  Let $L$ be a periodic self-adjoint linear elliptic operator in $\R^n$ with
coefficients periodic with respect to a lattice $\G$, e.g. Schr\"{o}dinger
operator $(i^{-1}\partial/\partial_x-A(x))^2+V(x)$ with periodic magnetic and
electric potentials $A,V$, or a Maxwell operator $\nabla\times\varepsilon
(x)^{-1}\nabla\times$ in a periodic medium. Let also $S$ be a finite part of
its spectrum separated by gaps from the rest of the spectrum. We address here
the question of existence of a finite set of exponentially decaying Wannier
functions $w_j(x)$ such that their $\G$-shifts $w_{j,\g}(x)=w_j(x-\g)$ for
$\g\in\G$ span the whole spectral subspace corresponding to $S$. It was shown
by D.~Thouless in 1984 that a topological obstruction sometimes exists to
finding exponentially decaying $w_{j,\g}$ that form an orthonormal (or any)
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 (Bloch bundle). It was shown in
2009 by one of the authors that it is always possible to find a finite number
$l$ of exponentially decaying Wannier functions $w_j$ such that their
$\G$-shifts form a tight (Parseval) frame in the spectral subspace. This
appears to be the best one can do when the topological obstruction is present.
Here we significantly improve the estimate on the number of extra Wannier
functions needed, showing that in physical dimensions the number $l$ can be
chosen equal to $m+1$, i.e. only one extra family of Wannier functions is
required. This is the lowest number possible in the presence of the topological
obstacle. The result for dimension four is also stated (without a proof), in
which case $m+2$ functions are needed.
  The main result of the paper was announced without a proof in Bull. AMS, July
2016.
