Reconstruction formulas for periodic potential functions of Hill's equation using nodal data
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.
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
Date of Publication:06/30/2005