# Expansion of the interaction term in the microscopic derivation of the Lindblad equation

I'm reading The Theory of Open Quantum Systems by Heinz-Peter Breuer and Francesco Petruccione, and in chapter 3, I can't understand the decomposition of the interaction Hamiltonian:

We assume that $$H=H_S + H_B + H_I$$, where we assume that $$H_I = \sum\limits_{\alpha}A_{\alpha}\otimes B_{\alpha} ~~~\textrm{(Eq. 3.119)},$$ where $$A_{\alpha}$$ acts on the system and $$B_{\alpha}$$ acts on the bath, only. If we assume that the system Hamiltonian, $$H_S$$ has a discrete energy spectra, then by the spectral theorem, we can write $$A_{\alpha} = \sum\limits_{\epsilon,\epsilon'} \langle \epsilon | A_{\alpha} | \epsilon '\rangle \cdot |\epsilon\rangle\langle \epsilon'| = \sum\limits_{\epsilon, \epsilon'}\Pi(\epsilon)A_{\alpha}\Pi(\epsilon').$$ However, in the book, Eq. 3.120 only considers $$\epsilon$$ and $$\epsilon'$$ values, for which $$\omega = \epsilon'-\epsilon$$ is fixed: $$A_{\alpha}(\omega) = \sum\limits_{\omega = \epsilon'-\epsilon}\Pi(\epsilon)A_{\alpha}\Pi(\epsilon').$$ What is the motivation for this expansion? Why are we restricting the values of $$\epsilon$$ and $$\epsilon'$$ to have a fixed difference?

All this is saying is that there are three important terms to consider: The Hamiltonian $$H^{\,}_S$$ for the system alone, the Hamiltonian $$H^{\,}_B$$ for the bath alone, and the terms $$H^{\,}_I$$ that couple the system and bath.
As far as the fixed energy difference, in the final expression for the Liouvillean / Lindbladian / master equation, you will sum over all energy differences $$\omega$$. So it's not that you're requiring a particular fixed energy difference. It's just that it's easier to collect terms based on how much energy is absorbed from / given to the bath.
I suggest reading ahead until you find the Lindblad equation with the sum over $$\omega$$. Things will probably be more clear there. For one thing, grouping terms by the amount of energy exchanged between the system and bath leads to a nicer form of the Lindblad/Liouville equation. You're basically working in the basis of the Hamiltonians for the system/bath individually, looking at the system-bath terms in this basis, regrouping them by the energy exchanged, and eventually, tracing out the bath. The point is that terms that exchange energy $$\omega$$ with the bath couple to one another in the Lindblad equation, and terms that exchange different amounts of energy are not coupled (to leading order).