### Title page for etd-0130108-181331

URN etd-0130108-181331 Yi-Chieh Hung boom2000tw@yahoo.com.tw This thesis had been viewed 5219 times. Download 1088 times. Applied Mathematics 2007 1 Master English A model of Sturm-Liouville operators defined on graphs and the associated Ambarzumyan problem 2008-01-11 38 Pokornyi's model graphs Ambarzumyan problems Sturm-Liouville operator In this thesis, we study the Pokornyi's model of aSturm-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\$. Wei-cheng Lian - chair Tzon-Tzer Lu) - co-chair Chun-Kong Law - advisor indicate accessible in a year 2008-01-30

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