The Molchanov–Vainberg Laplacian

Abstract:

It is well known that the Green function of the standard discrete Laplacian on $l^2(\mathbb {Z}^d)$, \[ \Delta _{st} \psi (n)=(2d)^{-1}\sum _{|n-m|=1}\psi (m), \] exhibits a pathological behavior in dimension $d \geq 3$. In particular, the estimate \begin{equation*} \langle \delta _0|(\Delta _{st}-E- \mathrm {i} 0)^{-1}\delta _n\rangle = O(|n|^{-\frac {d-1}{2}}) \end{equation*} 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, \begin{equation*} \Delta \psi (n) = 2^{-d} \sum _{|n-m|=\sqrt d}\psi (m), \end{equation*} and conjectured that the estimate \begin{equation*} \langle \delta _0|(\Delta -E-\mathrm {i} 0)^{-1}\delta _n \rangle = O(|n|^{-\frac {d-1}{2}}) \end{equation*} holds for all $0<|E|<1$. In this paper we prove this conjecture.