## Inverse Spectral Problem for the Diffusion Operator from a Particular Set of Eigenvalues

Cao Qing, Xu Xiaochuan,

 基金资助: 国家自然科学基金.  11901304南京信息工程大学人才启动基金

Received: 2020-05-26

 Fund supported: the NSFC.  11901304the Startup Foundation for Introducing Talent of NUIST

Abstract

In this paper, we study the inverse spectral problem for the diffusion operator on a finite interval with the Robin-Dirichlet boundary conditions, and prove that a particular set of eigenvalues can uniquely determine the diffusion operator, and give the reconstruction algorithm.

Keywords： Diffusion operator ; Inverse spectral problem ; Uniqueness theorem ; Reconstruction algorithm

Cao Qing, Xu Xiaochuan. Inverse Spectral Problem for the Diffusion Operator from a Particular Set of Eigenvalues. Acta Mathematica Scientia[J], 2021, 41(3): 577-582 doi:

## 1 引言

$$$y'' + \left( {{\rho ^2} - 2\rho p(x) - q(x)} \right)y = 0, \quad x \in (0, 1),$$$

$$$y'(0) - hy(0) = 0, \quad y(1) = 0,$$$

## 2 预备知识

$\varphi \left( {x, \rho } \right)$为方程(1.1) 满足$\varphi \left( {1, \rho } \right) = 0, \;\varphi '\left( {1, \rho } \right) = 1$的解. 由文献[6] 知, 对于固定的$x \in [0, 1]$, $\varphi \left( {x, \rho } \right) $$\varphi' \left( {x, \rho } \right)$$ \rho$的整函数, 并且对于充分大的$|\rho|$, 有

$\begin{eqnarray} {} \varphi \left( {x, \rho } \right)& = & - \frac{{\sin \left( {\rho \left( {1 - x} \right) - Q\left( 1 \right) + Q\left( x \right)} \right)}}{\rho }\left\{ {1 + \frac{{p\left( 1 \right) + p\left( x \right)}}{{2\rho }}} \right\}\\ && + \frac{{\cos \left( {\rho \left( {1 - x} \right) - Q\left( 1 \right) + Q\left( x \right)} \right)}}{{{\rho ^2}}}\left( {{\omega _1} - \omega \left( x \right)} \right)+ o\left( {\frac{1}{{{\rho ^2}}}{e^{\left| \tau \right|\left( {1 - x} \right)}}} \right), \end{eqnarray}$

$\begin{eqnarray} {} \varphi '\left( {x, \rho } \right)& = &\cos \left( {\rho \left( {1 - x} \right) - Q\left( 1 \right) + Q\left( x \right)} \right)\left\{ {1 + \frac{{p\left( 1 \right) - p\left( x \right)}}{{2\rho }}} \right\}\\ &&+ \frac{{\sin \left( {\rho \left( {1 - x} \right) - Q\left( 1 \right) + Q\left( x \right)} \right)}}{\rho }\left( {{\omega _1} - \omega \left( x \right)} \right) + o\left( {\frac{1}{\rho }{e^{\left| \tau \right|\left( {1 - x} \right)}}} \right), \end{eqnarray}$

$$$\Delta \left( \rho \right) = \varphi '\left( {0, \rho } \right) - h\varphi (0, \rho ).$$$

$\begin{eqnarray} {} \Delta \left( \rho \right) & = & \cos \left( {\rho - Q\left( 1 \right)} \right) + \frac{{\left( {p\left( 1 \right) - p\left( 0 \right)} \right)\cos \left( {\rho - Q\left( 1 \right)} \right)}}{{2\rho }} \\ &&+ \frac{{\sin \left( {\rho - Q\left( 1 \right)} \right)}}{\rho }\left( {{\omega _1} + h} \right) + o\left( {\frac{{{e^{\left| \tau \right|}}}}{\rho }} \right). \end{eqnarray}$

$$${\rho _n} = \left( {n + \frac{1}{2} + {\omega _0}} \right)\pi + \frac{{{\omega _1} + h}}{{n\pi }} + o\left( {\frac{1}{n}} \right), \quad \left| n \right| \to\infty.$$$

文献[6] 已经证明：若$M(\rho ) = \tilde M(\rho )$, 则$p\left( x \right) \mathop = \limits^{\rm a.e.} \tilde p\left( x \right), q\left( x \right) = \tilde q\left( x \right).$下面我们仅需证明: 若(3.2) 式成立, 则$M(\rho ) = \tilde M(\rho )$. 固定$j_1\in {\Bbb Z}$, 记$\rho _k^{{j_1}}: = {\rho _{{j_1}}}(p, q, {h_k}), k \in {\Bbb N}$.$M(\rho )$的定义知

$$$M(\rho _k^{{j_1}}) = \frac{{\varphi '(0, \rho _k^{{j_1}})}}{{\varphi (0, \rho _k^{{j_1}})}} = {h_k}, \quad k \in {\Bbb N},$$$

$$$\tilde{M}(\rho _k^{{j_1}}) = \frac{{\tilde{\varphi} '(0, \rho _k^{{j_1}})}}{{\tilde{\varphi} (0, \rho _k^{{j_1}})}} = {h_k}, \quad k \in {\Bbb N},$$$

${f_1}\left( \rho \right) $${f_2}\left( \rho \right) 为整函数. 由引理3.2知, {\big\{ {\rho _k^{{j_1}}} \big\}_{k \in {\Bbb N}}} 为有界无限点集, 所以必有聚点. 因此, 由引理3.3知 {f_1}\left( \rho \right) = {f_2}\left( \rho \right), \;\rho \in {\Bbb C}. 结合 M(\rho )$$ \tilde M(\rho )$的定义得

$f(z) $$\left\{ {z:\left| {z - {z_0}} \right| < R} \right\} 上解析, 并且在 \left\{ {z:\left| {z - {z_0}} \right| = R} \right\} 上至少存在一个奇点. 因为两集合 {\left\{ {{h_k}} \right\}_{k \in {\Bbb N}}}, {\big\{ {\rho _k^{{j_1}}} \big\}_{k \in {\Bbb N}}} 有界并且无限, 所以存在聚点(分别记为 h_*, \rho_* )满足 $${h_ * } = \mathop {\lim }\limits_{k \to \infty } {h_k}, \quad {\rho _ * } = \mathop {\lim }\limits_{k \to \infty } \rho _k^{{j_1}}.$$ $$a_0 = h_*, \quad {a_n} = \mathop {\lim }\limits_{k \to + \infty } \frac{{{h_k} - \sum\limits_{i = 0}^{n - 1} {{a_i}{{(\rho _k^{{j_1}} - {\rho _ * })}^i}} }}{{{{(\rho _k^{{j_1}} - {\rho _ * })}^n}}}{\rm{ }}n \in {\Bbb N}.$$ 由(3.3)、(3.5) 和(3.6) 式知 a_n\;(n\ge 0)$$ M(\rho)$的Taylor系数, 因此

$$$M(\rho ) = \sum\limits_{n = 0}^\infty {{a_n}{{(\rho - {\rho _ * })}^n}} , \quad\rho \in {D_1} = \left\{ {\rho :\left| {\rho - {\rho _ * }} \right| < {R_ * }} \right\},$$$

$$${R_ * } = \frac{1}{{\overline {\mathop {\lim }\limits_{n \to {\rm{ + }}\infty } } \sqrt[n]{{\left| {{a_n}} \right|}}}}.$$$

$$${c_{ - m}} = \mathop {\lim }\limits_{ \rho \to \rho \atop \rho \in {D_1}} {(\rho - \rho _{1k}^{})^m}M(\rho ),$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Jaulent M , Jean C .

The inverse s-wave scattering problem for a class of potentials depending on energy

Comm Math Phys, 1972, 28, 177- 220

Freiling G , Yurko V A . Inverse Sturm-Liouville Problems and their Applications. New York: NOVA Science Publishers, 2001

Levitan B M . Inverse Sturm-Liouville Problems. Utrecht: VNU Science Press, 1987

Marchenko V . Sturm-Liouville Operators and Applications. Boston: Birkhüser, 1986

McLaughlin J R , Rundell W .

A uniqueness theorem for an inverse Sturm-Liouville problem

Journal of Mathematics Physics, 1987, 28 (7): 1471- 1472

Buterin S A , Yurko V A .

Inverse problems for second-order differential pencils with Dirichlet boundary conditions

J Inverse Ill-Posed Probl, 2012, 20, 855- 881

Schrödinger算子二次微分束的半逆问题

Wang Y , Yang C , Huang Z .

Half inverse problem for a quadratic pencil of Schrödinger operators

Acta Math Sci, 2011, 31A (6): 1708- 1717

Yang C F .

Reconstruction of the diffusion operator from nodal data

Zeitschrift für Naturforschung, 2010, 65A, 100- 106

Koyunbakan H , Panakhov E S .

Half-inverse problem for diffusion operators on the finite interval

J Math Anal Appl, 2007, 326, 1024- 1030

Wang Y .

A uniqueness theorem for diffusion operators on the finite interval

Acta Math Sci, 2013, 33A (2): 333- 339

Zhong Y Q . Theory of Functions of Complex Varibles. Beijing: Higher Education Press, 1988

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