Parseval Frames of Exponentially Localized Magnetic Wannier Functions

Tight frames of exponentially decaying Wannier functions

Let L be a Schrödinger operator ( i ∂ ∂x −A(x))2+V (x) with periodic magnetic and electric potentials A,V , a Maxwell operator ∇× 1 ε(x)∇× in a periodic medium, or an arbitrary self-adjoint elliptic linear partial differential operator in R with coefficients periodic with respect to a lattice Γ. Let also S be a finite part of its spectrum separated by gaps from the rest of the spectrum. We cons...

Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

Excess of Parseval Frames

The excess of a sequence in a Hilbert space H is the greatest number of elements that can be removed yet leave a set with the same closed span. This paper proves that if F is a frame for H and there exist infinitely many elements gn ∈ F such that F \ {gn} is complete for each individual n and if there is a uniform lower frame bound L for each frame F \ {gn}, then for each ε > 0 there exists an ...

Maximally localized Wannier functions for GW quasiparticles

We review the formalisms of the self-consistent GW approximation to many-body perturbation theory and of the generation of optimally localized Wannier functions from groups of energy bands. We show that the quasiparticle Bloch wave functions from such GW calculations can be used within this Wannier framework. These Wannier functions can be used to interpolate the many-body band structure from t...

Sub-exponentially Localized Kernels and Frames Induced by Orthogonal Expansions

The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions.