The Periodic Schrِdinger Operators with Potentials in the Morrey Class

We consider the periodic Schrödinger operator −Δ+V(x) in Rd, 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, iflim supr→0supx∈Ωr21∣B(x,r)∣∫B(x,r)∣V(y)∣pdy1/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 Ld/2 or weak-Ld/2 space.