Discrete dynamical systems: Inverse problems and related topics

References

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

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964. 10.1063/1.3051875Suche 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. Suche 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.1997.5.4.309Suche 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. Suche 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.2008.2.1Suche 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. 10.17586/2220-8054-2015-6-1-63-78Suche 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/abd7cbSuche 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. 10.1088/0266-5611/26/4/045009Suche 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/s10958-017-3388-2Suche 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/s00220-007-0315-2Suche 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. Suche 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. 10.1088/0266-5611/13/5/002Suche 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.2001.9.6.547Suche 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. 10.1088/0266-5611/20/3/002Suche 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/R01Suche 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. Suche 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/rm9768Suche 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. Suche 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-2012-0040Suche 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. 10.1515/156939406776237474Suche 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-14379Suche 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-zSuche 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. Suche in Google Scholar

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

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

[27] H. Dym and H. P. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York, 1976. Suche 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/BF02788147Suche 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. Suche 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/S1085337504306135Suche in Google Scholar

[31] S. I. Kabanikhin, Inverse and Ill-Posed Problems. Theory and Applications, Walter de Gruyter, Berlin, 2012. 10.1515/9783110224016Suche 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/B978-008044268-6/50057-0Suche 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. 10.1163/1569394042530900Suche 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/trans2/103/02Suche 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. 10.1201/9781420036220Suche 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. Suche 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. Suche in Google Scholar

[38] P. Kurasov, Spectral Geometry of Graphs, Oper. Theory Adv. Appl. 293, Birkhäuser, Berlin, 2024. 10.1007/978-3-662-67872-5Suche 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. Suche 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. 10.17586/2220-8054-2016-7-5-842-853Suche 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.2017.8168031Suche 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.2017.12.013Suche 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.2019021Suche 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. Suche 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/DD46733.2019.9016622Suche 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.2019.05.002Suche 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. 10.17586/2220-8054-2020-11-1-25-29Suche 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.2020.123970Suche 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/DD49902.2020.9274657Suche 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/DD52349.2021.9598751Suche 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-2020-0112Suche 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. 10.1088/1742-6596/2092/1/012002Suche 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/DD55230.2022.9960960Suche 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. 10.17586/2220-8054-2022-13-1-24-29Suche 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/DD58728.2023.10325752Suche 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. Suche 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/s10958-024-07294-8Suche 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. Suche 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. Suche 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. Suche 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.5147Suche 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_12Suche 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_9-1Suche in Google Scholar

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

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

[66] J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York, 1943. 10.1090/surv/001Suche 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.1998.1728Suche 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. 10.2307/121061Suche 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/072Suche in Google Scholar

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

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

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