Tight frames of exponentially decaying Wannier functions

Let L be a Schrödinger operator ( i ∂ ∂x −A(x))2+V (x) with periodic magnetic and electric potentials A,V , a Maxwell operator ∇× 1 ε(x)∇× in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R with coefficients periodic with respect to a lattice Γ. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We consider the old question of existence of a finite set of exponentially decaying Wannier functions wj(x) such that their Γ-shifts wj,γ(x) = wj(x− γ) for γ ∈ Γ span the whole spectral subspace corresponding to S in some “nice” manner. It is known that a topological obstruction might exist to finding exponentially decaying wj,γ that form an orthonormal basis of the spectral subspace. This obstruction has the form of non-triviality of certain finite dimensional (with the dimension equal to the number of spectral bands in S) analytic vector bundle that we denote ΛS . It was shown by G. Nenciu in 1983 that in the presence of time reversal symmetry, and if S is a single band, the bundle is trivial and thus the desired Wannier functions do exist. In 2007, G. Panati proved that in dimensions n ≤ 3, even if S consists of several spectral bands, the time reversal symmetry removes the obstruction as well, if one uses the so called composite Wannier functions. It has not been known what could be achieved when the bundle is non-trivial (which can happen in presence of magnetic fields). We show that it is always possible