To describe the proximity in $\DZ^d$, we need the graph distance $\rtd(x,y)$ defined as the length of the shortest path from $x$ to $y$ formed by the lattice bonds.

Consider a lattice ball $
\ball_\ell(u) = \myset{x:\, \| x-u\| \leq \ell}.
$ Note that if $\|\cdot\|$ is the max-norm, then the ball of radius $\ell$ is actually a (lattice) cube of side length $2\ell$ with sides parallel to coordinate hyperplanes. Introduce the following boundaries relative to $\ball_\ell$: $$\bal
\pt^- \ball_\ell(u) &= \myset{x\in \DZ^d:\, d(x,u) = \ell} \\
\pt^+ \ball_\ell(u) &= \myset{x\in \DZ^d\setminus\ball_\ell(u):\, \rtd(x, \ball_\ell(u)) =1} \\
\pt \ball_\ell(u)   &= \myset{ (x,y)\in\pt^- \ball_\ell(u)\times\pt^+ \ball_\ell(u):\; \rtd(x,y)=1}.
\eal$$ Next, consider a pair of embedded balls $\ball_\ell(u)\subset\ball_L(w)$, $\ell<L$, and the complementary set $\ball^c_\ell(u) := \ball_L(w)\setminus \ball_\ell(u)$. Introduce LSOs $H_{\ball_L(w)}$, $H_{\ball_\ell(u)}$, $H_{\ball^c_\ell(u)}$ with Dirichlet boundary conditions and the resolvents $G_{\ball_L(w)}(E)$, $G_{\ball_\ell(u)}(E)$. We will denote by $G(x,y;E)$ the matrix elements of resolvents (Green functions) in the standard basis of functions $\delta_x:y\mapsto\delta_{xy}$. It follows from the second resolvent equation that, for any $x\in\ball_\ell(u)$, $y\in\ball^c_\ell(u)$ and any $E\not\in \Sigma(H_{\ball_L(w)})\cup \Sigma(H_{\ball_\ell(u)})$ \[eq:GRI.equal\] | G\_[\_L]{}(x,y;E) | & C\_\_[v \^[\^-]{}\_(x)]{} |G\_[\_(x)]{}(x,v;E) | \_[v’\^-\_(v)]{} |G\_[\_L]{}(v’, y; E) |\
& ( C\_\_[v: d(x,v)=]{} |G\_[\_(x)]{}(x,v;E) | ) \_[v: d(u,v)+1]{} |G\_[\_L]{}(v, y; E) | (the 