Zhongwei Shen
The Periodic Schrodinger Operators with Potentials in the 
C.Fefferman-Phong Class
(60K, amstex)

ABSTRACT.  We consider the periodic Schr\"odinger operator $-\Delta +V(x)$ 
in $R^d$, $d\ge 3$ with potential $V$ in the 
C.~Fefferman-Phong 
class. Let $\Omega$ be a periodic cell for $V$. We show that, 
for $p\in((d-1)/2, d/2]$, there exists a positive constant $\epsilon$ 
depending only on the shape of $\Omega$, $p$ and $d$ such that, 
if 
$$ 
\limsup_{r\to 0} 
\, \sup_{x\in \Omega} 
r^2\left\{\frac{1}{|B(x,r)|} 
\int_{B(x,r)} 
|V(y)|^p dy\right\}^{1/p} 
< \epsilon, 
$$ 
then the spectrum of $-\Delta +V$ is purely absolutely 
continuous. We obtain this result as a consequence 
of certain weighted $L^2$ 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.