99-455 Zhongwei Shen

The Periodic Schrodinger Operators with Potentials in the C.Fefferman-Phong 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.~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.

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