99-189 Zhongwei Shen

On Absolute Continuity of the Periodic Schrodinger Operators (58K, AMS-TeX) May 20, 99

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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.

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