In this paper, we prove a Szegö type limit theorem on ℓ 2 (Z d ). We take the self-adjoint operator H= - Δ + V on ℓ 2 (Z d ) , where (Δ u) (n) = ∑ | n - k | = 1 (u(k) - u(n)) and the operator V is the multiplication by a positive sequence { V(n) , n∈ Z d } with V(n) → ∞ as | n| → ∞. We take the orthogonal projection π λ onto the subspace, in ℓ 2 (Z d ) , spanned by eigenfunctions of H with eigenvalues ≤ λ. Let B be a zeroth order self-adjoint pseudo-difference operator associated with symbol b∈ S 1 , , ∞ (T d × Z d ). We then show for “nice functions” f, that limλ→∞Tr(f(πλBπλ))Tr(πλ)=limλ→∞1(2π)d∑V(n)≤λ∫Tdf(b(x,n))dx∑V(n)≤λ1. © 2018, Springer Nature Switzerland AG.