Mean-Field Evolution of Fermions with Singular Interaction

Abstract

We consider a system of N fermions in the mean-field regime interacting through an inverse power law potential $V(x)=|x|^{-\alpha }$, for $\alpha \in (0,1]$. We prove the convergence of a solution of the many-body Schrödinger equation to a solution of the time-dependent Hartree–Fock equation in the sense of reduced density matrices, for a specific class of initial data, namely translation-invariant states. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of Porta et al. (J. Stat. Phys. 166:1345–1364, 2017, [27]), where the case $\alpha =1$ is treated.

A Kinetic Energy Estimates

To bound the $L^{5/3}$ norm of the density $\rho _t$, we observe that the Lieb–Thirring inequality and the positivity of the interaction potential yield

$$\Vert \rho _t\Vert _{L^{5/3}}^{5/3}\le \mathrm{{tr}}\ (-\Delta )\omega _{N,t}\le \varepsilon ^{-2}\mathcal {E}_\mathrm{HF}(\omega _{N,t})$$

where $\mathcal {E}_\mathrm{HF}$ is the Hartree–Fock energy functional defined in (1.4). Conservation of energy implies

$$\Vert \rho _t\Vert _{L^{5/3}}^{5/3}\le \varepsilon ^{-2}\mathcal {E}_\mathrm{HF}(\omega _{N,t})=\varepsilon ^{-2}\mathcal {E}_\mathrm{HF}(\omega _{N}).$$

To close the estimate using the assumption on the kinetic energy of the initial sequence $\mathrm{{tr}}\ (-\varepsilon ^2\Delta )\omega _{N}\le C\,N$, we observe that the potential energy can be bounded by the kinetic energy. Indeed, the Hardy–Littlewood–Sobolev inequality yields

$$\begin{aligned} \frac{1}{N}\int \frac{1}{|x-y|^\alpha }\rho _0(x)\rho _0(y)\,dx\,dy\le \frac{C}{N}\Vert \rho _0\Vert ^2_{L^{6/(5-\alpha )}} \end{aligned}$$

By interpolation, using that $\frac{6}{6-\alpha }\in \left( 1,\frac{5}{3}\right)$ and $\Vert \rho _0\Vert _{L^1}=N$, we have

$$\begin{aligned} \begin{aligned} \frac{1}{N}\int \frac{1}{|x-y|^\alpha }\rho _0(x)\rho _0(y)\,dx\,dy&\le \frac{C}{N}\Vert \rho _0\Vert _{L^1}^{\frac{12-5\alpha }{6}}\Vert \rho _0\Vert _{L^{\frac{5}{3}}}^{\frac{5}{6}\alpha } \\&\le C\,N^{1-\frac{5}{6}\alpha }\Vert \rho _0\Vert _{L^{\frac{5}{3}}}^{\frac{5}{6}\alpha }\\&\le C\,N+N^{-\frac{2}{3}\alpha }\Vert \rho _0\Vert ^{\frac{5}{3}\alpha }_{L^{\frac{5}{3}}}, \end{aligned} \end{aligned}$$

where in the last line we have used Young’s inequality $ab\le \frac{a^p}{p}+\frac{b^q}{q}$, $p^{-1}+q^{-1}=1$, on the quantities $a=N^{1-\frac{\alpha }{2}}$ and $b=N^{-\frac{\alpha }{3}}\Vert \rho _0\Vert _{L^{\frac{5}{3}}}^{\frac{5}{6}\alpha }$ with $p=2/(2-\alpha )$ and $q=2/\alpha$. Thus, applying again the Lieb–Thirring inequality and recalling that $\varepsilon =N^{-1/3}$, we obtain

$$\begin{aligned} \frac{1}{N}\int \frac{1}{|x-y|^\alpha }\rho _0(x)\rho _0(y)\,dx\,dy\le C\,N+C\,\mathrm{{tr}}\ (-\varepsilon ^2\Delta )\omega _N \end{aligned}$$

which gives the bound

$$\begin{aligned} \frac{1}{N}\int \frac{1}{|x-y|^\alpha }\rho _0(x)\rho _0(y)\,dx\,dy\le C\,N \end{aligned}$$

thanks to assumption (i).