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

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.