## 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.

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## Notes

- 1.
At the end, we will need the high-momentum cutoff \(\ell ^{-\alpha }\) to be sufficiently large. To reach this goal, we will choose \(\ell \) sufficiently small. Alternatively, we could decouple the cutoff from the radius \(\ell \) introduced in (4.1), keeping \(\ell \in (0;1/2)\) fixed and choosing instead the exponent \(\alpha \) sufficiently large.

- 2.
At the end, we will need the low-momentum cutoff \(\ell ^{-\beta }\) to be sufficiently large (preserving however certain relations with the high-momentum cutoff). We will reach this goal by choosing \(\ell \) small enough. Alternatively, as already remarked in the footnote after (4.12), also here we could decouple the low-momentum cutoff from the radius \(\ell \) introduced in (4.1), by keeping \(\ell \in (0;1/2)\) fixed and varying instead the exponent \(\beta \).

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## Acknowledgements

We would like to thank P. T. Nam and R. Seiringer for several useful discussions and for suggesting us to use the localization techniques from [9]. C. Boccato has received funding from the European Research Council (ERC) under the programme Horizon 2020 (Grant Agreement 694227). B. Schlein gratefully acknowledges support from the NCCR SwissMAP and from the Swiss National Foundation of Science (Grant No. 200020_1726230) through the SNF Grant “Dynamical and energetic properties of Bose–Einstein condensates”.

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Communicated by R. Seiringer

## Properties of the Scattering Function

### 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

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

With the result of part (i), we obtain

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

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

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,

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

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

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

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Boccato, C., Brennecke, C., Cenatiempo, S. *et al.* Optimal Rate for Bose–Einstein Condensation in the Gross–Pitaevskii Regime.
*Commun. Math. Phys.* **376, **1311–1395 (2020). https://doi.org/10.1007/s00220-019-03555-9

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