11:00 am Wednesday, April 2, 2014
Mathematical Physics Seminar : Wannier function frames in presence of non-trivial Bloch bundles by Peter Kuchment (Texas A & M University) in RLM 12.166
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. These sets of functions are extensively used for computations in solid state physics, photonic crystals, etc. However, it is known (Thouless, 1984) that a topological obstruction sometimes exists 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, in spite of this obstacle, it is always possible to find a finite number of exponentially decaying composite Wannier functions such that their G-shifts form a 1-tight (Parseval) 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 required 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. In particular, in physical dimensions only one additional function (and its shifts) is needed.
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