Author details

i.e.,

i.e.,

7. Conclusion

Hilbert space.

of complex numbers such that P

when its entries ai,j are such that ai,j <sup>¼</sup> <sup>a</sup><sup>∗</sup>

ð Þ FNð Þ Dg ð Þ¼ v ivð Þ F<sup>N</sup>þ<sup>1</sup>g ð Þv

Fourier transform of g, plus boundary terms.

we consider the equality

36 Matrix Theory-Applications and Theorems

�iqj

ð<sup>L</sup>=<sup>2</sup> �L=2 þ e �iv<sup>Δ</sup> e �iqj v gj N <sup>j</sup>¼�Nþ<sup>2</sup> þ

> d dv e �iqj

h vð Þ¼�

The integration of this equality with appropriate weights gives

F h<sup>0</sup> ð Þ<sup>j</sup> ¼ iqj

dv e�iqj v

ffiffiffi 2 p L e �iqj v gj

� � � �

" # (73)

N�1

j¼�Nþ1

: (74)

L=2

v¼�L=2

: (76)

, (75)

� �<sup>N</sup> i¼1

� � � �

Therefore, the discrete Fourier transform of the derivative of a vector g is iv times the discrete

The Fourier transform of the derivative of a continuous function of variable v is easily found if

<sup>v</sup> ¼ �iqj e �iqj v

dv e�iqj

1 L ffiffiffi 2 p e �iqj v h vð Þ � � � �

Hence, as is usual, the Fourier transform of the derivative of a function h vð Þ of continuous variable v is equal to iqj times the Fourier transform of the function, plus boundary terms.

We proceed with a brief discussion of the relationship between the derivative matrix D<sup>N</sup> and an important concept in quantum mechanics; the concept of self-adjoint operators [8, 9]. In particular, we focus on the momentum operator, whose continuous coordinate representation (operation) is given by �i d=dq, i.e., a derivative times �i, in the case of infinite-dimensional

In the finite-dimensional complex vectorial space (where each vector define a sequence gi

related to an approximation of the derivative (see Section 3) which uses second order finite

j,i (

<sup>i</sup> gi � � � � 2

differences. Therefore, we can ask if the matrix �iD<sup>N</sup> is also Hermitian.

<sup>v</sup> dh vð Þ dv <sup>þ</sup> <sup>e</sup>

�iqj v h vð Þ � � � � �

L=2

v¼�L=2

< ∞). A transformation A is usually called Hermitian,

<sup>∗</sup> denote the complex conjugate). Our matrix D<sup>N</sup> is

ð<sup>L</sup>=<sup>2</sup> �L=2

ð Þ F h <sup>j</sup> þ

Armando Martínez Pérez and Gabino Torres Vega\*

\*Address all correspondence to: gabino@fis.cinvestav.mx

Physics Department, Cinvestav, México City, México
