Abstract |
In this thesis, we study two new inverse nodal problems introduced by Yang, Shen and Shieh respectively. Consider the classical Sturm-Liouville problem: $$ left{ egin{array}{c} -phi'+q(x)phi=la phi phi(0)cosalpha+phi'(0)sinalpha=0 phi(1)coseta+phi'(1)sineta=0 end{array} ight. , $$ where $qin L^1(0,1)$ and $al,ein [0,pi)$. The inverse nodal problem involves the determination of the parameters $(q,al,e)$ in the problem by the knowledge of the nodal points in $(0,1)$. In 1999, X.F. Yang got a uniqueness result which only requires the knowledge of a certain subset of the nodal set. In short, he proved that the set of all nodal points just in the interval $(0,b) (frac{1}{2}<bleq 1)$ is sufficient to determine $(q,al,e)$ uniquely. In this thesis, we show that a twin and dense subset of all nodal points in the interval $(0,b)$ is enough to determine $(q,al,e)$ uniquely. We improve Yang's theorem by weakening its conditions, and simplifying the proof. In the second part of this thesis, we will discuss an inverse nodal problem for the vectorial Sturm-Liouville problem: $$ left{egin{array}{c} -{f y}'(x)+P(x){f y}(x) = la {f y}(x) A_{1}{f y}(0)+A_{2}{f y}'(0)={f 0} B_{1}{f y}(1)+B_{2}{f y}'(1)={f 0} end{array} ight. . $$ Let ${f y}(x)$ be a continuous $d$-dimensional vector-valued function defined on $[0,1]$. A point $x_{0}in [0,1]$ is called a nodal point of ${f y}(x)$ if ${f y}(x_{0})=0$. ${f y}(x)$ is said to be of type (CZ) if all the zeros of its components are nodal points. $P(x)$ is called simultaneously diagonalizable if there is a constant matrix $S$ and a diagonal matrix-valued function $U(x)$ such that $P(x)=S^{-1}U(x)S.$ If $P(x)$ is simultaneously diagonalizable, then it is easy to show that there are infinitely many eigenfunctions which are of type (CZ). In a recent paper, C.L. Shen and C.T. Shieh (cite{SS}) proved the converse when $d=2$: If there are infinitely many Dirichlet eigenfunctions which are of type (CZ), then $P(x)$ is simultaneously diagonalizable. We simplify their work and then extend it to some general boundary conditions. |