Spectral asymptotics of periodic elliptic operators

Abstract.

We demonstrate that the structure of complex second-order strongly elliptic operators H on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$, $z\in{\bf T}^d$, of H with z-periodic boundary conditions acting on $L_2({\bf I}^d)$ where ${\bf I}=[0,1\rangle$. If the s emigroup S generated by H has a Hölder 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 ${\bf C}^d\backslash\{0\}$ 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.