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VIDEO LIBRARY |
Complex Analysis and Related Topics (satelllite of ICM-2022)
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On effective quantum dynamics A. E. Teretenkov |
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Abstract: 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 $$\mathfrak{P}(\Phi) \equiv \lim\limits_{T\to\infty}\frac1T\int\limits_0^T dse^{-\mathscr{L}_0s}\Phi e^{\mathscr{L}_0s}.$$ 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 $$\frac{d}{dt}\mathfrak{P}(\Phi_{t ;\lambda}) =\mathscr{L}_{\mathrm{eff}}(t ;\lambda)\mathfrak{P}(\Phi_{t ;\lambda}).$$ Then for a fixed $$\mathscr{L}_{\mathrm{eff}}(t ;\lambda) =\mathscr{L}_0+\lambda\mathfrak{P}(\mathscr{L}_I ) +\lambda^2\Biggl(\mathfrak{P}\Biggl(\mathscr{L}_I\frac{e^{t [\mathscr{L}_0, \cdot ]}-1}{[\mathscr{L}_0, \cdot ]}\mathscr{L}_I\Biggr) -t (\mathfrak{P}(\mathscr{L}_I ))^2\Biggr)+O(\lambda^3)$$ as This work was funded by Russian Federation represented by the Ministry of Science and Higher Education (grant No. 075-15-2020-788). Language: English References
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