Spectral asymptotics of periodic elliptic operators

Research Report MRR97-002

Spectral asymptotics of periodic elliptic operators

Ola Bratteli, Palle E.T. Jørgensen and Derek W. Robinson

Abstract: We demonstrate that the structure of complex second-order strongly elliptic operators H on $\bold R^d$ with coefficients invariant under translation by $\bold Z^d$ can be analyzed through decomposition in terms of versions

, $z\in\bold T^d$ , of H with z-periodic boundary conditions acting on $L_2(\bold I^d)$ where $\bold I=[0,1\rangle$ . If the semigroup S generated by H has a Hölder continuous integral kernel satisfying Gaussian bounds then the semigroups

generated by the

have kernels with similar properties and $z\mapsto S^z$ extends to a function on $\bold C^d\backslash\{0\}$ which is analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$ obtained by rescaling the coefficients of

by $c(x)\to c(mx)$ converges in trace norm to the semigroup ${\widehat S}^z$ generated by the homogenization ${\widehat H}_z$ of

. These convergence properties allow asymptotic analysis of the spectrum of H.

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