三组谱的Sturm-Liouville反问题
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摘要
考虑与三组谱关联的逆Sturm-Liouville问题,证明了若对于给定的两组数列,在一定条件下,可划分为三组数列,使其分别为区间[0, a]上三个Sturm-Liouville问题的部分特征值,则通过三组谱的部分特征值能唯一确定区间[0, a]上的势函数q(x).
The inverse Sturm-Liouville problem associated with three spectra is considered. It is shown that if the given two sequences can divided into three sequences in certain conditions, which can be the corresponding parts of eigenvalues of three Sturm-Liouville problems defined on the interval [0, a],then the potential function q(x) on the interval [0, a] can be uniquely determined by the corresponding parts of the three spectra.
The inverse Sturm-Liouville problem associated with three spectra is considered. It is shown that if the given two sequences can divided into three sequences in certain conditions, which can be the corresponding parts of eigenvalues of three Sturm-Liouville problems defined on the interval [0, a],then the potential function q(x) on the interval [0, a] can be uniquely determined by the corresponding parts of the three spectra.
引文
[1] Freiling G, Yurko V. Inverse Sturm-Liouville Problems and Their Applications[M]. New York:Nova Science pub., 2001.
[2] Pivovarchik V. An inverse problem by eigenvalues of four spectra[J]. Journal of Mathematical Analysis and Applications, 2012,396(2):715-723.
[3] Borg G. Eine unkehrung der Sturm-Liouvilleschen eigenwert aufgabe[J]. Acta Mathematica,1946,78(1):1-96.
[4] Hald O H. Discontinuous inverse eigenvalue problems[J]. Communications on Pure and Applied Mathematics, 1984,37(5):539-577.
[5] Wei G, Xu H K. On the missing eigenvalue problem for an inverse Sturm-Liouville problem[J].Journal De Math′ematiques Pures Et Appliqu′ees, 2009,91(5):468-475.
[6] Pivovarchik V. An inverse Sturm-Liouville problem by three spectra[J]. Integral Equations Operator Theory, 1998,78:1037-1038.
[7] Pivovarchik V. Inverse problem for the Sturm-Liouville equation on a simple graph[J]. Mathematische Nachrichten, 2010,280(13-14):1595-1619.
[8] Levinson N. The inverse Sturm-Liouville problem[J]. Mat. Tidsskr. B, 1949,1949:25-30.
[9]傅守忠,王忠,魏广生. Sturm-Liouville问题及其逆问题[M].北京:科学出版社, 2015.
[10]曹之江.常微分算子[M].上海:上海科技出版社, 1986.
[11]王玉华,魏广生. Sturm-Liouville算子部分谱信息的逆问题[J].纯粹数学与应用数学, 2011,27(2):267-272.
[12]王於平.参数边界条件下Sturm-Liouville算子的逆谱问题[J].应用泛函分析学报, 2017,19(3):294-298.
[13]赵迎春,孙炯.一类内部具有无穷多不连续点Sturm-Liouville算子的亏指数[J].数学物理学报,2018,38(03):484-495.
[14] Levin B J, Ljubarski J I. Interpolation by entire functions belonging to special classes and related expansions in series of exponentials[J]. Izv. Akad. Nauk SSSR Ser. Mat., 1975(3):657-702.
[15] Young R. An Introduction to Nonharmonic Fourier Series[M]. New York:Academic Press, 1980.
[2] Pivovarchik V. An inverse problem by eigenvalues of four spectra[J]. Journal of Mathematical Analysis and Applications, 2012,396(2):715-723.
[3] Borg G. Eine unkehrung der Sturm-Liouvilleschen eigenwert aufgabe[J]. Acta Mathematica,1946,78(1):1-96.
[4] Hald O H. Discontinuous inverse eigenvalue problems[J]. Communications on Pure and Applied Mathematics, 1984,37(5):539-577.
[5] Wei G, Xu H K. On the missing eigenvalue problem for an inverse Sturm-Liouville problem[J].Journal De Math′ematiques Pures Et Appliqu′ees, 2009,91(5):468-475.
[6] Pivovarchik V. An inverse Sturm-Liouville problem by three spectra[J]. Integral Equations Operator Theory, 1998,78:1037-1038.
[7] Pivovarchik V. Inverse problem for the Sturm-Liouville equation on a simple graph[J]. Mathematische Nachrichten, 2010,280(13-14):1595-1619.
[8] Levinson N. The inverse Sturm-Liouville problem[J]. Mat. Tidsskr. B, 1949,1949:25-30.
[9]傅守忠,王忠,魏广生. Sturm-Liouville问题及其逆问题[M].北京:科学出版社, 2015.
[10]曹之江.常微分算子[M].上海:上海科技出版社, 1986.
[11]王玉华,魏广生. Sturm-Liouville算子部分谱信息的逆问题[J].纯粹数学与应用数学, 2011,27(2):267-272.
[12]王於平.参数边界条件下Sturm-Liouville算子的逆谱问题[J].应用泛函分析学报, 2017,19(3):294-298.
[13]赵迎春,孙炯.一类内部具有无穷多不连续点Sturm-Liouville算子的亏指数[J].数学物理学报,2018,38(03):484-495.
[14] Levin B J, Ljubarski J I. Interpolation by entire functions belonging to special classes and related expansions in series of exponentials[J]. Izv. Akad. Nauk SSSR Ser. Mat., 1975(3):657-702.
[15] Young R. An Introduction to Nonharmonic Fourier Series[M]. New York:Academic Press, 1980.