Direct and Inverse Problems for a Third-order Differential Operator with Anti-periodic Boundary Conditions and a Non-local Potential  

反周期边值条件下具有非局部势的三阶微分算子的正反问题

作　　者：ZHANG Mingming LIU Yixuan 张明明;刘乙萱(中国民航大学理学院,天津300300)

机构地区：[1]School of Science,Civil Aviation University of China,Tianjin 300300,China

出　　处：《数学理论与应用》2025年第1期62-80,共19页Mathematical Theory and Applications

基　　金：supported by the Tianjin Municipal Science and Technology Program of China(No.23JCZDJC00070)。

摘　　要：This paper focuses on the direct and inverse problems for a third-order self-adjoint differential operator with non-local potential and anti-periodic boundary conditions.Firstly,we obtain the expressions for the characteristic function and resolvent of this third-order differential operator.Secondly,by using the expression for the resolvent of the operator,we prove that the spectrum for this operator consists of simple eigenvalues and a finite number of eigenvalues with multiplicity 2.Finally,we solve the inverse problem for this operator,which states that the non-local potential function can be reconstructed from four spectra.Specially,we prove the Ambarzumyan theorem and indicate that odd or even potential functions can be reconstructed by three spectra.本文考虑反周期边值条件下具有非局部势的三阶自伴微分算子的正反问题.首先,得到该三阶微分算子的特征判别式和预解算子的表达式.其次,利用预解算子的表达式,证明该微分算子的谱由单重特征值和有限多个重数是2的特征值组成.最后,求解该算子的反问题,结果表明非局部势函数可以由四组谱重构.特别地,我们证明Ambarzumyan定理,并指出具有奇偶对称性的势函数可以由三组谱重构.

关 键 词：Direct problem Inverse problem Non-local potential Anti-periodic boundary condition 

分 类 号：O175.2[理学—数学]