Derivation of the Quintic NLS from many-body quantum dynamics in T^2

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September 2015, 14(5): 1941-1960. doi: 10.3934/cpaa.2015.14.1941

Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$

Jianjun Yuan ^1,

1.	The College of Information and Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China

Received November 2014 Revised March 2015 Published June 2015

Abstract
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In this paper, we investigate the dynamics of a boson gas with three-body interactions in $T^2$. We prove that when the particle number $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.

Keywords: BBGKY hierarchy, Gross-Pitaevskii hierarchy, Schrödinger equation..

Mathematics Subject Classification: Primary: 35L15, 35L45; Secondary: 35Q4.

Citation: Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941

References:

[1]	R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrodinger equation in dimension one, Asymptot. Anal., 40 (2004), 93-108.
[2]	R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127 (2007), 1193-1220. doi: 10.1007/s10955-006-9271-z.
[3]	E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357. doi: 10.1215/S0012-7094-89-05915-2.
[4]	Thomas Chen and Natasa Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2010), 715-739. doi: 10.3934/dcds.2010.27.715.
[5]	Thomas Chen and Natasa Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.
[6]	Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies, Proc. Amer. Math. Soc., 141 (2013), 279-293. doi: 10.1090/S0002-9939-2012-11308-5.
[7]	T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. H. Poincare, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.
[8]	T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy, Commun. PDE, 39 (2014), 1658-1693. doi: 10.1080/03605302.2014.917380.
[9]	T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, CPAM, to appear, in arXiv:1307.3168.
[10]	X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions, Arch. Rational Mech. Anal., 203 (2012), 455-497. doi: 10.1007/s00205-011-0453-8.
[11]	X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478. doi: 10.1016/j.matpur.2012.02.003.
[12]	X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Rational Mech. Anal., 210 (2013), 365-408. doi: 10.1007/s00205-013-0645-5.
[13]	X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics, Arch. Rational Mech. Anal., 210 (2013), 909-954. doi: 10.1007/s00205-013-0667-z.
[14]	X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation, 41pp, arXiv:1308.3895.
[15]	X. Chen and J. Holmer, Focusing quantum many-body dynamics II: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3D, 48pp, arXiv:1407.8457.
[16]	X. Chen and J. Holmer, On the Klainerman-Machedon onjecture of the quantum BBGKY hierarchy with self-interaction, preprint, arxiv:1303.5385.
[17]	X. Chen and J. Holmer, Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy, 48pp, arXiv:1409.1425.
[18]	X. Chen and P. Smith, On the unconditional uniqueness of solutions to the infinite radial Chern- Simons-Schrödinger hierarchy, Analysis and PDE, 7 (2014), 1683-1712. doi: 10.2140/apde.2014.7.1683.
[19]	D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D, Discrete and Continuous Dynamical Systems-Series A, 19 (2007), 37-65. doi: 10.3934/dcds.2007.19.37.
[20]	A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Rat. Mech. Anal., 179 (2006), 265-283. doi: 10.1007/s00205-005-0388-z.
[21]	A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134.
[22]	L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.
[23]	L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.
[24]	L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), 1099-1156. doi: 10.1090/S0894-0347-09-00635-3.
[25]	P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal., 266 (2014), 4705-4764. doi: 10.1016/j.jfa.2014.02.006.
[26]	M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I, Commun. Math. Phys., 324 (2013), 601-636. doi: 10.1007/s00220-013-1818-7.
[27]	M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I, Commun. Math. Phys., 294 (2010), 273-301. doi: 10.1007/s00220-009-0933-y.
[28]	M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II, Adv. Math., 228 (2011) 1788-1815. doi: 10.1016/j.aim.2011.06.028.
[29]	Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity, 26pp, arXiv:1402.5347.
[30]	Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, American Journal of Mathematics, 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.
[31]	S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.
[32]	M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.
[33]	E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88 (2002), 170409, 1-4.
[34]	E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31. doi: 10.1007/s002200100533.
[35]	E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, 34 (2005), Oberwolfach Seminars Series, Birkhäuser.
[36]	I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys., 291 (2009), 31-61. doi: 10.1007/s00220-009-0867-4.
[37]	L. Pitaevskii, Vortex lines in an imperfect Bose-gas, Sov. phys. JETP, 13 (1961), 451-454.
[38]	P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97 (2011), 151-164. doi: 10.1007/s11005-011-0470-4.
[39]	H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569.
[40]	V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems, Ann. I.H.Poincaré-AN. doi: 10.1016/j.anihpc.2014.09.005.

show all references

References:

[1]	R. Adami, C. Bardos, F. Golse and A. Teta, Towards a rigorous derivation of the cubic nonlinear Schrodinger equation in dimension one, Asymptot. Anal., 40 (2004), 93-108.
[2]	R. Adami, F. Golse and A. Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys., 127 (2007), 1193-1220. doi: 10.1007/s10955-006-9271-z.
[3]	E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357. doi: 10.1215/S0012-7094-89-05915-2.
[4]	Thomas Chen and Natasa Pavlović, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2010), 715-739. doi: 10.3934/dcds.2010.27.715.
[5]	Thomas Chen and Natasa Pavlović, The quintic NLS as the mean field limit of a Boson gas with three-body interactions, J. Funct. Anal., 260 (2011), 959-997. doi: 10.1016/j.jfa.2010.11.003.
[6]	Thomas Chen and Natasa Pavlović, A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies, Proc. Amer. Math. Soc., 141 (2013), 279-293. doi: 10.1090/S0002-9939-2012-11308-5.
[7]	T. Chen and N. Pavlović, Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in $d = 3$ based on spacetime norms, Ann. H. Poincare, 15 (2014), 543-588. doi: 10.1007/s00023-013-0248-6.
[8]	T. Chen and K. Taliaferro, Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy, Commun. PDE, 39 (2014), 1658-1693. doi: 10.1080/03605302.2014.917380.
[9]	T. Chen, C. Hainzl, N. Pavlović and R. Seiringer, Unconditional uniqueness for the cubic Gross-Pitaevskii hierarchy via quantum de Finetti, CPAM, to appear, in arXiv:1307.3168.
[10]	X. Chen, Second order corrections to mean field evolution for weakly interacting Bosons in the case of three-body interactions, Arch. Rational Mech. Anal., 203 (2012), 455-497. doi: 10.1007/s00205-011-0453-8.
[11]	X. Chen, Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps, J. Math. Pures Appl., 98 (2012), 450-478. doi: 10.1016/j.matpur.2012.02.003.
[12]	X. Chen, On the rigorous derivation of the 3d cubic nonlinear Schrödinger equation with a quadratic trap, Arch. Rational Mech. Anal., 210 (2013), 365-408. doi: 10.1007/s00205-013-0645-5.
[13]	X. Chen and J. Holmer, On the rigorous derivation of the 2d cubic nonlinear Schrödinger equation from 3d quantum many-body dynamics, Arch. Rational Mech. Anal., 210 (2013), 909-954. doi: 10.1007/s00205-013-0667-z.
[14]	X. Chen and J. Holmer, Focusing quantum many-body dynamics: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation, 41pp, arXiv:1308.3895.
[15]	X. Chen and J. Holmer, Focusing quantum many-body dynamics II: The rigorous derivation of the 1d focusing cubic nonlinear Schrödinger equation from 3D, 48pp, arXiv:1407.8457.
[16]	X. Chen and J. Holmer, On the Klainerman-Machedon onjecture of the quantum BBGKY hierarchy with self-interaction, preprint, arxiv:1303.5385.
[17]	X. Chen and J. Holmer, Correlation structures, many-body scattering processes and the derivation of the Gross-Pitaevskii hierarchy, 48pp, arXiv:1409.1425.
[18]	X. Chen and P. Smith, On the unconditional uniqueness of solutions to the infinite radial Chern- Simons-Schrödinger hierarchy, Analysis and PDE, 7 (2014), 1683-1712. doi: 10.2140/apde.2014.7.1683.
[19]	D. De Silva, N. Pavlović, G. Staffilani and N. Tzirakis, Global well-posedness for a periodic nonlinear Schrodinger equation in 1D and 2D, Discrete and Continuous Dynamical Systems-Series A, 19 (2007), 37-65. doi: 10.3934/dcds.2007.19.37.
[20]	A. Elgart, L. Erdos, B. Schlein and H.-T. Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Rat. Mech. Anal., 179 (2006), 265-283. doi: 10.1007/s00205-005-0388-z.
[21]	A. Elgart and B. Schlein, Mean field dynamics of Boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545. doi: 10.1002/cpa.20134.
[22]	L. Erdős, B. Schlein and H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math., 167 (2007), 515-614. doi: 10.1007/s00222-006-0022-1.
[23]	L. Erdős, B. Schlein and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Ann. of Math., 172 (2010), 291-370. doi: 10.4007/annals.2010.172.291.
[24]	L. Erdős, B. Schlein and H.-T. Yau, Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., 22 (2009), 1099-1156. doi: 10.1090/S0894-0347-09-00635-3.
[25]	P. Gressman, V. Sohinger and G. Staffilani, On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy, J. Funct. Anal., 266 (2014), 4705-4764. doi: 10.1016/j.jfa.2014.02.006.
[26]	M. G. Grillakis and M. Machedon, Pair excitations and the mean .eld approximation of interacting Bosons, I, Commun. Math. Phys., 324 (2013), 601-636. doi: 10.1007/s00220-013-1818-7.
[27]	M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. I, Commun. Math. Phys., 294 (2010), 273-301. doi: 10.1007/s00220-009-0933-y.
[28]	M. G. Grillakis, M. Machedon and D. Margetis, Second order corrections to mean field evolution for weakly interacting Bosons. II, Adv. Math., 228 (2011) 1788-1815. doi: 10.1016/j.aim.2011.06.028.
[29]	Y. Hong, K. Taliaferro and Z. Xie, Unconditional uniqueness of the cubic Gross-Pitaevskii hierarchy with low regularity, 26pp, arXiv:1402.5347.
[30]	Kay Kirkpatrick, Benjamin Schlein and Gigliola Staffilani, Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics, American Journal of Mathematics, 133 (2011), 91-130. doi: 10.1353/ajm.2011.0004.
[31]	S. Klainerman and M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys., 279 (2008), 169-185. doi: 10.1007/s00220-008-0426-4.
[32]	M. Lewin, P. T. Nam and N. Rougerie, Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621. doi: 10.1016/j.aim.2013.12.010.
[33]	E. H. Lieb and R. Seiringer, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett., 88 (2002), 170409, 1-4.
[34]	E. H. Lieb and R. Seiringer and J. Yngvason, A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas, Comm. Math. Phys., 224 (2001), 17-31. doi: 10.1007/s002200100533.
[35]	E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and Its Condensation, 34 (2005), Oberwolfach Seminars Series, Birkhäuser.
[36]	I. Rodnianski and B. Schlein, Quantum fluctuations and rate of convergence towards mean field dynamics, Comm. Math. Phys., 291 (2009), 31-61. doi: 10.1007/s00220-009-0867-4.
[37]	L. Pitaevskii, Vortex lines in an imperfect Bose-gas, Sov. phys. JETP, 13 (1961), 451-454.
[38]	P. Pickl, A simple derivation of mean field limits for quantum systems, Lett. Math. Phys., 97 (2011), 151-164. doi: 10.1007/s11005-011-0470-4.
[39]	H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569.
[40]	V. Sohinger, A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbbT^3$ from the dynamics of many-body quantum systems, Ann. I.H.Poincaré-AN. doi: 10.1016/j.anihpc.2014.09.005.

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