I'm currently trying to understand Fermi's golden rule more rigorously.
The basic idea is that if $H$ is a self-adjoint operator and $|i\rangle, |f\rangle$ are states in the Hilbert space, then $|\langle f|e^{-iHt} |i\rangle|^2\propto e^{-\Gamma t}$ where $\Gamma$ is a constant decay rate. Now if you were to take Wiki's proof at face value and first consider finite-dimensional Hilbert spaces, you see that this only holds if $|i\rangle,|f\rangle$ have the same energies, i.e., they are eigenstates with respect to $H$ and have the same eigenvalues. Now unless you have degenerate energy levels, this formulation doesn't quite make sense, i.e., there's always a gap between energy levels in finite-dims. However, since in physics, we usually care about infinite-dim spaces such as $L^2(\mathbb{R})$, the spectrum of $H$ would decomposed into a continuous and a pure point part. In this case, Fermi's golden rule would sort of hold if the initial state and final states were in the continuous part of the Hilbert space (orthogonal to the eigenstates of $H$).
By the RAGE theorem (or usually the lemma leading to it), we know that if $|i\rangle \in \mathscr{H}^c$ (corresponding measure is continuous), then $$ \frac{1}{T} \int_0^T |\langle f|e^{-iHt}|i\rangle|^2\to0 $$ In fact, if $|i\rangle \in \mathscr{H}^{ac}$ (absolutely continuous), then $$ |\langle f|e^{-iHt}|i\rangle|^2\to0, \qquad t\to\infty $$ This seems very similar to Fermi's golden rule where we have an exponential decay rate. Therefore, I was wondering if there is a rigorous proof which shows that $|\langle f|e^{-iHt}|i\rangle|^2$ has an asymptotically exponential decay rate as $t\to \infty$, i.e., a rigorous statement of Fermi's golden rule?