On the spectrum of a class of second order periodic elliptic differential operators (Q1849470)

The author deals with the spectral properties of \[ L=-\sum^n_{p, l=1} {\partial\over \partial x_p} g_{pl}(x) {\partial\over \partial x_l}+ {1 \over i}\sum^n_{l=1} \left(a_l(x) {\partial\over \partial x_l}+ {\partial \over \partial x_l}a_l(x) \right)+V(x), \tag{1} \] where \(g_{pl}(x)= g_{lp}(x)\), \(p,l=1, \dots,n\), all the functions \(g_{pl}(x)\), \(a_l(x)\), and \(V(x)\) are smooth, real-valued, and \(2\pi\)-periodic in all variables. The differential equation (1) defines a selfadjoint operator in \(L^2(\mathbb{R}^n)\). The author assumes that \(L\) is invariant under the symmetry \(x_1\mapsto-x_1\). Under the assumptions that \(L\) is elliptic, \(g_{pl}(x)\), \(a_l(x)\), \(V(x)\) are smooth, real-valued, \(2\pi\)-periodic and obey a symmetry assumption, he proves that the spectrum of the operator \(L\) in \(L^2(\mathbb{R}^n)\) is purely absolutely continuous.