Self-Fourier Transforms

Just spent a long, long time working with the quantum harmonic oscillator, with the equation of motion -- Schrodinger's time-independent equation:

\begin{eqnarray}
-\frac{\hbar^2}{2m}\nabla^2 \psi + \frac{m \omega^2 x^2}{2}\psi &=& E\psi
\end{eqnarray}

Typical ways to solve this involve power series or algebraic factoring of the equation of motion, but I was trying to formulate the problem as either a Sturm-Liouville differential equation or as an invertible equation of motion with well-defined Green's function.

From the Green's function perspective, we find that this equation under Fourier transform yields

\begin{eqnarray}
-\frac{\hbar^2}{2m}k^2 \overline{\psi} + \frac{m \omega^2}{2}\nabla_k^2 \overline{\psi} &=& E\overline{\psi}
\end{eqnarray}

Which leads to problems in defining $G(k)$, due to the derivative on the left hand side, but has the exact same structure -- aside from coefficients -- as the EOM in x-space. The equations of motion is "self-fourier" and so we might expect the solution $\psi$ to be self-fourier as well.

Well known self-fourier functions are a Gaussian and a dirac comb, but I came upon this interesting article that pointed out if one uses an even superposition of a function $g$ and it's transform $\overline{g}$, that superposition will be self-fourier. For example

\begin{eqnarray}
f &=& g(x) + g(-x) +  \overline{g(x)} + \overline{g(-x)} \\
\overline{f} &=&  \overline{g(k)} + \overline{g(-k)}+g(k) + g(-k)
\end{eqnarray}

Candidates would be

\begin{eqnarray}
f &=& 1 + \delta(x) \\
f &=& \Pi(x) + Sinc(x) \\
f &=& \Lambda (x) + Sinc(x)^2 \\
f &=& e^{|x|}+ \frac{1}{1+4\pi^2x^2}
\end{eqnarray}

Pretty cool, right? And the only sensible of these Self-fourier functions, in the context of our harmonic oscillator is the Gaussian.

There is a way to transform our equation of motion into a simple Sturm-Liouville, but it still escapes me...