A differential—Difference bicomplex

Abstract

We develop the method of difference jets on a multidimensional integer lattice. Based on this, we construct a lattice differential-difference bicomplex in the class of functions of a locally finite order and prove that it is acyclic.It is well known that theoretical and applied investigations broadly use variational calculus on Euclidean spaces based on the variational complex (see, e.g., [1]) that is in turn part of a fundamental object, the variational bicomplex (see, e.g., [2]-[4]). On the other hand, because of the rapid growth of computational power, lattice models are becoming increasingly more popular in theoretical and mathematical physics, as are difference equations in practical physical and engineering problems, which makes the development of variational techniques on a lattice quite relevant. In the framework of this program, we here construct a differential-difference bicomplex on a multidimensional integer lattice in the class of functions of a locally finite order and prove that the complex is acyclic, i.e., its rows and columns are all exact. For this, we introduce the notion of a difference jet and develop the corresponding formalism. In Sec. 2.5, we discuss the dependence of the form of the differential-difference bicomplex on the choice of the class of functions on the space of difference jets. We use the standard notation: R is the field of real numbers, Z = {0, ±1, ±2, . . . } is the set of integers, Z + = {0, 1, 2, . . . } is the set of nonnegative integers, and N = {1, 2, . . . } is the set of natural numbers. 1. Jet bundle over a lattice 1.1. The lattice. Let D ∈ N and D = {1, 2, . . . , D}. A lattice (more precisely, a D-dimensional integer lattice) is the set L = Z D . Two compatible algebraic structures are defined on L: an Abelian group (by addition) and a Z-module. As a Z-module, L has the basis {e 1 , . . . , e D }, where e 1 = (1, 0, . . . , 0), . . . , e D = (0, . . . , 0, 1).We set |i| = μ∈D |i μ | ∈ Z + for i = (i 1 , . . . , i D ) = μ∈D i μ e μ ∈ L and B p = {i ∈ L | |i| ≤ p} for p ∈ Z + .