Generalized Wannier Functions

TL;DR

This paper constructs exponentially localized basis functions for the invariant space of Schrödinger operators with spectral gaps, providing estimates on decay rates and geometric insights into the localization surfaces.

Contribution

It introduces a new method to build localized basis functions for spectral subspaces of Schrödinger operators with gaps, including decay estimates and geometric analysis.

Findings

Constructed orthonormal basis functions localized around surfaces.

Derived estimates on exponential decay rates.

Analyzed the geometry of localization surfaces.

Abstract

We consider single particle Schrodinger operators with a gap in the en ergy spectrum. We construct a complete, orthonormal basis function set for the inv ariant space corresponding to the spectrum below the spectral gap, which are exponentially localized a round a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential dec ay rate and a discussion of the geometry of these surfaces is included.