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Reconstruction formulas for periodic potential functions of Hill's equation using nodal data

by Wu, Chun-Jen

Abstract (Summary)
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.
Bibliographical Information:

Advisor:Wei-Cheng Lian; Chun-Kong Law; Chiu-Ya Lan

School:National Sun Yat-Sen University

School Location:China - Taiwan

Source Type:Master's Thesis

Keywords:hill s equation inverse nodal problems periodic potential function reconstruction formula point

ISBN:

Date of Publication:06/30/2005

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