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The Molchanov-Vainberg Laplacian

Authors:	Philippe Poulin

Institution:	Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada H3A-2K6

Abstract:	It is well known that the Green function of the standard discrete Laplacian on $l^2(\mathbb{Z}^d)$ , $\displaystyle \Delta_{\mathit{st}} \psi(n)=(2d)^{-1}\sum_{\vert n-m\vert=1}\psi(m),$ exhibits a pathological behavior in dimension $d \geq 3$ . In particular, the estimate $\displaystyle \langle \delta_0\vert(\Delta_{\mathit{st}}-E- \mathrm{i} 0)^{-1}\delta_n\rangle = O(\vert n\vert^{-\frac{d-1}{2}})$ fails for $0 <\vert E\vert<1-2/d$ . This fact complicates the study of the scattering theory of discrete Schrödinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian, $\displaystyle \Delta \psi(n) = 2^{-d} \sum_{\vert n-m\vert=\sqrt d}\psi(m),$ and conjectured that the estimate $\displaystyle \langle \delta_0\vert(\Delta-E-\mathrm{i} 0)^{-1}\delta_n \rangle = O(\vert n\vert^{-\frac{d-1}{2}})$ holds for all $0<\vert E\vert<1$ . In this paper we prove this conjecture.

Keywords:	Discrete Laplacian Green function

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