Abstract
The mean-field regime of bosonic systems provides one of the simplest time-dependent effective theory, the Hartree equation. We introduce the Hartree equation, notions of convergence based on reduced density matrices and review the literature. We review in some more detail proofs of the Hartree equation based on the BBGKY hierarchy.
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Notes
- 1.
First, by testing the difference \(N^{-1} \gamma ^{(1)}_{N,t} - |\varphi _t \rangle \langle \varphi _t|\) against \(|\varphi _t \rangle \langle \varphi _t|\), weak convergence implies Hilbert-Schmidt convergence. Then, since \(|\varphi _t \rangle \langle \varphi _t|\) is a rank-one projection, the operator \(N^{-1} \gamma ^{(1)}_{N,t} - |\varphi _t \rangle \langle \varphi _t|\) has exactly one negative eigenvalue. (If there were two linearly independent eigenvectors \(\xi _1,\xi _2\) with negative eigenvalue, one could find a linear combination \(\xi \) such that \(\langle \xi , \gamma ^{(1)}_{N,t}\xi \rangle < 0\).) Since \(\text{ Tr }\, \gamma ^{(1)}_{N,t} - |\varphi _t \rangle \langle \varphi _t| = 0\), its absolute value is equal to the sum of all positive eigenvalues, and therefore the trace norm is twice the operator norm.
- 2.
i.e. the equation for \(\widetilde{\gamma }^{(k)}_{N,t}\) depends on \(\widetilde{\gamma }^{(k+1)}_{N,t}\).
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Benedikter, N., Porta, M., Schlein, B. (2016). Mean-Field Regime for Bosonic Systems. In: Effective Evolution Equations from Quantum Dynamics. SpringerBriefs in Mathematical Physics, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-319-24898-1_2
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