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 $Gamma$. |