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論文名稱 Title |
不同型態的Sturm-Liouville 算子之反演問題 Inverse Problems for Various Sturm-Liouville Operators |
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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 |
94 |
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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 |
2005-06-27 |
繳交日期 Date of Submission |
2005-07-04 |
關鍵字 Keywords |
Hill's 方程、節點問題、譜問題、Sturm-Liouville 問題 Sturm-Liouville problem, spectral problem, nodal problem, Hill's equation |
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統計 Statistics |
本論文已被瀏覽 5713 次,被下載 2033 次 The thesis/dissertation has been browsed 5713 times, has been downloaded 2033 times. |
中文摘要 |
在這篇論文裡,我們討論不同型態的 Sturm-Liouville 算子之節點反演問題和譜反演問題,特別針對 Hill 算子。 hspace*{0pt} 首先,我們證明了,由 Schr"odinger 算子伴隨分離式邊界條件所構成的空間 ${H=(q,al, e)in L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ 同胚於收集所有合格的、並使相關公式收斂的數列分割集 $Gamma$。 我們稱集合 $Gamma$ 中的元素為「擬節點數列」,這些數列使得 $q$ 的重構公式是 $L^{1}$ 收斂的。 hspace*{0pt} 接著,我們討論 Hill 方程的節點反演問題。使用的技巧是將 Hill 方程平移為 Dirichlet 的 Schr"odinger 方程。利用對 Schr"odinger 問題的瞭解,我們可得到節點的估計式,並可藉此導出能量函數的重構公式。 然後,我們證明了商空間 $Lambda/sim$ 同胚於空間 $Omega={qin L^{1}(0,1) : int_{0}^{1}q = 0, q(x)=q(x+1) mbox{on} mathbb{R}}$。此處的空間 $Lambda$ ,是收集所有合格的、並使相關的公式收斂的數列。 於是,我們解決了唯一性、重構公式和穩定性問題。 hspace*{0pt} 最後,我們證明了,在 Hill 方程中,如果能量函數在區間 $[0,1]$ 中是單谷的 ,則能量函數為常數函數若且唯若第一個不穩定區間長度為零。同樣的結果對凸能量函數也是成立的。 對於對稱的單谷型 能量函數,黃明傑證明了同樣的結果,而我們減弱其條件,並成功的推廣至週期弦方程。 這也算是 Borg 和 Hochstadt 一些定理的補強。 |
Abstract |
In this thesis, we study the inverse nodal problem and inverse spectral problem for various Sturm-Liouville operators, in particular, Hill's operators. We first show that the space of Schr"odinger operators under separated boundary conditions characterized by ${H=(q,al, e)in L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic to the partition set of the space of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q, al$ and $ e$ individually. The definition of $Gamma$, the space of quasinodal sequences, relies on the $L^{1}$ convergence of the reconstruction formula for $q$ by the exactly nodal sequence. Then we study the inverse nodal problem for Hill's equation, and solve the uniqueness, reconstruction and stability problem. We do this by making a translation of Hill's equation and turning it into a Dirichlet Schr"odinger problem. Then the estimates of corresponding nodal length and eigenvalues can be deduced. Furthermore, the reconstruction formula of the potential function and the uniqueness can be shown. We also show the quotient space $Lambda/sim$ is homeomorphic to the space $Omega={qin L^{1}(0,1) : int_{0}^{1}q = 0, q(x)=q(x+1) mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection of all admissible sequences $X={X_{k}^{(n)}}$ which form sequences that converge to $q$. Finally we show that if the periodic potential function $q$ of Hill's equation is single-well on $[0,1]$, then $q$ is constant if and only if the first instability interval is absent. The same is also valid for convex potentials. Then we show that similar statements are valid for single-barrier and concave density functions for periodic string equation. Our result extends that of M. J. Huang and supplements the works of Borg and Hochstadt. |
目次 Table of Contents |
1. Introduction 1.1 Sturm-Liouville problem and Hill's equation 1.2 Inverse nodal problems 1.3 Inverse spectral problems 1.4 Conclusion 2. The quasinodal map for the Schrodinger operator 2.1 The space of quasinodal sequences 2.2 Preliminaries 2.3 Proof of main theorem 2.4 Two examples 3. The inverse nodal problem for Hill's equation 3.1 Preliminaries 3.2 Uniqueness 3.3 Stability and existence 4. Firse instability intervals for periodic Sturm-Liouville problems 4.1 Preliminaries 4.2 Single-well potentials and single-barrier densities 4.3 Convex potentials and concave densities 5. Appendix: Sturm-Liouville theory for L^1 potential functions 5.1 Analyticity properties of the eigenfunction 5.2 Counting Lemma 5.3 Properties of eigenvalues and nodal points |
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