On the Phase-Space Distribution of Bloch Eigenmodes for Periodic Point Scatterers

TL;DR

This paper proves that for a 3D lattice of point scatterers, certain eigenfunctions become evenly spread in space while localized in momentum, revealing new insights into quantum behavior in periodic structures.

Contribution

It establishes the first 3D result showing eigenfunction equidistribution in position space and localization in momentum space for Floquet-Bloch modes.

Findings

Existence of a positive density subsequence of eigenvalues

Eigenfunctions exhibit equidistribution in position space

Eigenfunctions show localization in momentum space

Abstract

Consider the 3-dimensional Laplacian with a potential described by point scatterers placed on the integer lattice. We prove that for Floquet-Bloch modes with fixed quasi-momentum satisfying a certain Diophantine condition, there is a subsequence of eigenvalues of positive density whose eigenfunctions exhibit equidistribution in position space and localisation in momentum space. This result complements the result of Uebersch\"{a}r and Kurlberg who show momentum localisation for zero quasi-momentum in 2-dimensions, and is the first result in this direction in 3-dimensions.