Tight frames of exponentially decaying Wannier functions

Abstract

Let L be a Schroedinger operator with periodic magnetic and electric potentials, a Maxwell operator in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R^n with coefficients periodic with respect to a lattice G. 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 such that their G-shifts 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 Wannier functions 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. We show that it is always possible to find a finite number of exponentially decaying composite Wannier functions such that their G-shifts form a 1-tight frame in the spectral subspace. This appears to be the best, one can do when the topological obstruction is present. The number of the functions is the smallest dimension of a trivial bundle containing an equivalent copy of the "obstacle bundle". In particular, it coincides with the number of spectral bands if and only if the bundle is trivial, in which case an orthonormal basis of exponentially decaying composite Wannier functions is known to exist. An estimate on this number is provided.