Abstract

In modern physics literature, the need for formulas that permit, in a quantum onedimensional 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 LippmannSchwinger 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 doublebarrier 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 Toperator. 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 LippmannSchwinger 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.

References

 Lousse V., Vigneron J.P. Use of Fano resonances for bistable optical transfer through photonic crystal films, Phys. Rev. B., 2004, vol. 69, 155106 (11 p).
 Broer W., Hoenders B.J. Natural modes and resonances in a dispersive stratified Nlayer medium, J. Phys. A: Math. Theor., 2009, vol. 42, 245207 (18 p).
 Gain J., Sarkar M.D., Kundu S. Energy and effective mass dependence of electron tunnelling through multiple quantum barriers in different heterostructures, 2010, 8 p., arXiv: 1002.1931. http://arxiv.org/abs/1002.1931
 Pendry J.B. Low energy electron diffraction, London: Academic Press, 1974.
 Datta S. Kvantovyi transport: ot atoma k tranzistoru (Quantum transport: from the atom to the transistor), MoscowIzhevsk: Regular and Chaotic Dynamics, Institute of Computer Science, 2009, 532 p.
 Reed M., Simon B. Metody sovremennoi matematicheskoi fiziki. I. Funktsionalnyi analiz (Methods of modern mathematical physics, I. Functional analysis), Moscow: Mir, 1977, 360 p.
 Baranova L.Y., Chuburin Y.P. Quasilevels of the twoparticle discrete Schrödinger operator with a perturbed periodic potential, J. Phys. A.: Math. Theor., 2008, vol. 41, 435205 (11 p).
 Fadeev L.D., Yakubovskii О.А. Lektsii po kvantovoi mekhanike dlya studentovmatematikov (Lectures on quantum mechanics for students of mathematics), Leningrad: Leningrad State University, 1980, 200 p.
 Reed M., Simon B. Metody sovremennoi matematicheskoi fiziki. III. Teoriya rasseyaniya (Methods of modern mathematical physics, III. Scattering theory), Moscow: Mir, 1982, 446 p.
 Reed M., Simon B. Metody sovremennoi matematicheskoi fiziki. IV. Analiz operatorov (Methods of of modern mathematical physics, IV. Analysis of operators), Moscow: Mir, 1982, 428 p.
 Taylor J. Teoriya rasseyaniya. Kvantovaya teoriya nerelyativistskikh stolknovenii (Scattering theory: the quantum theory of nonrelativistic collisions), Moscow: Mir, 1975, 567 p.
 Tinyukova T.S. The LippmannSchwinger equation for quantum wires, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2011, no. 1, pp. 99104 (in Russian).
