# Derivation of the time-dependent Gross-Pitaevskii equation for the dipolar gases

## TL;DR

This paper rigorously derives the time-dependent dipolar Gross-Pitaevskii equation from the N-body Schrödinger equation, establishing convergence of solutions and density matrices for dipolar Bose gases under certain scaling regimes.

## Contribution

It provides a mathematical derivation and proof of convergence for the dipolar GP equation from many-body quantum dynamics, extending previous results to dipolar interactions.

## Key findings

- Norm approximation of many-body solutions
- Convergence of one-body density matrix to the GP solution
- Validity of results for specific interaction scaling regimes

## Abstract

We derive the time-dependent dipolar Gross-Pitaevskii (GP) equation from the N-body Schr{\"o}dinger equation. More precisely we show a norm approximation for the solution of the many body equation as well as the convergence of its one-body reduced density matrix towards the orthogonal projector onto the solution of the dipolar GP equation. We consider the interpolation regime where interaction potential is scaled like $N^{3\beta--1} w(N^\beta (x -- y))$, the range of validity of $\beta$ depends on the stability of the ground state problem. In particular we can prove the convergence on the one-body density matrix assuming $\widehat{w} $\ge$ 0$ and $\beta < 3/8$.