Abstract
We compute the energy needed to add or remove one particle from a homogeneous system of N bosons on a torus. We focus on the mean-field limit when N becomes large and the strength of particle interactions is proportional to \(N^{-1}\), which allows us to justify Bogoliubov’s approximation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Strictly speaking, for \(a_0^*a_0^*a_0a_0\) we should rewrite it as \((a_0^*a_0)^2-a_0^*a_0\) before doing the substitution.
References
Bach, V.: Ionization energies of bosonic Coulomb systems. Lett. Math. Phys. 21, 139–149 (1991)
Boccato, C., Brennecke, C., Cenatiempo, S., Schlein, B.: Complete Bose–Einstein condensation in the Gross–Pitaevskii regime (2017). Preprint arXiv:1703.04452
Bogoliubov, N.N.: On the theory of superfluidity. J. Phys. (USSR) 11, 23 (1947)
Dereziński, J., Napiórkowski, M.: Excitation spectrum of interacting bosons in the mean-field infinite-volume limit. Ann. Henri Poincaré 15, 2409–2439 (2014)
Erdös, L., Schlein, B., Yau, H.-T.: Ground-state energy of a low-density Bose gas: a second-order upper bound. Phys. Rev. A 78, 053627 (2008)
Giuliani, A., Seiringer, R.: The ground state energy of the weakly interacting Bose gas at high density. J. Stat. Phys. 135, 915–934 (2009)
Grech, P., Seiringer, R.: The excitation spectrum for weakly interacting bosons in a trap. Commun. Math. Phys. 322, 559–591 (2013)
Lewin, M., Nam, P.T., Rougerie, N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)
Lewin, M., Nam, P.T., Serfaty, S., Solovej, J.P.: Bogoliubov spectrum of interacting Bose gases. Commun. Pure Appl. Math. 68, 413–471 (2015)
Lieb, E.H., Solovej, J.P.: Ground state energy of the one-component charged Bose gas. Commun. Math. Phys. 217, 127–163 (2001)
Lieb, E.H., Solovej, J.P.: Ground state energy of the two-component charged Bose gas. Commun. Math. Phys. 252, 485–534 (2004)
Nam, P.T., Seiringer, R.: Collective excitations of Bose gases in the mean-field regime. Arch. Ration. Mech. Anal. 215, 381–417 (2015)
Rougerie, N., Spehner, D.: Interacting bosons in a double-well potential : localization regime (2016). Preprint arXiv:1612.05758
Seiringer, R.: The excitation spectrum for weakly interacting bosons. Commun. Math. Phys. 306, 565–578 (2011)
Solovej, J.P.: Upper bounds to the ground state energies of the one- and two-component charged Bose gases. Commun. Math. Phys. 266, 797–818 (2006)
Yau, H.-T., Yin, J.: The second order upper bound for the ground energy of a Bose gas. J. Stat. Phys. 136, 453–503 (2009)
Acknowledgements
I would like to thank Mathieu Lewin and Robert Seiringer for helpful discussions and the referee for interesting remarks.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nam, P.T. Binding energy of homogeneous Bose gases. Lett Math Phys 108, 141–159 (2018). https://doi.org/10.1007/s11005-017-0992-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-017-0992-5