Discrete dynamical systems: Inverse problems and related topics

References

[1] N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh, 1965. Search in Google Scholar

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. 10.1063/1.3051875Search in Google Scholar

[3] S. A. Avdonin and M. I. Belishev, Boundary control and dynamical inverse problem for nonselfadjoint Sturm–Liouville operator (BC-method), Control Cybernet. 25 (1996), no. 3, 429–440. Search in Google Scholar

[4] S. A. Avdonin, M. I. Belishev and Y. S. Rozhkov, The BC-method in the inverse problem for the heat equation, J. Inverse Ill-Posed Probl. 5 (1997), no. 4, 309–322. 10.1515/jiip.1997.5.4.309Search in Google Scholar

[5] S. A. Avdonin, A. Choke, A. S. Mikhaylov and V. S. Mikhaylov, Discretization of the wave equation on a metric graph, preprint (2022), https://arxiv.org/abs/2210.09274; to appear in Math. Methods Appl. Sci. Search in Google Scholar

[6] S. A. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging 2 (2008), no. 1, 1–21. 10.3934/ipi.2008.2.1Search in Google Scholar

[7] S. A. Avdonin, A. S. Mikhaylov and V. S. Mikhaylov, On some applications of the boundary control method to spectral estimation and inverse problems, Nanosystems Phys. Chem. Math. 6 (2015), no. 1, 63–78. 10.17586/2220-8054-2015-6-1-63-78Search in Google Scholar

[8] S. A. Avdonin, A. S. Mikhaylov, V. S. Mikhaylov and J. C. Park, Inverse problem for the Schrödinger equation with non-self-adjoint matrix potential, Inverse Problems 37 (2021), no. 3, Article ID 035002. 10.1088/1361-6420/abd7cbSearch in Google Scholar

[9] S. A. Avdonin and V. S. Mikhaylov, The boundary control approach to inverse spectral theory, Inverse Problems 26 (2010), no. 4, Article ID 045009. 10.1088/0266-5611/26/4/045009Search in Google Scholar

[10] S. A. Avdonin, V. S. Mikhaylov and K. B. Nurtazina, On inverse dynamical and spectral problems for the wave and Schrödinger equations on finite trees. The leaf peeling method, J. Math. Sci. (N. Y.) 224 (2017), 1–10. 10.1007/s10958-017-3388-2Search in Google Scholar

[11] S. A. Avdonin, V. S. Mikhaylov and A. Rybkin, The boundary control approach to the Titchmarsh–Weyl 𝑚-function. I. The response operator and the 𝐴-amplitude, Comm. Math. Phys. 275 (2007), no. 3, 791–803. 10.1007/s00220-007-0315-2Search in Google Scholar

[12] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Sov. Math. Dokl. 36 (1988), no. 3, 481–484. Search in Google Scholar

[13] M. I. Belishev, Boundary control in reconstruction of manifolds and metrics (the BC method), Inverse Problems 13 (1997), no. 5, R1–R45. 10.1088/0266-5611/13/5/002Search in Google Scholar

[14] M. I. Belishev, On relations between spectral and dynamical inverse data, J. Inverse Ill-Posed Probl. 9 (2001), no. 6, 547–565. 10.1515/jiip.2001.9.6.547Search in Google Scholar

[15] M. I. Belishev, Boundary spectral inverse problem on a class of graphs (trees) by the BC method, Inverse Problems 20 (2004), no. 3, 647–672. 10.1088/0266-5611/20/3/002Search in Google Scholar

[16] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), no. 5, R1–R67. 10.1088/0266-5611/23/5/R01Search in Google Scholar

[17] M. I. Belishev, Boundary control and inverse problems: a one-dimensional version of the boundary control method (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 354 (2008), no. 37, 19–80; translation in J. Math. Sci. (N. Y.) 155 (2008), no. 3, 343–378. Search in Google Scholar

[18] M. I. Belishev, Boundary control and tomography of Riemannian manifolds (the BC-method) (in Russian), Uspekhi Mat. Nauk 72 (2017), no. 4(436), 3–66. 10.4213/rm9768Search in Google Scholar

[19] M. I. Belishev and T. S. Khabibullin, Characterization of data in dynamical inverse problem for the 1d wave equation with matrix potential, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 493 (2020), no. 50, 48–72. Search in Google Scholar

[20] M. I. Belishev and V. S. Mikhaylov, Unified approach to classical equations of inverse problem theory, J. Inverse Ill-Posed Probl. 20 (2012), no. 4, 461–488. 10.1515/jip-2012-0040Search in Google Scholar

[21] M. I. Belishev and A. F. Vakulenko, Inverse problems on graphs: Recovering the tree of strings by the BC-method, J. Inverse Ill-Posed Probl. 14 (2006), no. 1, 29–46. 10.1515/156939406776237474Search in Google Scholar

[22] C. Berg, Y. Chen and M. E. H. Ismail, Small eigenvalues of large Hankel matrices: The indeterminate case, Math. Scand. 91 (2002), no. 1, 67–81. 10.7146/math.scand.a-14379Search in Google Scholar

[23] C. Berg and R. Szwarc, Closable Hankel operators and moment problems, Integral Equations Operator Theory 92 (2020), no. 1, Paper No. 5. 10.1007/s00020-020-2561-zSearch in Google Scholar

[24] A. S. Blagovestchenskii, The local method of solution of the nonstationary inverse problem for an inhomogeneous string (in Russian), Trudy Mat. Inst. Steklov. 115 (1971), 28–38. Search in Google Scholar

[25] A. S. Blagovestchenskii, Inverse Problems of Wave Processes, VSP, Utrecht, 2001. 10.1515/9783110940893Search in Google Scholar

[26] L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, 1968. Search in Google Scholar

[27] H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York, 1976. Search in Google Scholar

[28] F. Gesztesy and B. Simon, 𝑚-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267–297. 10.1007/BF02788147Search in Google Scholar

[29] E. K. Ifantis and K. N. Vlachou, Solution of the semi-infinite Toda lattice for unbounded sequences, Lett. Math. Phys. 59 (2002), no. 1, 1–17. Search in Google Scholar

[30] E. K. Ifantis and K. N. Vlachou, Exact solutions of the semi-infinite Toda lattice with applications to the inverse spectral problem, Abstr. Appl. Anal. (2004), no. 5, 435–451. 10.1155/S1085337504306135Search in Google Scholar

[31] S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Walter de Gruyter, Berlin, 2012. 10.1515/9783110224016Search in Google Scholar

[32] S. I. Kabanikhin and M. A. Shishlenin, Comparative analysis of boundary control and Gel’fand–Levitan methods of solving inverse acoustic problem, Inverse Problems in Engineering Mechanics IV, Elsevier, Amsterdam (2003), 503–512. 10.1016/B978-008044268-6/50057-0Search in Google Scholar

[33] S. I. Kabanikhin and M. A. Shishlenin, Boundary control and Gel’fand–Levitan–Krein methods in inverse acoustic problem, J. Inverse Ill-Posed Probl. 12 (2004), no. 2, 125–144. 10.1163/1569394042530900Search in Google Scholar

[34] I. S. Kac and M. G. Krein, On the spectral functions of the string, Amer. Math. Soc. Transl. Ser. (2) 103 (1974), 19–102. 10.1090/trans2/103/02Search in Google Scholar

[35] A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monogr. Surveys Pure Appl. Math. 123, Chapman & Hall/CRC, Boca Raton, 2001. 10.1201/9781420036220Search in Google Scholar

[36] M. G. Krein, On the transfer function of a one-dimensional boundary problem of the second order (in Russian), Doklady Akad. Nauk SSSR (N. S.) 88 (1953), 405–408. Search in Google Scholar

[37] M. G. Krein, On a method of effective solution of an inverse boundary problem (in Russian), Doklady Akad. Nauk SSSR (N. S.) 94 (1954), 987–990. Search in Google Scholar

[38] P. Kurasov, Spectral Geometry of Graphs, Oper. Theory Adv. Appl. 293, Birkhäuser, Berlin, 2024. 10.1007/978-3-662-67872-5Search in Google Scholar

[39] A. S. Mikhaylov and V. S. Mikhaylov, Connection of the different types of inverse data for the one-dimensional Schrödinger operator on the half-line, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 451 (2016), no. 46, 134–155. Search in Google Scholar

[40] A. S. Mikhaylov and V. S. Mikhaylov, Dynamical inverse problem for the discrete Schrödinger operator, Nanosystems Phys. Chem. Math. 7 (2016), no. 5, 842–854. 10.17586/2220-8054-2016-7-5-842-853Search in Google Scholar

[41] A. S. Mikhaylov and V. S. Mikhaylov, Quantum and acoustic scattering on R + and one representation of a scattering matrix, 2017 Days on Diffraction, IEEE Press, Piscataway (2017), 237–240. 10.1109/DD.2017.8168031Search in Google Scholar

[42] A. S. Mikhaylov and V. S. Mikhaylov, The boundary control method and de Branges spaces. Schrödinger equation, Dirac system and discrete Schrödinger operator, J. Math. Anal. Appl. 460 (2018), no. 2, 927–953. 10.1016/j.jmaa.2017.12.013Search in Google Scholar

[43] A. S. Mikhaylov and V. S. Mikhaylov, Dynamic inverse problem for Jacobi matrices, Inverse Probl. Imaging 13 (2019), no. 3, 431–447. 10.3934/ipi.2019021Search in Google Scholar

[44] A. S. Mikhaylov and V. S. Mikhaylov, Forward and backward dynamic problems for a finite Krein–Stieltjes string. Approximation of constant density by point masses, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 483 (2019), no. 49, 128–141; translation in J. Math. Sci. 252 (2021), no. 5, 654–663. Search in Google Scholar

[45] A. S. Mikhaylov and V. S. Mikhaylov, Forward and inverse dynamic problems for a Krein string. Approximation by point-mass densities, 2019 Days on Diffraction, IEEE Press, Piscataway (2019), 125–131. 10.1109/DD46733.2019.9016622Search in Google Scholar

[46] A. S. Mikhaylov and V. S. Mikhaylov, Inverse dynamic problem for a Krein–Stieltjes string, Appl. Math. Lett. 96 (2019), 195–201. 10.1016/j.aml.2019.05.002Search in Google Scholar

[47] A. S. Mikhaylov and V. S. Mikhaylov, Finite Toda lattice and classical moment problem, Nanosystems Phys. Chem. Math. 11 (2020), 25–29. 10.17586/2220-8054-2020-11-1-25-29Search in Google Scholar

[48] A. S. Mikhaylov and V. S. Mikhaylov, Inverse problem for dynamical system associated with Jacobi matrices and classical moment problems, J. Math. Anal. Appl. 487 (2020), no. 1, Article ID 123970. 10.1016/j.jmaa.2020.123970Search in Google Scholar

[49] A. S. Mikhaylov and V. S. Mikhaylov, Solution of Toda lattices for semi-bounded initial data, 2020 Days on Diffraction, IEEE Press, Piscataway (2020), 85–88. 10.1109/DD49902.2020.9274657Search in Google Scholar

[50] A. S. Mikhaylov and V. S. Mikhaylov, Hilbert spaces of functions associated with Jacobi matrices, 2021 Days on Diffraction, IEEE Press, Piscataway (2021), 10.1109/DD52349.2021.9598751. 10.1109/DD52349.2021.9598751Search in Google Scholar

[51] A. S. Mikhaylov and V. S. Mikhaylov, Inverse problems for finite Jacobi matrices and Krein–Stieltjes strings, J. Inverse Ill-Posed Probl. 29 (2021), no. 4, 611–628. 10.1515/jiip-2020-0112Search in Google Scholar

[52] A. S. Mikhaylov and V. S. Mikhaylov, On application of the boundary control method to classical moment problems, J. Phys. Conf. Ser. 2092 (2021), Article ID 012002. 10.1088/1742-6596/2092/1/012002Search in Google Scholar

[53] A. S. Mikhaylov and V. S. Mikhaylov, On a special Hilbert space of functions associated with the multidimensional wave equation in a bounded domain, 2022 Days on Diffraction, IEEE Press, Piscataway (2022), 101–105. 10.1109/DD55230.2022.9960960Search in Google Scholar

[54] A. S. Mikhaylov and V. S. Mikhaylov, On the construction of de Branges spaces for dynamical systems associated with finite Jacobi matrices, Nanosystems Phys. Chem. Math. 13 (2022), no. 1, 24–29. 10.17586/2220-8054-2022-13-1-24-29Search in Google Scholar

[55] A. S. Mikhaylov and V. S. Mikhaylov, Dynamic inverse problem for complex Jacobi matrices, 2023 Days on Diffraction, IEEE Press, Piscataway (2023), 10.1109/DD58728.2023.10325752. 10.1109/DD58728.2023.10325752Search in Google Scholar

[56] A. S. Mikhaylov and V. S. Mikhaylov, Dynamic inverse problem for complex Jacobi matrices, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 521 (2023), no. 53, 136–153. Search in Google Scholar

[57] A. S. Mikhaylov and V. S. Mikhaylov, Construction of solutions of Toda lattices by the classical moment problem, J. Math. Sci. 283 (2024), 573–585. 10.1007/s10958-024-07294-8Search in Google Scholar

[58] A. S. Mikhaylov and V. S. Mikhaylov, Inverse problem for semi-infinite Jacobi matrices and associated Hilbert spaces of analytic functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), to appear. Search in Google Scholar

[59] A. S. Mikhaylov and V. S. Mikhaylov, On the relationships between hyperbolic and parabolic inverse discrete problems, J. Appl. Indust. Math., to appear. Search in Google Scholar

[60] A. S. Mikhaylov, V. S. Mikhaylov and M. A. Shishlenin, Inverse dynamic problem for parabolic equation on finite interval, in progress. Search in Google Scholar

[61] A. S. Mikhaylov, V. S. Mikhaylov and S. A. Simonov, On the relationship between Weyl functions of Jacobi matrices and response vectors for special dynamical systems with discrete time, Math. Methods Appl. Sci. 41 (2018), no. 16, 6401–6408. 10.1002/mma.5147Search in Google Scholar

[62] J. Moser, Finitely many mass points on the line under the influence of an exponential potential–an integrable system, Dynamical Systems, Theory and Applications, Lecture Notes in Phys. 38, Springer, Berlin (1975), 467–497. 10.1007/3-540-07171-7_12Search in Google Scholar

[63] R. V. Romanov, Canonical systems and de Branges spaces, preprint (2014), https://arxiv.org/abs/1408.6022. 10.1007/978-3-0348-0692-3_9-1Search in Google Scholar

[64] V. G. Romanov, Inverse Problems of Mathematical Physics, “Nauka”, Moscow, 1984. Search in Google Scholar

[65] K. Schmüdgen, The Moment Problem, Springer, Cham, 2017. 10.1007/978-3-319-64546-9Search in Google Scholar

[66] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York, 1943. 10.1090/surv/001Search in Google Scholar

[67] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203. 10.1006/aima.1998.1728Search in Google Scholar

[68] B. Simon, A new approach to inverse spectral theory. I. Fundamental formalism, Ann. of Math. (2) 150 (1999), no. 3, 1029–1057. 10.2307/121061Search in Google Scholar

[69] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surveys Monogr. 72, American Mathematical Society, Providence, 2000. 10.1090/surv/072Search in Google Scholar

[70] M. Toda, Theory of Nonlinear Lattices, 2nd ed., Springer Ser. Solid-State Sci. 20, Springer, Berlin, 1989. 10.1007/978-3-642-83219-2Search in Google Scholar

[71] V. S. Vladimirov, Equations of Mathematical Physics, 3rd ed., “Nauka”, Moscow, 1976. Search in Google Scholar

[72] D. R. Yafaev, Unbounded Hankel operators and moment problems, Integral Equations Operator Theory 85 (2016), no. 2, 289–300. Search in Google Scholar

[73] D. R. Yafaev, Correction to: Unbounded Hankel Operators and Moment Problems (Integral Equations Operator Theory 85 (2016), no. 2, 289–300, Integral Equations Operator Theory 91 (2019), no. 5, Paper No. 44. 10.1007/s00020-016-2289-ySearch in Google Scholar