Inverse spectral problem for the matrix Sturm-Liouville operator with the general separated self-adjoint boundary conditions
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Abstract
In this work, we study the matrix Sturm-Liouville operator with the separated self-adjoint boundary conditions of general type, in terms of two unitary matrices. Some properties of the eigenvalues and the normalization matrices are given. Uniqueness theorems for determining the potential and the unitary matrices in the boundary conditions from the Weyl matrix, two characteristic matrices or one spectrum and the corresponding normalization matrices are proved.
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Xu, X.-C. . (2019). Inverse spectral problem for the matrix Sturm-Liouville operator with the general separated self-adjoint boundary conditions. Tamkang Journal of Mathematics, 50(3), 321–336. https://doi.org/10.5556/j.tkjm.50.2019.3360
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References
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[15] M.S. Harmer, Inverse scattering for the matrix Schr ̈odinger operator and Schr ̈odinger operator on graphs with general self-adjoint boundary conditions, ANZIAM J. 44 (2002), 161-168.
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[17] V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen. 32(4) (1999), 595-630.
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[20] V.N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal. 32 (2000), 801-819.
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[24] C.-T. Shieh, Isospectral sets and inverse problems for vector-valued Sturm-Liouville equations, In- verse Problems 23 (2007), 2457-2468.
[25] F. Visco-Comandini, M. Mirrahimi, M. Sorine, Some inverse scattering problems on star-shaped graphs, J. Math. Anal. Appl. 378 (2011), 343-358.
[26] X.-C. Xu, C.-F. Yang, Determination of the self-adjoint matrix Schr ̈odinger operators without the bound state data, Inverse Problems 34 (2018), 065002 (20pp).
[27] X.-C. Xu, C.-F. Yang, Inverse scattering problems on a noncompact star graph, Inverse Problems 34 (2018), 115004 (12pp).
[28] C.-F. Yang, Trace formula for the matrix Sturm-Liouville operator, Analysis and Mathematical Physics 6 (2016), 31-41.
[29] C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator on a d-star graph, J. Math. Anal. Appl. 365 (2010), 742-749.
[30] C.-F. Yang, V.A. Yurko Recovering differential operators with nonlocal boundary conditions, Anal- ysis and Mathematical Physics 6 (2016), 315-326.
[31] V.A. Yurko, Inverse nodal problems for Sturm-Liouville operators on star-type graphs, J. Inverse Ill-Posed Probl. 16 (2008), 715-722.
[32] V.A. Yurko, Inverse problems for the matrix Sturm-Liouville equation on a finite interval, Inverse Problems 22(4) (2006), 1139-1149.
[33] V.A. Yurko, Inverse problems for matrix Sturm-Liouville operators, Russian Journal of Mathemat- ical Physics 13 (2006), 111-118.
[34] V.A. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075-1086.
[35] V.A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2002.
[2] T. Aktosun, R. Weder, High-energy analysis and Levinson’s theorem for the self-adjoint matrix Schr ̈odinger operator on the half line, J. Math. Phys. 54 (2013), 012108.
[3] N.P. Bondarenko, Inverse scattering on the line for the matrix Sturm-Liouville equation, J. Differ- ential Equations 262 (2017), 2073-2105.
[4] N.P. Bondarenko, A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph, Analysis and Mathematical Physics 8 (2018), 155-168.
[5] N.P. Bondarenko, An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator, Tamk. J. Math. 50 (2019), 71-102.
[6] S.A. Buterin, G. Freiling, Inverse spectral-scattering problem for the Sturm-Liouville operator on a noncompact star-type graph, Tamk. J. Math. 44 (2013), 327-49.
[7] R. Carlson, An inverse problem for the matrix Schr ̈odinger equation, J. Math. Anal. Appl. 267 (2002), 564-575.
[8] T.-H. Chang, C.-T. Shieh, Uniqueness of the potential function for the vectorial Sturm-Liouville equation on a finite interval, Boundary Value Problems 40 (2011), 8pp.
[9] D. Chelkak, E. Korotyaev, Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval, Journal of Functional Analysis 257 (2009) 1546-1588.
[10] Y.H. Cheng, Reconstruction of the Sturm-Liouville operator on a p -star graph with nodal data, Rocky Mountain J. Math. 42 (2012), 1431-1446.
[11] Y.-H. Cheng, C.-T. Shieh, C.K. Law, A vectorial inverse nodal problem, Proc. Amer. Math. Soc. 133 (2005), 1475-1484.
[12] G. Freiling, V.A. Yurko, Inverse Sturm-Liouville Problems and Their Applications, NOVA Science Publishers, New York, 2001.
[13] G. Freiling, V.A. Yurko, An inverse problem for the non-selfadjoint matrix Sturm-Liouville equation on the half-line, J. Inverse Ill-Posed Probl. 15 (2007), 785-798.
[14] G. Freiling, V.A. Yurko, Inverse problems for Sturm-Liouville operators on noncompact trees, Re- sult.Math. 50 (2007), 195-212.
[15] M.S. Harmer, Inverse scattering for the matrix Schr ̈odinger operator and Schr ̈odinger operator on graphs with general self-adjoint boundary conditions, ANZIAM J. 44 (2002), 161-168.
[16] M.S. Harmer, Inverse scattering on matrices with boundary conditions, J. Phys. A: Math. Gen. 38 (2005) 4875-4885.
[17] V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires, J. Phys. A: Math. Gen. 32(4) (1999), 595-630.
[18] P. Kuchment, Quantum graphs. I. Some basic structures, Waves Random Media 14 (2004), S107- S128.
[19] B.M. Levitan, I.S. Sargsjan, Introduction to Spectral Theory, Nauka, Moscow, 1970; English transl.: AMS Transl. of Math. Monographs. 39, Providence, RI, 1975.
[20] V.N. Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal. 32 (2000), 801-819.
[21] W. Rundell, P. Sacks, Inverse eigenvalue problem for a simple star graph, J. Spectr. Theory 5 (2015), 363-380.
[22] C.-L. Shen, Some inverse spectral problems for vectorial Sturm-Liouville equations, Inverse Problems 17 (2001), 1253-1294.
[23] C.-L. Shen, C.-T. Shieh, Two inverse eigenvalue problems for vectorial Sturm-Liouville equations, Inverse Problems 14 (1998), 1331-1343.
[24] C.-T. Shieh, Isospectral sets and inverse problems for vector-valued Sturm-Liouville equations, In- verse Problems 23 (2007), 2457-2468.
[25] F. Visco-Comandini, M. Mirrahimi, M. Sorine, Some inverse scattering problems on star-shaped graphs, J. Math. Anal. Appl. 378 (2011), 343-358.
[26] X.-C. Xu, C.-F. Yang, Determination of the self-adjoint matrix Schr ̈odinger operators without the bound state data, Inverse Problems 34 (2018), 065002 (20pp).
[27] X.-C. Xu, C.-F. Yang, Inverse scattering problems on a noncompact star graph, Inverse Problems 34 (2018), 115004 (12pp).
[28] C.-F. Yang, Trace formula for the matrix Sturm-Liouville operator, Analysis and Mathematical Physics 6 (2016), 31-41.
[29] C.-F. Yang, Inverse spectral problems for the Sturm-Liouville operator on a d-star graph, J. Math. Anal. Appl. 365 (2010), 742-749.
[30] C.-F. Yang, V.A. Yurko Recovering differential operators with nonlocal boundary conditions, Anal- ysis and Mathematical Physics 6 (2016), 315-326.
[31] V.A. Yurko, Inverse nodal problems for Sturm-Liouville operators on star-type graphs, J. Inverse Ill-Posed Probl. 16 (2008), 715-722.
[32] V.A. Yurko, Inverse problems for the matrix Sturm-Liouville equation on a finite interval, Inverse Problems 22(4) (2006), 1139-1149.
[33] V.A. Yurko, Inverse problems for matrix Sturm-Liouville operators, Russian Journal of Mathemat- ical Physics 13 (2006), 111-118.
[34] V.A. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), 1075-1086.
[35] V.A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2002.