## Analysis & PDE

• Anal. PDE
• Volume 10, Number 3 (2017), 589-633.

### Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D

#### Abstract

We consider the focusing 3D quantum many-body dynamic which models a dilute Bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is attractive and given by $a3β−1V (aβ ⋅)$, where $∫ V ≤ 0$ and $a$ matches the Gross–Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D $N$-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates. We study the corresponding BBGKY hierarchy, which contains a diverging coefficient as the strength of the confining potential tends to $∞$. We prove that the limiting structure of the density matrices counterbalances this diverging coefficient. We establish the convergence of the BBGKY sequence and hence the propagation of chaos for the focusing quantum many-body system. We derive rigorously the 1D focusing cubic NLS as the mean-field limit of this 3D focusing quantum many-body dynamic and obtain the exact 3D-to-1D coupling constant.

#### Article information

Source
Anal. PDE, Volume 10, Number 3 (2017), 589-633.

Dates
Revised: 29 September 2016
Accepted: 14 January 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843450

Digital Object Identifier
doi:10.2140/apde.2017.10.589

Mathematical Reviews number (MathSciNet)
MR3641881

Zentralblatt MATH identifier
1368.35245

#### Citation

Chen, Xuwen; Holmer, Justin. Focusing quantum many-body dynamics, II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D. Anal. PDE 10 (2017), no. 3, 589--633. doi:10.2140/apde.2017.10.589. https://projecteuclid.org/euclid.apde/1510843450

#### References

• R. Adami, F. Golse, and A. Teta, “Rigorous derivation of the cubic NLS in dimension one”, J. Stat. Phys. 127:6 (2007), 1193–1220.
• Z. Ammari and F. Nier, “Mean field limit for bosons and infinite dimensional phase-space analysis”, Ann. Henri Poincaré 9:8 (2008), 1503–1574.
• Z. Ammari and F. Nier, “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states”, J. Math. Pures Appl. $(9)$ 95:6 (2011), 585–626.
• M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose–Einstein condensation in a dilute atomic vapor”, Science 269:5221 (1995), 198–201.
• W. Beckner, “Multilinear embedding-convolution estimates on smooth submanifolds”, Proc. Amer. Math. Soc. 142:4 (2014), 1217–1228.
• N. Benedikter, G. de Oliveira, and B. Schlein, “Quantitative derivation of the Gross–Pitaevskii equation”, Comm. Pure Appl. Math. 68:8 (2015), 1399–1482.
• L. Chen, J. O. Lee, and B. Schlein, “Rate of convergence towards Hartree dynamics”, J. Stat. Phys. 144:4 (2011), 872–903.
• T. Chen and N. Pavlović, “On the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies”, Discrete Contin. Dyn. Syst. 27:2 (2010), 715–739.
• T. Chen and N. Pavlović, “The quintic NLS as the mean field limit of a boson gas with three-body interactions”, J. Funct. Anal. 260:4 (2011), 959–997.
• 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. Henri Poincaré 15:3 (2014), 543–588.
• T. Chen and K. Taliaferro, “Derivation in strong topology and global well-posedness of solutions to the Gross–Pitaevskii hierarchy”, Comm. Partial Differential Equations 39:9 (2014), 1658–1693.
• T. Chen, N. Pavlović, and N. Tzirakis, “Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies”, Ann. Inst. H. Poincaré Anal. Non Linéaire 27:5 (2010), 1271–1290.
• T. Chen, N. Pavlović, and N. Tzirakis, “Multilinear Morawetz identities for the Gross–Pitaevskii hierarchy”, pp. 39–62 in Recent advances in harmonic analysis and partial differential equations, edited by A. R. Nahmod et al., Contemp. Math. 581, Amer. Math. Soc., Providence, RI, 2012.
• T. Chen, C. Hainzl, N. Pavlović, and R. Seiringer, “Unconditional uniqueness for the cubic Gross–Pitaevskii hierarchy via quantum de Finetti”, Comm. Pure Appl. Math. 68:10 (2015), 1845–1884.
• X. Chen, “Classical proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in $\mathbb {R}^{n+1}$ with application”, Differential Integral Equations 24:3–4 (2011), 209–230.
• X. Chen, “Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps”, J. Math. Pures Appl. $(9)$ 98:4 (2012), 450–478.
• X. Chen, “Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions”, Arch. Ration. Mech. Anal. 203:2 (2012), 455–497.
• X. Chen, “On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap”, Arch. Ration. Mech. Anal. 210:2 (2013), 365–408.
• X. Chen and J. Holmer, “On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics”, Arch. Ration. Mech. Anal. 210:3 (2013), 909–954.
• X. Chen and J. Holmer, “Correlation structures, many-body scattering processes, and the derivation of the Gross–Pitaevskii hierarchy”, Int. Math. Res. Not. 2016:10 (2016), 3051–3110.
• X. Chen and J. Holmer, “Focusing quantum many-body dynamics: the rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation”, Arch. Ration. Mech. Anal. 221:2 (2016), 631–676.
• X. Chen and J. Holmer, “On the Klainerman–Machedon conjecture for the quantum BBGKY hierarchy with self-interaction”, J. Eur. Math. Soc. 18:6 (2016), 1161–1200.
• \ChenX and J. Holmer, “The rigorous derivation of the 2D cubic focusing NLS from quantum many-body evolution”, Int. Math. Res. Not. 2016 (2016), [article id.] rnw113.
• X. Chen and P. Smith, “On the unconditional uniqueness of solutions to the infinite radial Chern–Simons–Schrödinger hierarchy”, Anal. PDE 7:7 (2014), 1683–1712.
• S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Stable $^{85}$Rb Bose–Einstein Condensates with Widely Turnable Interactions”, Phys. Rev. Lett. 85:9 (2000), 1795–1798.
• K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Bose–Einstein condensation in a gas of sodium atoms”, Phys. Rev. Lett. 75:22 (1995), 3969–3973.
• R. Desbuquois, L. Chomaz, T. Yefsah, J. Leonard, J. Beugnon, C. Weitenberg, and J. Dalibard, “Superfluid behaviour of a two-dimensional Bose gas”, Nat Phys 8:9 (2012), 645–648.
• E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, “Dynamics of collapsing and exploding Bose–Einstein condensates”, Nature 412:6844 (2001), 295–299.
• A. Elgart, L. Erdős, B. Schlein, and H.-T. Yau, “Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons”, Arch. Ration. Mech. Anal. 179:2 (2006), 265–283.
• L. Erdős and H.-T. Yau, “Derivation of the nonlinear Schrödinger equation from a many body Coulomb system”, Adv. Theor. Math. Phys. 5:6 (2001), 1169–1205.
• L. Erdős, B. Schlein, and H.-T. Yau, “Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate”, Comm. Pure Appl. Math. 59:12 (2006), 1659–1741.
• L. Erdős, B. Schlein, and H.-T. Yau, “Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems”, Invent. Math. 167:3 (2007), 515–614.
• L. Erdős, B. Schlein, and H.-T. Yau, “Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential”, J. Amer. Math. Soc. 22:4 (2009), 1099–1156.
• L. Erdős, B. Schlein, and H.-T. Yau, “Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate”, Ann. of Math. $(2)$ 172:1 (2010), 291–370.
• J. Fröhlich, A. Knowles, and S. Schwarz, “On the mean-field limit of bosons with Coulomb two-body interaction”, Comm. Math. Phys. 288:3 (2009), 1023–1059.
• A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, “Realization of Bose–Einstein condensates in lower dimensions”, Phys. Rev. Lett. 87:13 (2001), art. id. 130402, 4 pp.
• P. Gressman, V. Sohinger, and G. Staffilani, “On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy”, J. Funct. Anal. 266:7 (2014), 4705–4764.
• M. Grillakis and M. Machedon, “Pair excitations and the mean field approximation of interacting bosons, I”, Comm. Math. Phys. 324:2 (2013), 601–636.
• M. G. Grillakis and D. Margetis, “A priori estimates for many-body Hamiltonian evolution of interacting boson system”, J. Hyperbolic Differ. Equ. 5:4 (2008), 857–883.
• M. G. Grillakis, M. Machedon, and D. Margetis, “Second-order corrections to mean field evolution of weakly interacting bosons, I”, Comm. Math. Phys. 294:1 (2010), 273–301.
• M. Grillakis, M. Machedon, and D. Margetis, “Second-order corrections to mean field evolution of weakly interacting bosons, II”, Adv. Math. 228:3 (2011), 1788–1815.
• Z. Hadzibabic, P. Kruger, M. Cheneau, B. Battelier, and J. Dalibard, “Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas”, Nature 441:7097 (2006), 1118–1121.
• Y. Hong, K. Taliaferro, and Z. Xie, “Unconditional uniqueness of the cubic Gross–Pitaevskii hierarchy with low regularity”, SIAM J. Math. Anal. 47:5 (2015), 3314–3341.
• L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, “Formation of a matter-wave bright soliton”, Science 296:5571 (2002), 1290–1293.
• K. Kirkpatrick, B. Schlein, and G. Staffilani, “Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics”, Amer. J. Math. 133:1 (2011), 91–130.
• S. Klainerman and M. Machedon, “On the uniqueness of solutions to the Gross–Pitaevskii hierarchy”, Comm. Math. Phys. 279:1 (2008), 169–185.
• A. Knowles and P. Pickl, “Mean-field dynamics: singular potentials and rate of convergence”, Comm. Math. Phys. 298:1 (2010), 101–138.
• H. Koch and D. Tataru, “$L^p$ eigenfunction bounds for the Hermite operator”, Duke Math. J. 128:2 (2005), 369–392.
• 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.
• E. H. Lieb, R. Seiringer, and J. Yngvason, “One-dimensional behavior of dilute, trapped Bose gases”, Comm. Math. Phys. 244:2 (2004), 347–393.
• E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars 34, Birkhäuser, Basel, 2005.
• A. Michelangeli and B. Schlein, “Dynamical collapse of boson stars”, Comm. Math. Phys. 311:3 (2012), 645–687.
• P. Pickl, “A simple derivation of mean field limits for quantum systems”, Lett. Math. Phys. 97:2 (2011), 151–164.
• M. Reed and B. Simon, Methods of modern mathematical physics, IV: Analysis of operators, Academic Press, New York, 1978.
• I. Rodnianski and B. Schlein, “Quantum fluctuations and rate of convergence towards mean field dynamics”, Comm. Math. Phys. 291:1 (2009), 31–61.
• B. Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, 2005.
• V. Sohinger, “A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on $\mathbb{T}^3$ from the dynamics of many-body quantum systems”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32:6 (2015), 1337–1365.
• V. Sohinger and G. Staffilani, “Randomization and the Gross–Pitaevskii hierarchy”, Arch. Ration. Mech. Anal. 218:1 (2015), 417–485.
• H. Spohn, “Kinetic equations from Hamiltonian dynamics: Markovian limits”, Rev. Modern Phys. 52:3 (1980), 569–615.
• E. M. Stein and R. Shakarchi, Real analysis: measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis 3, Princeton University Press, 2005.
• S. Stock, Z. Hadzibabic, B. Battelier, M. Cheneau, and J. Dalibard, “Observation of phase defects in quasi-two-dimensional Bose–Einstein condensates”, Phys. Rev. Lett. 95 (2005), art. id. 190403, 4 pp.
• K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, “Formation and propagation of matter-wave soliton trains”, Nature 417:6885 (2002), 150–153.
• S. Thangavelu, Lectures on Hermite and Laguerre expansions, Mathematical Notes 42, Princeton University Press, 1993.