Anatomy of Maximally Localized Wannier Functions: A Multiresolution Analysis Perspective

This paper presents a deep dive into the anatomy of Wannier functions associated with phased-periodic boundary value problems involving Hermitian differential operators and generalizations thereof. Novel first principle relationships have been established between the theory of Wannier functions and the multiresolution analysis epitomized by wavelets and frames. Constructing envelop functions, the prospect of introducing Wannier-Hilbert transform and inverse transform has been indicated. The new findings are expressed in terms of three theorems. The insight presented in this paper alongside previous results of the authors (diagonalization of PDEs) promise wide-ranging applications in photonic, plasmonic, and quantum device applications.