Absolute Continuity of the Spectrum of a Periodic Schr\"odinger Operator

L. I. Danilov

Matematičeskie zametki, Tome 73 (2003) no. 1, pp. 49-62

Résumé

We prove the absolute continuity of the spectrum of the Schrödinger operator in $L^2(\mathbb R^n)$, $n\ge3$, with periodic (with a common period lattice $\Lambda$) scalar $V$ and vector $A\in C^1(\mathbb R^n,\mathbb R^n)$ potentials for which either $A\in H_{\operatorname{loc}}^q(\mathbb R^n;\mathbb R^n)$, $2q>n-2$, or the Fourier series of the vector potential $A$ converges absolutely, $V\in L_w^{p(n)}(K)$, where $K$ is an elementary cell of the lattice $\Lambda$, $p(n)=n/2$ for $n=3,4,5,6$, and $p(n)=n-3$ for $n\ge7$, and the value of $\lim_{t\to+\infty}\|\theta_tV\|_{L_w^{p(n)}(K)}$ is sufficiently small, where $\theta_t(x)=0$, if $|V(x)|\le t$ and $\theta_t(x)=1$ otherwise, $x\in K$ and $t>0$.