Orthogonal localized wave functions of an electron in a magnetic field

We prove the existence of a set of two-scale magnetic Wannier orbitals, w{sub mn}({bold 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 w{sub 00}({bold r}) localized at the origin has a nearly Gaussian shape of exp({minus}r{sup 2}/4l{sup 2})/{radical}(2{pi}) for r{approx_lt}{radical}(2{pi})l, where l is the magnetic length. This region makes a dominating contribution to the normalization integral. Outside this region function w{sub 00}({bold r}) is small, oscillates, and falls off with the Thouless critical exponent for magnetic orbitals, r{sup {minus}2}. These functions form a complete basis for many-electron problems. {copyright} {ital 1997} {ital The American Physical Society}