Localised module frames and Wannier bases from groupoid Morita equivalences

TL;DR

This paper develops a groupoid operator algebra framework to construct localized frames and Wannier bases for spectral subspaces, revealing topological obstructions and stability properties relevant to aperiodic materials.

Contribution

It introduces a novel approach connecting groupoid Morita equivalences with localized Wannier bases and topological invariants in spectral analysis.

Findings

Frames of translates are orthonormal bases iff the module is free.

Noncommutative Chern numbers obstruct fast-decaying Wannier bases.

Wannier bases stability under Delone set deformations.

Abstract

Following the operator algebraic approach to Gabor analysis, we construct frames of translates for the Hilbert space localisation of the Morita equivalence bimodule arising from a groupoid equivalence between Hausdorff groupoids, where one of the groupoids is \'{e}tale and with a compact unit space. For finitely generated and projective submodules, we show these frames are orthonormal bases if and only if the module is free. We then apply this result to the study of localised Wannier bases of spectral subspaces of Schr\"{o}dinger operators with atomic potentials supported on (aperiodic) Delone sets. The noncommutative Chern numbers provide a topological obstruction to fast-decaying Wannier bases and we show this result is stable under deformations of the underlying Delone set.