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Based on spectral data, the potential function

Author: ChengWei
Tutor: WeiGuangSheng
School: Shaanxi Normal University
Course: Basic mathematics
Keywords: spectrum spectral data uniqueness existence reconstruction of the potential function
CLC: O175.8
Type: Master's thesis
Year: 2011
Downloads: 16
Quote: 0
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Abstract


Inverse Sturm-Liouville problems is an important part of Sturm-Liouville the-ory and is also the foundation of the theory of inversion, for which it is through study of spectral information to discuss how to identify and reconstruct Sturm-Liouville system. This paper is based on the spectral data to discuss the existence and uniqueness of Sturm-Liouville systems, and the reconstruction of the potential function under certain conditions. The results are as following:(1) Given an function f∈L2[0,π], we obtain the following expansion, and therefore have the expressions of h, H. q(x). Consequently, the converges of series such as (?)yj(0),(?)1/(kj)yj(π) and (?)(yjuj)’ can be proved by more precisely asymptotic expressions of the spectrum, spectral functions and normal constants. Finally we proved that potential function can be uniquely determined by the spectral data.(2) We proved the existence of solutions, which is based on the spectral data {λj,kj}. and then obtain the existence and uniqueness of the potential function.(3) When the finite spectral data is lost, how to reconstruct the potential func-tion can give the methods and steps of calculation, and using this method to obtain the expression of potential, that is, q(x)=(2a2)/((1-ax)2), in the case ofλj=j2 (j≥0) kj＝(-1)j(j≥1).

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Boundary Value Problems
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