Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions

Abstract

We present a microscopic derivation of the defocusing two-dimensional cubic nonlinear Schrödinger equation as a mean field equation starting from an interacting $N$-particle system of Bosons. We consider the interaction potential to be given either by $W_{\beta }(x)=N^{-1+2\beta }W(N^{\beta }x)$, for any $\beta >0$, or to be given by $V_N(x)=e^{2N}V(e^{N}x)$, for some spherical symmetric, positive and compactly supported $W,V\in L^{\infty }({\mathbb{R}}^2,\mathbb{R})$. In both cases we prove the convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the corresponding nonlinear Schrödinger equation in trace norm. For the latter potential $V_N$ we show that it is crucial to take the microscopic structure of the condensate into account in order to obtain the correct dynamics.

63 pages, improved presentation, minor changes

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Cited by in corpus (17)

• Derivation of the Time Dependent Gross-Pitaevskii Equation in Two Dimensions
• Higher order corrections to the mean-field description of the dynamics of interacting bosons
• Beyond Bogoliubov Dynamics
• Derivation of the Time Dependent Two Dimensional Focusing NLS Equation
• Derivation of the Maxwell-Schrödinger Equations from the Pauli-Fierz Hamiltonian
• Derivation of the Landau-Pekar equations in a many-body mean-field limit
• Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials
• Mean-field limits of particles in interaction with quantized radiation fields
• Derivation of the 1d NLS equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 1d Gross-Pitaevskii equation from the 3d quantum many-body dynamics of strongly confined bosons
• Derivation of the 2d Gross-Pitaevskii equation for strongly confined 3d bosons
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• Two dimensional NLS ground states with attractive Coulomb potential and point interaction
• Derivation of the Bogoliubov Time Evolution for a Large Volume Mean-field Limit
• The Dynamics of the Hubbard Model through Stochastic Calculus and Girsanov Transformation
• Bose-Einstein condensation for two dimensional bosons in the Gross-Pitaevskii regime
• On the emergence of quantum Boltzmann fluctuation dynamics near a Bose-Einstein Condensate