Green's functions of generalized Laplacians

The Green's function of a typical discrete Laplacian, Delta, in dimension d is given by G(n - m,z) = integral(Td) e(i(n-m)center dot x)/Phi(x) - z dx, where m,n is an element of Z(d), z is an element of C+, and Phi(x) is the symbol of Delta. Using the stationary phase method we study the decay of G(n, e + i0) when |n| -> infinity for values of energy, e, inside the range of Phi(x), where Phi(x) is analytic. We focus on two specific examples: the standard Laplacian and the Molchanov-Vainberg Laplacian.