Đạo Hàm Định Lượng và Tán Xạ của Phương Trình NLS Bậc Ba 3D Trong Không Gian Năng Lượng

Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127, 1194–1220 (2007) Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor. Science 269, 198–201 (1995) Bourgain, J.: Scattering in the energy space and below for 3D NLS. J. Anal. Math. 75, 267–297 (1998) Bourgain, J.: New Global Well-posedness Results for Non-linear Schrödinger Equations, AMS Publications, (1999) Benedikter, N., Oliveira, G., Schlein, B.: Quantitative Derivation of the Gross-Pitaevskii Equation. Comm. Pure. Appl. Math. 68, 1399–1482 (2015) Boccato, C., Cenatiempo, S., Schlein, B.: Quantum Many-body Fluctuations around Nonlinear Schrödinger Dynamics. Ann. Henri Poincaré 18, 113–191 (2017) Brennecke, C., Schlein, B.: Gross-Pitaevskii Dynamics for Bose-Einstein Condensates. Analysis & PDE 12, 1513–1596 (2019) Chen, T., Hainzl, C., Pavlović, N., Seiringer, R.: Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti. Commun. Pure Appl. Math. 68, 1845–1884 (2015) Chen, T., Pavlović, N.: On the Cauchy Problem for Focusing and Defocusing Gross-Pitaevskii Hierarchies. Discrete Contin. Dyn. Syst. 27, 715–739 (2010) Chen, T., Pavlović, N.: The Quintic NLS as the Mean Field Limit of a Boson Gas with Three-Body Interactions. J. Funct. Anal. 260, 959–997 (2011) Chen, T., Pavlović, N.: A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies. Proc. Amer. Math. Soc. 141, 279–293 (2013) Chen, T., Pavlović, N.: Higher order energy conservation and global wellposedness of solutions for Gross-Pitaevskii hierarchies. Commun. PDE 39, 1597–1634 (2014) Chen, T., Pavlović, N.: Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in \(d=3\) based on spacetime norms. Ann. H. Poincare 15, 543–588 (2014) Chen, T., Taliaferro, K.: Derivation in Strong Topology and Global Well-posedness of Solutions to the Gross-Pitaevskii Hierarchy. Commun. PDE 39, 1658–1693 (2014) Chen, T., Pavlović, N., Tzirakis, N.: Energy Conservation and Blowup of Solutions for Focusing Gross-Pitaevskii Hierarchies. Ann. I. H. Poincaré 27, 1271–1290 (2010) Chen, X.: Classical Proofs Of Kato Type Smoothing Estimates for The Schrödinger Equation with Quadratic Potential in \( R^{n+1}\) with Application. Differential and Integral Equations 24, 209–230 (2011) Chen, X.: Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons in the Case of Three-body Interactions. Arch. Rational Mech. Anal. 203, 455–497 (2012) Chen, X.: Collapsing Estimates and the Rigorous Derivation of the 2d Cubic Nonlinear Schrödinger Equation with Anisotropic Switchable Quadratic Traps. J. Math. Pures Appl. 98, 450–478 (2012) Chen, X.: On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with A Quadratic Trap. Arch. Rational Mech. Anal. 210, 365–408 (2013) Chen, X., Holmer, J.: On the Rigorous Derivation of the 2D Cubic Nonlinear Schrödinger Equation from 3D Quantum Many-Body Dynamics. Arch. Rational Mech. Anal. 210, 909–954 (2013) Chen, X., Holmer, J.: On the Klainerman-Machedon Conjecture of the Quantum BBGKY Hierarchy with Self-interaction. J. Eur. Math. Soc. (JEMS) 18, 1161–1200 (2016) Chen, X., Holmer, J.: Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation. Arch. Rational Mech. Anal. 221, 631–676 (2016) Chen, X., Holmer, J.: Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D. Analysis & PDE 10, 589–633 (2017) Chen, X., Holmer, J.: Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy, Int. Math. Res. Notices 2016, 3051-3110 Chen, X., Holmer, J.: The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution, Int. Math. Res. Notices 2017, 4173–4216 Chen, X., Holmer, J.: The Derivation of the Energy-critical NLS from Quantum Many-body Dynamics. Invent. Math. 217, 433–547 (2019) Chen, X., Holmer, J.: The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on \({\mathbb{T}}^{4}\). Forum Math. Pi 10(e3), 1–49 (2022) Chen, X., Shen, S., Zhang, Z.: The Unconditonal Uniqueness for \(H^{1}\) -Supercritical NLS, to appear in Annals of PDE. (arXiv:2104.06592) Chen, X., Shen, S., Wu, J., Zhang, Z.: The Derivation of the Compressible Euler Equation from Quantum Many-body Dynamics, arXiv:2112.14897, 48pp Chen, X., Smith, P.: On the Unconditional Uniqueness of Solutions to the Infinite Radial Chern-Simons-Schrödinger Hierarchy. Analysis & PDE 7, 1683–1712 (2014) Chong, J.: Dynamics of Large Boson Systems with Attractive Interaction and a Derivation of the Cubic Focusing NLS in \( {\mathbb{R}}^{3}\). J. Math. Phys. 62, 042106 (2021) Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on \({\mathbb{R}}^{3}\). Comm. Pure Appl. Math. 57, 987–1014 (2004) Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn, D.M., Ketterle, W.: Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969–3973 (1995) Duyckaerts, T., Holmer, J., Roudenko, S.: Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math. Res. Lett. 15, 1233–1250 (2008) Erdös, L., Schlein, B., Yau, H.T.: Derivation of the Cubic non-linear Schrödinger Equation from Quantum Dynamics of Many-body Systems. Invent. Math. 167, 515–614 (2007) Erdös, L., Schlein, B., Yau, H.T.: Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential. J. Amer. Math. Soc. 22, 1099–1156 (2009) Erdös, L., Schlein, B., Yau, H.T.: Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate. Annals Math. 172, 291–370 (2010) Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), pp. 363–401 Gressman, P., Sohinger, V., Staffilani, G.: On the Uniqueness of Solutions to the Periodic 3D Gross-Pitaevskii Hierarchy. J. Funct. Anal. 266, 4705–4764 (2014) Grillakis, M.G., Margetis, D.: A Priori Estimates for Many-Body Hamiltonian Evolution of Interacting Boson System. J. Hyperb. Diff. Eqs. 5, 857–883 (2008) Grillakis, M.G., Machedon, M.: Pair excitations and the mean field approximation of interacting Bosons, I. Comm. Math. Phys. 324, 601–636 (2013) Grillakis, M.G., Machedon, M.: Pair excitations and the mean field approximation of interacting Bosons, II. Commun. PDE 42, 24–67 (2017) Grillakis, M.G., Machedon, M., Margetis, D.: Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons. I. Commun. Math. Phys. 294, 273–301 (2010) Grillakis, M.G., Machedon, M., Margetis, D.: Second Order Corrections to Mean Field Evolution for Weakly Interacting Bosons. II. Adv. Math. 228, 1788–1815 (2011) Hadac, M., Herr, S., Koch, H.: Well-posedness and scattering for the KP-II equation in a critical space, Ann. Inst. H. Poincar é Anal. Non Linéaire 26 (2009), 917-941. See also M. Hadac, S. Herr, H. Koch, Erratum to “Well-posedness and scattering for the KP-II equation in a critical space,” Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 971–972 Herr, S., Sohinger, V.: The Gross-Pitaevskii Hierarchy on General Rectangular Tori. Arch. Rational Mech. Anal. 220, 1119–1158 (2016) Herr, S., Sohinger, V.: Unconditional Uniqueness Results for the Nonlinear Schrödinger Equation. Commun. Contemp. Math. 21, 1850058 (2019) Hong, Y., Taliaferro, K., Xie, Z.: Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity. SIAM J. Math. Anal. 47, 3314–3341 (2015) Hong, Y., Taliaferro, K., Xie, Z.: Uniqueness of solutions to the 3D quintic Gross-Pitaevskii hierarchy. J. Functional Analysis 270(1), 34–67 (2016) Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schr ödinger equation in the radial case. Invent. Math. 166(3), 645–675 (2006) Killip, R., Vişan, M.: Scale invariant Strichartz estimates on tori and applications. Math. Res. Lett. 23, 445–472 (2016) Kirkpatrick, K., Schlein, B., Staffilani, G.: Derivation of the Two Dimensional Nonlinear Schrödinger Equation from Many Body Quantum Dynamics. Amer. J. Math. 133, 91–130 (2011) Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993) Klainerman, S., Machedon, M.: On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy. Commun. Math. Phys. 279, 169–185 (2008) Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58, 217–284 (2005) Koch, H., Tataru, D.: A priori bounds for the 1D cubic NLS in negative Sobolev spaces, Int. Math. Res. Not. IMRN (2007), no. 16, Art. ID rnm053, 36 pp Koch, H., Tataru, D., Vişan, M.: Dispersive Equations and Nonlinear Waves, Oberwolfach Seminars 45 (2014), Birkhäuser Knowles, A., Pickl, P.: Mean-field Dynamics: Singular Potentials and Rate of Convergence. Comm. Math. Phys. 298, 101–138 (2010) Kuz, E.: Exact Evolution versus Mean Field with Second-order Correction for Bosons Interacting via Short-range Two-body Potential. Differential and Integral Equations 30, 587–630 (2017) Lewin, M., Nam, P.T., Rougerie, N.: Derivation of H artree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014) Lin, J.E., Strauss, W.A.: Decay and scattering of solutions of a nonlinear Schrödinger equation. J. Funct. Anal. 30, 245–263 (1978) Mendelson, D., Nahmod, A., Pavlović, N., Staffilani, G.: An infinite sequence of conserved quantities for the cubic Gross-Pitaevskii hierarchy on \({\mathbb{R}}\). Trans. Amer. Math. Soc 371, 5179–5202 (2019) Mendelson, D., Nahmod, A., Pavlović, N., Rosenzweig, M., Staffilani, G.: A Rigorous Derivation of the Hamiltonian Structure for the Nonlinear Schrödinger Equation. Adv. Math. 365, 107054 (2020) Mendelson, D., Nahmod, A., Pavlović, N., Rosenzweig, M., Staffilani, G.: Poisson Commuting Energies for a System of Infinitely Many Bosons, 97pp, arXiv:1910.06959 Miao, C., Xu, G., Zhao, L.: Global well-posedness and scattering for the defocusing \(H^{1/2}\)-subcritical Hartree equation in \( {\mathbb{R}}^{d}\). Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1831–1852 (2009) Nam, P.T., Salzmann, R.: Derivation of 3D energy-critical nonlinear Schrödinger equation and Bogoliubov excitations for Bose gases. Commun. Math. Phys. 375, 495–571 (2020) Nugent, E.: Novel Traps for Bose-Einstein Condensates, PhD thesis, University of Oxford, (2009) Savard, T.A., O’Hara, K.M., Thomas, J.E.: Laser-noise-induced heating in far-off resonance optical traps. PRA 56, R1098 (1997) Shen, S.: The Rigorous Derivation of the \({\mathbb{T}}^{2}\) Focusing Cubic NLS from 3D. J. Funct. Anal. 280, 108934 (2021) Sohinger, V.: Local Existence of Solutions to Randomized Gross-Pitaevskii Hierarchies. Trans. Amer. Math. Soc. 368, 1759–1835 (2016) Sohinger, V.: 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, 1337–1365 (2015) Sohinger, V., Staffilani, G.: Randomization and the Gross-Pitaevskii hierarchy. Arch. Rational Mech. Anal. 218, 417–485 (2015) Stamper-Kurn, D.M., Andrews, M.R., Chikkatur, A.P., Inouye, S., Miesner, H.-J., Stenger, J., Ketterle, W.: Optical Confinement of a Bose-Einstein Condensate. Phys. Rev. Lett. 80, 2027–2030 (1998) Wiener, N.: The Quadratic Variation of a Function and its Fourier Coefficients. Journal of Mathematics and Physics 3, 72–94 (1924) Xie, Z.: Derivation of a Nonlinear Schrödinger Equation with a General Power-type Nonlinearity in \(d=1,2\). Differ. Integral Equ. 28, 455–504 (2015)