## Journal of Mathematics of Kyoto University

### 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.

#### Article information

Source
J. Math. Kyoto Univ., Volume 46, Number 4 (2006), 713-754.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250281601

Digital Object Identifier
doi:10.1215/kjm/1250281601

Mathematical Reviews number (MathSciNet)
MR2320348

Zentralblatt MATH identifier
1142.35013

#### Citation

Murata, Minoru; Tsuchida, Tetsuo. Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients. J. Math. Kyoto Univ. 46 (2006), no. 4, 713--754. doi:10.1215/kjm/1250281601. https://projecteuclid.org/euclid.kjm/1250281601