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Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
Journal of Inverse and Ill-posed Problems ( IF 1.1 ) Pub Date : 2020-06-01 , DOI: 10.1515/jiip-2019-0003
Ran Zhang 1 , Xiao-Chuan Xu 2 , Chuan-Fu Yang 1 , Natalia Pavlovna Bondarenko 3
Affiliation  

Abstract In this work, we consider the inverse spectral problem for the impulsive Sturm–Liouville problem on ( 0 , π ) {(0,\pi)} with the Robin boundary conditions and the jump conditions at the point π 2 {\frac{\pi}{2}} . We prove that the potential M ⁢ ( x ) {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 ) {M(x)} is given on ( 0 , ( 1 + α ) ⁢ π 4 ) {(0,\frac{(1+\alpha)\pi}{4})} ; (ii) the potential M ⁢ ( x ) {M(x)} is given on ( ( 1 + α ) ⁢ π 4 , π ) {(\frac{(1+\alpha)\pi}{4},\pi)} , where 0 < α < 1 {0<\alpha<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.

中文翻译:

从一组特征值确定脉冲 Sturm-Liouville 算子

摘要 在这项工作中,我们考虑了 ( 0 , π ) {(0,\pi)} 上的脉冲 Sturm-Liouville 问题的逆谱问题,具有 Robin 边界条件和点 π 2 {\frac{ \pi}{2}} 。我们证明了整个区间上的势 M ⁢ ( x ) {M(x)} 以及边界条件和跳跃条件中的参数可以从两种情况下的一组特征值中确定: (i) 势 M ⁢ ( x ) {M(x)} 在 ( 0 , ( 1 + α ) ⁢ π 4 ) {(0,\frac{(1+\alpha)\pi}{4})} 上给出;(ii) 势 M ⁢ ( x ) {M(x)} 在 ( ( 1 + α ) ⁢ π 4 , π ) {(\frac{(1+\alpha)\pi}{4},\ pi)} ,其中 0 < α < 1 {0<\alpha<1} ,分别。还表明,势和所有参数可以通过一个谱和一些内点的本征函数信息唯一地恢复。
更新日期:2020-06-01
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