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URN etd-0717100-110512 Author Yan-Hsiou Cheng Author's Email Address jengyh@ibm7.math.nsysu.edu.tw Statistics This thesis had been viewed 5116 times. Download 1730 times. Department Applied Mathematics Year 1999 Semester 2 Degree Master Type of Document Language English Title On Some New Inverse nodal problems Date of Defense 2000-06-02 Page Count 39 Keyword Sturm-Liouville nodal set inverse nodal problem 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.Advisory Committee Tzon-Tzer Lu - co-chair

Hua-Huai Chern - co-chair

Jhishen Tsay - co-chair

Tzy-Wei Hwang - co-chair

Chun-Kong Law - advisor

Files indicate in-campus access immediately and off_campus access in a year

graduate.pdf Date of Submission 2000-07-17