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 and 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.

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

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.

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

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

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