Exact Finite Differences. The Derivative on Non Uniformly Spaced Partitions

Abstract

We define a finite-differences derivative operation, on a non uniformly spaced partition, which has the exponential function as an exact eigenvector. We discuss some properties of this operator and we propose a definition for the components of a finite-differences momentum operator. This allows us to perform exact discrete calculations.

Keywords

• discrete quantum mechanics
• self-adjoint operators
• exact finite differences
• momentum operator
• non equally spaced mesh points, translations