Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\mathbf{R}}^d$ with coefficients invariant under translation by ${\mathbf{Z}}^d$ can be analyzed through decomposition in terms of versions $H_z$, $z\in {\mathbf{T}}^d$, of $H$ with $z$-periodic boundary conditions acting on $L_2({\mathbf{I}}^d)$ where $\mathbf{I}=[0,1>$. If the semigroup $S$ generated by $H$ has a H"older continuous integral kernel satisfying Gaussian bounds then the semigroups $S^z$ generated by the $H_z$ have kernels with similar properties and $z\mapsto S^z$ extends to a function on ${\mathbf{C}}^d\setminus \{0\}$ which is analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$ obtained by rescaling the coefficients of $H_z$ by $c(x)\to c(mx)$ converges in trace norm to the semigroup ${\widehat{S}}^z$ generated by the homogenization ${\widehat{H}}_z$ of $H_z$. These convergence properties allow asymptotic analysis of the spectrum of $H$.