On effective quantum dynamics

By effective dynamics we mean the dynamics which arises after averaging with respect to a free Hamiltonian. Such averaging leads to effective evolution generators, which can be obtained by some generalization of projective methods. A thermodynamical analog of such an approach was developed in [1], and a particular case of quadratic Hamiltonians was discussed in [2]. Here we consider a general case of bounded generators.
Let $\mathscr{L}_0$ and $\Phi$ be bounded operators in a Banach space, then let us define the averaging map $\mathfrak{P}$ as
$$\mathfrak{P}(\Phi) \equiv \lim\limits_{T\to\infty}\frac1T\int\limits_0^T dse^{-\mathscr{L}_0s}\Phi e^{\mathscr{L}_0s}.$$
We have developed a systematic perturbative expansion for an effective generator, first terms of which can be defined by the following theorem.
Theorem 1. Let $\Phi_{t ;\lambda}$ be a semi-group with a bounded generator of the form $\mathscr{L}_0 +\lambda\mathscr{L}_I$ . Let the effective generator $\mathscr{L}_{\mathrm{eff}}(t ;\lambda)$ be defined by
$$\frac{d}{dt}\mathfrak{P}(\Phi_{t ;\lambda}) =\mathscr{L}_{\mathrm{eff}}(t ;\lambda)\mathfrak{P}(\Phi_{t ;\lambda}).$$

Then for a fixed $t$
$$\mathscr{L}_{\mathrm{eff}}(t ;\lambda) =\mathscr{L}_0+\lambda\mathfrak{P}(\mathscr{L}_I ) +\lambda^2\Biggl(\mathfrak{P}\Biggl(\mathscr{L}_I\frac{e^{t [\mathscr{L}_0, \cdot ]}-1}{[\mathscr{L}_0, \cdot ]}\mathscr{L}_I\Biggr) -t (\mathfrak{P}(\mathscr{L}_I ))^2\Biggr)+O(\lambda^3)$$
as $\lambda\to 0$.
This work was funded by Russian Federation represented by the Ministry of Science and Higher Education (grant No. 075-15-2020-788).