Geodesic
On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential
Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 451-455 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2009},
     volume = {85},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a12/}
}
TY  - JOUR
AU  - M. M. Faddeev
TI  - On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential
JO  - Matematičeskie zametki
PY  - 2009
SP  - 451
EP  - 455
VL  - 85
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a12/
LA  - ru
ID  - MZM_2009_85_3_a12
ER  - 
%0 Journal Article
%A M. M. Faddeev
%T On Spectral Properties of the Discrete Schrödinger Operator with Pure Imaginary Finite Potential
%J Matematičeskie zametki
%D 2009
%P 451-455
%V 85
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2009_85_3_a12/
%G ru
%F MZM_2009_85_3_a12
M. M. Faddeev. On Spectral Properties of the Discrete Schrödinger 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/

[1] S. Albeverio, S. Kuzhel, “Pseudo-Hermiticity and theory of singular perturbations”, Lett. Math. Phys., 67:3 (2004), 223–238 | DOI | MR | Zbl

[2] S. Albeverio, S.-M. Fei, P. Kurasov, “Point interactions: $\mathscr{PT}$-Hermiticity and reality of the spectrum”, Lett. Math. Phys., 59:3 (2002), 227–242 | DOI | MR | Zbl

[3] C. M. Bender, S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathscr{PT}$ symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243–5246 | DOI | MR | Zbl

[4] V. E. Lyantse, “Nesamosopryazhennyi raznostnyi operator”, Dokl. AN SSSR, 173:6 (1967), 1260–1263 | MR | Zbl

[5] S. N. Naboko, “Ob usloviyakh podobiya unitarnym i samosopryazhennym operatoram”, Funkts. analiz i ego pril., 18:1 (1984), 16–27 | MR | Zbl

[6] L. A. Sakhnovich, “Neunitarnye operatory s absolyutno nepreryvnym spektrom”, Izv. AN SSSR. Ser. matem., 33:1 (1969), 52–64 | MR | Zbl

[7] B. Ya. Levin, “Preobrazovanie tipa Fure i Laplasa pri pomoschi reshenii differentsialnogo uravneniya vtorogo poryadka”, Dokl. AN SSSR, 106:2 (1956), 187–190 | MR | Zbl

[8] V. V. Blaschak, “O razlozhenii po glavnym funktsiyam nesamosopryazhennogo differentsialnogo operatora vtorogo poryadka na vsei osi”, Dokl. AN USSR, 4 (1965), 416–419

[9] D. Damanik, D. Hundertmark, R. Killip, B. Simon, “Variational estimates for discrete Schrödinger operators with potentials of indefinite sign”, Comm. Math. Phys., 238:3 (2003), 545–562 | DOI | MR | Zbl