= Solution
By the two <Fourier transform> rules in part (b),
$$
\mathcal F[f'']=-k^2\widetilde f,\qquad\mathcal F[x^2f]=i^2\widetilde f''=-\widetilde f''.
$$
Consequently transforming the <differential equation> gives
$$
-k^2\widetilde f+\widetilde f''=\mu\widetilde f,
$$
or
$$
\boxed{\widetilde f''(k)-k^2\widetilde f(k)=\mu\widetilde f(k).}
$$
The inverse <Fourier transform> gives the reverse implication under the same regularity hypotheses. This expresses the Fourier symmetry of the <quantum harmonic oscillator> operator.
Back to article page
