In this paper, we will obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on the three-dimensional torus ${\mathbb{T}}^{3}$ from the many-body limit of interacting bosonic systems. This type of result was previously obtained on ${\mathbb{R}}^{3}$ in the work of Erdős, Schlein, and Yau [54–57], and on ${\mathbb{T}}^{2}$ and ${\mathbb{R}}^{2}$ in the work of Kirkpatrick, Schlein, and Staffilani [78]. Our proof relies on an unconditional uniqueness result for the Gross–Pitaevskii hierarchy at the level of regularity $\alpha =1$, which is proved by using a modification of the techniques from the work of T. Chen, Hainzl, Pavlović and Seiringer [20] to the periodic setting. These techniques are based on the Quantum de Finetti theorem in the formulation of Ammari and Nier [6,7] and Lewin, Nam, and Rougerie [83]. In order to apply this approach in the periodic setting, we need to recall multilinear estimates obtained by Herr, Tataru, and Tzvetkov [74].Having proved the unconditional uniqueness result at the level of regularity $\alpha =1$, we will apply it in order to finish the derivation of the defocusing cubic nonlinear Schrödinger equation on ${\mathbb{T}}^{3}$, which was started in the work of Elgart, Erdős, Schlein, and Yau [50]. In the latter work, the authors obtain all the steps of Spohn's strategy for the derivation of the NLS [108], except for the final step of uniqueness. Additional arguments are necessary to show that the objects constructed in [50] satisfy the assumptions of the unconditional uniqueness theorem. Once we achieve this, we are able to prove the derivation result. In particular, we show Propagation of Chaos for the defocusing Gross–Pitaevskii hierarchy on ${\mathbb{T}}^{3}$ for suitably chosen initial data.

Keywords: Gross–Pitaevskii hierarchy, Nonlinear Schrödinger equation, Quantum de Finetti Theorem, Propagation of Chaos

@article{AIHPC_2015__32_6_1337_0, author = {Sohinger, Vedran}, title = {A rigorous derivation of the defocusing cubic nonlinear {Schr\"odinger} equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1337--1365}, publisher = {Elsevier}, volume = {32}, number = {6}, year = {2015}, doi = {10.1016/j.anihpc.2014.09.005}, mrnumber = {3425265}, zbl = {1328.35220}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.005/} }

TY - JOUR AU - Sohinger, Vedran TI - A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems JO - Annales de l'I.H.P. Analyse non linéaire PY - 2015 SP - 1337 EP - 1365 VL - 32 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.005/ DO - 10.1016/j.anihpc.2014.09.005 LA - en ID - AIHPC_2015__32_6_1337_0 ER -

%0 Journal Article %A Sohinger, Vedran %T A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems %J Annales de l'I.H.P. Analyse non linéaire %D 2015 %P 1337-1365 %V 32 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.005/ %R 10.1016/j.anihpc.2014.09.005 %G en %F AIHPC_2015__32_6_1337_0

Sohinger, Vedran. A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $ {\mathbb{T}}^{3}$ from the dynamics of many-body quantum systems. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 6, pp. 1337-1365. doi : 10.1016/j.anihpc.2014.09.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.09.005/

[1] Towards a rigorous derivation of the cubic nonlinear Schrödinger equation in dimension one, Asymptot. Anal. 40 no. 2 (2004), 93 -108 | MR | Zbl

, , , ,[2] Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 no. 6 (2007), 1193 -1220 | MR | Zbl

, , ,[3] Bose–Einstein quantum phase transition in an optical lattice model, Phys. Rev. A 70 (2004), 023612 | Zbl

, , , , ,[4] Bose–Einstein condensation as a quantum phase transition in an optical lattice, Mathematical Physics of Quantum Mechanics, Lecture Notes in Physics vol. 690 (2006), 199 -215 | MR | Zbl

, , , , ,[5] Systèmes hamiltoniens en théorie quantique des champs: dynamique asymptotique et limite classique, University of Rennes I (February 2013)

,[6] Mean field limit for bosons and infinite dimensional phase-space analysis, Ann. Henri Poincaré 9 (2008), 1503 -1574 | MR | Zbl

, ,[7] Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl. 95 (2011), 585 -626 | MR | Zbl

, ,[8] Rate of Convergence towards the Hartree-von Neumann limit in the mean-field regime, Lett. Math. Phys. 98 no. 1 (Oct. 2011), 1 -31 | MR

,[9] Observations of Bose–Einstein condensation in a dilute atomic vapor, Science 269 (1995), 198 -201

, , , , ,[10] More on convergence in unitary metric spaces, Proc. Am. Math. Soc. 83 (1981), 44 -48 | MR | Zbl

,[11] Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), 275 -293 | MR | Zbl

, , ,[12] Mean-field evolution of Fermionic systems, http://arxiv.org/abs/1305.2768 (2013) | MR

, , ,[13] Mean-field dynamics of Fermions with relativistic dispersion, http://arxiv.org/abs/1311.6270 (2013) | MR | Zbl

, , ,[14] Quantitative derivation of the Gross–Pitaevskii equation, http://arxiv.org/abs/1208.0373 (2012) | MR | Zbl

, , ,[15] Plancks Gesetz und Lichtquantenhypothese, Z. Phys. 26 (1924), 178 | JFM

,[16] Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations, Geom. Funct. Anal. 3 (1993), 107 -156 | EuDML | MR | Zbl

,[17] Periodic nonlinear Schrödinger equation and invariant measures, Commun. Math. Phys. 166 no. 1 (1994), 1 -26 | MR | Zbl

,[18] Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173 no. 3 (2008), 449 -475 | MR | Zbl

, ,[19] Rate of convergence towards Hartree dynamics, J. Stat. Phys. 144 no. 4 (2011), 872 -903 | MR | Zbl

, , ,[20] Unconditional uniqueness for the cubic Gross–Pitaevskii hierarchy via quantum de Finetti, http://arxiv.org/abs/1307.3168 (2013) | MR | Zbl

, , , ,[21] On the well-posedness and scattering for the Gross–Pitaevskii hierarchy via quantum de Finetti, Lett. Math. Phys. 104 no. 7 (2014), 871 -891 | MR | Zbl

, , , ,[22] On the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies, Discrete Contin. Dyn. Syst. 27 no. 2 (2010), 715 -739 | MR | Zbl

, ,[23] Recent results on the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies, Math. Model. Nat. Phenom. 5 no. 4 (2010), 54 -72 | EuDML | MR | Zbl

, ,[24] The quintic NLS as the mean field limit of a Boson gas with three-body interactions, J. Funct. Anal. 260 no. 4 (2011), 959 -997 | MR | Zbl

, ,[25] A new proof of existence of solutions for focusing and defocusing Gross–Pitaevskii hierarchies, Proc. Am. Math. Soc. 141 no. 1 (2013), 279 -293 | MR | Zbl

, ,[26] Higher order energy conservation and global well-posedness for Gross–Pitaevskii hierarchies, Commun. Partial Differ. Equ. 39 no. 9 (2014), 1597 -1634 | MR | Zbl

, ,[27] Derivation of the cubic NLS and Gross–Pitaevskii hierarchy from many-body dynamics in $d=2,3$ based on spacetime norms, Ann. Henri Poincaré 15 no. 3 (2014), 543 -588 | MR | Zbl

, ,[28] Energy conservation and blowup of solutions for focusing and defocusing Gross–Pitaevskii hierarchies, Ann. Inst. Henri Poincaré (C), Anal. Non Linéaire 27 no. 5 (2010), 1271 -1290 | Numdam | MR | Zbl

, , ,[29] Multilinear Morawetz identities for the Gross–Pitaevskii hierarchy, Recent Advances in Harmonic Analysis and Partial Differential Equations, Contemp. Math. vol. 581 , Amer. Math. Soc., Providence, RI (2012), 39 -62 | MR | Zbl

, , ,[30] Positive semidefiniteness and global well-posedness of solutions to the Gross–Pitaevskii hierarchy, http://arxiv.org/abs/1305.1404 (2013) | MR

, ,[31] The Grillakis–Machedon–Margetis second order corrections to mean field evolution for weakly interacting bosons in the case of 3-body interactions, http://arxiv.org/abs/0911.4153 (2009)

,[32] Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions, Arch. Ration. Mech. Anal. 203 (2012), 455 -497 | MR | Zbl

,[33] Collapsing estimates and the Rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl. (9) 98 no. 4 (2012), 450 -478 | MR | Zbl

,[34] On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Ration. Mech. Anal. 210 no. 2 (2013), 365 -408 | MR | Zbl

,[35] On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics, Arch. Ration. Mech. Anal. 210 no. 3 (2013), 909 -954 | MR | Zbl

, ,[36] On the Klainerman–Machedon conjecture of the quantum BBGKY hierarchy with self-interaction, J. Eur. Math. Soc. (2013) | EuDML | MR | Zbl

, ,[37] Focusing quantum many-body dynamics: the Rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation, http://arxiv.org/abs/1308.3895 (2013) | Zbl

, ,[38] Focusing quantum many-body dynamics II: the Rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D, http://arxiv.org/abs/1407.8457 (2014) | MR

, ,[39] Correlation structures, many-body scattering processes and the derivation of the Gross–Pitaevskii hierarchy, http://arxiv.org/abs/1409.1425 (2014) | MR

, ,[40] On the unconditional uniqueness of solutions to the infinite radial Chern–Simons–Schrödinger hierarchy, http://arxiv.org/abs/1406.2649 (2014) | MR | Zbl

, ,[41] On the Cauchy problem for Gross–Pitaevskii hierarchies, J. Math. Phys. 52 no. 3 (2011), 032103 | MR | Zbl

, ,[42] A Course in Operator Theory, Graduate Studies in Mathematics vol. 21 , American Mathematical Society, Providence, RI (1991) | MR

,[43] Bose–Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75 no. 22 (1995), 3969 -3973

, , , , , , ,[44] B. de Finetti, Funzione caratteristica di un fenomeno aleatorio, Mem. R. Accad. Lincei 4, 86–133.

[45] La pr'evision: ses lois logiques, ses sources subiectives, Ann. Inst. Henri Poincaré 7 no. 1 (1937) | EuDML | JFM | Numdam | MR

,[46] On the limits of sequences of normal states, Commun. Pure Appl. Math. 20 (1967), 413 -429 | MR

,[47] Finite exchangeable sequences, Ann. Probab. 8 (1980), 745 -764 | MR | Zbl

, ,[48] Classes of equivalent random quantities, Usp. Mat. Nauk 8 (1953), 125 -130 | MR

,[49] Quantentheorie des einatomigen idealen Gases, Sitz.ber. Preuss. Akad. Wiss. (Berl.), Phys.-math. Kl. 1 no. 3 (1925), 18 -25 | JFM

,[50] Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), 265 -283 | MR | Zbl

, , , ,[51] Mean field dynamics of boson stars, Commun. Pure Appl. Math. 60 no. 4 (2007), 500 -545 | MR | Zbl

, ,[52] Quantum dynamics with mean field interactions: a new approach, J. Stat. Phys. 134 no. 5 (2009), 859 -870 | MR | Zbl

, ,[53] Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate, Commun. Pure Appl. Math. 59 no. 12 (2006), 1659 -1741 | MR | Zbl

, , ,[54] Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 no. 3 (2007), 515 -614 | MR | Zbl

, , ,[55] Rigorous derivation of the Gross–Pitaevskii equation, Phys. Rev. Lett. 98 no. 4 (2007), 040404

, , ,[56] Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential, J. Am. Math. Soc. 22 no. 4 (2009), 1099 -1156 | MR | Zbl

, , ,[57] Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate, Ann. Math. (2) 172 no. 1 (2010), 291 -370 | MR | Zbl

, , ,[58] Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 no. 6 (2001), 1169 -1205 | MR | Zbl

, ,[59] Mean-field and classical limit of many body Schrödinger dynamics for bosons, Commun. Math. Phys. 271 no. 3 (2007), 681 -697 | MR | Zbl

, , ,[60] Atomism and quantization, J. Phys. A 40 no. 12 (2007), 3033 -3045 | MR | Zbl

, , ,[61] On the mean-field limit of bosons with Coulomb two-body interaction, Commun. Math. Phys. 288 no. 3 (2009), 1023 -1059 | MR | Zbl

, , ,[62] Mean-field limit of quantum Bose gases and nonlinear Hartree equation, Sem. É.D.P. 2003–2004, Exp. No. XIX, Sémin. Équ. Dériv. Partielles , École Polytech, Palaiseau (2004) | EuDML | Numdam | MR | Zbl

, ,[63] On a classical limit of quantum theory and the non-linear Hartree equation, Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud. vol. 21 , Kluwer Acad. Publ., Dordrecht (2000), 189 -207 | MR | Zbl

, , ,[64] On a classical limit of quantum theory and the non-linear Hartree equation, GAFA 2000, Tel Aviv, 1999, special volume, part I, Geom. Funct. Anal. (2000), 57 -78 | MR | Zbl

, , ,[65] On the point-particle (Newtonian) limit of the non-linear Hartree equation, Commun. Math. Phys. 225 no. 2 (2002), 223 -274 | MR | Zbl

, , ,[66] The classical field limit of scattering theory for nonrelativistic many-boson systems I, Commun. Math. Phys. 66 no. 1 (1979), 37 -76 | MR | Zbl

, ,[67] The classical field limit of scattering theory for nonrelativistic many-boson systems II, Commun. Math. Phys. 68 no. 1 (1979), 45 -68 | MR | Zbl

, ,[68] On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy, J. Funct. Anal. 266 no. 7 (2014), 4705 -4764 | MR | Zbl

, , ,[69] Pair excitations and the mean field approximation of interacting bosons, I, Commun. Math. Phys. 324 no. 2 (2013), 601 -636 | MR | Zbl

, ,[70] Second order corrections to mean field evolution of weakly interacting bosons, I, Commun. Math. Phys. 294 (2010), 273 -301 | MR | Zbl

, , ,[71] Second order corrections to mean field evolution of weakly interacting bosons, II, Adv. Math. 228 no. 3 (2011), 1788 -1815 | MR | Zbl

, , ,[72] Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961), 454 -466 | MR | Zbl

,[73] The classical limit for quantum mechanical correlation functions, Commun. Math. Phys. 35 (1974), 265 -277 | MR

,[74] Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in ${H}^{1}\left({\mathbb{T}}^{3}\right)$ , Duke Math. J. 159 no. 2 (2011), 329 -349 | Zbl

, , ,[75] Symmetric measures on Cartesian products, Trans. Am. Math. Soc. 80 (1955), 470 -501 | MR | Zbl

, ,[76] Unconditional uniqueness of the cubic Gross–Pitaevskii hierarchy with low regularity, http://arxiv.org/abs/1402.5347 (2014) | MR

, , ,[77] Locally normal symmetric states and an analogue of de Finetti's theorem, Z. Wahrscheinlichkeitstheor. Verw. Geb. 33 (1975/1976), 343 -351 | MR | Zbl

, ,[78] Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems: the periodic case, Am. J. Math. 133 no. 1 (2011), 91 -130 | MR | Zbl

, , ,[79] Space-time estimates for null forms and the local existence theorem, Commun. Pure Appl. Math. 46 (1993), 169 -177 | MR | Zbl

, ,[80] On the uniqueness of solutions to the Gross–Pitaevskii hierarchy, Commun. Math. Phys. 279 no. 1 (2008), 169 -185 | MR | Zbl

, ,[81] Mean-field dynamics: singular potentials and rate of convergence, Commun. Math. Phys. 298 no. 1 (2010), 101 -139 | MR | Zbl

, ,[82] Rate of convergence towards semi-relativistic Hartree dynamics, Ann. Henri Poincaré 14 no. 2 (2013), 313 -346 | MR | Zbl

,[83] Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math. 254 (2014), 570 -621 | MR | Zbl

, , ,[84] Remarks on the quantum de Finetti theorem for bosonic systems, http://arxiv.org/abs/1310.2200 (2013) | MR | Zbl

, , ,[85] The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases, http://arxiv.org/abs/1405.3220 (2014) | MR

, , ,[86] The Hartree equation for infinitely many particles. I. Well-posedness theory, Commun. Math. Phys. (2013) | MR

, ,[87] The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D, http://arxiv.org/abs/1310.0604 (2013) | MR

, ,[88] Proof of Bose–Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88 (2002), 170409-1 -170409-4

, ,[89] The quantum-mechanical many-body problem: the Bose gas, Perspectives in Analysis, Math. Phys. Stud. vol. 27 , Springer, Berlin (2005), 97 -183 | MR | Zbl

, , ,[90] The Mathematics of the Bose Gas and Its Condensation, Oberwolfach Seminars vol. 34 , Birkhäuser Verlag, Basel (2005) | MR | Zbl

, , , ,[91] Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional, Phys. Rev. A 61 (2000), 043602

, , ,[92] A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional Bose gas. Dedicated to Joel L. Lebowitz, Commun. Math. Phys. 224 no. 1 (2001), 17 -31 | MR | Zbl

, , ,[93] Mean-field quantum dynamics with magnetic fields, J. Math. Phys. 53 no. 2 (2012), 022105 | MR | Zbl

,[94] Equivalent definitions of asymptotic 100% BEC, Il Nuovo Cimento B 123 no. 2 (2008), 181 -192. | DOI

,[95] Dynamical collapse of boson stars, Commun. Math. Phys. 311 no. 3 (2012), 645 -687 | MR | Zbl

, ,[96] Derivation of the time dependent Gross–Pitaevskii equation with external fields, J. Stat. Phys. 140 no. 1 (2010), 76 -89 | MR | Zbl

,[97] A simple derivation of mean field limits for quantum systems, Lett. Math. Phys. 97 no. 2 (2011), 151 -164 | MR | Zbl

,[98] Vortex lines in an imperfect Bose gas, Sov. Phys. JETP 13 (1961), 451 -454

,[99] Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, San Diego, CA (1975) | MR

, ,[100] Normal and locally normal states, Commun. Math. Phys. 19 (1970), 219 -234 | MR | Zbl

,[101] Quantum fluctuations and rate of convergence towards mean field dynamics, Commun. Math. Phys. 291 no. 1 (2009), 31 -61 | MR | Zbl

, ,[102] Functional Analysis, McGraw–Hill Series in Higher Mathematics , McGraw–Hill Book Co., New York, NY (1991) | MR | Zbl

,[103] Derivation of effective evolution equations from microscopic quantum dynamics, , , , (ed.), Evolution Equations, Clay Mathematics Proceedings vol. 17 (2013) | MR

,[104] Convergence in trace ideals, Proc. Am. Math. Soc. 83 (1981), 39 -43 | MR | Zbl

,[105] Trace Ideals and Their Applications, Mathematical Surveys and Monographs vol. 120 , American Mathematical Society, Providence, RI (2005) | MR | Zbl

,[106] Local existence of solutions to randomized Gross–Pitaevskii hierarchies, Trans. Am. Math. Soc. (2014) | MR | Zbl

,[107] Randomization and the Gross–Pitaevskii hierarchy, http://arxiv.org/abs/1308.3714 (2013) | MR | Zbl

, ,[108] Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys. 52 no. 3 (1980), 569 -615 | MR

,[109] Symmetric states of infinite tensor products of ${C}^{\u204e}$ algebras, J. Funct. Anal. 3 (1969), 48 -68 | MR | Zbl

,[110] Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions, http://arxiv.org/abs/1401.6080 (2014) | MR | Zbl

,[111] Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity, http://arxiv.org/abs/1305.7240 (2013) | MR

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