Optimal Rate for Bose–Einstein Condensation in the Gross–Pitaevskii Regime

Abstract

We consider systems of bosons trapped in a box, in the Gross–Pitaevskii regime. We show that low-energy states exhibit complete Bose–Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in Boccato et al. (Commun Math Phys 359(3):975–1026, 2018), removing the assumption of small interaction potential.

Properties of the Scattering Function

In this appendix we give a proof of Lemma 4.1 containing the basic properties of the solution of the Neumann problem (4.1).

Proof of Lemma 4.1

Part (i) and the bounds $0\le f_\ell , w_\ell \le 1$ in part (ii) follow from [6, Lemma A.1]. We prove (4.6). We set $r=|x|$ and $m_\ell (r)=rf_\ell (r)$. We rewrite (4.1) as

$$\begin{aligned} -\,m_\ell ''(r)+\frac{1}{2}V(r)m_\ell (r)=\lambda _\ell m_\ell (r). \end{aligned}$$

(A.1)

Let $R > 0$ be the radius of the support of V, so that $V(x) = 0$ for all $x \in {\mathbb {R}}^3$ with $|x| > R$. For $r\in (R,N\ell ]$ we can solve (A.1) explicitly; since the boundary conditions $f_\ell (N\ell ) = 1$ and $(\partial _r f_\ell ) (N\ell ) = 0$ translate into $m_\ell (N\ell ) = N\ell$ and $m'_\ell (N\ell ) = 1$, we find

$$\begin{aligned} m_\ell (r)=\lambda _\ell ^{-1/2}\sin (\lambda _\ell ^{1/2}(r-N\ell ))+N\ell \cos (\lambda _\ell ^{1/2}(r-N\ell )). \end{aligned}$$

(A.2)

With the result of part (i), we obtain

$$\begin{aligned} m_\ell (r)=r-{\mathfrak {a}}_0+\frac{3}{2}\frac{\mathfrak {a}_0}{N\ell }r-\frac{1}{2}\frac{\mathfrak {a}_0}{(N\ell )^3}r^3+{\mathcal {O}}(\mathfrak {a}_0^2(N\ell )^{-1}) \end{aligned}$$

(A.3)

for all $r \in (R,N\ell ]$ (the error is uniform in r). Using the scattering equation we can write

$$\begin{aligned} \begin{aligned} \int V(x) f_\ell (x) dx&=4\pi \int _0^{N\ell } dr\, rV(r)m_\ell (r)=8\pi \int _0^{N\ell }dr\, (rm_\ell ''(r)+\lambda _\ell rm_\ell (r)). \end{aligned} \end{aligned}$$

Integrating by parts, we observe that the first contribution on the r.h.s. vanishes (because $m_\ell (N\ell ) = N\ell$, $m'_\ell (N\ell ) = 1$ and $m_\ell (0) = 0$). With the result of part (i) and with (A.3), we get

$$\begin{aligned} \begin{aligned} 8\pi \lambda _\ell \int _0^{N\ell }dr\, rm_\ell (r)&= 8\pi \lambda _\ell \left( \frac{(N\ell )^3}{3}+{\mathcal {O}}\big (\mathfrak {a}_0(N\ell )^2\big )\right) = 8\pi \mathfrak {a}_0 +{\mathcal {O}} \big (\mathfrak {a}_0^2 / \ell N\big ) \end{aligned} \end{aligned}$$

which proves (4.6).

We consider now part (iii). Combining (A.3) for $r \in (R,N\ell ]$ with $w_\ell (r) \le 1$ for $r \le R$, we obtain the first bound in (4.7). To show the second bound in (4.7), we observe that, for $r \in (R,N\ell ]$, (A.2) and the estimate in part (i) imply that $|f'_\ell (r)| \le C r^{-2}$, for a constant $C > 0$ independent of N and $\ell$, provided $N \ell \ge 1$. For $r < R$ we write, integrating by parts,

$$\begin{aligned} \begin{aligned} f'_\ell (r)&=\frac{m_\ell '(r)r-m_\ell (r)}{r^2}=\frac{1}{r^2}\int _0^rds\, s\,m_\ell ''(s). \end{aligned} \end{aligned}$$

With (A.1) and since $0 \le f_\ell \le 1$, we obtain

$$\begin{aligned} \begin{aligned} |f'_\ell (r)|&=\Big |\frac{1}{r^2}\int _0^rds\, s\,\Big [\frac{1}{2}V(s)m_\ell (s)-\lambda _\ell m_\ell (s)\Big ]\Big |\\&= \frac{1}{r^2}\Big [\frac{1}{8\pi }\int _{|x|< r} dx\, \,V(x)f_\ell (x)+\lambda _\ell \int _{|x| < r} dx \, f_\ell (x) \Big ] \le C (\Vert V \Vert _3 + 1) \end{aligned} \end{aligned}$$

for a constant $C > 0$ independent of N and $\ell$, if $N\ell \ge 1$ and for all $0< r < R$. This concludes the proof of the second bound in (4.7).

To show part (iv), we use (4.4) and we observe that, by (4.5), (4.6) and $f_\ell \le 1$, there exists a constant $C > 0$ such that

$$\begin{aligned} \begin{aligned} |{\widehat{w}}_\ell (p/N)|&\le \frac{N^2}{p^2} \left[ \big ({{\widehat{V}}}(./N)*{\widehat{f}}_{N,\ell }\big )(0) + C \ell ^{-3} \big ({\widehat{\chi }}_\ell *{\widehat{f}}_{N,\ell }\big )(0) \right] \\&\le \frac{N^2}{p^2}\left[ \int V(x)f_\ell (x) dx + C \ell ^{-3} \int \chi _\ell (x) f_\ell (Nx) dx \right] \le \frac{CN^2}{p^2} \end{aligned} \end{aligned}$$

for all $N \in {\mathbb {N}}$ and $\ell > 0$, if $N \ell \ge 1$. $\quad \square$