Quantum dynamics in a thin film, I. Propagation of localized perturbations

• Published in: Russian journal of mathematical physics 2008-03, Vol.15 (1), p.1-16
• Main Authors: Brüning, J., Dobrokhotov, S. Yu, Nekrasov, R. V., Tudorovskiy, T. Ya
• Format: Article
• Language: English
• Subjects: Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical
• ISSN: 1061-9208; 1555-6638
• DOI: 10.1134/S1061920808010019

In the one-particle approximation, the quantum behavior of a (quasi-)particle is studied in a thin waveguide having the form of a thin curvilinear film (in three-dimensional space) placed in external magnetic and electric fields. Objects of this type arise in the actively developing physics of nano-structures and, in particular, in the theory of ballistic transport of electrons. The corresponding quantum-mechanical equation is a Pauli-type equation with nonrelativistic Rashba spin-orbital interaction for a two-dimensional vector function. Asymptotic solutions of the Cauchy problem with special localized initial data and those of the spectral problem are obtained. The construction of asymptotic solutions is carried out in two stages. At the first stage, in the framework of the adiabatic approximation, using the “operator separation of variables” (the “generalized adiabatic principle”) for a rather broad class of quantum states, the original three-dimensional equation is reduced to a two-dimensional surface (the limit film), and then diverse solutions of this reduced equation are constructed. The first part of the paper is devoted to the reduction and the solutions of the Cauchy problem. Spectral problems will be treated in the second part.