基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 859-870.

基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性

郭燕,徐小川^*()

南京信息工程大学数学与统计学院南京 210044；南京信息工程大学江苏省应用数学中心南京 210044；南京信息工程大学江苏省系统建模与数据分析国际合作联合实验室南京 210044

收稿日期:2023-06-15 修回日期:2024-01-15 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *徐小川, E-mail:xcxu@nuist.edu.cn
基金资助:
国家自然科学基金(11901304)

Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data

Guo Yan,Xu Xiaochuan^*()

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044; Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science and Technology, Nanjing 210044; Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044

Received:2023-06-15 Revised:2024-01-15 Online:2024-08-26 Published:2024-07-26
Supported by:
NSFC(11901304)

摘要/Abstract

摘要：

该文研究有限区间(0,1)上具有 Robin 边界条件和不连续点$x=d\in(0,\frac{1}{2}]$ 的 Sturm-Liouville 算子逆谱问题. 假设已知的数据为一组子谱、势函数在 $(d,1)$ 上的信息以及右边界条件和不连续条件中的部分参数, 该文证明恢复 $(0,d)$ 上的势函数和左边界条件参数的逆谱问题局部可解性和稳定性, 其中已知的势函数信息和右边界条件参数允许存在一定的误差.

关键词: Sturm-Liouville 算子, 不连续条件, 局部可解性, 稳定性

Abstract:

This paper studies inverse spectral problems for the Sturm-Liouville operator on $(0,1)$ with the Robin boundary conditions and a discontinuity at $x=d\in(0,\frac{1}{2}]$. Suppose that the known data contains one subspectrum, the potential function on $(d,1)$ as well as partial parameters in the right boundary condition and the discontinuous conditions. The paper proves the local solvability and stability for the inverse problems of recovering the potential function on $(0,d)$ and the parameter in left boundary condition, where the known potential and the parameter in the right boundary condition are allowed to contain errors.

Key words: Sturm-Liouville operator, Discontinuity condition, Local solvability, Stability

中图分类号:

O175.7

郭燕, 徐小川. 基于混合谱数据的不连续Sturm-Liouville逆谱问题局部可解性和稳定性[J]. 数学物理学报, 2024, 44(4): 859-870.

Guo Yan, Xu Xiaochuan. Local Solvability and Stability of the Inverse Spectral Problems for the Discontinuous Sturm-Liouville Problem with the Mixed Given Data[J]. Acta mathematica scientia,Series A, 2024, 44(4): 859-870.

参考文献 28

[1]	Borg G. Eine Umkehrung der Sturm-Liouvilleschen eigenwertaufgabe. Acta Math, 1946, 78(1): 1-96
[2]	Freiling G, Yurko V A. Inverse Sturm-Liouville Problems and Their Applications. New York: NOVA Science Publishers, 2001
[3]	Gesztesy F, Simon B. Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum. Trans Amer Math Soc, 2000, 352(6): 2765-2787
[4]	Hochstadt H, Lieberman B. An inverse Sturm-Liouville problem with mixed given data. SIAM J Appl Math, 1978, 34: 676-680
[5]	Horváth M. Inverse spectral problems and closed exponential systems. Ann Math, 2005, 162: 885-918
[6]	Horváth M. On the inverse spectral theory of Schrödinger and Dirac operators. Trans Amer Math Soc, 2001, 353(10): 4155-4171
[7]	Levitan B M. Inverse Sturm-Liouville Problems. Utrecht: VNU Science Press, 1987
[8]	Marchenko V A. Sturm-Liouville Operators and Applications. Boston: Publisher Birkhüser, 1986
[9]	Pöschel J, Trubowitz E. Inverse Spectral Theory. London: Academic Press, 1987
[10]	Shieh C T, Buterin S A, Ignatiev M. On Hochstadt-Lieberman theorem for Sturm-Liouville operators. Far East J Appl Math, 2011, 52: 131-146
[11]	王於平, 杨传富, 黄振友. Schrödinger 算子二次微分束的半逆问题. 数学物理学报, 2011, 31A(6): 1708-1717
	Wang Y P, Yang C F, Huang Z Y. Half inverse problem for a quadratic pencil of Schrödinger operators. Acta Math Sci, 2011, 31A(6): 1708-1717
[12]	Krueger R J. Inverse problems for nonabsorbing media with discontinuous material properties. J Math Phys, 1982, 23: 396-404
[13]	Anderssen R S. The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of tortional eigenfrequencies of the Earth. Geophys J R Astron Soc, 1997, 50: 303-309
[14]	Litvinenko O N, Soshnikov V I. The Theory of Heterogenious Lines and Their Applications in Radio Engineering (in Russian). Moscow: Radio, 1964
[15]	Hald O. Discontinuous inverse eigenvalue problem. Commun Pure Appl Math, 1984, 37(5): 539-577
[16]	Ozkan A S, Keskin B. Uniqueness theorems for an impulsive Sturm-Liouville boundary value problem. Appl Math J Chin Univ, 2012, 27(4): 428-434
[17]	Shieh C T, Yurko V A. Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J Math Anal Appl, 2008, 347: 266-272
[18]	Wang Y P. Inverse problems for discontinuous Sturm-Liouville operators with mixed spectral data. Inverse Probl Sci Eng, 2015, 23(7): 1180-1198
[19]	Yang C F. Inverse problems for the Sturm-Liouville operator with discontinuity. Inverse Probl Sci Eng, 2014, 22(2): 232-244
[20]	Yang C F, Bondarenko N P. Reconstruction and solvability for discontinuous Hochstadt-Lieberman problems. J Spectral Theory, 2020, 10: 1445-1469
[21]	Yang C F, Bondarenko N P. Local solvability and stability of inverse problems for Sturm-Liouville operators with a discontinuity. J Differential Equations, 2020, 268: 6173-6188
[22]	Xu X C, Yang C F. Inverse spectral problems for the Sturm-Liouville operator with discontinuity. J Differential Equations, 2017, 262: 3093-3106
[23]	Yang C F. Traces of Sturm-Liouville operators with discontinuities. Inverse Probl Sci Eng, 2014, 22: 803-813
[24]	Yurko V A. Integral transforms connected with discontinuous boundary value problems. Integral Transforms Spec Funct, 2000, 10: 141-164
[25]	Yang C F, Bondarenko N P, Xu X C. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems and Imaging, 2020, 14: 153-169
[26]	Guo Y, Ma L J, Xu X C, et al. Weak and strong stability of the inverse Sturm-Liouville problem. Math Meth Appl Sci, 2023, 46(14): 15684-15705
[27]	Bondarenko N P. A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph. Anal Math Phys, 2018, 8: 155-168
[28]	Levin B Y. Lectures on Entire Functions. Providence RI: Amer Math Soc, 1996