Infinite degeneracy of Landau levels from the Euclidean symmetry with central extension revisited

The one dimensional Hydrogen atom revisited

The one dimensional Schroedinger hydrogen atom is an interesting mathematical and physical problem to study bound states, eigenfunctions and quantum degeneracy issues. This 1D physical system gave rise to some intriguing controversy over more than four decades. Presently, still no definite consensus seems to have been reached. We reanalyzed this apparently controversial problem, approaching it from a Fourier transform representation method combined with some fundamental (basic) ideas found in self-adjoint extensions of symmetric operators. In disagreement with some previous claims, we found th…

Relativistic corrections to Landau levels in the presence of a parallel linear electric field

We consider an electron moving under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. Two frequencies characterize the system: the cyclotron frequency $ω_c$ corresponding to motion along the x-y plane and associated with the usual Landau levels, and a second frequency $ω_z$ corresponding to motion along the z-direction. In previous work, the non-relativistic energies of this system were obtained, and it was shown that an extra degeneracy (beyond the Landau degeneracy) occurs…

New degeneracies and modification of Landau levels in the presence of a parallel linear electric field

We consider a three-dimensional system where an electron moves under a constant magnetic field (in the z-direction) and a \textit{linear} electric field parallel to the magnetic field above the z=0 plane and anti-parallel below the plane. The linear electric field leads to harmonic oscillations along the z-direction. There are therefore two frequencies characterizing the system: the usual cyclotron frequency $ω_c$ corresponding to motion along the x-y plane and associated with Landau levels and a second frequency $ω_z$ for motion along the z-direction. Most importantly, when the ratio $W=ω_c/ω…

Landau Levels in the Presence of Topological Defects

We study the Landau levels associated with electrons moving in a magnetic field in the presence of a continuous distribution of disclinations, a magnetic screw dislocation and a dispiration. We focus on the influence of these topological defects on the spectrum of the electron(or hole) in a magnetic field in the framework of the geometric theory of defects in solids of Katanaev and Volovich. The presence of the defects breaks the degeneracy of the Landau levels in different ways depending on the defect. Exact expressions for energies and eigenfuctions are found for all cases.

Landau levels in a gravitational field: The Levi-Civita and Kerr spacetimes case

We have recently found that the gravitational field of a static spherical mass removes the Landau degeneracy of the energy levels of a particle moving around the mass inside a magnetic field by splitting the energy of the Landau orbitals. In this paper we present the second part of our investigation of the effect of gravity on Landau levels. We examine the effect of the gravitational fields created by an infinitely long massive cylinder and a rotating spherical mass. In both cases, we show that the degeneracy is again removed thanks to the splitting of the particle's orbitals. The first case w…

二维费米子平均场极限下的多重朗道能级填充

Multiple Landau level filling for a mean field limit of 2D fermions

本文研究大磁场下有界区域内二维相互作用费米子的多重朗道能级填充问题，推导出最后一个部分填充能级的平均场与半经典极限描述。