= Solution
Part (b) shows that the transform of $g_n$ is $q_n(k)e^{-k^2/2}$ with $q_n$ of degree $n$. Part (c) shows that it solves the same <eigenvalue> equation as $g_n$, with the same $\mu_n$. The assumed uniqueness of the polynomial solution of each degree therefore forces $q_n=c_np_n$; hence $\widetilde g_n=c_ng_n$.
The constants can also be determined. Substituting $g_n=p_ne^{-x^2/2}$ into the equation gives the <Hermite differential equation>
$$
p_n''-2xp_n'-(1+\mu_n)p_n=0.
$$
Comparison of the leading coefficient yields $\mu_n=-(2n+1)$. If the leading coefficient of $p_n$ is $a_n$, part (b) gives leading coefficient $\sqrt{2\pi}(-i)^na_n$ for $q_n$. Thus
$$
\boxed{\widetilde g_n(k)=\sqrt{2\pi}(-i)^ng_n(k),\qquad c_n=\sqrt{2\pi}(-i)^n.}
$$
Thus <Hermite functions are Fourier eigenfunctions> with the paper's unnormalized <Fourier transform>; dividing that transform by $\sqrt{2\pi}$ gives the unitary eigenvalues $(-i)^n$.
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