  We present a new method for calculating the Green functions for a lattice
scalar field theory in $D$ dimensions with arbitrary potential $V(\phi)$. The
method for non-perturbative evaluation of Green functions for $D \! = \! 1$ is
generalized to higher dimensions. We define ``hole functions''
$A^{(i)}~(i=0,1,2,\cdots,N \! -\! 1)$ from which one can construct $N$-point
Green functions. We derive characteristic equations of $A^{(i)}$ that form a
{\it finite closed} set of coupled local equations. It is shown that the Green
functions constructed from the solutions to the characteristic equations
satisfy the Dyson-Schwinger equations. To fix the boundary conditions of
$A^{(i)}$, a prescription is given for selecting the vacuum state at the
boundaries.
