05-20 Philippe Poulin

The Molchanov-Vainberg Laplacian (228K, Postscript) 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 $$<\delta_0|(\Delta-E-i0)^{-1}\delta_n> = O(|n|^{-(d-1)/2)})$$ fails for $0<|E|<2d-4$. 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, $$\Delta\psi(n) = 2^{-d}d!\sum_{|n-m|=\sqrt{d}}\psi(m)$$ and conjecture that the estimate $$<\delta_0|(\Delta-E-i0)^{-1}>=O(|n|^{-(d-1)/2})$$ holds for all $0<|E|<1$. In this paper we prove this conjecture.

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