Fourier transform has been used to determine eigenstates with path integral method also in Journal of Mathematical Physics, 59(5), [052104] (pdf). Suppose we have a stationary system with Hamiltonian eigenstates $f_k$, $k \in \mathbb{N}$, and energies $E_k$. Let $$\Psi(x,0) = \sum_{k=0}^\infty c_k f_k(x)$$ and assume that $\Psi(x,0)$ is normed. Now $$\Psi(x,t) = \sum_k \exp(-i E_k t) c_k f_k(x)$$ Let $g(t) := \Psi(x_0,t)$ for some fixed point $x_0$. Now $$\hat{g}(\omega) = \sqrt{2\pi} \sum_k c_k f_k(x_0) \delta(\omega+E_k)$$ where the unitary definition of the Fourier transform is used. So in principle we should see the energies of the eigenstates from the Fourier spectrum of $g$. We can also get the coefficients $c_k$ and functions $f_k$ from the heights of the delta peaks in $\hat{g}(\omega)$ and normalization of the functions $f_k$.
So far I have tested that this method gives correct energies for the harmonic oscillator. Any comments about this method?
