$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Preliminary results
$$
[D_tD_t e^{i\omega t}]^n = -\frac{4}{\Delta t^2}\sin^2\left(
\frac{\omega\Delta t}{2}\right)e^{i\omega n\Delta t}
$$
By \( \omega\rightarrow k \),
\( t\rightarrow x \), \( n\rightarrow q \)) it follows that
$$
[D_xD_x e^{ikx}]_q = -\frac{4}{\Delta x^2}\sin^2\left(
\frac{k\Delta x}{2}\right)e^{ikq\Delta x}
$$
