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

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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. 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. 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 \(\beta \).

References

  1. 1.

    Boccato, C., Brennecke, C., Cenatiempo, S., Schlein, B.: Complete Bose–Einstein condensation in the Gross–Pitaevskii regime. Commun. Math. Phys. 359(3), 975–1026 (2018)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Boccato, C., Brennecke, C., Cenatiempo, S., Schlein, B.: The excitation spectrum of Bose gases interacting through singular potentials (To appear on J. Eur. Math. Soc). Preprint arXiv:1704.04819

  3. 3.

    Boccato, C., Brennecke, C., Cenatiempo, S., Schlein, B.: Bogoliubov theory in the Gross–Pitaevskii limit. Acta Math. 222(2), 219–335 (2019)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bogoliubov, N.N.: On the theory of superfluidity. Izv. Akad. Nauk. USSR 11, 77 (1947). (Engl. Transl. J. Phys. (USSR) 11 (1947), 23)

    MathSciNet  Google Scholar 

  5. 5.

    Brennecke, C., Schlein, B.: Gross–Pitaevskii dynamics for Bose–Einstein condensates. Anal. PDE 12(6), 1513–1596 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose gases. Adv. Math. 254, 570–621 (2014)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Lewin, M., Nam, P.T., Rougerie, N.: The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Trans. Am. Math. Soc. 368(9), 6131–6157 (2016)

    Article  Google Scholar 

  9. 9.

    Lewin, M., Nam, P.T., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. 68(3), 413–471 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lieb, E.H., Seiringer, R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002)

    ADS  Article  Google Scholar 

  11. 11.

    Lieb, E.H., Seiringer, R.: Derivation of the Gross–Pitaevskii equation for rotating Bose gases. Commun. Math. Phys. 264(2), 505–537 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Lieb, E.H., Seiringer, R., Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000)

    ADS  Article  Google Scholar 

  13. 13.

    Lieb, E.H., Solovej, J.P.: Ground state energy of the one-component charged Bose gas. Commun. Math. Phys. 217, 127–163 (2001). (Errata: Commun. Math. Phys. 225 (2002), 219–221)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Nam, P.T., Rougerie, N., Seiringer, R.: Ground states of large bosonic systems: the Gross–Pitaevskii limit revisited. Anal. PDE 9(2), 459–485 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Seiringer, R.: The excitation spectrum for weakly interacting Bosons. Commun. Math. Phys. 306, 565–578 (2011)

    ADS  MathSciNet  Article  Google Scholar 

Download references

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

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

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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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