Geodesic
On Spectral Properties of the Discrete Schr\"odinger Operator with Pure Imaginary Finite Potential
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 451-455

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In this paper, we consider the spectral properties of the discrete Schrödinger operator in the space of square integrable two-sided sequences with a pure imaginary potential of finite rank with zero mean value. We show that if such potentials are small, then the spectrum of the operator under study coincides with the spectrum of the unperturbed operator, and the operator itself is similar to a self-adjoint operator.
Keywords: discrete Schrödinger operator, spectral problem, $\mathscr{PT}$-symmetric potential, similarity to a self-adjoint operator. 
@article{MZM_2009_85_3_a12,
     author = {M. M. Faddeev},
     title = {On {Spectral} {Properties} of the {Discrete} {Schr\"odinger} {Operator} with {Pure} {Imaginary} {Finite} {Potential}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {451--455},
     publisher = {mathdoc},
     volume = {85},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a12/}
}
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M. M. Faddeev. On Spectral Properties of the Discrete Schr\"odinger Operator with Pure Imaginary Finite Potential. Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 451-455. http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a12/