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Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$
1. | The College of Information and Technology, Nanjing University of Chinese Medicine, Nanjing 210046, China |
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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