# Inverse Problems for Various Sturm-Liouville Operators

Abstract (Summary)

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.
Bibliographical Information:

Advisor:Chiu-Ya Lan; Chun-Kong Law; Chao-Liang Shen; Jenn-Nan Wang; Wei-Cheng Lian; Tzy-Wei Hwang; Chung-Tsun Shieh; Chao-Nien Chen; Tzon-Tzer Lu

School:National Sun Yat-Sen University

School Location:China - Taiwan

Source Type:Master's Thesis

Keywords:sturm liouville problem spectral nodal hill s equation

ISBN:

Date of Publication:07/04/2005