Zhongwei Shen
On Absolute Continuity of the Periodic Schrodinger Operators
(58K, AMS-TeX)

ABSTRACT.  This paper concerns the Schrodinger operator $-\Delta +V$ in 
$R^d$, $d\ge 3$, with periodic potential $V$. Under the assumption 
$V\in L^{d/2}_{loc} (R^d)$, it is shown that the spectrum of $-\Delta +V$ 
is purely absolutely continuous. The condition on the potential $V$ 
is optimal in the context of $L^p$ spaces. The proof relies on certain 
uniform Sobolev inequalities on the d-torus. We also establish the 
absolute continuity of $-\Delta +V$ with certain periodic potential 
$V$ in the weak-L^{d/2} space.