## Differential and Integral Equations

- Differential Integral Equations
- Volume 30, Number 7/8 (2017), 587-630.

### Exact Evolution versus mean field with second-order correction for Bosons interacting via short-range two-body potential

#### Abstract

We consider the evolution of $N$ bosons, where $N$ is large, with two-body interactions of the form $N^{3\beta}v(N^{\beta}\mathbf{\cdot})$, $0\leq\beta\leq 1$. The parameter $\beta$ measures the strength of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [25, 26]. For $0\leq \beta < 1/2$, we derive an error bound of the form $p(t)/N^\alpha$, where $\alpha>0$ and $p(t)$ is a polynomial, which implies a specific rate of convergence as $N\rightarrow\infty$.

#### Article information

**Source**

Differential Integral Equations, Volume 30, Number 7/8 (2017), 587-630.

**Dates**

Accepted: May 2016

First available in Project Euclid: 4 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1493863396

**Mathematical Reviews number (MathSciNet)**

MR3646465

**Zentralblatt MATH identifier**

06738563

**Subjects**

Primary: 81V70: Many-body theory; quantum Hall effect 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general) 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Kuz, Elif. Exact Evolution versus mean field with second-order correction for Bosons interacting via short-range two-body potential. Differential Integral Equations 30 (2017), no. 7/8, 587--630. https://projecteuclid.org/euclid.die/1493863396