Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 018, 28 pages      arXiv:0902.2464      http://dx.doi.org/10.3842/SIGMA.2009.018
Contribution to the Proceedings of the VIIth Workshop ''Quantum Physics with Non-Hermitian Operators''

Inverse Spectral Problems for Tridiagonal N by N Complex Hamiltonians

Gusein Sh. Guseinov
Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey

Received November 18, 2008, in final form February 09, 2009; Published online February 14, 2009

Abstract
In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.

Key words: Jacobi matrix; difference equation; generalized spectral function; spectral data.

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References

  1. Boley D., Golub G.H., A survey of matrix inverse eigenvalue problems, Inverse Problems 3 (1987), 595-622.
  2. Ikramov Kh.D., Chugunov V.N., Inverse matrix eigenvalue problems, J. Math. Sciences 98 (2000), 51-136.
  3. Chu M.T., Golub G.H., Inverse eigenvalue problems: theory, algorithms, and applications, Oxford University Press, New York, 2005.
  4. Marchenko V.A., Expansion in eigenfunctions of non-selfadjoint singular second order differential operators, Mat. Sb. 52 (1960), 739-788 (in Russian).
  5. Rofe-Beketov F.S., Expansion in eigenfunctions of infinite systems of differential equations in the non-selfadjoint and selfadjoint cases, Mat. Sb. 51 (1960), 293-342 (in Russian).
  6. Guseinov G.Sh., Determination of an infinite non-selfadjoint Jacobi matrix from its generalized spectral function, Mat. Zametki 23 (1978), 237-248 (English transl.: Math. Notes 23 (1978), 130-136).
  7. Guseinov G.Sh., The inverse problem from the generalized spectral matrix for a second order non-selfadjoint difference equation on the axis, Izv. Akad. Nauk Azerb. SSR Ser. Fiz.-Tekhn. Mat. Nauk (1978), no. 5, 16-22 (in Russian).
  8. Kishakevich Yu.L., Spectral function of Marchenko type for a difference operator of an even order, Mat. Zametki 11 (1972), 437-446 (English transl.: Math. Notes 11 (1972), 266-271).
  9. Kishakevich Yu.L., On an inverse problem for non-selfadjoint difference operators, Mat. Zametki 11 (1972), 661-668 (English transl.: Math. Notes 11 (1972), 402-406).
  10. Bender C.M., Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), 947-1018, hep-th/0703096.
  11. Znojil M., Matching method and exact solvability of discrete PT-symmetric square wells, J. Phys. A: Math. Gen. 39 (2006), 10247-10261, quant-ph/0605209.
  12. Znojil M., Maximal couplings in PT-symmetric chain models with the real spectrum of energies, J. Phys. A: Math. Theor. 40 (2007), 4863-4875, math-ph/0703070.
  13. Znojil M., Tridiagonal PT-symmetric N by N Hamiltonians and fine-tuning of their observability domains in the strongly non-Hermitian regime, J. Phys. A: Math. Theor. 40 (2007), 13131-13148, arXiv:0709.1569.
  14. Allakhverdiev B.P., Guseinov G.Sh., On the spectral theory of dissipative difference operators of second order, Mat. Sb. 180 (1989), 101-118 (English transl.: Math. USSR Sbornik 66 (1990), 107-125).
  15. Guseinov G.Sh., Completeness of the eigenvectors of a dissipative second order difference operator, J. Difference Equ. Appl. 8 (2002), 321-331.
  16. van Moerbeke P., Mumford D., The spectrum of difference operators and algebraic curves, Acta Math. 143 (1979), 93-154.
  17. Sansuc J.J., Tkachenko V., Spectral parametrization of non-selfadjoint Hill's operators, J. Differential Equations 125 (1996), 366-384.
  18. Egorova I., Golinskii L., Discrete spectrum for complex perturbations of periodic Jacobi matrices, J. Difference Equ. Appl. 11 (2005), 1185-1203, math.SP/0503627.
  19. Atkinson F.V., Discrete and continuous boundary problems, Academic Press, New York, 1964.
  20. Akhiezer N.I., The classical moment problem and some related questions in analysis, Hafner, New York, 1965.
  21. Berezanskii Yu.M., Expansion in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.
  22. Nikishin E.M., Sorokin V.N., Rational approximations and orthogonality, Translations of Mathematical Monographs, Vol. 92, American Mathematical Society, Providence, R.I., 1991.
  23. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, American Mathematical Society, Providence, R.I., 2000.

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