“Layerwise” scattering for a difference Schrödinger operator

In modern physics literature, the need for formulas that permit, in a quantum one-dimensional problem, to reduce a calculation of the reflection (transmission) probability for the potential consisting of some “barriers” to the reflection and transmission probabilities over these “barriers” repeatedly occurred. In this paper, we study the scattering problem for the difference Schrödinger operator with the potential which is the sum of $N$ functions (describing the “barriers” or “layers”) with pairwise disjoint supports. With the help of the Lippmann–Schwinger equation, we proved the theorem which reduces the calculation of the reflection and transmission amplitudes for this potential, to the calculation of the ones for these barriers. For $N=2$ simple explicit formulas which realized this reduction were obtained. The particular cases for the even first barrier and two identical even (after appropriate shifts) barriers were studied. Of course, the similar results hold for the reflection (transmission) probabilities. We obtained the simple equation for the double-barrier structure resonances in terms of the amplitudes of each of the two barriers.
In the paper, we also present the alternative scheme of the proof of the obtained results which are based on the series expansion of the $T$-operator. This approach substantiates the physical understanding of the scattering by a multilayer structure as multiple scattering on separate layers. To proof the theorems, the known method of reduction of the Lippmann–Schwinger equation to the “modified” equation in a Hilbert space is used. Of course, all the results remain valid for the “continuous” Schrödinger operator, and the choice of the discrete approach is due to its growing popularity in the quantum theory of solids.