 99455 Zhongwei Shen
 The Periodic Schrodinger Operators with Potentials in the
C.FeffermanPhong Class
(60K, amstex)
Nov 30, 99

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Abstract. We consider the periodic Schr\"odinger operator $\Delta +V(x)$
in $R^d$, $d\ge 3$ with potential $V$ in the
C.~FeffermanPhong
class. Let $\Omega$ be a periodic cell for $V$. We show that,
for $p\in((d1)/2, d/2]$, 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 dtorus.
It improves an early result by the author for potentials
in $L^{d/2}$ or weak$L^{d/2}$ space.
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