In 1947, Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely, to test the validity of Bogoliubov’s predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes, we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.
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In the regime where , it is easy to see that if we take the thermodynamic limit in the second and third line of (3), the contribution to the integral coming from momenta larger than for any α > 0 is of smaller order in δ w.r.t. . Moreover, for momenta smaller than with 0 < α < 1/2 we can substitute in the same integral by , again up to errors of smaller order in δ w.r.t. . Hence Bogoliubov’s substitution of by occurs at the level of the integral over momenta smaller than with 0 < α < 1/2.
The method employed by Benfatto in Ref. 8 is the Wilsonian Renormalization Group, combined with the ideas of constructive renormalization group, in the form developed by the roman school of Benfatto, Gallavotti et al. since the late seventies. See also Refs. 21 and 22 for similar theoretical physics results obtained by means of dimensional regularization.
Indeed, if we consider N bosons interacting through a potential scaling as N2V(Nx), and initially trapped in a volume of order one, it is known that at time zero the system exhibits Bose-Einstein condensation into the minimizer of the Gross-Pitaevskii energy functional. If we now let the system evolve by removing the trapping, one expects condensation to be preserved at any time in the limit N → ∞, and the condensate wave function to evolve according to the Gross-Pitaevskii equation , with initial data φt = 0 = φ. This fact has been well established mathematically [see the references in Ref. 51 (Chap. 5), and the recent result53], and confirms the use of the Gross-Pitaevskii equation to effectively describe the time evolution of Bose-Einstein condensates.
Usually the second quantization formalism is used to represent the system in a grand-canonical picture, where the particle number can vary. On the contrary here the particle number is fixed to be N, but the excitation number is not fixed, and can vary up to N.
