| 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. |
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