Tight frames of exponentially decaying Wannier functions

Let L be a Schroedinger operator (1/i ∇ - A(x))2 + V(x) with periodic magnetic and electric potentials A, V, a Maxwell operator ∇ x 1/ε(x) ∇x in a periodic medium or an arbitrary self-adjoint elliptic linear partial differential operator in Rn with coefficients periodic with respect to a lattice Γ. Also let S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We consider the old question of the 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 element of 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 a certain finite-dimensional (with the dimension of the fiber equal to the number m of spectral bands in S) analytic vector bundle ΛS over the n-dimensional torus. It was shown by G Nenciu in 1983 that in the presence of time reversal symmetry (which implies the absence of magnetic fields), 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 occur, for instance, in the presence of magnetic fields or for Chern insulators). Let τ be the type of the bundle ΛS, i.e. the number of open sub-domains over which it is trivial (for the trivial bundle τ = 1, and τ never exceeds 2n, where n is the dimension of the coordinate space). We show that it is always possible to find a finite number l ≤ τm (and thus m ≤ l ≤ 2nm) of exponentially decaying composite Wannier functions wj such that their Γ-shifts form a 1-tight frame in the spectral subspace. Here the 1-tight frame is a redundant analog of an orthogonal basis, which appears in many applications, e.g. to signal processing and communication theory. This appears to be the best one can do when the topological obstruction is present. The number l is the smallest dimension of a trivial bundle containing an equivalent copy of ΛS. In particular, l = m if and only if ΛS is trivial, in which case an orthonormal basis of exponentially decaying composite Wannier functions is known to exist