Orthogonal localized wave functions of an electron in a magnetic field

We prove the existence of a set of two-scale magnetic Wannier orbitals, ${\mathrm{w}}_{\mathrm{mn}}$(r), in the infinite plane. The quantum numbers of these states are the positions (m,n) of their centers which form a von Neumann lattice. Function ${\mathrm{w}}_{00}$(r) localized at the origin has a nearly Gaussian shape of exp(-${\mathrm{r}}^2$/${4\mathrm{l}}^2$)/$\sqrt{2\pi }$ for r≲$\sqrt{2\pi }$l, where l is the magnetic length. This region makes a dominating contribution to the normalization integral. Outside this region function ${\mathrm{w}}_{00}$(r) is small, oscillates, and falls off with the Thouless critical exponent for magnetic orbitals, ${\mathrm{r}}^{\mathrm{-}2}$. These functions form a complete basis for many-electron problems.