 Introduction 1.1 Inverse nodal problems 1.2 Hill's equation 1.3 Continuous potentials 1.4 Main results 2 Proof of Main Theorems 2.1 A reconstruction formula for continuous potentials 2.2 Some asymptotic formulas for $L^1$ potentials 2.3 A reconstruction formula for $L^1$ potentials 2.4 The case of anti-periodic boundary conditions A Comparsion with Coskun's theorem |
參考文獻 References |
1. Y.T. Chen, Y.H. Cheng, C.K. Law and J. Tsay, $L^1$ convergence of the reconstruction formula for the potential function, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2319-2324. 2. Y.H. Cheng and C.K. Law, On the quasinodal map for the Sturm-Liouville problem, to appear in Proc. Royal Soc. Edinburgh series A. 3. H. Coskun and B. J. Harris, Estimates for the periodic and semi-periodic eigenvalues of the Hill's equation, Proc. Royal Soc. Edinburgh, 130A (2000), 991-998. 4. 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. 5. W. Magnus and S. Winkler, Hill's equation, Dover, New York. (1979) 6. C.K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems 17 (2001), 1493-1512. 7. H. Coskun and B. J. Harris, Estimates for the periodic and semi-periodic eigenvalues of the Hill's equation, Proceedings of the Royal Society of Edinburgh, 130A (2000), 991-998. 8. J.R. McLaughlin, Inverse spectral theory using nodal data - a uniqueness result, J. Diff. Eqns. 73 (1988), 354-362. 9. X.F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997), 203-213. |
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