The Molchanov-Vainberg Laplacian

It is well known that the Green function of the standard discrete Laplacian on $l^2(\mathbb{Z}^d)$,

$\Delta_{\mathit{st}} \psi(n)=(2d)^{-1}\sum_{\vert n-m\vert=1}\psi(m),$

exhibits a pathological behavior in dimension $d \geq 3$. In particular, the estimate

$\langle \delta_0\vert(\Delta_{\mathit{st}}-E- \mathrm{i} 0)^{-1}\delta_n\rangle = O(\vert n\vert^{-\frac{d-1}{2}})$

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,

$\Delta \psi(n) = 2^{-d} \sum_{\vert n-m\vert=\sqrt d}\psi(m),$

and conjectured that the estimate

$\langle \delta_0\vert(\Delta-E-\mathrm{i} 0)^{-1}\delta_n \rangle = O(\vert n\vert^{-\frac{d-1}{2}})$

holds for all $0<\vert E\vert<1$. In this paper we prove this conjecture.