数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1561-1568.doi: 10.1016/S0252-9602(11)60342-1

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AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

杨传富|杨孝平   

  1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China
  • 收稿日期:2008-12-10 修回日期:2010-04-02 出版日期:2011-07-20 发布日期:2011-07-20
  • 基金资助:

    This work was supported by Natural Science Foundation of Jiangsu Province of China (BK 2010489) and the Outstanding Plan-Zijin Star Foundation of Nanjing University of Science and Technology (AB 41366), and NUST Research Funding (AE88787), and the National Natural Science Foundation of China (11071119).

AMBARZUMYAN’S THEOREM WITH EIGENPARAMETER IN THE BOUNDARY CONDITIONS

 YANG Chuan-Fu, YANG Xiao-Ping   

  1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, China
  • Received:2008-12-10 Revised:2010-04-02 Online:2011-07-20 Published:2011-07-20
  • Supported by:

    This work was supported by Natural Science Foundation of Jiangsu Province of China (BK 2010489) and the Outstanding Plan-Zijin Star Foundation of Nanjing University of Science and Technology (AB 41366), and NUST Research Funding (AE88787), and the National Natural Science Foundation of China (11071119).

摘要:

In this paper, the classical Ambarzumyan’s theorem for the regular Sturm-Liouville problem is extended to the case in which the boundary conditions are eigenpa-rameter dependent. Specifically, we show that if the spectrum of the operator −D2+q with
eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.

关键词: Sturm-Liouville equation, eigenparameter-dependent boundary condition, Ambarzumyan’s theorem, inverse problem, asymptotics of eigenvalue

Abstract:

In this paper, the classical Ambarzumyan’s theorem for the regular Sturm-Liouville problem is extended to the case in which the boundary conditions are eigenpa-rameter dependent. Specifically, we show that if the spectrum of the operator −D2+q with
eigenparameter dependent boundary conditions is the same as the spectrum belonging to the zero potential, then the potential function q is actually zero.

Key words: Sturm-Liouville equation, eigenparameter-dependent boundary condition, Ambarzumyan’s theorem, inverse problem, asymptotics of eigenvalue

中图分类号: 

  • 34A55