TY - JOUR

T1 - A simple proof of convergence to the Hartree dynamics in Sobolev trace norms

AU - Anapolitanos, Ioannis

AU - Hott, Michael

N1 - Publisher Copyright:
© 2016 Author(s).

PY - 2016/12/1

Y1 - 2016/12/1

N2 - The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the kth marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see the work of Michelangeli and Schlein [Commun. Math. Phys. 311(3), 645-687 (2012)] and Lührmann [J. Math. Phys. 53(2), 022105 (2012)]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional aN introduced by Pickl [Lett. Math. Phys. 97(2), 151-164 (2011)].

AB - The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the kth marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see the work of Michelangeli and Schlein [Commun. Math. Phys. 311(3), 645-687 (2012)] and Lührmann [J. Math. Phys. 53(2), 022105 (2012)]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional aN introduced by Pickl [Lett. Math. Phys. 97(2), 151-164 (2011)].

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U2 - 10.1063/1.4968820

DO - 10.1063/1.4968820

M3 - Article

AN - SCOPUS:85008486108

SN - 0022-2488

VL - 57

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 12

M1 - 122108

ER -