Geodesic
Limit sets for the discrete spectrum of complex Jacobi matrices
Sbornik. Mathematics, Tome 196 (2005) no. 6, pp. 817-844

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The discrete spectrum of complex Jacobi matrices that are compact perturbations of the discrete Laplacian is studied. The precise stabilization rate (in the sense of order) of the matrix elements ensuring the finiteness of the discrete spectrum is found. An example of a Jacobi matrix with discrete spectrum having a unique limit point is constructed. These results are discrete analogues of Pavlov's well-known results on Schrödinger operators with complex potential on a half-axis. 
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     author = {L. B. Golinskii and I. E. Egorova},
     title = {Limit sets for the discrete spectrum of complex {Jacobi} matrices},
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L. B. Golinskii; I. E. Egorova. Limit sets for the discrete spectrum of complex Jacobi matrices. Sbornik. Mathematics, Tome 196 (2005) no. 6, pp. 817-844. http://geodesic.mathdoc.fr/item/SM_2005_196_6_a2/