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博碩士論文 etd-0130108-181331 詳細資訊
Title page for etd-0130108-181331
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
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
38
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
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
統計
Statistics
本論文已被瀏覽 5797 次,被下載 1309
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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