Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues

Ran Zhang 2 , Xiao-Chuan Xu 3 , Chuan-Fu Yang 1  and Natalia Pavlovna Bondarenko 4
  • 1 Department of Applied Mathematics, Nanjing University of Science and Technology, School of Science, Nanjing, P. R. China
  • 2 Department of Applied Mathematics, Nanjing University of Science and Technology, School of Science, Nanjing, P. R. China
  • 3 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, 210044, Jiangsu, Nanjing, P. R. China
  • 4 Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara, Russia
Ran Zhang
  • Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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, Xiao-Chuan Xu
  • School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, P. R. China
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, Chuan-Fu Yang
  • Corresponding author
  • Department of Applied Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, P. R. China
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and Natalia Pavlovna Bondarenko
  • Department of Applied Mathematics and Physics, Samara National Research University, Moskovskoye Shosse 34, Samara, 443086; and Department of Mechanics and Mathematics, Saratov State University, Astrakhanskaya 83, Saratov 410012, Russia
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Abstract

In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on (0,π) with the Robin boundary conditions and the jump conditions at the point π2. We prove that the potential M(x) on the whole interval and the parameters in the boundary conditions and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potential M(x) is given on (0,(1+α)π4); (ii) the potential M(x) is given on ((1+α)π4,π), where 0<α<1, respectively. It is also shown that the potential and all the parameters can be uniquely recovered by one spectrum and some information on the eigenfunctions at some interior point.

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