Maximally localized Wannier functions: existence and exponential localization

We describe recent results proved in [32] in collaboration with G. Panati, concerning a periodic Schrodinger operator and the maximally localized (composite) Wannier functions corresponding to a relevant family of its Bloch bands. More precisely, we discuss the minimization problem for the associated localization functional introduced in [22] and we review some rigorous results about the existence and exponential localization of its minimizers, in dimension d <= 3. The proof combines ideas and methods from the Calculus of Variations and the regularity theory for harmonic maps between Riemannian manifolds.