Hochstadt-Lieberman定理的重构问题
The Reconstructing Problem for Hochstadt-Lieberman Theorem
摘要:
Hochstadt-Lieberman唯一性定理表明，对于定义在[0, 1]区间上的Sturm-Liouville问题，若[0, 1/2]区间上的势函数已知，则一组Dirichlet-Dirichlet特征值即可唯一确定整个区间上的势函数。本文应用亚纯函数的Mittag-Leffler展开定理，给出了重构该问题势函数的一种新方法，同时给出了该问题的解存在的充要条件。
Abstract:
In this paper we are concerned with the Hochstadt-Lieberman uniqueness theorem which states that, when the potential is known a priori on [0, 1/2], the full Dirichlet-Dirichlet spectrum of a Sturm-Liouville problem defined on the interval [0, 1] uniquely determines its potential. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition Theorem of meromorphic functions associated with the solution of Sturm-Liouville equantions. We also give a necessary and sufficient condition for the existence of the solution.
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