Absolute continuity of periodic Schrödinger operators with potentials in the Kato class (Q5954306)

Let \(V\) be a real-valued measurable function on \({\mathbb R}^d, d \geq 2\). \(V\) is said to belong to the Kato class \(K_d\) if NEWLINE\[NEWLINE\lim_{r\to 0}\sup_{x\in{\mathbb R}^d}\int_{|x-y|<r}{|V(y)|\over{|x-y|^{d-2}}} dy = 0,\quad d\geq 3,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\lim_{r\to 0}\sup_{x\in{\mathbb R}^d}\int_{|x-y|<r} |V(y)|\ln\{|x-y|^{-1}\} dy = 0, \quad d=2.NEWLINE\]NEWLINE Let \(A=(a_{ij})_{d\times d}\) be a symmetric positive definite matrix with real constant entries. It is proved that for a periodic potential \(V\in K_d, d=2,3\) the spectrum of the operator \(\sum_{i,j}D_ja_{ij}D_i + V\) is absolutely continuous.