On Pairs of Difference Operators Satisfying: [P,Q] = Id

TL;DR

This paper investigates finite difference operators satisfying the Heisenberg commutation relation, classifying their types and constructing new examples, thereby deepening understanding of their algebraic and operator-theoretic properties.

Contribution

It classifies finite difference operators satisfying [P,Q]=Id into two types and constructs new examples, extending previous realizations and linking operator properties to their algebraic structure.

Findings

Central difference scheme generalization yields unitary equivalent representations

Two operator classes are distinguished based on self-adjointness properties

New classes of difference operators satisfying the commutation relation are constructed

Abstract

Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider $P$ and $X$ to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of $[P,X]=Id$.