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

SIGMA 5 (2009), 018, 28 pages      arXiv:0902.2464      https://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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