Generalised Bogoliubov approximation 

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The mean field Bose gas

Our results for the mean field Bose gas are weaker than for the perfect gas, as we do not establish the existence of a limit for the sequence of kinetic occupation measures ˜mλl . However, we do prove that kinetic generalised BEC must occur if generalised condensation occurs in the generalised eigenstates, and both phenomena are absent below the critical value μc of the chemical potential. Note that our result does not say what happens at the critical point. The only assumption on the external potential that we shall use in this section is the non negativity. As we emphasised in Section 1.2.2, the mean field gas is superstable, which imply that the pressure is well defined for any real value of the chemical potential μ. Hence, the fixed density constraint ρ = ρλl (μ) has always a unique solution μl for any given l, and this is still true in the thermodynamic limit. In particular, we can without loss of generality consider that μ is kept fixed in the thermodynamic limit, instead of fixing the mean density ρ as in the perfect gas, see Section 1.2.1.

Localisation and BEC in single kinetic states

Having established the occurrence of kinetic generalised BEC in presence of an external potential in the previous chapter, the next question is to determine its type. As we discussed in Section 1.4, it is in general more difficult to find out the type of generalised BEC than to simply show the occurrence of generalised condensation, even when one considers BEC in the eigenstates. The main idea of this chapter is to notice that in our models, the density of states is fast decreasing, which is generally believed to force the corresponding eigenstates to become localised in the limit. However, the kinetic states are planes waves, hence delocalised in space. Hence, since these two states are “asymptotically orthogonal”, this should prevent condensation to occur in any kinetic states. We first use the strong localisation property of the Luttinger-Sy model to prove in a simple way that the kinetic generalised BEC in this model is of the type III. We then extend that result to a more general class of localised systems, and establish that, for a more realistic random model and a general family of weak external potential, the required localisation criterion indeed holds.
The results of this chapter have been published in Journal of Mathematical Physics [30], a copy of this article is reproduced in Appendix E. In Section 1.4, we briefly reviewed what could be rigorously proved for the condensation in single eigenstates φl i, and in particular, we emphasised that the classification of the generalised BEC in the eigenstates required a fine knowledge of the spectrum, namely the speed at which the gap between any two eigenvalues vanishes in the limit l → ∞. Clearly, this information cannot be extracted from the limiting density of states ν of the Schr¨odinger operator with a external potential. Hence, it is in general very complicated to classify the generalised BEC in the eigenstates, in particular in the random case where the required knowledge is quite hard to obtain even in simple examples like the Luttinger-Sy model.

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A general localisation criterion

As we pointed out at the beginning of this chapter, the localisation estimate that we obtained in the Luttinger-Sy model, see Lemma 3.1.1 and equation (3.2) is uniform with respect to i (the index of the eigenstates). However, it turns out that in more general models, while we still expect localisation to happen, it does not seem possible to obtain a uniform estimate. Hence, we must find a way to deal with the infinite sum in (3.1), which is the aim of this section. Let us introduce the notation ρl i := 1 Vl hNl(φl i)iλl .

Table of contents :

Introduction
1 BEC in the eigenstates 
1.1 Definitions and notation
1.2 The phase transition and generalised BEC
1.2.1 The perfect Bose gas
1.2.2 The mean field Bose gas
1.3 The density of states for specific models
1.3.1 Random potentials: the Lifshitz tails
1.3.2 Weak external potentials
1.4 BEC in single eigenstates
2 Generalised BEC in the kinetic states 
2.1 Some general results
2.2 The perfect Bose gas
2.2.1 The random case
2.2.2 Weak external potentials
2.3 The mean field Bose gas
3 Localisation and BEC in single kinetic states 
3.1 The Luttinger-Sy model
3.2 A general localisation criterion
3.3 Proof of localisation in some specific models
3.3.1 The Stollmann model
3.3.2 Weak external potentials
4 Generalised Bogoliubov approximation 
4.1 Heuristic discussion
4.2 Model and definitions
4.3 The approximated pressure
4.3.1 Exactness of the generalised Bogoliubov approximation
4.3.2 The main proof
4.3.3 Some technical results
4.4 From the pressure to the condensate density
A Brownian motions
B Probabilistic estimates
C Some multiscale analysis results
C.1 Sketch of the proof of Proposition 3.3.1
C.2 Sketch of the proof of the eigenfunction decay inequality
D Coherents states: lower and upper symbols
E Publications

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