The inverse spectral problem is the problem of
understanding the potential function of the Sturm-Liouville
operator from the set of eigenvalues plus some additional
spectral data. The theory of transformation operators, first
introduced by Marchenko, and then reinforced by Gelfand and
Levitan, is a powerful method to deal with the different stages
of the inverse spectral problem: uniqueness, reconstruction,
stability and existence. In this thesis, we shall give a survey
on the theory of transformation operators. In essence, the theory
says that the transformation operator $X$ mapping the solution of
a Sturm-Liouville operator $varphi$ to the solution of a
Sturm-Liouville operator, can be written as
$$Xvarphi=varphi(x)+int_{0}^{x}K(x,t)varphi(t)dt,$$ where the
kernel $K$ satisfies the Goursat problem
$$K_{xx}-K_{tt}-(q(x)-q_{0}(t))K=0$$ plus some initial boundary
conditions. Furthermore, $K$ is related by a function $F$ defined
by the spectral data ${(lambda_{n},alpha_{n})}$ where
$alpha_{n}=(int_{0}^{pi}|varphi_{n}(t)|^{2})^{frac{1}{2}}$
through the famous Gelfand-Levitan equation
$$K(x,y)+F(x,y)+int_{o}^{x}K(x,t)F(t,y)dt=0.$$ Furthermore, all
the above relations are bilateral, that is $$qLeftrightarrow
KLeftrightarrow FLeftarrow {(lambda_{n},alpha_{n})}.$$
hspace*{0.25in}We shall give a concise account of the above
theory, which involves Riesz basis and order of entire functions.
Then, we also report on some recent applications on the
uniqueness result of the inverse spectral problem.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0704105-182611 |
| Date | 04 July 2005 |
| Creators | LEE, YU-HAO |
| Contributors | C.K.Low, none, none |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0704105-182611 |
| Rights | unrestricted, Copyright information available at source archive |
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