Opérateurs aux différences finies, calcul pseudo-différentiel et représentations des groupes de Lie

Abstract

We investigate the canonical commutation relations on a one-dimensional lattice. In the continuum, the commutation relations can be interpreted in terms of representations of the three-dimensional Heisenberg group. Here the Heisenberg group is replaced by a (variable) three-dimensional solvable Lie group with parameter hδ, where h is a constant, and δ is the distance between two neighbouring points in the lattice. We apply the pseudo-differential calculus developed in Manchon [Acta App. Math. 2 (1993) 159–183] to obtain a pseudo-differential calculus for finite difference operators. We also investigate the behaviour when δ → 0 (the continuum limit). It is relevant to consider an extension of degree 2 of the three-dimensional Heisenberg group rather than the Heisenberg group itself. By means of multiresolution analysis, we give a precise formulation of this fact by embedding square-integrable functions on the lattice into pairs of square-integrable functions on the real line. Then we investigate the behaviour in the continuum limit of pseudo-differential operators with symbols living on the dual of the Lie algebra.