Effect of magnetic field on electron spectrum in spherical nano-structures

The effect of a magnetic field on the energy spectrum and on the wave functions of an electron in spherical nano-structures such as single quantum dot and spherical layer is investigated. It is shown that the magnetic field removes the spectrum degeneration with respect to the magnetic quantum number. An increasing magnetic field induction entails a monotonous character of electron energy for the states with $m \geqslant 0$ and a non-monotonous one for the states with $m<0$. The electron wave functions of the ground state and several excited states are studied considering the effect of the magnetic field. It is shown that $1s$ and $1p$ states are degenerated in the spherical layer driven by a strong magnetic field. In the limit case, a series of size-quantized levels produce the Landau levels which are typical of bulk crystals.Comment: 8 pages, 4 figure

Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in $\mathbb{R}^n$. The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces.Comment: 27 page