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.

1. 引言

$Lu=-{u}^{″}+qu$ (1)

$u\left(0\right)=0=u\left(1\right)$ , (2)

Martinyuk及Pivoarchik [2] 曾对以上唯一性定理给出了重构势函数的方法。本文的目的是对Hochstadt-Lieberman唯一性定理提供一种新的重构势函数的方法。通过应用Mittag-Leffler展开定理，将“较大的”全纯函数分解为两个“较小的”全纯函数，此分解为我们更好地使用Levin-Lyubarski插值公式重构全纯函数 ${u}_{-}\left(1/2,\lambda \right)$${{u}^{\prime }}_{-}\left(1/2,\lambda \right)$ 提供了环境。此外，该重构方法亦给出了该问题的解存在且唯一的充要条件。

2. 势函数的重构

${u}_{-}\left(x,\lambda \right)$ 为方程(1)满足初始条件 ${u}_{-}\left(0\right)=0$${{u}^{\prime }}_{-}\left(0\right)=1$ 的解。由文 [4] 可得：

$\begin{array}{c}{u}_{-}\left(x,\lambda \right)=\frac{\mathrm{sin}\lambda x}{\lambda }+{\int }_{0}^{x}K\left(x,t\right)\frac{\mathrm{sin}\lambda t}{\lambda }\text{d}t\\ =\frac{\mathrm{sin}\lambda x}{\lambda }-K\left(x,x\right)\frac{\mathrm{cos}\lambda x}{{\lambda }^{2}}+{\int }_{0}^{x}{K}_{t}\left(x,t\right)\frac{\mathrm{cos}\lambda x}{{\lambda }^{2}}\text{d}t\end{array}$ , (3)

$K\left(x,t\right)=\stackrel{˜}{K}\left(x,t\right)-\stackrel{˜}{K}\left(x,-t\right)$ , ${K}_{t}\left(x,t\right)=\frac{\partial K\left(x,t\right)}{\partial t}$ ,

$\stackrel{˜}{K}\left(x,t\right)$ 满足以下积分方程：

$\stackrel{˜}{K}\left(x,t\right)=\frac{1}{2}{\int }_{0}^{\frac{x+t}{2}}q\left(s\right)\text{ds}+{\int }_{0}^{\frac{x+t}{2}}\text{d}\alpha {\int }_{0}^{\frac{x-t}{2}}q\left(\alpha +\beta \right)\stackrel{˜}{K}\left(\alpha +\beta ,\alpha -\beta \right)\text{d}\beta$ ,

$K\left(x,x\right)=\frac{1}{2}{\int }_{0}^{x}q\left(t\right)\text{d}t$ , $K\left(x,0\right)=0$ . (4)

$\begin{array}{l}{u}_{-}\left(\frac{1}{2},\lambda \right)=\frac{1}{\lambda }\mathrm{sin}\left(\frac{\lambda }{2}\right)-\frac{{K}_{-}}{{\lambda }^{2}}\mathrm{cos}\left(\frac{\lambda }{2}\right)+\frac{{\psi }_{-,0}\left(\lambda \right)}{{\lambda }^{2}};\\ {{u}^{\prime }}_{-}\left(\frac{1}{2},\lambda \right)=\mathrm{cos}+\frac{{K}_{-}}{\lambda }\mathrm{sin}\left(\frac{\lambda }{2}\right)+\frac{{\psi }_{-,1}\left(\lambda \right)}{{\lambda }^{2}}\end{array}$ (5)

${u}_{+}\left(x,\lambda \right)=-\frac{\mathrm{sin}\lambda \left(1-x\right)}{\lambda }-{\int }_{x}^{1}K\left(x,t\right)\frac{\mathrm{sin}\lambda \left(1-t\right)}{\lambda }\text{d}t$ . (6)

${u}_{+}\left(1/2,\lambda \right)$${{u}^{\prime }}_{+}\left(1/2,\lambda \right)$ 有如下渐近式：

$\begin{array}{l}{u}_{+}\left(\frac{1}{2},\lambda \right)=-\frac{1}{\lambda }\mathrm{sin}\left(\frac{\lambda }{2}\right)-\frac{{K}_{+}}{{\lambda }^{2}}\mathrm{cos}\left(\frac{\lambda }{2}\right)+\frac{{\psi }_{+,0}\left(\lambda \right)}{{\lambda }^{2}}\\ {{u}^{\prime }}_{+}\left(\frac{1}{2},\lambda \right)=\mathrm{cos}\left(\frac{\lambda }{2}\right)+\frac{{K}_{+}}{\lambda }\mathrm{sin}\left(\frac{\lambda }{2}\right)+\frac{{\psi }_{+,1}\left(\lambda \right)}{{\lambda }^{2}}\end{array}$ (7)

$\Delta \left(\lambda \right)={u}_{-}\left(1,\lambda \right)$ (8)

$\Delta \left(\lambda \right)=\frac{1}{\lambda }\mathrm{sin}\lambda -\frac{{K}_{-}+{K}_{+}}{{\lambda }^{2}}\mathrm{cos}\lambda +\frac{\stackrel{^}{\psi }\left(\lambda \right)}{{\lambda }^{2}}$ , (9)

${\lambda }_{n}=n\text{π}+\frac{{K}_{-}+{K}_{+}}{n\text{π}}+\frac{{\alpha }_{n}}{n}$ (10)

, (11)

, (12)

(13)

(14)

, (15)

, (16)

, (17)

. (18)

, (19)

. (20)

(21)

，且在展开式(20)中为唯一的。

(22)

,

, (23)

,

,

. (24)

, (25)

, (26)

. (27)

(28)

, (29)

(30)

，式(30)结合(27)，可知存在满足

. (31)

,

，从而可得(28)。定理得证。

(32)

(33)

,.

,

,

NOTES

*通讯作者。

[1] Hochstadt, H. and Lieberman, B. (1978) An Inverse Sturm-Liouville Problem with Mixed Given Data. SIAM Journal on Applied Mathematics, 34, 676-680.
https://doi.org/10.1137/0134054

[2] Martinyuk, O. and Pivovarchik, V. (2010) On the Hochstadt-Lieberman Theorem. Inverse Problems, 26, Article ID: 035011, 6 p.
https://doi.org/10.1088/0266-5611/26/3/035011

[3] Levin, B.J. (1980) Distribution of Zeros of Entire Functions. American Mathematical Society, Providence, RI.

[4] Marchenko, V. (1986) Sturum-Liouville Operators and Applications. Birkhäuser, Ba-sel.

[5] Ablowitz, M.J. and Fokas, A.S. (2003) Complex Variables Introduction and Applications. 2nd Edition, Cambridge Univer-sity Press, Cambridge.

[6] Levin, B. and Yu, I. (1975) Lyubarskii, Interpolation by Entire Functions of Special Classes and Related Expansions in Series of Exponents. Izvestiya Rossiiskoi Akademii Nauk USSR, 39, 657-702. (In Russian)
https://doi.org/10.1070/im1975v009n03abeh001493

[7] Gesztesy, F. and Simon, B. (2000) Inverse Spectral Analysis with Partial Information on the Potential II: The Case of Discrete Spectrum. Transactions of the American Mathematical Society, 352, 2765-2787.

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