Derivation of the time dependent Gross-Pitaevskii equation for a class of non purely positive potentials

Abstract

We present a microscopic derivation of the time-dependent Gross-Pitaevskii equation starting from an interacting N-particle system of Bosons. We prove convergence of the reduced density matrix corresponding to the exact time evolution to the projector onto the solution of the respective Gross-Pitaevskii equation. Our work extends a previous result by one of us (P.P.[44]) to interaction potentials which need not to be nonnegative, but may have a sufficiently small negative part. One key estimate in our proof is an operator inequality which was first proven by Jun Yin, see [49].

34 pages

References in corpus (3)

• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• Derivation of the Time Dependent Gross Pitaevskii Equation with External Fields
• Absence of bound states implies non-negativity of the scattering length

Cited by in corpus (2)

• Classical Dynamics Generated by Long-Range Interactions for Lattice Fermions and Quantum Spins
• Derivation of the time-dependent Gross-Pitaevskii equation for the dipolar gases