Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients

Abstract

We give the asymptotics of Green functions $G_{\lambda \pm i0}(x, y)$ as $|x-y| \to \infty$ for an elliptic operator with periodic coefficients on $\mathbf{R}^{d}$ in the case where $d \geq 2$ and the spectral parameter $\lambda$ is close to and greater than the bottom of the spectrum of the operator. The main tools are the Bloch representation of the resolvent and the stationary phase method. As a by-product, we also show directly the limiting absorption principle. In the one dimensional case, we show that Green functions are written as products of exponential functions and periodic functions for any $\lambda$ in the interior of the spectrum or the resolvent set.