The Gaussian beams summation method in the quantum problems of electronic motion in a magnetic field

Abstract

In this paper the Gaussian beams summation method, developed earlier for acoustic wave propagation, is generalized and applied to electron motion in a magnetic field and arbitrary potential in the case of shortwave approximation. It provides semiclassical uniform approximation for Green's function for stationary two-dimensional quantum problems. The approximation is valid near the caustics of an arbitrary geometrical structure and focal points. This approach is tested for two special cases of waveguide excitation by a point source for electron motion in a magnetic field with linear or parabolic potentials.

Localised States of Fabry-Perot Type in Graphene Nano-Ribbons

“…where e n (s) is the unit vector normal to the trajectory. Introducing ν = n/ √ h = O(1), we seek an asymptotic solution to the Dirac system related to (2) where S 0 (s) and S 1 (s) are chosen similar to [55], [66] as they give a linear approximation for solution to the Hamilton-Jacobi equation 55(see [55], [66])). The parameter for monolayers…”

Complex WKB Approximations in Graphene Electron-Hole Waveguides in Magnetic Field

“…The generalisation of the method of Gaussian beams summation for electron motion in magnetic field was done in (18), (19). In this section, the approximation of electron-hole Green's tensor near to caustics and focal points as an integral over Gaussian beams is described briefly.…”

“…In this section, the approximation of electron-hole Green's tensor near to caustics and focal points as an integral over Gaussian beams is described briefly. According to (17), (18), (19), and taking into account ray asymptotics of electron-hole Green's tensor (see (10)), the integral over all Gaussian beams irradiated from the point source x (0) is represented as follows…”

“…Application of this method to problems of electron motion in magnetic field in graphene required a generalization of the approach originally developed for acoustic wave propagation problems. The first step in this direction has been done in ( (18), (19)). The chapter is organized as follows.…”

“…This solution is singular at s = νπR, ν ∈ N (see (19)). The set of singular points are the circle with ν = 2n + 1 (see Fig.…”