On the range of use of the tight-binding approximation method for a complex-valued potential

We consider a one-dimensional Schrodinger operator with a periodic potential constructed as the sum of the shifts of a given complex-valued potential q that decreases at infinity. Mathematical justification of the tight-binding approximation method is presented. Let λ0 be an isolated eigenvalue of the Schrodinger operator with a potential q. Then, for the corresponding operator with a periodic potential, a continuous spectrum exists in the neighborhood of λ0. The asymptotic behavior of this part of the spectrum as the period increases infinitely is studied for the cases of one- and two-dimensional invariant subspaces corresponding to λ0.