Limits of applicability of the tight binding approximation for complex-valued potential function

We consider a one-dimensional Schrödinger operator with periodic potential that is constructed as a sum of shifts of a given complex-valued potential $q\in L^1(\mathbf R)$. A mathematical basis of the tight binding approximation in this case is given. Let $\lambda_0$ be an isolated eigenvalue of Schrödinger operator with potential $q$. Then for the operator with periodic potential there exists a continuos spectrum that lies near $\lambda_0$. An asymptotic behavior of this part of the spectrum for the cases of one- and two-dimensional invariant subspace corresponding to $\lambda_0$ when the period tends to infinity is studied.