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2 M. LEWIN, P. T. NAM, S. SERFATY, AND J. P. SOLOVEJ
1. Introduction
In a famous paper [9], Bogoliubov was able to predict the excitation spectrum
of a quantum gas satisfying the Bose statistics and he used this to
understand its superfluid behavior. Since Bogoliubov’s work, there has
been several attempts to formulate Bogoliubov’s theory in a mathematically
rigorous way. This was especially successful for completely integrable
1D systems [21, 35, 32, 12, 11, 59, 60], for the ground state energy of one and
two-component Bose gases [41, 42, 55], and for the Lee-Huang-Yang formula
of dilute gases [19, 22, 64]. Recently, Seiringer [52] and Grech-Seiringer [24]
have for the first time justified Bogoliubov’s theory for the excitation spectrum
of trapped Bose gases, with a general short range interaction, in the
mean-field regime. See, e.g., [65] for a recent review on the subject and [15]
for a discussion of translation-invariant systems.
The purpose of this article is to give general conditions under which Bogoliubov’s
theory is valid, that is, predicts the lowest part of the spectrum
of the many-body Hamiltonian of bosons, in the mean-field regime. Our
results cover a very large class of interacting Boses gases and they generalize
the recent works [52, 24]. In particular, our method applies to Coulomb
systems.
We consider a system of N quantum particles, described by the Hamiltonian
N� 1 �
HN = Txi + w(xi −xj),
N −1
i=1
1≤i