On infinite system of resonance and eigenvalues with exponential asymptotics generated by distant perturbations

We consider an one-dimensional Schrödinger operator with four distant potentials separated by large distance. All distances are proportional to a sam large parameter. The initial potentials are of kink shapes, which are glued mutually so that the final potential vanishes at infinity and between the second and third initial potentials and it is equal to one between the first and the second potentials as well as between the third and fourth potentials. The potentials are not supposed to be real and can be complex-valued. We show that under certain, rather natural and easily realizable conditions on the four initial potentials, the considered operator with distant potentials possesses infinitely many resonances and/or eigenvalues of form $\lambda=k_n^2$, $n\in\mathbb{Z}$, which accumulate along a given segment in the essential spectrum. The distance between neighbouring numbers $k_n$ is of order the reciprocal of the distance between the potentials, while the imaginary parts of these quantities are exponentially small. For the numbers $k_n$ we obtain the representations via the limits of some explicitly calculated sequences and the sum of infinite series. We also prove explicit effective estimates for the convergence rates of the sequences and for the remainders of the series.