High-energy localized eigenstates of an electronic resonator in a magnetic field

We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called 'bouncing ball vibrations'. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrodinger equation (electronic Gaussian beam-wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr-Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.