June  2019, 13(3): 431-447. doi: 10.3934/ipi.2019021

Dynamic inverse problem for Jacobi matrices

1. 

St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian, Academy of Sciences, 27, Fontanka, 191023 St. Petersburg, Russia

2. 

Saint Petersburg State University, St.Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia

 

Received  October 2017 Revised  October 2018 Published  March 2019

We consider the inverse dynamic problem for a dynamical system with discrete time associated with a semi-infinite Jacobi matrix. We derive discrete analogs of Krein equations and answer a question on the characterization of dynamic inverse data. As a consequence we obtain a necessary and sufficient condition for a measure on a real line to be a spectral measure of a semi-infinite discrete Schrödinger operator.

Citation: Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
References:
[1]

S. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem. The case of multiple poles, Math. Contr. Sign. Syst., 22 (2011), 245-265.  doi: 10.1007/s00498-010-0052-5.  Google Scholar

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Acad. Press, 1964.   Google Scholar
[3]

S. AvdoninA. Bulanova and D. Nicolsky, Boundary control approach to the spectral estimation problem. The case of simple poles, Sampling Theory in Signal and Image Processing, 8 (2009), 225-248.   Google Scholar

[4]

S. Avdonin and V. S. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

[5]

S. AvdoninV. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl m-function, Comm. Math. Phys., 275 (20017), 791-803.  doi: 10.1007/s00220-007-0315-2.  Google Scholar

[6]

M. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.  Google Scholar

[7]

M. Belishev, Boundary control and tomography of Riemannian manifolds (the BC-method), Uspekhi Matem. Nauk., 72 (2017), 3–66, (in Russian). doi: 10.4213/rm9768.  Google Scholar

[8]

M. Belishev, Boundary control and inverse problems: A one-dimensional version of the boundary control method, J. Math. Sci. (N.Y.), 155 (2008), 343-378.  doi: 10.1007/s10958-008-9220-2.  Google Scholar

[9]

M. Belishev and S. Ivanov, Characterization of data of dynamical inverse problem for two-velocity system, J. Math. Sci. (N.Y.), 109 (2002), 1814-1834.  doi: 10.1023/A:1014484022838.  Google Scholar

[10]

M. Belishev and V. Mikhaylov, Unified approach to classical equations of inverse problem theory, Journal of Inverse and Ill-Posed Problems, 20 (2012), 461-488.  doi: 10.1515/jip-2012-0040.  Google Scholar

[11]

M. Belishev and V. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 26 pp. doi: 10.1088/0266-5611/30/12/125013.  Google Scholar

[12]

M. Belishev and A. Pestov, Characterization of the inverse problem data for one-dimensional two-velocity dynamical system, St. Petersburg Mathematical Journal, 26 (2015), 411-440.  doi: 10.1090/s1061-0022-2015-01344-7.  Google Scholar

[13]

F. Gesztesy and B. Simon, m-functions and inverse spectral analisys for finite and semi-infinite Jacobi matrices, J. d'Analyse Math., 73 (1997), 267-297.  doi: 10.1007/BF02788147.  Google Scholar

[14]

V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Studies, Springer, v. 127, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[15]

S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Inverse and Ill-posed Problems Series, 55. Walter de Gruyter GmbH & Co. KG, Berlin, 2012.  Google Scholar

[16]

A. Kachalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall, 2001. doi: 10.1201/9781420036220.  Google Scholar

[17]

A. Mikhaylov and V. Mikhaylov, Dynamical inverse problem for the discrete Schrödinger operator, Nanosystems: Physics, Chemistry, Mathematics, 7 (2016), 842-854.  doi: 10.17586/2220-8054-2016-7-5-842-853.  Google Scholar

[18]

A. Mikhaylov and V. Mikhaylov, Boundary Control method and de Branges spaces. Schrödinger operator, Dirac system, discrete Schrödinger operator, Journal of Mathematical Analysis and Applications, 460 (2018), 927-953.  doi: 10.1016/j.jmaa.2017.12.013.  Google Scholar

[19]

A. Mikhaylov and V. Mikhaylov, Relationship between different types of inverse data for the one-dimensional Schrödinger operator on a half-line, J. Math. Sci. (N.Y.), 226 (2017), 779-794.  doi: 10.1007/s10958-017-3566-2.  Google Scholar

[20]

A. MikhaylovV. Mikhaylov and S. Simonov, On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time, Mathematical Methods in the Applied Sciences, 41 (2018), 6401-6408.  doi: 10.1002/mma.5147.  Google Scholar

[21]

B. Simon, The classical moment problem as a self-adjoint finite difference operator, Advances in Math., 137 (198), 82–203. doi: 10.1006/aima.1998.1728.  Google Scholar

show all references

References:
[1]

S. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem. The case of multiple poles, Math. Contr. Sign. Syst., 22 (2011), 245-265.  doi: 10.1007/s00498-010-0052-5.  Google Scholar

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Acad. Press, 1964.   Google Scholar
[3]

S. AvdoninA. Bulanova and D. Nicolsky, Boundary control approach to the spectral estimation problem. The case of simple poles, Sampling Theory in Signal and Image Processing, 8 (2009), 225-248.   Google Scholar

[4]

S. Avdonin and V. S. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems, 26 (2010), 045009, 19 pp. doi: 10.1088/0266-5611/26/4/045009.  Google Scholar

[5]

S. AvdoninV. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh-Weyl m-function, Comm. Math. Phys., 275 (20017), 791-803.  doi: 10.1007/s00220-007-0315-2.  Google Scholar

[6]

M. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.  Google Scholar

[7]

M. Belishev, Boundary control and tomography of Riemannian manifolds (the BC-method), Uspekhi Matem. Nauk., 72 (2017), 3–66, (in Russian). doi: 10.4213/rm9768.  Google Scholar

[8]

M. Belishev, Boundary control and inverse problems: A one-dimensional version of the boundary control method, J. Math. Sci. (N.Y.), 155 (2008), 343-378.  doi: 10.1007/s10958-008-9220-2.  Google Scholar

[9]

M. Belishev and S. Ivanov, Characterization of data of dynamical inverse problem for two-velocity system, J. Math. Sci. (N.Y.), 109 (2002), 1814-1834.  doi: 10.1023/A:1014484022838.  Google Scholar

[10]

M. Belishev and V. Mikhaylov, Unified approach to classical equations of inverse problem theory, Journal of Inverse and Ill-Posed Problems, 20 (2012), 461-488.  doi: 10.1515/jip-2012-0040.  Google Scholar

[11]

M. Belishev and V. Mikhaylov, Inverse problem for one-dimensional dynamical Dirac system (BC-method), Inverse Problems, 30 (2014), 125013, 26 pp. doi: 10.1088/0266-5611/30/12/125013.  Google Scholar

[12]

M. Belishev and A. Pestov, Characterization of the inverse problem data for one-dimensional two-velocity dynamical system, St. Petersburg Mathematical Journal, 26 (2015), 411-440.  doi: 10.1090/s1061-0022-2015-01344-7.  Google Scholar

[13]

F. Gesztesy and B. Simon, m-functions and inverse spectral analisys for finite and semi-infinite Jacobi matrices, J. d'Analyse Math., 73 (1997), 267-297.  doi: 10.1007/BF02788147.  Google Scholar

[14]

V. Isakov, Inverse Problems for Partial Differential Equations, Appl. Math. Studies, Springer, v. 127, 1998. doi: 10.1007/978-1-4899-0030-2.  Google Scholar

[15]

S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Inverse and Ill-posed Problems Series, 55. Walter de Gruyter GmbH & Co. KG, Berlin, 2012.  Google Scholar

[16]

A. Kachalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall, 2001. doi: 10.1201/9781420036220.  Google Scholar

[17]

A. Mikhaylov and V. Mikhaylov, Dynamical inverse problem for the discrete Schrödinger operator, Nanosystems: Physics, Chemistry, Mathematics, 7 (2016), 842-854.  doi: 10.17586/2220-8054-2016-7-5-842-853.  Google Scholar

[18]

A. Mikhaylov and V. Mikhaylov, Boundary Control method and de Branges spaces. Schrödinger operator, Dirac system, discrete Schrödinger operator, Journal of Mathematical Analysis and Applications, 460 (2018), 927-953.  doi: 10.1016/j.jmaa.2017.12.013.  Google Scholar

[19]

A. Mikhaylov and V. Mikhaylov, Relationship between different types of inverse data for the one-dimensional Schrödinger operator on a half-line, J. Math. Sci. (N.Y.), 226 (2017), 779-794.  doi: 10.1007/s10958-017-3566-2.  Google Scholar

[20]

A. MikhaylovV. Mikhaylov and S. Simonov, On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time, Mathematical Methods in the Applied Sciences, 41 (2018), 6401-6408.  doi: 10.1002/mma.5147.  Google Scholar

[21]

B. Simon, The classical moment problem as a self-adjoint finite difference operator, Advances in Math., 137 (198), 82–203. doi: 10.1006/aima.1998.1728.  Google Scholar

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