Partial quantum ergodicity for Dirichlet-truncated periodic Schrödinger operators with longer periods

Determine whether partial quantum ergodicity holds for Dirichlet-truncated periodic Schrödinger operators on ℓ^2(Z^d) whose potentials are periodic with respect to a full-rank lattice L having at least one period length q_l ≥ 3; specifically, ascertain whether for the Dirichlet restriction H_{Λ_N} of H = A_{Z^d} + V to Λ_N and for diagonal observables a_N whose block-averaged sums over each periodic block Λ_1 + k are independent of k, the quantum variance (1/|Λ_N|) Σ_{j=1}^{|Λ_N|} |⟨ψ_N^{(j)}, a_N ψ_N^{(j)}⟩ − ⟨a_N⟩|^2 converges to 0 as N → ∞, where {ψ_N^{(j)}} is an eigenbasis of H_{Λ_N}.

Background

The paper proves quantum ergodicity for Dirichlet truncations of the adjacency matrix on Z^d and establishes partial quantum ergodicity for Dirichlet-truncated periodic Schrödinger operators when all period lengths satisfy q_l ∈ {1,2}. The authors emphasize that extending the partial quantum ergodicity result to larger period lengths requires additional ideas.

This unresolved question concerns whether the delocalization property (in the averaged sense encoded by the vanishing quantum variance for block-averaged observables) persists for Dirichlet truncations when the periodicity in the potential is longer than two in at least one direction, a case not covered by the paper’s Theorem on partial quantum ergodicity.