Acta mathematica scientia,Series B ›› 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

 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).

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

CLC Number: 

  • 34A55
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