Inverse spectral problems for Jacobi matrix with finite perturbed parameters
DOI:
Keywords:
Jacobi matrix, direct and inverse spectral problems, generalized eigenvector, spectral function, m-function, Weyl solutionAbstract
For Jacobi matrices with finitely perturbed parameters, we get an explicit re\-presentation of the Weyl function, and solve inverse spectral problems, that is, we recover Jacobi matrices from spectral data. For the spectral data, we take the following: the spectral density of the absolutely continuous spectrum, with or without all the eigenvalues; the numerical parameters of the representation of one component of the vector-eigenfunction in terms of Chebyshev polynomials. We prove that these inverse problems have a unique solution, or only a finite number of solutions.References
N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. (Russian edition: Fizmatgiz, Moscow, 1961) MathSciNet
Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968. MathSciNet
Yu. M. Berezanskii, Integration of nonlinear difference equations by the method of the inverse spectral problem, Dokl. Akad. Nauk SSSR 281 (1985), no. 1, 16-19. MathSciNet
Yu. M. Berezanski, The integration of semi-infinite Toda chain by means of inverse spectral problem, Rep. Math. Phys. 24 (1986), no. 1, 21-47. MathSciNet CrossRef
Yu. M. Berezansky, Integration of the modified double-infinite Toda lattice with the help of inverse spectral problem, Ukrain. Mat. Zh. 60 (2008), no. 4, 453-469. MathSciNet CrossRef
Yurij Berezansky, The integration of double-infinite Toda lattice by means of inverse spectral problem and related questions, Methods Funct. Anal. Topology 15 (2009), no. 2, 101-136. MathSciNet
Yurij M. Berezansky, Linearization of double-infinite Toda lattice by means of inverse spectral problem, Methods Funct. Anal. Topology 18 (2012), no. 1, 19-54. MathSciNet
Yurij M. Berezansky, Mykola E. Dudkin, The direct and inverse spectral problems for the block Jacobi type unitary matrices, Methods Funct. Anal. Topology 11 (2005), no. 4, 327-345. MathSciNet
Yurij M. Berezansky, Mykola E. Dudkin, The complex moment problem and direct and inverse spectral problems for the block Jacobi type bounded normal matrices, Methods Funct. Anal. Topology 12 (2006), no. 1, 1-31. MathSciNet
Yu. M. Berezanskii, M. I. Gekhtman, M. E. Shmoish, Integration of certain chains of nonlinear difference equations by the method of the inverse spectral problem, Ukrain. Mat. Zh. 38 (1986), no. 1, 84-89, 134. MathSciNet
Yu. M. Berezanskii, M. I. Gekhtman, Inverse problem of spectral analysis and nonabelian chains of nonlinear equations, Ukrain. Mat. Zh. 42 (1990), no. 6, 730-747. MathSciNet CrossRef
Yu. M. Berezanskii, A. A. Mokhon′ko, Integration of some nonlinear differential-difference equations using the spectral theory of normal block-Jacobi matrices, Funktsional. Anal. i Prilozhen. 42 (2008), no. 1, 1-21, 95. MathSciNet CrossRef
A. Boutet de Monvel, I. Egorova, E. Khruslov, Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems 13 (1997), no. 2, 223-237. MathSciNet CrossRef
Andries E. Brouwer, Willem H. Haemers, Spectra of graphs, Springer, New York, 2012. MathSciNet CrossRef
I. Egorova, The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom. 9 (2002), no. 2, 188-205. MathSciNet
Iryna Egorova, Johanna Michor, Gerald Teschl, Scattering theory for Jacobi operators with a steplike quasi-periodic background, Inverse Problems 23 (2007), no. 3, 905-918. MathSciNet CrossRef
L. D. Faddeev, L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Springer-Verlag, Berlin, 1987. Translated from the Russian by A. G. Reyman [A. G. Reiman] MathSciNet CrossRef
Fritz Gesztesy, Helge Holden, Johanna Michor, Gerald Teschl, Soliton equations and their algebro-geometric solutions. Vol. II, Cambridge University Press, Cambridge, 2008. $(1+1)$-dimensional discrete models MathSciNet CrossRef
Fritz Gesztesy, Barry Simon, $m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math. 73 (1997), 267-297. MathSciNet CrossRef
L. B. Golinskii, Schur flows and orthogonal polynomials on the unit circle, Mat. Sb. 197 (2006), no. 8, 41-62. MathSciNet CrossRef
Ya. I. Ivasiuk, Direct spectral problem for the generalized Jacobi Hermitian matrices, Methods Funct. Anal. Topology 15 (2009), no. 1, 3-14. MathSciNet
M. Krein, Infinite $J$-matrices and a matrix-moment problem, Doklady Akad. Nauk SSSR (N.S.) 69 (1949), 125-128. MathSciNet
V. O. Lebid, L. P. Nizhnik, Spectral analysis of locally finite graphs with one infinite ray, Reports of the National Academy of Sciences of Ukraine (2014), no. 3, 29-35. (Ukrainian)
V. O. Lebid, L. O. Nyzhnyk, Spectral analysis of some graphs with infinite rays, Ukrainian Math. J. 66 (2015), no. 9, 1333-1345. MathSciNet Translation of Ukrain. Mat. Zh. 66 (2014), no. 9, 1193-1204 CrossRef
V. A. Marchenko, Operatory Shturma-Liuvillya i ikh prilozheniya, Izdat. ``Naukova Dumka'', Kiev, 1977. MathSciNet
Bojan Mohar, Wolfgang Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc. 21 (1989), no. 3, 209-234. MathSciNet CrossRef
O. A. Mokhon′ko, Nonisospectral flows on semiinfinite unitary block Jacobi matrices, Ukrain. Mat. Zh. 60 (2008), no. 4, 521-544. MathSciNet CrossRef
Leonid Nizhnik, Inverse nonlocal Sturm-Liouville problem, Inverse Problems 26 (2010), no. 12, 125006, 9. MathSciNet CrossRef
Leonid Nizhnik, Inverse spectral nonlocal problem for the first order ordinary differential equation, Tamkang J. Math. 42 (2011), no. 3, 385-394. MathSciNet CrossRef
L. P. Nizhnik, Inverse eigenvalue problems for nonlocal Sturm-Liouville operators on a star graph, Methods Funct. Anal. Topology 18 (2012), no. 1, 68-78. MathSciNet
L. P. Nizhnik, Spectral analysis of metric graphs with infinite rays, Methods Funct. Anal. Topology 20 (2014), no. 4, 391-396. MathSciNet
Vyacheslav Pivovarchik, Inverse problem for the Sturm-Liouville equation on a simple graph, SIAM J. Math. Anal. 32 (2000), no. 4, 801-819 (electronic). MathSciNet CrossRef
Barry Simon, Szegos theorem and its descendants. Spectral theory for $L^2$ perturbations of orthogonal polynomials, Princeton University Press, Princeton, NJ, 2011. MathSciNet
Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, American Mathematical Society, Providence, RI, 2000. MathSciNet
V. Yurko, Inverse spectral problems for Sturm-Liouville operators on graphs, Inverse Problems 21 (2005), no. 3, 1075-1086. MathSciNet CrossRef
N. V. Zhernakov, Direct and inverse problems for a periodic Jacobian matrix, Ukrain. Mat. Zh. 38 (1986), no. 6, 785-788, 816. MathSciNet
