The Hill's equation is the Schrodinger equation $$-y'+qy=la y$$ with a periodic one-dimensional
potential function $q$ and coupled with periodic boundary
conditions $y(0)=y(1)$, $y'(0)=y'(1)$ or anti-periodic boundary conditions $y(0)=-y(1)$, $y'(0)=-y'(1)$.
We study the inverse nodal problem for Hill's
equation, in particular the reconstruction problem. Namely, we want to reconstruct the potential function using only nodal data ( zeros of eigenfunctions ). In this thesis, we give a reconstruction formula for $q$ using the periodic nodal data or using anti-periodic nodal data
We show that the convergence is pointwise for all $x in (0,1)$ where $q$ is continuous; and pointwise for $a.e.$ $x in (0,1)$ as well as $L^1$ convergence when $qin L^1(0,1)$. We do this by making a translation so that the problem becomes a Dirichlet problem. The idea comes from the work of Coskun and Harris.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0630105-012021 |
| Date | 30 June 2005 |
| Creators | Wu, Chun-Jen |
| Contributors | Wei-Cheng Lian, Chun-Kong Law, Chiu-Ya Lan |
| 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-0630105-012021 |
| Rights | unrestricted, Copyright information available at source archive |
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