Green’s formula for difference operators

Abstract

We construct an analogue of the classic Green's formula for linear partial differential operators for difference operators on a multidimensional integer lattice.Green's formula for the Laplace equationand its generalizations to other equations of classical mathematical physics are widely used in the theory of partial differential equations. Essentially, a similar Green's formula holds for any linear system of partial differential equations with variable coefficients (see, e.g., [1]). In the algebro-geometric approach (group analysis), it also allows a natural generalization to nonlinear systems of partial differential equations (see, e.g., [2], [3]). In what follows, we prove a similar formula for systems of difference equations on a multidimensional integer lattice. We introduce the following notation: 1. F = R, C is the number field, 2. Z = {0, ±1, ±2, . . . }, N = {1, 2, 3, . . . }, 3. A is a unital associative commutative F-algebra, and 4. M and N are A-modules. Integer latticeFor a given natural number D ∈ N, we set D = {1, 2, . . . , D}. A lattice (a D-dimensional integer lattice) is the set L = Z D . The set L carries two algebraic structures: the structures of an Abelian group with respect to addition (group of shifts)and of a free Z-module with λm = (λµ 1 , . . . , λµ D ) for all λ ∈ Z and all m = (µ 1 , . . . , µ D ) ∈ L with a basis e 1 = (1, 0, . . . , 0), . . . , e D = (0, . . . , 0, 1) such that m = α∈D µ α e α for all m = (