The periodic Schrödinger operators with potentials in the Morrey class

We consider the periodic Schrödinger operator - Δ + V(x) in ℝ^d, d ≥ 3 with potential V in the Morrey class. Let Ω be a periodic cell for V. We show that, for p ∈ ((d - 1)/2,d/2], there exists a positive constant ε depending only on the shape of Ω, p and d such that, if lim sup sup r²/_{r→0 xεΩ} {1/ B(x,r) ∫_B(xr) V(y) ^pdy}^1/p <ε, then the spectrum of - Δ + V is purely absolutely continuous. We obtain this result as a consequence of certain weighted L² Sobolev inequalities on the d-torus. It improves an early result by the author for potentials in L^d/2 or weak-L^d/2 space.