臺灣博碩士論文加值系統

希爾方程也就是水丁格方程：
-y''+qy=la y在勢函數q具有週期性質的條件下的邊值問題。
不失一般性，我們可以假設邊界條件為週期：
y(0)=y(1)，y''(0)=y''(1) 或半週期：y(0)=-y(1)，y''(0)=-y''(1)。
我們有興趣研究希爾方程的反節點問題，特別是關於勢函數q的重構問題。
我們想利用節點（特徵方程的零點）來重構勢函數q。
本論文中，我們給出了兩個勢函數的重構。
分別是：
（1）在周期邊界條件下
（2）在半周期邊界條件下
另外，我們也證明了上式數列的收斂性：若勢函數為連續函數則逐點收斂；若為 L^1 函數則是處處逐點收斂，同時亦是 L^1 收斂。我們利用了平移轉換到狄利克雷問題解決這些問題，這個想法來自寇斯肯和哈雷斯。

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.

1 Introduction
1.1 Inverse nodal problems
1.2 Hill''s equation
1.3 Continuous potentials
1.4 Main results
2 Proof of Main Theorems
2.1 A reconstruction formula for continuous potentials
2.2 Some asymptotic formulas for $L^1$ potentials
2.3 A reconstruction formula for $L^1$ potentials
2.4 The case of anti-periodic boundary conditions
A Comparsion with Coskun''s theorem

1. Y.T. Chen, Y.H. Cheng, C.K. Law and J. Tsay, $L^1$
convergence of the reconstruction formula for the potential
function, Proc. Amer. Math. Soc. 130 (2002), no. 8,
2319-2324.
2. Y.H. Cheng and C.K. Law, On the quasinodal map for the
Sturm-Liouville problem, to appear in Proc. Royal Soc.
Edinburgh series A.
3. H. Coskun and B. J. Harris, Estimates for the periodic
and semi-periodic eigenvalues of the Hill''s equation, Proc. Royal Soc. Edinburgh, 130A (2000), 991-998.
4. C.K. Law, C.L. Shen and C.F. Yang, The inverse nodal
problem on the smoothness of the potential function, Inverse Problems 15 (1999), 253-263; Errata, 17 (2001), 361-364.
5. W. Magnus and S. Winkler, Hill''s equation, Dover, New York. (1979)
6. C.K. Law and J. Tsay, On the well-posedness of the
inverse nodal problem, Inverse Problems 17 (2001),
1493-1512.
7. H. Coskun and B. J. Harris, Estimates for the periodic
and semi-periodic eigenvalues of the Hill''s equation,
Proceedings of the Royal Society of Edinburgh, 130A (2000),
991-998.
8. J.R. McLaughlin, Inverse spectral theory using nodal data - a uniqueness result, J. Diff. Eqns. 73 (1988), 354-362.
9. X.F. Yang, A solution of the inverse nodal problem, Inverse
Problems 13 (1997), 203-213.