Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators H on R-d with coefficients invariant under translation by Z(d) c an be analyzed through decomposition in terms of versions H-z, z is an elem ent of T-d, of H with z-periodic boundary conditions acting on L-2(I-d) whe re I = [0, 1). if the semigroup S generated by H has a Holder continuous in tegral kernel satisfying Gaussian bounds then the semigroups S-z generated by the H, have kernels with similar properties and z bar right arrow St ext ends to a function on C-d\{0} which is analytic with respect to the trace n orm. The sequence of semigroups S-(m),S-z obtained by rescaling the coeffic ients of H-z by c(x) --> c(mx) converges in trace norm to the semigroup (S) over cap(z) generated by the homogenization (H) over cap(z) of H-z. These convergence properties allow asymptotic analysis of the spectrum of H.