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URN etd-0704105-132441 Author Yan-Hsiou Cheng Author's Email Address No Public. Statistics This thesis had been viewed 5077 times. Download 1728 times. Department Applied Mathematics Year 2004 Semester 2 Degree Ph.D. Type of Document Language English Title Inverse Problems for Various Sturm-Liouville Operators Date of Defense 2005-06-27 Page Count 94 Keyword Sturm-Liouville problem spectral problem nodal problem Hill's equation Abstract In this thesis, we study the inverse nodal problem and inverse

spectral problem for various Sturm-Liouville operators, in

particular, Hill's operators.

We first show that the space of Schr"odinger operators under

separated boundary conditions characterized by ${H=(q,al, e)in

L^{1}(0,1) imes [0,pi)^{2} : int_{0}^{1}q=0}$ is homeomorphic

to the partition set of the space of all admissible

sequences $X={X_{k}^{(n)}}$ which form sequences that

converge to $q, al$ and $ e$ individually. The definition of

$Gamma$, the space of quasinodal sequences, relies on the $L^{1}$

convergence of the reconstruction formula for $q$ by the exactly

nodal sequence.

Then we study the inverse nodal problem for Hill's equation, and

solve the uniqueness, reconstruction and stability problem. We do

this by making a translation of Hill's equation and turning it

into a Dirichlet Schr"odinger problem. Then the estimates of

corresponding nodal length and eigenvalues can be deduced.

Furthermore, the reconstruction formula of the potential function

and the uniqueness can be shown. We also show the quotient space

$Lambda/sim$ is homeomorphic to the space $Omega={qin

L^{1}(0,1) :

int_{0}^{1}q = 0, q(x)=q(x+1)

mbox{on} mathbb{R}}$. Here the space $Lambda$ is a collection

of all admissible

sequences $X={X_{k}^{(n)}}$ which form sequences that

converge to $q$.

Finally we show that if the periodic potential function $q$ of

Hill's equation is single-well on $[0,1]$, then $q$ is constant if

and only if the first instability interval is absent. The same is

also valid for convex potentials. Then we show that similar

statements are valid for single-barrier and concave density

functions for periodic string equation. Our result extends that of

M. J. Huang and supplements the works of Borg and Hochstadt.Advisory Committee Chao-Liang Shen - chair

Tzon-Tzer Lu - co-chair

Jenn-Nan Wang - co-chair

Chiu-Ya Lan - co-chair

Chung-Tsun Shieh - co-chair

Wei-Cheng Lian - co-chair

Chao-Nien Chen - co-chair

Tzy-Wei Hwang - co-chair

Chun-Kong Law - advisor

Files indicate accessible in a year

etd-0704105-132441.pdf Date of Submission 2005-07-04