 0520 Philippe Poulin
 The MolchanovVainberg Laplacian
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Jan 14, 05

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Abstract. It is well known that the Green function of the standard discrete Laplacian on a lattice exhibits a pathological behavior in dimension $d>2$. In particular, the estimate
$$<\delta_0(\DeltaEi0)^{1}\delta_n> = O(n^{(d1)/2)})$$
fails for $0<E<2d4$. This fact complicates the study of the scattering theory of discrete Schrodinger operators. Molchanov and Vainberg suggested the following alternative to the standard discrete Laplacian,
$$\Delta\psi(n) = 2^{d}d!\sum_{nm=\sqrt{d}}\psi(m)$$
and conjecture that the estimate
$$<\delta_0(\DeltaEi0)^{1}>=O(n^{(d1)/2})$$
holds for all $0<E<1$. In this paper we prove this conjecture.
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