Riv. Mat. Univ. Parma, Vol. 8, No. 1, 2017
Benjamin Schlein^{[a]}
Derivation of effective evolution equations from manybody quantum mechanics
Pages: ( in press )
Received: 31 December 2016
Accepted in revised form: 23 May 2017
Mathematics Subject Classification (2010): 82C10.
Keywords: Quantum dynamics, BoseEinstein condensation, GrossPitaevskii dynamics, HartreeFock equation.
Author address:
[a]: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
Abstract:
In these notes, based on a minicourse held at the summer school ''Methods and Models of Kinetic Theory''
that took place in Porto Ercole in June 2016, we review some of the recent developments in the derivation
of effective evolution equations starting from manybody quantum mechanics.
We discuss the derivation of the Hartree equation in the bosonic meanfield limit,
of the GrossPitaevskii equation describing the dynamics of initially trapped BoseEinstein
condensates and of the HartreeFock equation for fermions in a joint meanfield and semiclassical limit.
References
 [1]

R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one,
J. Stat. Phys. 127 (2007), 1193–1220.
MR2331036
 [2]
 Z. Ammari and F. Nier,
Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,
J. Math. Pures Appl. 95 (2011), no. 6, 585–626.
MR2802894
 [3]
 A. Athanassoulis, T. Paul, F. Pezzotti and M. Pulvirenti,
Strong semiclassical approximation of Wigner functions for the Hartree dynamics,
Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), no. 4, 525–552.
MR2904998
 [4]
 V. Bach, S. Breteaux, S. Petrat, P. Pickl and T. Tzaneteas,
Kinetic energy estimates for the accuracy of the timedependent HartreeFock approximation with Coulomb interaction,
J. Math. Pures Appl. 105 (2016), no. 1, 1–30.
MR3427937
 [5]
 C. Bardos, F. Golse, A. D. Gottlieb and N. J. Mauser,
Mean field dynamics of fermions and the timedependent HartreeFock equation,
J. Math. Pures Appl. (9) 82 (2003), no. 6, 665–683.
MR1996777
 [6]
 C. Bardos, F. Golse and N. J. Mauser,
Weak coupling limit of the \(N\)particle Schrödinger equation,
Methods Appl. Anal. 7 (2000), no. 2, 275–293.
MR1869286
 [7]
 N. Benedikter, G. de Oliveira and B. Schlein,
Quantitative derivation of the GrossPitaevskii equation, Comm. Pure Appl. Math. 68 (2015), no. 8, 1399–1482.
MR3366749
 [8]
 N. Benedikter, V. Jakšić, M. Porta, C. Saffirio and B. Schlein,
Meanfield evolution of fermionic mixed states, Comm. Pure Appl. Math. 69 (2016), n. 12, 2250–2303.
MR3570479
 [9]
 N. Benedikter, M. Porta, C. Saffirio and B. Schlein,
From the Hartree dynamics to the Vlasov equation,
Arch. Ration. Mech. Anal. 221 (2016), no. 1, 273–334.
MR3483896
 [10]
 N. Benedikter, M. Porta and B. Schlein,
Meanfield evolution of fermionic systems, Comm. Math. Phys 331 (2014), 1087–1131.
MR3248060
 [11]
 N. Benedikter, M. Porta and B. Schlein,
Meanfield dynamics of fermions with relativistic dispersion, J. Math. Phys. 55 (2014), 021901, 10 pp.
MR3202863
 [12]
 X. Chen and J. Holmer,
Focusing quantum manybody dynamics: the rigorous derivation of the \(1D\) focusing cubic nonlinear Schrödinger equation,
Arch. Ration. Mech. Anal. 221 (2016), no. 2, 631–676.
MR3488534
 [13]
 L. Chen, J. O. Lee and B. Schlein, Rate of convergence towards Hartree dynamics,
J. Stat. Phys. 144 (2011), no. 4, 872–903.
MR2826623
 [14]
 T. Chen, C. Hainzl, N. Pavlović and R. Seiringer,
Unconditional uniqueness for the cubic GrossPitaevskii hierarchy via quantum de Finetti,
Comm. Pure Appl. Math. 68 (2015), no. 10, 1845–1884.
MR3385343
 [15]
 T. Chen and N. Pavlović,
The quintic NLS as the meanfield limit of a boson gas with threebody interactions,
J. Funct. Anal. 260 (2011), no. 4, 959–997.
MR2747009
 [16]
 A. Elgart, L. Erdős, B. Schlein and H.T. Yau,
Nonlinear Hartree equation as the mean field limit of weakly coupled fermions,
J. Math. Pures Appl. (9) 83 (2004), no. 10, 1241–1273.
MR2092307
 [17]
 A. Elgart and B. Schlein,
Mean field dynamics for boson stars, Comm. Pure Appl. Math. 60 (2007), no. 4, 500–545.
MR2290709
 [18]
 L. Erdős, B. Schlein and H.T. Yau,
Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of manybody systems,
Invent. Math. 167 (2007), no. 3, 515–614.
MR2276262
 [19]
 L. Erdős, B. Schlein and H.T. Yau,
Derivation of the GrossPitaevskii equation for the dynamics of BoseEinstein condensate,
Ann. of Math. (2) 172 (2010), no. 1, 291–370.
MR2680421
 [20]
 L. Erdős, B. Schlein and H.T. Yau,
Rigorous derivation of the GrossPitaevskii equation with a large interaction potential,
J. Amer. Math. Soc. 22 (2009), 1099–1156.
MR2525781
 [21]
 L. Erdős and H.T. Yau,
Derivation of the nonlinear Schrödinger equation from a manybody Coulomb system,
Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205.
MR1926667
 [22]
 J. Fröhlich and A. Knowles,
A microscopic derivation of the timedependent HartreeFock equation with Coulomb twobody interaction,
J. Stat. Phys. 145 (2011), no. 1, 23–50.
MR2841931
 [23]
 J. Fröhlich, A. Knowles and A. Pizzo,
Atomism and quantization, J. Phys. A 40 (2007), no. 12, 3033–3045.
MR2313859
 [24]
 J. Fröhlich, A. Knowles and S. Schwarz,
On the meanfield limit of bosons with Coulomb twobody interaction,
Comm. Math. Phys. 288 (2009), no. 3, 1023–1059.
MR2504864
 [25]
 F. Golse, C. Mouhot and T. Paul,
On the mean field and classical limits of quantum mechanics, Comm. Math. Phys. 343 (2016), 165–205.
MR3475664
 [26]
 P. Grech and R. Seiringer, The excitation spectrum for weakly interacting bosons in a trap,
Comm. Math. Phys. 322 (2013), no. 2, 559–591.
MR3077925
 [27]
 J. Ginibre and G. Velo,
The classical field limit of scattering theory for nonrelativistic manyboson systems, I and II,
Comm. Math. Phys. 66 (1979), no. 1, 37–76 and 68 (1979), no. 1, 45–68.
MR0530915,
MR0539736
 [28]
 K. Hepp,
The classical limit for quantum mechanical correlation functions,
Comm. Math. Phys. 35 (1974), 265–277.
MR0332046
 [29]
 K. Kirkpatrick, B. Schlein and G. Staffilani,
Derivation of the two dimensional nonlinear Schrödinger equation from manybody quantum dynamics,
Amer. J. Math. 133 (2011), no. 1, 91–130.
MR2752936
 [30]
 S. Klainerman and M. Machedon,
On the uniqueness of solutions to the GrossPitaevskii hierarchy,
Comm. Math. Phys. 279 (2008), no. 1, 169–185.
MR2377632
 [31]
 A. Knowles and P. Pickl,
Meanfield dynamics: singular potentials and rate of convergence,
Comm. Math. Phys. 298 (2010), no. 1, 101–138.
MR2657816
 [32]
 E. H. Lieb and R. Seiringer,
Proof of BoseEinstein condensation for dilute trapped gases,
Phys. Rev. Lett. 88 (2002), 170409.
DOI: https://doi.org/10.1103/PhysRevLett.88.170409
 [33]
 E. H. Lieb, R. Seiringer and J. Yngvason,
Bosons in a trap: A rigorous derivation of the GrossPitaevskii energy functional,
Phys. Rev. A 61 (2000), 043602.
DOI: https://doi.org/10.1103/PhysRevA.61.043602
 [34]
 P.L. Lions and T. Paul,
Sur les mesures de Wigner (French), Rev. Mat. Iberoamericana 9 (1993), 553–618.
MR1251718
 [35]
 H. Narnhofer and G. L. Sewell,
Vlasov hydrodynamics of a quantum mechanical model,
Comm. Math. Phys. 79 (1981), no. 1, 9–24.
MR0609224
 [36]
 S. Petrat and P. Pickl,
A new method and a new scaling for deriving fermionic meanfield dynamics,
Math. Phys. Anal. Geom. 19 (2016), no. 1, Art. 3, 51 pp.
MR3461406
 [37]
 P. Pickl,
Derivation of the time dependent GrossPitaevskii equation with external fields,
Rev. Math. Phys. 27 (2015), no. 1, 1550003, 45 pp.
MR3317556
 [38]
 M. Porta, S. Rademacher, C. Saffirio and B. Schlein,
Meanfield evolution of fermions with Coulomb interaction,
J. Stat. Phys. 166 (2017), no. 6, 1345–1364.
MR3612230
 [39]
 I. Rodnianski and B. Schlein,
Quantum fluctuations and rate of convergence towards mean field dynamics,
Comm. Math. Phys. 291 (2009), no. 1, 31–61.
MR2530155
 [40]
 H. Spohn,
Kinetic equations from Hamiltonian dynamics: Markovian limits,
Rev. Modern Phys. 52 (1980), no. 3, 569–615.
MR0578142
 [41]
 H. Spohn,
On the Vlasov hierarchy,
Math. Methods Appl. Sci. 3 (1981), no. 4, 445–455.
MR0657065
Home Riv.Mat.Univ.Parma