My question stems from a derivation given in chapter 12 of R. McWeeny's Methods of Molecular Quantum Mechanics for solving linear response equations via variational perturbation theory. (Linear response correlation functions for a little more context).
My issue stems from a strange choice the author makes in defining the perturbation part of the Hamiltonian. They write the perturbation as $\mathbf{H}'(t)=F(t)\mathbf{A}$ where they describe $F(t)$ as a time dependent strength function and $A$ as some sort of shape function for the perturbation.
He then expresses the Hamiltonian using a Fourier Transform as $$\mathbf{H}'(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}f(\omega)\frac{1}{2}(\mathbf{A}_\omega e^{-i\omega t}+\mathbf{A}_{-\omega} e^{i\omega t}) d\omega$$ with $\mathbf{A}=\frac{1}{2}(\mathbf{A}_\omega+\mathbf{A}_{-\omega})$, assuming $\mathbf{A}_{-\omega}=\mathbf{A}^\dagger_\omega$, and $f(\omega)=f(-\omega)$ with $f(\omega)$ the Fourier transform of $F(t)$.
Now here is where I get confused. He goes on to define $\mathbf{H}'(\omega)=\frac{1}{2}(\mathbf{A}_\omega e^{-i\omega t}+\mathbf{A}_{-\omega}e^{i\omega t})$. To me the notation is already misleading enough, as it seems to imply that this is the Fourier Transform of $\mathbf{H}'$, but he also directly replaces instances of $\mathbf{H}'(t)$ with $\mathbf{H}'(\omega)$ to solve them. For instance, he gives an initial expression for the first order coefficients of the perturbed wavefunction as: $$c_n(t)=\frac{1}{i\hbar}\int_{-\infty}^t \left<n|\mathbf{A}|0\right> F(t')\exp(i\omega_{0n}t')dt'$$ Then on the next page, he gives another expression by substituting $\mathbf{H}'(\omega)$ (with a convergence factor $\epsilon$ also introduced): $$c_n(t)=-\frac{1}{2\hbar}\big[\frac{\left<n|\mathbf{A}|0\right>}{\omega_{0n}-\omega-i\epsilon}\exp(i(\omega_{0n}-\omega-i\epsilon))+(\omega\to -\omega)\big]$$
Which would be the expression obtained if $\mathbf{H}'(\omega)$ directly replaced $\mathbf{H}'(t)$. I had initially glossed over this section because it didn't have much bearing on the later sections, but this use of $\mathbf{H}'(\omega)$ has cropped up again and I'm still uncertain how/why it is used.
Is there an explanation for what the author is doing with this factor or whether there is some error in the formulas? My guess was that maybe the second expression for the coefficients should be a function of $\omega$ instead of time, but I can't confirm for certain if that makes sense.
In searching online, I found another use of this format, though it seems to possibly be derived from my source. Has anyone else encountered this before?