RECONSTRUCTION OF COMPLEX JACOBI MATRICES FROM SPECTRAL DATA

Year 2009, Volume: 38 Issue: 3, 317 - 327, 01.03.2009

Gusein Sh. Guseinov

https://izlik.org/JA69UR48SK

Abstract

RECONSTRUCTION OF COMPLEX JACOBI MATRICES FROM SPECTRAL DATA

https://izlik.org/JA69UR48SK

Abstract

In this paper, we introduce spectral data for finite order complex Jacobi matrices and investigate the inverse problem of determining the matrix from its spectral data. Necessary and sufficient conditions for the solvability of the inverse problem are established. An explicit procedure of reconstruction of the matrix from the spectral data is given.

Keywords

Jacobi matrix , Difference equation , Spectral data , Inverse spectral problem , 2000 AMS Classification: 15 A 29

References

• Akhiezer, N. I. The Classical Moment Problem and Some Related Questions in Analysis (Hafner, New York, 1965).
• Beckermann, B. Complex Jacobi matrices, J. Comput. Appl. Math. 127, 17—65, 2001.
• Berezanskii, Yu.M. Expansion in Eigenfunctions of Selfadjoint Operators (Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968).
• Berezanskii, Yu.M. The integration of semi-infinite Toda chain by means of inverse spectral problem, Rep. Math. Phys. 24, 21-47, 1986.
• Brüning, J., Chelkak, D. and Korotyaev, E. Inverse spectral analysis for finite matrix-valued Jacobi operators, preprint, arXiv: math/0607809.
• Charris, J. A. and Soriano, F. H. Complex and distributional weights for sieved ultraspherical polynomials, Internat. J. Math. and Math. Sci. 19, 229-242, 1996.
• Chihara, T. S. An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978).
• Chu, M.T. and Golub, G.H. Inverse Eigenvalue Problems: Theory, Algorithms, and Ap plications (Oxford University Press, New York, 2005).
• Guseinov, G. Sh. Determination of an infinite non-selfadjoint Jacobi matrix from its gen eralized spectral function, Mat. Zametki 23, 237-248, 1978; English transl. in Math. Notes 23, 130-136, 1978.
• Guseinov, G. Sh. An inverse spectral problem for complex Jacobi matrices, Communications in Nonlinear Science and Numerical Simulation. (To appear)
• Korotyaev, E. and Kutsenko, A. Lyapunov functions of periodic matrix-valued Jacobi op erators, in Spectral Theory of Differential Operators: M. Sh. Birman 80 th Anniversary Collection (American Mathematical Society Translations - Series 2, Vol. 225, Amer. Math. Soc., Providence, RI, 2008).
• Korotyaev, E. and Kutsenko, A. Borg-type uniqueness Theorems for periodic Jacobi opera tors with matrix-valued coefficients, Proc. Amer. Math. Soc. 137, 1989—1996, 2009.
• Marcellan, F. and Alvarez-Nodarse, R. On the “Favard theorem” and its extensions, J. Comput. Appl. Math. 127, 231-254, 2001.
• Moser, J.K. Various aspects of integrable Hamiltonian systems, Dynamical Systems, Progr. Math., Vol. 8, Birkhauser, Boston, 233-289, 1980.
• Nikishin, E.M. and Sorokin, V.N. Rational Approximations and Orthogonality (Transla tions of Mathematical Monographs, Vol. 92, American Mathematical Society, Providence, R.I., 1991).
• Teschl, G. Jacobi Operators and Completely Integrable Nonlinear Lattices (Mathematical Surveys and Monographs, Vol. 72, American Mathematical Society, Providence, R.I., 2000).
• Toda, M. Theory of Nonlinear Lattices (Springer-Verlag, New York, 1981).
• Wall, H. S. Analytic Theory of Continued Fractions (Chelsea, New York, 1973).