向量型 Sturm-Liouville问题的特征值重数及逆结点问题

Acta mathematica scientia,Series A ›› 2023, Vol. 43 ›› Issue (3): 669-679.

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The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem

Liu Xiaoyun¹(),Shi Guoliang²,Yan Jun^2,^*()

¹School of Mathematics and Information Science, Anyang Institute of Techology, Henan Anyang 455000
²School of Mathematical Sciences, Tianjin University, Tianjin 300354

Received:2021-08-23 Revised:2023-02-06 Online:2023-06-26 Published:2023-06-01
Contact: Jun Yan E-mail:xyl.hb@163.com;jun.yan@tju.edu.cn
Supported by:
National Natural Science Foundation of China(12001153);National Natural Science Foundation of China(62065015);Natural Science Foundation of Hebei Province(F2022407007);Science and Technology Research Project of Colleges and Universities in Hebei Province(ZC2023122)

Abstract

Abstract:

The $m$ -dimensional vectorial Sturm-Liouville problem with Dirichlet boundary conditions on $(0,1)$ is studied. We firstly discuss the relationship between the matrix-valued potential and the multiplicities of eigenvalues. We prove that if the multiplicities of eigenvalues of $\int_{0}^{1}Q(x){\rm d}x$ are at most $k$ $(1\leq k\leq m-1)$ , with finitely many exceptions, the multiplicities of eigenvalues of the vectorial problem are also at most $k$ . Then, the inverse nodal problem is investigated with a different method. We show that if there exists an infinite eigenfunctions sequence $\{y_{n_{j},r}(x,\lambda_{n_{j},r})\}_{j=1}^{\infty }$ which are all vectorial functions of type $(CZ)$ , then $Q$ is simultaneously diagonalizable.

Key words: Vectorial Sturm-Liouville problems, Multiplicities, Estimation of eigenvalues, Inverse nodal problem

CLC Number:

O175.3

Liu Xiaoyun,Shi Guoliang,Yan Jun. The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem[J].Acta mathematica scientia,Series A, 2023, 43(3): 669-679.

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References 20

[1]	Marchenko V A. Sturm-Liouville Operators and Applications. Operator Theory Advances & Applications 22. Basle: Birkhauser Verlag, 1986
[2]	Zakhariev B N, Suzko A A. Direct and Inverse Problems. Berlin-Heidelberg: Springer, 1990
[3]	Calvert J M, Davison W D. Oscillation theory and computational procedures for matrix Sturm-Liouville eigenvalue problems with an application to the hydrogen molecular ion. Journal of Physics a Mathematical & General, 1968, 2(3): 278-292
[4]	Shen C L, Shieh C T. On the multiplicity of eigenvalues of a vectorial Sturm-Liouville differential equations and some related spectral problems. Proc Amer Math Soc, 1999, 127: 2943-2952 doi: 10.1090/proc/1999-127-10
[5]	Kong Q. Multiplicities of eigenvalues of a vector-valued Sturm-Liouville problem. Mathematika, 2002, 49: 119-127 doi: 10.1112/mtk.v49.1-2
[6]	Yang C F, Huang Z Y, Yang X P. The multiplicity of spectra of a vectorial Sturm-Liouville differential equation of dimension two and some applications. Rocky Mountain J Math, 2007, 37(4): 1379-1398
[7]	Veliev O A. Non-self-adjoint Sturm-Liouville operators with matrix potentials. Mathematical Notes, 2007, 81(4): 496-506 doi: 10.1134/S0001434607030273
[8]	Seref F, Veliev O A. On sharp asymptotic formulas for the Sturm-Liouville operator with a matrix potential. Mathematical Notes, 2016, 100(1/2): 291-297 doi: 10.1134/S0001434616070245
[9]	McLaughlin J R. Inverse spectral theory using nodal points as data-a uniqueness result. J Differential Equations, 1988, 73(2): 354-362 doi: 10.1016/0022-0396(88)90111-8
[10]	Chen X, Cheng Y H, Law C K. Reconstructing potentials from zeros of one eigenfunction. Trans Amer Math So, 2011, 363: 4831-4851
[11]	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: 2765-2787 doi: 10.1090/tran/2000-352-06
[12]	Guo Y X, Wei G S. Inverse problems: dense nodal subset on an interior subinterval. J Differential Equations, 2013, 255(7): 2002-2017 doi: 10.1016/j.jde.2013.06.006
[13]	Yang X F. A new inverse nodal problem. J Differential Equations, 2001, 169: 633-653 doi: 10.1006/jdeq.2000.3911
[14]	Yurko V. Inverse nodal problems for Sturm-Liouville operators on star-type graphs. J Inverse Ill-Posed Probl, 2008, 16(7): 715-722
[15]	Shen C L, Shieh C T. An inverse nodal problem for vectorial Sturm-Liouville equations. Inverse Problems, 2000, 16(2): 349-356 doi: 10.1088/0266-5611/16/2/306
[16]	Cheng Y H, Shieh C T, Law C K. A vectorial inverse nodal problem. Proc Amer Math Soc, 2005, 133(5): 1475-1484 doi: 10.1090/proc/2005-133-05
[17]	Agranovich Z S, Marchenko V A. The Inverse Problem of Scattering Theory. New York-London: Gordon and Breach Science Publishers, 1963
[18]	Shen C L. Some inverse spectral problems for vectorial Sturm-Liouville equations. Inverse Problems, 2001, 17(5): 1253-1294 doi: 10.1088/0266-5611/17/5/303
[19]	Naimark M A. Linear Differential Operators, Part I. New York: Frederick Ungar Publishing Co, 1967
[20]	Weidmann J. Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics 1258. Berlin: Springer Verlag, 1987

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The Multiplicities of Eigenvalues and Inverse Nodal Problem of a Vectorial Sturm-Liouville Problem

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