The Molchanov–vainberg Laplacian

It is well known that the Green function of the standard discrete Laplacian on l2(Zd), ∆stψ(n) = (2d) −1 ∑ |n−m|=1 ψ(m), exhibits a pathological behavior in dimension d ≥ 3. In particular, the estimate 〈δ0|(∆st − E − i0)δn〉 = O(|n|− d−1 2 ) fails for 0 < |E| < 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, ∆ψ(n) = 2−d ∑ |n−m|= √ d ψ(m), and conjectured that the estimate 〈δ0|(∆− E − i0)δn〉 = O(|n|− d−1 2 ) holds for all 0 < |E| < 1. In this paper we prove this conjecture.