臺灣博碩士論文加值系統

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 ~$q=0$ 。

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 $Gamma$.

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

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