The derivation of the \(\mathbb{T}^{3}\) energy-critical NLS from quantum many-body dynamics

DOI10.1007/s00222-019-00868-3zbMath1422.35148arXiv1803.08082OpenAlexW2951172696MaRDI QIDQ2315188

Justin Holmer, Xuwen Chen

Publication date: 1 August 2019

Published in: Inventiones Mathematicae (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1803.08082

zbMATH Keywords

BBGKY hierarchyquintic nonlinear Schrödinger equationquantum many-body dynamicsGross-Pitaevskii (GP) hierarchy

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) A priori estimates in context of PDEs (35B45) NLS equations (nonlinear Schrödinger equations) (35Q55) Many-body theory; quantum Hall effect (81V70) Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals (35A23) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02)

Related Items

Ground state energy of the low density Bose gas with three-body interactions, The mean-field limit of the Lieb-Liniger model, Quantitative derivation and scattering of the 3D cubic NLS in the energy space, The unconditional uniqueness for the energy-supercritical NLS, Uniqueness of solutions to the spectral hierarchy in kinetic wave turbulence theory, A rigorous derivation of the Hamiltonian structure for the Vlasov equation, The derivation of the compressible Euler equation from quantum many-body dynamics, The condensation of a trapped dilute Bose gas with three-body interactions, On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein condensate, Unconditional local well-posedness for periodic NLS, The rigorous derivation of the \(\mathbb{T}^2\) focusing cubic NLS from 3D, Mini-workshop: Relativistic fluids at the intersection of mathematics and physics. Abstracts from the mini-workshop held December 13--19, 2020 (online meeting), Derivation of 3D energy-critical nonlinear Schrödinger equation and Bogoliubov excitations for Bose gases, Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations, Scaling limits of bosonic ground states, from many-body to non-linear Schrödinger, Counterexamples to \(L^p\) collapsing estimates, Derivation of the nonlinear Schrödinger equation with a general nonlinearity and Gross–Pitaevskii hierarchy in one and two dimensions, Convergence rate towards the fractional Hartree equation with singular potentials in higher Sobolev trace norms, Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Cites Work

• Unnamed Item
• The Gross-Pitaevskii hierarchy on general rectangular tori
• Focusing quantum many-body dynamics: the rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation
• On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction
• Pair excitations and the mean field approximation of interacting Bosons. I
• Dynamical collapse of Boson stars
• The energy-critical defocusing NLS on \({\mathbb{T}}^{3}\)
• Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps
• Randomization and the Gross-Pitaevskii hierarchy
• Focusing quantum many-body dynamics. II: The rigorous derivation of the 1D focusing cubic nonlinear Schrödinger equation from 3D
• Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states
• The mathematics of the Bose gas and its condensation.
• Second-order corrections to mean field evolution of weakly interacting bosons. I
• The quintic NLS as the mean field limit of a boson gas with three-body interactions
• Classical proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in \(\mathbb R^{n+1}\) with application.
• Second-order corrections to mean field evolution of weakly interacting bosons. II
• Energy conservation and blowup of solutions for focusing Gross-Pitaevskii hierarchies
• Second-order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions
• Scale-invariant Strichartz estimates on tori and applications
• On the mean-field limit of bosons with Coulomb two-body interaction
• Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems
• Uniqueness of solutions to the 3D quintic Gross-Pitaevskii hierarchy
• A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on \(\mathbb{T}^3\) from the dynamics of many-body quantum systems
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies
• Mean-field dynamics: Singular potentials and rate of convergence
• Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate
• Mean field limit for bosons and infinite dimensional phase-space analysis
• Quantum fluctuations and rate of convergence towards mean field dynamics
• Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity
• Derivation of the nonlinear Schrödinger equation from a many body Coulomb system
• On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy
• Derivation of a nonlinear Schrödinger equation with a general power-type nonlinearity in \(d=1,2\).
• The proof of the \(l^2\) decoupling conjecture
• Exact evolution versus mean field with second-order correction for bosons interacting via short-range two-body potential.
• A central limit theorem in many-body quantum dynamics
• One-and-a-half quantum de Finetti theorems
• Global well-posedness and scattering for the energy-critical Schrödinger equation in \(\mathbb R^{3}\)
• Blow-up profile of rotating 2D focusing Bose gases
• On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics
• Derivation of Hartree's theory for generic mean-field Bose systems
• Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in \(d=3\) based on spacetime norms
• On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy
• Rigorous derivation of the cubic NLS in dimension one
• Local existence of solutions to randomized Gross-Pitaevskii hierarchies
• The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
• Higher Order Energy Conservation and Global Well-Posedness of Solutions for Gross-Pitaevskii Hierarchies
• Unconditional Uniqueness of the cubic Gross--Pitaevskii Hierarchy with Low Regularity
• Unconditional Uniqueness for the Cubic Gross-Pitaevskii Hierarchy via Quantum de Finetti
• Pair excitations and the mean field approximation of interacting Bosons, II
• Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential
• Derivation of the two-dimensional nonlinear Schrödinger equation from many body quantum dynamics
• Rate of Convergence to Mean Field for Interacting Bosons
• Space‐time estimates for null forms and the local existence theorem
• Correlation Structures, Many-Body Scattering Processes, and the Derivation of the Gross–Pitaevskii Hierarchy
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-Body Evolution
• An infinite sequence of conserved quantities for the cubic Gross–Pitaevskii hierarchy on ℝ
• A new proof of existence of solutions for focusing and defocusing Gross-Pitaevskii hierarchies
• Quantitative Derivation of the Gross‐Pitaevskii Equation
• Mean-field evolution of fermionic systems