I have a model Hamiltonian that looks like \begin{equation} H = H_{metal} + H_{particle}(t) + \sum_{k}V_{k}(t)a^{\dagger}c_{k} + c.c. \end{equation} where we suppressed the spin indices for simplicity. $a^{\dagger}$ and $c_{k}$ are the operators of the particle's and metal electrons with energies $\varepsilon_{a}(t)$ and $\varepsilon_{k}$, respectively. $V_{k}(t) = V_{k}u(t)$ is the electronic matrix elements describing the coupling between the electrons of the particle and the metal, and $u(t)$ is some function of time. In the wide-band limit, the coupling strength $V_{k}$ is related to the broadening of the particle's orbital level through \begin{equation} \Delta =\pi |V_{k}|^{2}\rho(\varepsilon_{k}) \end{equation} where we assumed that $\Delta$ is energy independent, and $\rho(\varepsilon_{k})$ is metal's density of states. The particle's orbital occupation can be derived using the equations of motion approach and other techniques resulting in
\begin{equation} \langle n_{a}(t) \rangle = \langle a^{\dagger}(t)a(t) \rangle =\frac{\Delta}{\pi} \int d\varepsilon f(\varepsilon)\left| \int_{t_{0}}^{t} dt'u(t')\exp\left\{-i\int_{t'}^{t}dt'' \left[\varepsilon_{a}(t'') -i\Delta(t'') -\varepsilon \right] \right\}\right|^{2} \end{equation} This expression can be evaluated numerically for any forms of $\varepsilon_{a}(t'')$ and the $\Delta(t'')$. I am interested in the (near) adiabatic limit in which $\varepsilon_{a}(t'')$ and the $\Delta(t'')$ do not deviate much from their adiabatic values so that the terms in the exponential can be simplified and the occupation can be evaluated as ($t_{0}=-\infty$) \begin{equation} \begin{aligned} \langle n_{a}^{0}(t) \rangle &= \frac{\Delta(t)}{\pi} \int d\varepsilon f(\varepsilon)\left| \int_{-\infty}^{t} dt' \exp \left\{-i \left[\varepsilon_{a}(t) -i\Delta(t) -\varepsilon \right](t-t') \right\} \right|^{2} \\ & = \frac{\Delta(t)}{\pi} \int d\varepsilon f(\varepsilon)\left| \frac{1}{ \Delta(t) -i[\varepsilon-\varepsilon_{a}(t)]} \right|^{2} \\ & =\int d\varepsilon f(\varepsilon)\frac{1}{\pi}\frac{\Delta(t)}{ \Delta^{2}(t) +[\varepsilon-\varepsilon_{a}(t)]^{2}} \\ & =\int d\varepsilon f(\varepsilon)\rho_{a}(\varepsilon,t), \end{aligned} \end{equation} which for me is really a transparent way to get the expression of adiabatic projected density of states $\rho_{a}(\varepsilon,t)$. Now here is the issue. I have come across some articles of people using the following expression for the adiabatic occupation probability (let's call it method X)
\begin{equation} \begin{aligned} \langle n_{a}^{0}(t) \rangle &= \frac{1}{\pi} \int d\varepsilon f(\varepsilon)\mathrm{Re}\left[ \int_{-\infty}^{t} dt' \exp \left\{-i \left[\varepsilon_{a}(t) -i\Delta(t) -\varepsilon \right](t-t') \right\} \right] \\ & = \frac{1}{\pi} \int d\varepsilon f(\varepsilon)\mathrm{Re}\left[ \frac{1}{ \Delta(t) -i[\varepsilon-\varepsilon_{a}(t)]} \right] \\ & =\int d\varepsilon f(\varepsilon)\frac{1}{\pi}\frac{\Delta(t)}{ \Delta^{2}(t) +[\varepsilon-\varepsilon_{a}(t)]^{2}} \\ & =\int d\varepsilon f(\varepsilon)\rho_{a}(\varepsilon,t), \end{aligned} \end{equation} which essentially gives the same result. My question is how to prove these two methods are equivalent? Or are they equivalent? I find that if we introduce some hypothetical variable in the integrand inside the exponential say $F(t)$,
\begin{equation} \begin{aligned} \langle n_{a}^{0}(t) \rangle &= \frac{\Delta(t)}{\pi} \int d\varepsilon f(\varepsilon)\left| \int_{-\infty}^{t} dt' \exp \left\{-i \left[\varepsilon_{a}(t) -i\Delta(t) -iF(t) -\varepsilon \right](t-t') \right\} \right|^{2} \\ & = \frac{\Delta(t)}{\pi} \int d\varepsilon f(\varepsilon)\left| \frac{1}{ \Delta(t) + F(t)-i[\varepsilon-\varepsilon_{a}(t)]} \right|^{2} \\ & =\int d\varepsilon f(\varepsilon)\frac{1}{\pi}\frac{\Delta(t)}{ [\Delta(t)+F(t)]^{2} +[\varepsilon-\varepsilon_{a}(t)]^{2}} \end{aligned} \end{equation}
while method X yields \begin{equation} \begin{aligned} \langle n_{a}^{0}(t) \rangle &= \frac{1}{\pi} \int d\varepsilon f(\varepsilon)\mathrm{Re}\left[ \int_{-\infty}^{t} dt' \exp \left\{-i \left[\varepsilon_{a}(t) -i\Delta(t) -iF(t) -\varepsilon \right](t-t') \right\} \right] \\ & = \frac{1}{\pi} \int d\varepsilon f(\varepsilon)\mathrm{Re}\left[ \frac{1}{ \Delta(t) +F(t) -i[\varepsilon-\varepsilon_{a}(t)]} \right] \\ & =\int d\varepsilon f(\varepsilon)\frac{1}{\pi}\frac{\Delta(t)+F(t)}{ [\Delta(t) +F(t)]^2 +[\varepsilon-\varepsilon_{a}(t)]^{2}} \end{aligned} \end{equation} which is clearly different from the other method. What am I missing?