 06151 Philippe Poulin
 Green's Functions of Generalized Laplacians
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May 10, 06

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Abstract. The Green's function of any discretization, $\Delta$, of the Laplacian in dimension $d$ is given by
$$G(nm,z)=\int_{\mathbb{T}^d}\frac{e^{\mathrm{i}(nm)\cdot x}}{\Phi(x)z}\mathrm{d}x,$$
where $m,n\in\mathbb{Z}^d$, $z\in\mathbb{C}_+$, and $\Phi(x)$ is the symbol of $\Delta$. Using the stationary phase method we study the decay of $G(n,e+\mathrm{i}0)$ when $n\to\infty$ for values of energy, $e$, inside the range of $\Phi(x)$, where $\Phi(x)$ is analytic. We focus on two specific examples: the standard Laplacian and the MolchanovVainberg Laplacian.
This paper details the author's contribution to the MolchanovFest (Montreal, summer 2005).
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