4. The commutator between coordinate and derivative

Let us determine the commutator, from a local point of view first, between the coordinate—the points of the partition P Nð Þ—and the finite differences derivative. We begin with the derivative of q,

$$(\mathbf{D}q)\_j = \frac{q\_{j+1} - q\_{j-1}}{2\chi(v,\Delta)} = \frac{\Delta}{\chi(v,\Delta)} \approx 1 - \frac{v^2}{6}\Delta^2. \tag{51}$$

Hence, the finite differences derivative of the product qg qð Þ is

$$(\mathbf{D}\mathbf{q}\mathbf{g})\_{\circ} = q\_{\circ+1}(\mathbf{D}\mathbf{g})\_{\circ} + \mathbf{g}\_{\circ-1}(\mathbf{D}\mathbf{q})\_{\circ} = q\_{\circ+1}(\mathbf{D}\mathbf{g})\_{\circ} + \mathbf{g}\_{\circ-1}\frac{\Delta}{\chi(\upsilon,\Delta)}\tag{52}$$

i.e.,

expressions which are very similar to the continuous variable results. Again, these expressions coincide in the limit Δ ! 0, and they reduce to the corresponding expressions for continuous

The inverse operation to the finite differences derivative, at a given point, is the summation

gjþ<sup>1</sup> � gj�<sup>1</sup> � �

¼ g xð Þ� g að Þ. Note that the inverse at the local level is a bit different from the expressions obtained by means of the inverse matrix S (see below) of the derivative matrix D. When

Because the exponential function is an eigenfunction of the finite differences derivative and

2χð Þ v;Δ De

� ghqj�<sup>1</sup> � � � �

� ghqj�<sup>1</sup> � � � �

� h qj � � � �

χð Þ v;Δ

� h qj � � � �

2χð Þ v;Δ

<sup>χ</sup> <sup>v</sup>; h qjþ<sup>1</sup> � �

vqm � <sup>e</sup>

vq ð Þ<sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

vqn � <sup>e</sup>

j¼n e vqjþ<sup>1</sup> � <sup>e</sup> vqj�<sup>1</sup> ð Þ

vqn�<sup>1</sup> ,

<sup>2</sup><sup>χ</sup> <sup>v</sup>; h qjþ<sup>1</sup> � �

2χð Þ v; Δ

� h qj � � � �

<sup>¼</sup> gmþ<sup>1</sup> <sup>þ</sup> gm � gn � gn�<sup>1</sup>: (46)

<sup>a</sup> dy dg y ð Þ ð Þ=dy

(47)

(48)

<sup>a</sup> dx v evx <sup>¼</sup> <sup>e</sup>vx � <sup>e</sup>va. How-

variables.

with weights 2χð Þ v;Δ

30 Matrix Theory-Applications and Theorems

Xm j¼n

according to Eq. (46), we can say that

3.8. The chain rule

where

Xm j¼n

ð Þ Dghq ð Þ ð Þ <sup>j</sup> ¼

¼

3.6. The local inverse operation of the derivative

<sup>2</sup>χð Þ <sup>v</sup>;<sup>Δ</sup> ð Þ <sup>D</sup><sup>g</sup> <sup>j</sup> <sup>¼</sup> <sup>X</sup><sup>m</sup>

3.7. An eigenfunction of the summation operation

j¼n

dealing with matrices there are no boundary terms to worry about.

<sup>2</sup>χð Þ <sup>v</sup>; <sup>Δ</sup> v evqj <sup>¼</sup> <sup>X</sup><sup>m</sup>

ever, here, we have to deal with two values at each boundary.

The chain rule also has a finite differences version. That version is

ghqjþ<sup>1</sup> � � � �

ghqjþ<sup>1</sup> � � � �

¼ ð Þ Dg hð Þ <sup>j</sup>

<sup>2</sup><sup>χ</sup> <sup>v</sup>; h qjþ<sup>1</sup> � �

This equality is the equivalent to the usual result for continuous functions, Ð <sup>x</sup>

j¼n

vqmþ<sup>1</sup> <sup>þ</sup> <sup>e</sup>

¼ e

in agreement with the corresponding continuous variable equality Ð <sup>x</sup>

$$(\mathbf{D}\_{\varepsilon}q\mathbf{g})\_{j} - q\_{j+1}(\mathbf{D}\_{\varepsilon}\mathbf{g})\_{j} = \mathbf{g}\_{j-1} \frac{\Delta}{\chi(v,\Delta)}.\tag{53}$$

This is the finite differences version of the commutator between the coordinate q and the finite differences derivative D. This equality will become the identity operator in the small Δ limit, as expected. An equivalent expression is

$$(\mathrm{D}\mathbf{q}\mathbf{g})\_{\circ} - q\_{\circ - 1}(\mathrm{D}\mathbf{g})\_{\circ} = \mathbf{g}\_{\circ + 1} \frac{\Delta}{\chi(\upsilon, \Delta)}.\tag{54}$$

This is the finite differences version of the commutator between coordinate and derivative; the right hand side of this equality becomes gj in the small Δ limit, i.e., it becomes the identity operator.
