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論文名稱 Title |
圖上 Sturm-Liouville 算子的模型及其相關之 Ambarzumyan 問題 A model of Sturm-Liouville operators defined on graphs and the associated Ambarzumyan problem |
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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 |
38 |
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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 |
2008-01-11 |
繳交日期 Date of Submission |
2008-01-30 |
關鍵字 Keywords |
Sturm-Liouville 算子、圖、Ambarzumyan 問題、Pokornyi 模型 Pokornyi's model, graphs, Ambarzumyan problems, Sturm-Liouville operator |
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統計 Statistics |
本論文已被瀏覽 5797 次,被下載 1309 次 The thesis/dissertation has been browsed 5797 times, has been downloaded 1309 times. |
中文摘要 |
本論文將研究圖上~Sturm-Liouville 算子的~Pokornyi 模型,這個模型由~Pokornyi 和~Pryadiev 在~2004~年提出 ,藉由考慮在某一介質內,彈簧連結系統受振盪時抵抗的最小勢能得到。 此彈簧系統定義在圖~$Gamma$ 上,包括不含端點的線集合 ~$R(Gamma)={gamma_i:i=1,dots,n}$ 和內部點集合~$J(Gamma)$,令 ~$partialGamma$ 為邊界點集合,對任一 ~${f v}in J(Gamma)$ 我們令 ~$Gamma({f v})={gamma_iin R(Gamma):~{f v}$ 是$gamma_i$ 的一個端點 $}$。有關特徵值問題的模型如下 egin{eqnarray*} -(p_iy_i')'+q_iy_i&=&lambda y_i,~~~~~qquad on~gamma_i上, y_i({f v})&=&y_j({f v}),~~~~~~~~forall {f v}in J(Gamma)~ ~gamma_i,gamma_jin Gamma({f v}), sum_{gamma_iin Gamma({f v})}p_i({f v})frac{dy({f v})}{dgamma_i}+q({f v})y({f v})&=&lambda y({f v}),qquad ~~forall {f v}in J(Gamma), end{eqnarray*} 配合邊界點上的 ~Neumann 或 ~Dirichlet 邊界條件。這個模型也是~Kuchment 定義在量子圖上的一個特例。 我們將推導出此模型並討論其譜性質,也會依此模型解一些 ~Ambarzumyan 問題。特別地,我們將證明在 ~$n$ 條長度為 ~$a$ 的星狀圖上,若有 ~$p_iequiv1$, ${frac{(m+frac{1}{2})^2)pi^2}{a^2}:min Ncup{0}}$ 為其 ~Neumann 特徵值且 ~$0$ 為最小之特徵值,又在內點 ~${f v}$~處 ~$q_i({f v})=0$,則在圖$Gamma$~上 ~$q=0$ 。 |
Abstract |
In this thesis, we study the Pokornyi's model of a Sturm-Liouville operator defined on graphs. The model, proposed by Pokornyi and Pryadiev in 2004, is derived from the consideration of minimal energy of a system of interlocking springs oscillating in a medium with resistance. Here the system of springs is defined as a graph $Gamma$ with edges $R(Gamma)={gamma_i:i=1,dots,n}$ and set of internal vertices $J(Gamma)$. Let $partialGamma$ denote the set of boundary vertices of $Gamma$. For each vertex ${f v}in J(Gamma)$, we let $Gamma({f v})={gamma_iin R(Gamma):~{f v}$ is an endpoint of $ gamma_i}$. The related eigenvalue problem of the model is as follows: egin{eqnarray*} -(p_iy_i')'+q_iy_i&=&lambda y_i,~~~~~qquad mbox{on}~gamma_i, y_i({f v})&=&y_j({f v}),~~~~~~~~forall {f v}in J(Gamma)~ mbox{and}~gamma_i,gamma_jin Gamma({f v}), sum_{gamma_iin Gamma({f v})}p_i({f v})frac{dy({f v})}{dgamma_i}+q({f v})y({f v})&=&lambda y({f v}),qquad ~~forall {f v}in J(Gamma), end{eqnarray*} equipped with Neumann or Dirichlet boundary conditions. This model is also a special case of some quantum graphs defined by Kuchment . par We shall derive the model and discuss the spectral properties. We shall also solve several Ambarzumyan problems on the model. In particular, we show that for a $n$-star shaped graph of uniform length $a$ with $p_iequiv1$, if ${frac{(m+frac{1}{2})^2)pi^2}{a^2}:min Ncup{0}}$ are Neumann eigenvalues, $0$ is the least Neumann eigenvalue, and $q_i({f v})=0$ for ${f v}in J(Gamma)$, then $q=0$ on $Gamma$. |
目次 Table of Contents |
1 Introduction 5 2 Pokornyi's model 12 3 Direct Problem for the Ambarzumyan theorems 18 4 Inverse Problem for the Ambarzumyan theorems 22 5 Further Discussion 27 |
參考文獻 References |
[1] V.A. Ambarzumyan, ÄUber eine Frage der Eigenwerttheorie, Z. Phys., 53,(1929) 690-695. [2] R. Carlson and V.N. Pivovarchik, Ambarzumian's theorem for trees, Electronic J. Di®. Eqns., Vol. 2007(2007), no. 142, 1-9. [3] H.H. Chern, C.K. Law, and H.J. Wang, Extension of Ambarzumyan's theorem to general boundary conditions, J. Math. Anal. Appl., 263, no. 2 (2001) 333-342; Corrigendum, 309, no.2 (2005) 764-768. [4] N. Gerasimenko and B. Pavlov, Scattering problems on non-compact graphs, Theor. Math. Phys., 74, (1988) 230-240. [5] T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett., 79, (1997) 4794-4797. [6] P. Kuchment, Graph models for waves in thin structure, Waves in Random Media, 12, (2002) R1-R24. [7] P. Kuchment, Quantum graphs: I. Some basic structures, Waves in Random media, 14, (2004) S107-S128. [8] P. Kuchment, Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A: Math. Gen., 38, (2005) 4887-4900. [9] C.K. Law and C.T. Shieh, Ambarzumyan-type theorems for the Sturm-Liouville operator on star-shaped graphs, preprint(2007). [10] B.M. Levitan and I.S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluvwer Academic Publishers, Dordrecht, 1991. [11] V.N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal., 32, no.4 (2000) 801-819. [12] V.N. Pivovarchik, Ambarzumian's theorem for a Sturm-Liouville boundary value problem on a star-shaped graph, Funct. Anal. & Appli., 39, no.2 (2005)148-151. [13] V.N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a star-shaped graph, Math. Nachrichten, Vol.280 (2007), no.13-14, 1595-1619. [14] Yu.V. Pokornyi and A.V. Borvskikh, De®erential equations on networks (geometric graphs), J. Mathematical Sciences, 119, no.6 (2004) 691-718. [15] Yu.V. Pokornyi and V.L. Pryadiev, The qualitative Sturm-Liouville theory on spatial networks, J. Mathematical Sciences, 119, no.6 (2004) 788-835. [16] M. Solomyak, On the spectrum of the Laplacian on regular metric trees, Waves Random Media, 14, (2004) S143-153. [17] M.L. Wu, Ambarzumyan Theorem for the Sturm-Liouville Operator De‾ned on Graphs, Unpublished master Thesis, National Sun Yat-sen University, Kaohsiung, (2007). |
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