Consider the Sturm-Liouville system :
8 > > > > > < > > > > > :
− y00 + q(x)y = y
y(0) cos + y0(0) sin = 0
y(1) cos + y0(1) sin = 0
,
where q 2 L 1 (0, 1) and , 2 [0, £¾).
Let 0 < x(n)1 < x(n)2 < ... < x(n)n − 1 < 1 be the nodal points of n-th eigenfunction
in (0,1). The inverse nodal problem involves the determination of the parameters
(q, , ) in the system by the knowledge of the nodal points . This problem was
first proposed and studied by McLaughlin. Hald-McLaughlin gave a reconstruc-
tion formula of q(x) when q 2 C 1 . In 1999, Law-Shen-Yang improved a result
of X. F. Yang to show that the same formula converges to q pointwisely for a.e.
x 2 (0, 1), when q 2 L 1 .
We found that there are some mistakes in the proof of the asymptotic formulas
for sn and l(n)j in Law-Shen-Yang¡¦s paper. So, in this thesis, we correct the
mistakes and prove the reconstruction formula for q 2 L 1 again. Fortunately, the
mistakes do not affect this result.Furthermore, we show that this reconstruction formula converges to q in
L 1 (0, 1) . Our method is similar to that in the proof of pointwise convergence.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0612101-123650 |
| Date | 12 June 2001 |
| Creators | Chen, Ya-ting |
| Contributors | Tzy-Wei Hwang, Chung-Tsun Shieh, Chao-Nien Chen, Jhishen Tsay, Chun-Kong Law |
| 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-0612101-123650 |
| Rights | unrestricted, Copyright information available at source archive |
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