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

• Autores: Minoru Murata, Tetsuo Tsuchida
• Localización: Journal of mathematics of Kyoto University, ISSN 0023-608X, Vol. 46, Nº 4, 2006, págs. 713-754
• Idioma: inglés
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• Resumen

  • 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.