Thomas Jaeck
email: thomas.jaeck@ucdconnect.ie

Joseph V Pulé

Valentin A Zagrebnov
email: valentin.zagrebnov@cpt.univ-mrs.fr

On the nature of Bose-Einstein condensation enhanced by localization

Keywords: Generalized Bose-Einstein Condensation, Random Potentials, Integrated Density of States, Multiscale Analysis, Diagonal Particle Interactions PACS: 05.30.Jp, 03.75.Fi, 67.40.-w AMS: 82B10, 82B23, 81V70

In a previous paper we established that for the perfect Bose gas and the mean-field Bose gas with an external random or weak potential, whenever there is generalized Bose-Einstein condensation in the eigenstates of the single particle Hamiltonian, there is also generalized condensation in the kinetic energy states. In these cases Bose-Einstein condensation is produced or enhanced by the external potential. In the present paper we establish a criterion for the absence of condensation in single kinetic energy states and prove that this criterion is satisfied for a class of random potentials and weak potentials. This means that the condensate is spread over an infinite number of states with low kinetic energy without any of them being macroscopically occupied.

Introduction

It can be easily seen from the explicit formula for the occupation numbers in the noninteracting (perfect) Bose gas that, the condition for Bose-Einstein Condensation (BEC) to occur, is that the density of states of the one particle Schrödinger operator decreases fast enough near the bottom of the spectrum. In the absence of any external potential, it is known that this happens only in three dimension or higher. This is still true if one introduces a mean-field interaction between particles. It has been known for some time that the behavior of the density of states can be altered by the addition of suitable external potentials, in particular weak potentials or random potentials. The subject of this paper is the study of models of the Bose gas in the presence of such external potentials. The first case has been extensively studied, see e.g. [1,2], where sufficient conditions on the external potential were derived for the occurrence of BEC. In the random case, it has been shown in [3] that the so-called Lifshitz tails, which are a general feature of disordered systems, see for instance [START_REF] Pastur | Spectra of Random and Almost-Periodic Operators[END_REF], are able to produce BEC. In both cases, it is possible to obtain condensation even in dimensions 1 or 2. While BEC has historically been associated with the macroscopic occupation of the ground state only, it was pointed out in [START_REF] Girardeau | [END_REF] that this phenomena is more thermodynamically stable if it is interpreted as as the macroscopic accumulation of particles into an arbitrarily narrow band of energy above the ground state, or generalized BEC. While it is clear that condensation in the ground state implies generalized BEC, there exist many situations in which the converse is not true. For instance, it was shown in [2] that in the case of the weak potential, the condensate can be in one state, in infinitely many states or even not in any state at all, depending on the external potential. These situations correspond respectively to type I, II, III generalized BEC in the classification established by the Van den Berg, Lewis and Pulé, see e.g. [6]. In the random case, far less is known. The only case for which a rigorous proof of the exact type of BEC has been established is the Luttinger-Sy model, see [7], where it was shown that the ground-state only is macroscopically occupied. As far as we know, for more complicated systems this is still an open question. The difficulty lies in the fact that the characterization of the distribution of the condensate in individual states requires much more detailed knowledge about the spectrum than the occurrence of generalized condensate. Indeed, for the latter, it is enough to know the asymptotic behavior of the density of states, while for the former, one needs in addition to know how fast the gap between two eigenvalues vanishes in limit.

In the physics literature the phenomenon of BEC is generally understood to be the macroscopic occupation of the lowest kinetic energy (momentum) state, commonly referred as zero-mode condensation. We refer the reader to [9] for a discussion of the motivation for this type of condensation. This leads naturally to two questions in the case condensation is produced or enhanced by the addition of external potentials. The first one comes from the fact the condensates referred to here are to be found in the eigenstates of the one-particle Schrödinger operator and not the kinetic energy (momentum) states. Therefore, it is not immediately clear, and in fact counter intuitive in the random case because of the lack of translation invariance, that condensation occurs in the kinetic energy states as well. This problem has been addressed in a previous paper, see [9], where we have shown under fairly general assumptions on the external potential (random or weak) that the amount of generalized BEC in the eigenstates in turn creates a generalized condensate in the kinetic states, and moreover in the perfect gas the densities of condensed particles are iden-tical. These results were proved for the perfect Bose gas, and can be partially extended to the mean-field Bose gas. Hence, the (generalized) condensation produced in these models by the localization property of the one-particle Schrödinger operator can be correctly described as of "Bose-Einstein" type in the traditional sense. This opens up the possibility of formulating a generalized version of the c-number Bogoliubov approximation ( [9], [10]). In the case of the weak external potential, perhaps this result is not so surprising since the model is asymptotically translation invariant, but in the random case, it is less obvious since the system is translation invariant only in the sense that translates of the potential are equally probable and therefore for a given configuration the system is not translation invariant.

Having established generalized BEC in the kinetic states, the next question is about the fine structure of that condensate. In our paper [9], we conjectured that the kinetic generalized BEC is of type III, that is, no single kinetic state is macroscopically occupied, even though the amount of generalized condensation is non-zero. Our motivation came from the fact that the fast decrease of the density of states is usually associated with the corresponding eigenstates becoming localized in the infinite volume limit. Hence, since the kinetic states (plane waves) and the (localized) general eigenstates are "asymptotically orthogonal", it should follow that no condensation in any single-mode kinetic energy state could occur, independently of whether the (localized) ground state is macroscopically occupied or not. In [9], we were able to prove this conjecture in a simple example, the Luttinger-Sy model. Our proof in that case used the absence of tunneling effect specific to that model, which we can interpret as "perfect localization".

In this paper we give a proof of the conjecture under a fairly weak localization hypothesis and then we consider a family of continuous random models and a general class of weak external potential for which we are able to establish this localization property. Our results hold for both the perfect and mean-field Bose gas, and for any dimension. Note that, in addition to clarifying the nature of these condensates in low dimensions, we obtain an unexpected conclusion. Indeed, we show that the presence of randomness or a weak potential, however small, prevents condensation from occurring in any kinetic state, even if the corresponding free Bose gas (without external potential) exhibits zero-mode condensation (isotropic system in dimension 3, for example). This emphasizes the importance of the concept of generalized BEC.

The structure of the paper is as follows: in Section 2 we give the general setting for which our results are applicable and discuss generalized condensation in the kinetic energy states, while in Section 3 we derive a criterion for the absence of condensation into any single kinetic energy state. In Section 4 we establish that this criterion is satisfied for a class of random potentials (Subsection 4.1) and for weak (scaled) potentials (Subsection 4.2).

Notation and models

Let {Λ l := (-l/2, l/2) d } l 1 be a sequence of hypercubes of side l in R d , centered at the origin of coordinates with volumes V l = l d . We consider a system of identical bosons, of mass m, contained in Λ l . For simplicity, we use a system of units such that = m = 1. First we define the self-adjoint one-particle kinetic-energy operator of our system by:

h 0 l := -1 2 ∆ D , (2.1)
acting in the Hilbert space H l := L 2 (Λ l ), where ∆ is the usual Laplacian. The subscript D stands for Dirichlet boundary conditions. We denote by {ψ l k , ε l k } k 1 the set of normalized eigenfunctions and eigenvalues corresponding to h 0 l . By convention, we order the eigenvalues (counting multiplicity) as 0 < ε l 1 ε l 2 ε l 3 . . . . Note that all kinetic states satisfy the following bound

|ψ l k (x)| l -d/2 . (2.2)
Next we define the Hamiltonian with an external potential

h l := h 0 l + v l , (2.3)
also acting in H l , where v l : Λ l → [0, ∞) is positive and bounded. Let {φ l i } i 1 and {E l i } i 1 be respectively the sets of normalized eigenfunctions and corresponding eigenvalues of h l . Again, we order the eigenvalues (counting multiplicity) so that E l 1 E l 2 E l 3 . . . . Note that the non-negativity of the potential implies that E l 1 > 0. We shall also assume that the lower end of the spectrums of h 0 l and h l coincide in the limit l → ∞, that is, lim l→∞ E l 1 = 0. This assumption will be proved for the models considered in this paper. Now, we turn to the many-body problem. Let F l := F l (H l ) be the symmetric Fock space constructed over H l . Then H l := dΓ(h l ) denotes the second quantization of the one-particle Schrödinger operator h l in F l . Note that the operator H l acting in F l has the form:

H l = i 1 E l i a * (φ l i )a(φ l i ), (2.4)
where a * (ϕ), a(ϕ) are the creation and annihilation operators (satisfying the boson Canonical Commutation Relations) for the one-particle state ϕ ∈ H l . Then, the grand-canonical Hamiltonian of the perfect Bose gas in an external potential is given by:

H l -µN l = i 1 (E l i -µ) N l (φ l i ) (2.5)
where N l (φ) := a * (φ)a(φ) is the operator for the number of particles in the normalized state φ, N l := i N l (φ l i ) is the operator for the total number of particles in Λ l and µ is the chemical potential.

The results in this paper hold for the mean-field Bose gas whose many particle Hamiltonian, H l (µ), is obtained by adding a mean-field term to (2.5):

H l (µ) := H l -µN l + λ 2V l N 2 l , (2.6)
where λ is a non-negative parameter. Of course the results are valid also for the perfect Bose gas (λ = 0).

We recall that the thermodynamic equilibrium Gibbs statel associated with the Hamiltonian H l (µ) is defined by:

A l := Tr F l {exp(-βH l (µ l ))A} Tr F l exp(-βH l (µ l ))
where the value of µ, µ l is determined by fixing the mean density ρ > 0:

1 V l N l l = ρ. (2.7)
When referring specifically to the perfect Bose gas state, we shall use the notation -0 l . For simplicity, in the sequel we shall omit the explicit mention of the dependence on the thermodynamic parameters (β, µ) unless it is necessary to refer to them.

A normalized single particle state ϕ is macroscopically occupied if

lim l→∞ 1 V l N l (ϕ) l > 0
and in particular there is condensation in the ground state if

lim l→∞ 1 V l N l (φ l 1 ) l > 0.
The concept of generalized condensation consists in considering the possible macroscopic occupation of an arbitrary small band of energies at the bottom of the spectrum. To be more precise, we say that there is generalized condensation in the states

φ l i if lim δ↓0 lim l→∞ 1 V l i:E l i δ N l (φ l i ) l > 0.
It is clear that the usual one-mode condensation implies generalized condensation, however the converse is not true. Indeed, as was first established by the Dublin School in the eighties [6], it is possible to classify generalized condensation into three types. Type I condensation, when a finite number of states are macroscopically occupied (which includes the most commonly known notion of BEC as condensation in the ground state only), type II condensation, when condensation occurs in an infinite number of states, and finally type III, when, although the amount of generalized condensation is non-zero, no individual state is macroscopically occupied. One can easily show that in the perfect Bose gas, under fairly general assumptions, for both random and weak positive potentials, there is indeed generalized condensation in a suitable range of density (or temperature).

In [9] we discussed the possibility of generalized condensation not in the states φ l i but in the kinetic energy states ψ l k . For both random and weak positive potentials, we established that for models which are diagonal in the occupation numbers of the eigenstates of the Hamiltonian (2.3), the density of generalized BEC in the kinetic states is never less then that in the eigenstates of the single particle Hamiltonian. To be more precise, we proved that

lim δ↓0 lim l→∞ 1 V l k:ε l k <δ N l (ψ l k ) l lim δ↓0 lim l→∞ 1 V l i:E l i <δ N l (φ l i ) l .
We also showed that in the case of the perfect gas, the two quantities in the above inequality are equal. Here, we shall give a "localization criterion" on the states φ l i so that the condensation in the kinetic energy states ψ l k is of type III, that is no kinetic energy state is macroscopically occupied.

It is easy to see that the mean-field gas satisfies the following commutation relation [H l (µ), N l (φ l j )] = 0, for all j.

(2.8)

This property implies that a * (φ l i )a(φ l j ) l = 0 if i = j and allows us to obtain a simple relation between the mean occupation numbers for the ψ l k 's and φ l k 's:

1 V l N l (ψ l k ) l = 1 V l a * (ψ l k )a(ψ l k ) l = 1 V l i,j (φ l i , ψ l k )(φ l j , ψ l k ) a * (φ l i )a(φ l j ) l (2.9) = 1 V l i |(φ l i , ψ l k )| 2 N l (φ l i ) l .
Finally, we want to point out that it may be possible to extend the results of this paper to a more general class of interacting Bose gases. More precisely, consider a class of "diagonal" interactions defined by

U l := λ V l i,j a i,j N l (φ i )N l (φ j )
with suitable assumptions on the coefficients a i,j in order to make the associated manyparticle Hamiltonian well-defined, that is self-adjoint and bounded below. Note that the mean-field gas (2.6) is a particular case of this class, in which a i,j = δ i,j (with a shift in the chemical potential). It is easy to see that the condition (2.8) is satisfied. However we shall also need the monotonicity of the mean occupation numbers N l (φ l i ) l (see Lemma 3.1), which so far we are unable to prove beyond the mean-field case.

In the next section we use the expansion (2.9) to obtain a localization criterion for the absence of single mode condensation in the kinetic energy states.

Localization and kinetic single-state BEC

First we shall prove the following lemma which is trivial for the perfect gas. For the meanfield Bose gas it was proved by Fannes and Verbeure, [8], using correlations inequalities. Here we present an alternative proof, based only on a convexity argument. Lemma 3.1 For the mean-field Bose gas, i.e. for a bosonic system with Hamiltonian (2.6), the function i → N l (φ i ) l is non-increasing.

Proof:

Let us define f : R + → R by f (t) := β -1 ln Tr e -βH l (µ;t) , where H l (µ; t)

:= H l (µ) + t(N l (φ l m ) -N l (φ l n )),
for some 1 m < n. It follows that

f ′ (0) = N l (φ l n ) -N l (φ l m ) l
and since the function f is convex, we have the following inequality

N l (φ l n ) -N l (φ l m ) l f ′ (t), (3.1)
for any t 0. Now we set t = 1 2 (E l n -E l m ). Note that with this choice t 0, since we have assumed that m < n. From the explicit expression (2.6) for H l (µ), we have

H l (µ; t) = i =m,n (E l i -µ)N l (φ l i ) + λ 2V l N 2 l + ( E l m + E l n 2 -µ)N l (φ l m ) + ( E l m + E l n 2 -µ)N l (φ l n ).
Since the mean-field term in (2.6) is symmetric with respect to a permutation of any two eigenstate indices i, j, it follows that H l (µ; t) is symmetric with respect to the exchange of m and n. Hence

f ′ (t) = Tr N l (φ l n ) -N l (φ l m ) e -βH l (µ;t) Tr e -βH l (µ;t) = 0,
which in view of (3.1) gives

N l (φ l n ) -N l (φ l m ) l 0,
and the lemma follows since m < n are arbitrary.

Let us introduce the notation

ρ l i := 1 V l N l (φ l i ) l .
With this notation we can write the standard fixed density condition (2.7) for a given density ρ, as

i ρ l i = ρ,
and so for any N ∈ N,

N i=1 ρ l i ρ.
Letting

ρ i := lim sup l→∞ ρ l i ,
and taking the infinite volume limit, we then get

N i=1 ρ i = lim sup l→∞ N i=1 ρ l i ρ.
Letting N tend to infinity, this gives ∞ i=1 ρ i ρ, and hence, for any ε > 0, there exists i 0 < ∞ such that ρ i 0 < ε. Splitting up the sum in (2.9) and using the monotonicity property (see Lemma 3.1), property (2.2) and the fact that the kinetic eigenfunctions ψ l k are normalized, we obtain

1 V l N l (ψ l k ) l = i i 0 |(φ l i , ψ l k )| 2 ρ l i + i>i 0 |(φ l i , ψ l k )| 2 ρ l i i i 0 |(φ l i , ψ l k )| 2 ρ l i + ρ l i 0 i>i 0 |(φ l i , ψ l k )| 2 ρ i i 0 |(φ l i , ψ l k )| 2 + ρ l i 0 ρ i i 0 l -d/2 ||φ l i || 1 2 + ρ l i 0 . Therefore if l -d/2 ||φ l i || 1 → 0 as l → ∞ for each i, then lim sup l→∞ 1 V l N l (ψ l k ) l ε,
and since ε is arbitrary

lim l→∞ 1 V l N l (ψ l k ) l = 0.
The above argument leads us to define the following localization criterion for the absence of single mode condensation in the kinetic energy states.

Definition 3.1 We call an eigenfunction φ l

i localized if it satisfies the following condition

lim l→∞ 1 l d/2 Λ l dx |φ l i (x)| = 0. (3.2)
Note that this localization condition is not as strong as the usual localization property, in the following sense. While, localization is frequently understood to be associated with the persistence of a pure point spectrum in the limit l → ∞, at least near the bottom of the spectrum, the presence of a pure point spectrum is not necessary for the condition (3.2) to hold for all eigenfunctions. Indeed it may happen that (3.2) is satisfied and the infinite volume Schrödinger operator has only absolutely continuous spectrum.

In [9] we conjectured that the kinetic generalized BEC observed in the random models is in fact of type III, and gave a proof in a simple case, the Luttinger-Sy model. In the above argument we proved that our conjecture is correct under the fairly weak localization hypothesis (3.2). We formulate this result in the following theorem. Theorem 3.1 Assume that the eigenfunctions φ l i are localized in the sense of (3.2) for all i. Then, for the mean-field Bose gas, no kinetic state ψ l k can be macroscopically occupied, that is

lim l→∞ 1 V l N l (ψ l k ) l = 0, (3.3)
which implies in particular that any possible kinetic generalized BEC in these models is of type III.

In this paper, we provide two classes of externals potential for which we can prove localization in the sense of (3.2). The first one is a class of random external potentials, the second involves weak external potentials.

4 Proof of the localization condition [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF]). In the standard literature localization refers to the eigenfunctions of the infinite volume Hamiltonian and requires these functions, with energies in some band, to decay very fast, in many cases exponentially. This implies that the spectrum is pure point in that band. In our case we are dealing with eigenfunctions in finite volume with energies tending to zero as the volume increases and so these bear no relation to the infinite volume eigenfunctions. In particular, our localization condition (3.2) does not imply that the spectrum is discrete in the thermodynamic limit. While we only need the L 1 norm not to diverge too fast, because our eigenfunctions depend crucially on the volume and in particular, because we do not work at a fixed energy but with volume dependent eigenvalues, we have to deal with the additional problem of controlling the finite-volume behavior. However, we find that in fact the multiscale analysis developed for the infinite volume case can be adapted to establish our localization condition.

The model studied in this section is taken from [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF]. It consists of impurities located at points of the lattice Z d , with appropriate assumptions over the single-impurity potential, mainly designed to obtain independence between regions which are sufficiently far away from each other. Let us make it more explicit by giving some definitions. In the rest of this section, we shall denote by Λ l (x) the cubic box of side l centered at x. The single-site potential f , Λ 1 (0) → R has the following properties:

1. f is bounded; 2. there is σ > 0 such that f (x) σ for all x ∈ Λ 1 (0).
The randomness in this model is given by varying the strength of each impurity. For this purpose, we define a single-site (probability) measure µ, with supp(µ) = [0, a] for a finite a. We will assume that µ is Hölder-continuous, that is for some α > 0,

sup {s,t} µ([s, t]) : 0 t -s η η α , ∀ 0 η 1 (4.1)
The random potential is then defined by

v ω (x) := k∈Z d q ω (k) f (x -k), (4.2)
where the q ω (k)'s are i.i.d. random variables distributed according to µ. We denote by (Ω, F , P) the associated probability space, and by ω ∈ Ω a particular realization of the random potential. Note that by property 2 and the fact that a < ∞, there exists a nonrandom M < ∞ such that v ω (x) < M for any x and all ω.

The one-particle random Schrödinger operator in finite volume is then given as in (2.3) by

h ω l = h 0 l + v ω l , (4.3)
where v ω l is the restriction of v ω to Λ l . The eigenfunctions and eigenvalues of h ω l are denoted by φ ω,l i and E ω,l i respectively. We denote by h ω l (x) the restriction of the Schrödinger operator - 1 2 ∆ + v ω to the region Λ l (x), with Dirichlet boundary conditions.

Before we establish the localization criterion (3.2) we prove our assumption that the eigenvalues of h ω l tend to zero as l tends to ∞: Proof: Let ν denote the limiting integrated density of states for the Hamiltonians h ω l , that is for any Borel subset

A ⊂ R + , ν(A) := lim l→∞ 1 V l ♯{i : E ω,l i ∈ A}. (4.5)
Since by ergodicity ν is nonrandom (see for example Theorem 5.18 in [START_REF] Pastur | Spectra of Random and Almost-Periodic Operators[END_REF]), it is clearly sufficient to prove that for every E > 0, ν([0, E]) > 0. To do this we start from the following inequality (see equation ( 4) in [16]),

ν([0, E]) 1 V L E ♯{i : E ω,L i E} 1 V L P{ω : E ω,L 1 E}. (4.6)
which is satisfied for any L > 0. From the min-max principle, we obtain

E ω,L 1 ε L 1 + Λ L dx |ψ L 1 | 2 v ω (x) (4.7)
where ε L 1 is the first kinetic eigenvalue and ψ L 1 the corresponding eigenfunction. Since

|ψ L 1 (x)| 2 1/V L , we have E ω,L 1 ε L 1 + 1 V L Λ L dx v ω (x) ε L 1 + A V L k∈Z d ∩Λ L q ω (k). (4.8)
where

A := Λ 1 dx f (x). Letting L := π(E/2) -1/2 so that ε L 1 = E/2, (4.6) and (4.8) give ν([0, E]) 1 V L P{ω : k∈Z d ∩Λ L q ω (k) EV L /2A}. (4.9)
Since the right-hand side of the last inequality is strictly positive, the lemma is proved.

The rest of this subsection is devoted to proving that this model satisfies our localization assumption (3.2). For this purpose we need a result from multiscale analysis which exists in various forms in the literature (see references in [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF]). For convenience here we follow the version in [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF].

Adhering to the terminology of [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF], we first define so-called "good boxes": Definition 4.1 Given x ∈ Z d , a scale l ∈ 2N + 1, an energy E, a rate of decay γ > 0, we call the box Λ l (x) (γ, E)-good for a particular realization ω of the random potential (4.2) if E / ∈ σ(h ω l (x)) and

||χ out l (h ω l (x) -E) -1 χ int l || e -γl . (4.10)
Here σ(h ω l (x)) denotes the spectrum of h ω l (x), the norm in (4.10) refers to the operator norm in L 2 (Λ l (x)), and χ int l , χ out l are the characteristic functions of the regions Λ int l (x), Λ out l (x) respectively, which we define as follows

Λ int l (x) := Λ l/3 (x), Λ out l (x) := Λ l (x) \ Λ l-2 (x).
Our proof depends crucially on the following important multiscale analysis result extracted from [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF]. We refer the reader to Theorem 3.2.2 and Corollary 3.2.6 for the general multiscale analysis argument and to Theorems 2.3.2, 2.2.3 and 2.4.1 for proving that this particular model satisfies the necessary conditions required for multiscale analysis. l k l α k-1 + 6 for k 2, and constants r > 0 and γ > 0 such that if I := [0, r],

P ω : for all E ∈ I, either Λ l k (x) or Λ l k (y) is (γ, E)-good 1 -(l k ) -2ζ , (4.11)
for all k 1 and for all x, y ∈ Z d , satisfying |x -y| > l k .

For our proof we need also the Eigenfunction Decay Inequality. We state it in a convenient form for our purpose, and refer the reader to [START_REF] Stollmann | Caught by disorder. Bound states in random media[END_REF] (Lemma 3.3.2) for a detailed proof. Note that this inequality has to be understood for a given realization ω. Proposition 4.2 Let h ω l be defined as above, and φ ω,l i to be one eigenfunction with eigenvalue

E ω,l i . Let x ∈ Λ l , such that Λ l k (x) ⊂ Λ l . If E ω,l i
does not belong to the spectrum of h ω l k (x), then the following inequality holds

χ int l k (x)φ ω,l i κ χ out l k (x)(h ω l k (x) -E ω,l i ) -1 χ int l k (x) , (4.12)
where the norms are L 2 (Λ l )-norm, and κ is a constant depending only on M.

We are now ready to prove that for our model the localization condition (3.2) is satisfied.

Lemma 4.2 Assume that h ω l is as in (4.3) with random potential given by (4.2). Then almost surely, for all i,

lim l→∞ 1 V 1/2 l Λ l dx |φ ω,l i (x)| = 0. (4.

13)

Proof: We first choose 0 < δ < 1/7 and ζ > (2d + 1)/2δ and then we take the constants α, γ and r and the sequence {l k } to be those obtained in Proposition 4.1 for this value of ζ.

For a given scale l large enough we pick k = k(l) satisfying

1 ln α ln δ ln l ln l 1 < k < 1 ln α ln (1 -δ) ln l ln(l 1 + 6) .
The fact that δ < 1/7 ensures that there exists such an integer k. Then, by Proposition 4.1, we have

l δ < l k < l 1-δ . (4.14)
Now let us define A(ω, l) to be the event in which, for all E ∈ I, for any x, y ∈ Λ l ∩ Z d such that |x -y| > l k , either Λ l k (x) or Λ l k (y) are (γ, E)-good.

We shall first use the Borel-Cantelli lemma to show that almost surely A(ω, l) occurs for all l large enough. Let us define X l := ω : A(ω, l) is not true at scale l .

Then we can write

X l := ω : ∃ E ∈ I, ∃ x, y ∈ Λ l ∩ Z with |x -y| > l k ,
such that both Λ l k (x) and Λ l k (y) are not (γ, E)-good = x,y∈Λ l ∩Z |x-y|>l k ω : ∃ E ∈ I, such that both Λ l k (x) and Λ l k (y) are not (γ, E)-good , and by Proposition 4.1 we obtain

P(X l ) l 2d (l k ) -2ζ l -2(δζ-d) ,
where the last step follows from (4.14). Since 2(δζ -d) > 1, it follows that l P(X l ) < ∞.

By the Borel-Cantelli lemma, almost surely there exists L(ω) < ∞ such that the event A(ω, l) occurs for all l > L(ω).

Since by Lemma 4.1 with probability one, E ω,l i tends to 0 as l tends to ∞, E ω,l i ∈ I for l large enough almost surely. Therefore there exists Ω ⊂ Ω with P( Ω) = 1 such that for each ω ∈ Ω there is L 1 (ω) < ∞ such that for all l > L 1 (ω) and for any x, y ∈ Λ l ∩ Z d satisfying |x -y| > l k , either Λ l k (x) or Λ l k (y) are (γ, E ω,l i )-good.

Now we take ω ∈ Ω and l > L 1 (ω) and partition the box Λ l (0) into Λ 1 l := Λ l-l k (0) and Λ 2 l := Λ l (0) \ Λ 1 l . We then split up the integral in (4.13) into the interior cube Λ 1 l and the corridor Λ

2 l Λ l dx |φ ω,l i (x)| = Λ 1 l dx |φ ω,l i (x)| + Λ 2 l dx |φ ω,l i (x)| . (4.15)
In the second term, we can use the Schwarz inequality and the fact that the eigenfunctions are L 2 (Λ l )-normalized to obtain

Λ 2 l dx |φ ω,l i (x)| |Λ 2 l | 1/2 2 d l (d-1)/2 l 1/2 k 2 d l (d-δ)/2 . (4.16)
For the first term in (4.15), we shall use the eigenfunction decay inequality (4.12) of Proposition 4.2. We cover the "interior cube" Λ 1 l by disjoints subcubes Λ j of side l k /3. Let us call {x j } their respective centers. Then for each j the cube Λ l k (x j ) is included in Λ l and Λ j coincides with Λ int l k (x j ). Using the Schwarz inequality and Proposition 4.2, we obtain for any j the estimate:

Λ j dx |φ ω,l i (x)| l d/2 Λ l dx|χ int l k (x j )φ ω,l i (x)| 2 1/2 l d/2 κ χ int l k (x)(h ω l k (x j ) -E ω,l i ) -1 χ out l k (x) 1/2 .
Hence, for any j such that Λ j is (γ, E ω,l i )-good, one has the following upper bound

Λ j dx |φ ω,l i (x)| l d/2 e -1 2 γl k l d/2 e -1 2 γl δ . (4.17)
Now, we distinguish two cases.

The first one corresponds to the situation where all cubes Λ l k (x j ) are (γ, E ω,l i )-good. It then follows directly from (4.16) and (4.17) that

l -d/2 Λ l dx |φ ω,l i (x)| 2 d l (d-δ)/2 l d/2 + l -d/2 x j ∈Λ 1 l d/2 e -1 2 γl δ 2 d l -δ/2 + 3 d (l -l δ ) d l δd e -1 2 γl δ . ( 4

.18)

The second case corresponds to the situation when there exists at least one subcube Λ l k (x j ) which is not (γ, E ω,l i )-good. Let us denote by x the center of one such bad cube. Since ω ∈ Ω and l > L 1 (ω), for x, y ∈ Λ l ∩ Z d satisfying |x -y| > l k , either Λ l k (x) or Λ l k (y) are (γ, E ω,l i )good. It therefore follows that, outside of a box of side 2l k centered at x, all other Λ l k (x j ) are (γ, E ω,l i )-good. We treat the good boxes as above, and deal with Λ 2l k (x) by using the Schwarz inequality as we did for Λ 2 l , to obtain:

Λ 1 l dx |φ ω,l i (x)| = Λ 1 l \Λ 2l k (x) dx |φ ω,l i (x)| + Λ 2l k (x) dx |φ ω,l i (x)| x j ∈Λ 1 l \Λ 2l k (x) l d/2 e -1 2 γl δ + |Λ 2l k (x)| d/2 l d/2 3 d (l -l δ ) d l δd e -1 2 γl δ + (2l) d(1-δ)/2 .
The first step in our proof is to obtain an estimate for the eigenvalue E l i . To this end, let us denote by h (n) l the restriction of the Schrödinger operator to the region B(y n , δl), with Dirichlet boundary conditions. Then we have

h l h (n) l (4.24)
in quadratic form sense (c.f. [START_REF] Reed | Methods of Mathematical Physics, IV: Analysis of Operators[END_REF], Chapter VIII, Proposition 4). From the inequality (4.23), we obtain

h (n) l h(n) l := 1 2 ∆ D + C x -y n l αn , (4.25)
where the last operator acts on L 2 B(y n , δl)). Let U : L 2 B(y n , δl) → L 2 B(0, δl 1-γn ) be the unitary transformation defined by

(Uϕ)(x) := l γn/2 ϕ(l γn (x -y n )),
where

γ n := α n /(2 + α n ). By direct computation, one can check that h(n) l = l -2γn U ĥ(n) l U -1 where ĥ(n) l := (-1 2 ∆ + C|x| αn ), acting on L 2 B(0, δl 1-γn ) . Let 0 < D l 1 D l 2 .
. . be the eigenvalues of ĥ(n) l and 0 < D 1 D 2 . . . the eigenvalues of ĥ(n) where ĥ(n) := (- The rest of our proof relies on the methods developed in [START_REF] Mac Aonghusa | [END_REF]. We start with some definitions. Let Ω t , for some t > 0, to be the set of all continuous trajectories (paths) {ξ(s)} t s=0 in R d with ξ(0) = 0, and let w t denote the normalized Wiener measure on this set. For a given x ∈ R d , we define the following characteristic function χ x,l (ξ) := 1 ξ : ξ(s) ∈ Λ l -x, for all 0 s t .

We now use the following identity (c.f. [13]), (e -th l φ l i )(x) = Since s ′ (Λ l -ξ(s ′ ) + ξ(s)) ⊂ Λ l for all s, we can now extend the domain of integration over y to Λ l and use the fact that the Wiener measure w t is normalized to obtain Next, we obtain an upper bound for the last integral in (4.28). We have dz e -tK|x-y j | α j .

For each j, B(y j ,δ) dz e -tK|x-y j | α j t -d/α j K d/α j R d dz e -|z| α j K t -d/α j ,

Lemma 4 . 1

 41 With probability one, for each i lim l→∞

Proposition 4 . 1 1 , satisfying l 1 2 and l α k- 1

 4111 Assume that h ω l is as above with random potential given by (4.2). Then for any ζ > 0 and any α ∈ 1, 2 -(4d/(4d + ζ)) , there exist a sequence {l k }, k

1 2 ∆i

 2 + C|x| αn ), acting on L 2 (R d ). Since for each i, D l i → D i as l → ∞, there are constants Di such that D lDi for all l. Using this and the operator inequalities (4.24) and (4.25) we finally getE l i D l i l -2γnDi l -2γn . (4.26)

  dx e -tv((x + ξ(s))/l) .Letting y = x + ξ(s) in the second integral we getl -d/2 Λ l dx|φ l i (x)| c d l -d/2 e tE l i (E l i ) (s)∈ s ′ (Λ l -ξ(s ′ ))}dy e -tv(y/l) .

l -d/ 2 Λ 2 Λ 1

 221 l dx|φ l i (x)| c d e tE l i (E l i ) d/4 l -d/2 1 t t 0 ds Λ l dy e -tv((y)/l) (4.28) = c d e tE l i (E l i ) d/4 l d/dz e -tv(z) .

  Now, we insert into the right-hand side of (4.27) the following bound proved in[11],|φ l i (x)| c d (E l i ) d/4, where c d := (e/π) d/4 and we obtain from (4.27) the following estimate -tv((x + ξ(s))/l) χ x,l (ξ),where the last step follows from Jensen's inequality. Therefore, integrating over Λ l with respect to x, and then changing the order of integration, yields

								t
	|φ l i (x)|	c d e tE l i (E l i ) d/4	Ω t	-w t (dξ) e	0	dsv((x + ξ(s))/l)	χ x,l (ξ)
		= c d e tE l i (E l i ) d/4	Ω t	w t (dξ) e -	1 t	0	t	ds t v((x + ξ(s))/l)	χ x,l (ξ)
	c d e tE l i (E l i ) d/4 ds e l -d/2 Ω t w t (dξ) t 1 t 0 Λ l dx |φ l i (x)| c d l -d/2 e tE l i (E l i ) d/4 Λ l dx Ω t w t (dξ)	1 t
					t
		Ω t	w t (dξ) e -	0	dsv((x + ξ(s))/l)	φ l i (x + ξ(t)) χ x,l (ξ),
	from which, since E l i is the eigenvalue of h l corresponding to φ l i , we get
	|φ l i (x)|	e tE l

i Ω t w t (dξ) e -t 0 dsv((x + ξ(s))/l) |φ l i (x + ξ(t))| χ x,l (ξ). (4.27) t 0 ds e -tv((x + ξ(s))/l) χ x,l (ξ) c d l -d/2 e tE l i (E l i ) d/4 Ω t w t (dξ) 1 t t 0 ds {x∈ s ′ (Λ l -ξ(s ′ ))}

Acknowledgments

One of the authors (Th.Jaeck) is supported by funding from the UCD Ad Astra Research Scholarship.

From that last bound and from (4.16), we get

Weak external potentials

In this section we consider a scaled external potential. Let v be a non-negative, continuous real-valued function defined on the closed unit cube Λ 1 ⊂ R d which satisfies the following two conditions:

1. There is a finite, nonempty subset of Λ 1 , D := {y j } n j=1 such that v(x) = 0 if and only if x ∈ D.

2. For each y j ∈ D there are strictly positive numbers {α j }, {c j } such that lim

We order the y j 's in such a way that 0 < α 1 α 2 . . . α n .

The one-particle Schrödinger operator with a weak (scaled) external potential in a box Λ l is defined by:

We recall that the eigenfunctions and eigenvalues of h l are denoted by φ l i and E l i respectively. The aim of this section is to prove that our localization condition (3.2) holds for this class of weak potentials.

Lemma 4.3 Let h l be as in (4.21). Then, for all i

.22)

Proof: We start by noting that in view of the condition (4.20), for any ε > 0 small enough, there exists δ > 0 such that for all j = 1, . . . , n

for all x ∈ B(y j , δ), the ball of radius δ centered at y j . Note also that by continuity there exists a constant κ > 0 such that v(x) κ, for all x ∈ Λ 1 \ n j=1 B(y j , δ) . We let K := min(κ, c 1 -ε, . . . , c n -ε) and C := max(c 1 + ε, . . . , c n + ε).

where K := K d/α 1 max j R d dz e -|z| α j , which, in view of (4.29), gives the following bound

Now, fixing t = (E l i ) -1 , we get from the last inequality and (4.28)

Since by (4.26), E l i → 0 as l → ∞, and since we have ordered the α i 's such that α 1 < α 2 < • • • < α n , there exist new constants A i such that the following bound holds for l large enough and the lemma follows since γ n < 1.


Thomas Jaeck

Valentin A Zagrebnov

Exactness of the Bogoliubov approximation in random external potentials

Keywords: Generalized Bose-Einstein Condensation, Random Potentials, Bogoliubov c-number Approximation, Bogoliubov Quasi-Averages, Berezin-Lieb Inequalities PACS: 05.30.Jp, 03.75.Fi, 67.40.-w AMS: 82B10, 82B23, 81V70

We investigate the validity of the Bogoliubov c-number approximation in the case of interacting Bose-gas in a homogeneous random media. To take into account the possible occurence of type III generalized Bose-Einstein condensation (i.e. the occurrence of condensation in an infinitesimal band of low kinetic energy modes without macroscopic occupation of any of them) we generalize the c-number substitution procedure to this band of modes with low momentum. We show that, as in the case of the one-mode condensation for translation-invariant interacting systems, this procedure has no effect on the exact value of the pressure in the thermodynamic limit, assuming that the cnumbers are chosen according to a suitable variational principle. We then discuss the relation between these c-numbers and the (total) density of the condensate.

Motivation

In 1947 Bogoliubov [1] proposed an ansatz that for large Boson systems the particle creation and annihilation operators: a * 0 , a 0 , corresponding to the zero momentum k = 0, can be replaced by complex numbers. This procedure is known as the Bogoliubov c-number approximation. It is based on the idea that these creation and annihilation operators, when divided by the square root of the volume, V , of the region Λ containing the system, can be expressed as space averages:

a # 0 √ V := 1 V Λ dx a # (x),
where a # (x) are the usual local creation and annihilation operators. Therefore for translationinvariant ergodic states these operators should converge to multiples of the identity:

a # 0 √ V → α # ,
in some weak sense, see e.g. [START_REF] Fannes | [END_REF]. These ideas were exploited to construct a various truncations of the full interacting boson Hamiltonian. We refer the reader to, e.g., [14] for a review of these models and to [22], [23] for some recent applications.

The most spectacular result derived from this ansatz was its application to a homogeneous model of a weakly interacting Bose gas [1], [14], which provided explicitly a spectrum of collective excitations satisfying the Landau criteria of superfluidity. Note that this microscopic theory of superfluidity is also based on two other Bogoluibov's ansatse: the occurrence in the weakly interacting boson system of condensation in the k = 0 mode and the truncation of the full Hamiltonian, keeping only the "dominant" terms, that is those that involves at least two particles from the condensate.

Recently, see [17], [18], the Bogoliubov approximation has been used to study interacting bosons systems in homogeneous external random potentials, where the notion of the groundstate as well as the existence of the condensation are quite subtle. The aim of our paper is to study the validity of the Bogoliubov scheme for this kind of models.

The first rigorous result concerning the Bogoliubov c-number approximation was due to Ginibre [4]. For a homogeneous boson gas with a two-body superstable interaction he proved that the Bogoliubov ansatz, supplemented by a self-consistency condensate equation which maximizes the approximated pressure with respect to the c-number α, gives the right pressure in the thermodynamic limit. A transparent and elegant proof of this and other related results has been recently given by Lieb et al [5], using, in particular, the Berezin-Lieb and the Bogoliubov convexity inequalities. The paper [5] also investigates a delicate point: namely, whether the value of the variational parameter α max maximizing the approximating pressure coincides with the total condensate density. There it was shown that the maximizer α max corresponds to the zero-mode condensate density if the gauge symmetry breaking term (quasi-average sources) of the form: √ V (η a * 0 + η a 0 ) is added to the full Hamiltonian. The idea of breaking the gauge-symmetry of the quantum Gibbs state for k = 0 is due to Bogoliubov, [2]. This forces the totality of the condensate to be concentrated in the zero-mode (ground state). It must be switched off (η = |η| e i arg η , |η| → 0 with a fixed gauge φ := arg η) after the thermodynamic limit to produce a limiting Gibbs state. The expectation defined by this state is called the Bogoliubov quasi-average with respect to this source. The quasiaverages of the operators a # 0 / √ V coincide with |α max | e ±iφ , i.e., |α max | 2 is equal to the total condensate density [5]. It has been argued in [2], [5] that this quasi-average is the only physically reliable quantity to describe Bose Einstein condensation.

We emphasize this point, because it has been known since [6] that the Bose-Einstein condensation in the gauge-invariant systems does not necessarily imply a macroscopic occupation of the ground state only. Indeed, the condensate can be spread over many (possibly infinitely many) quantum states, and in some cases, none of these states are macroscopically occupied. In all cases however, the total amount of condensate in an arbitrary small band of energy in the vicinity of the ground-state is the same, a phenomenon known as generalized condensation in the terminology of van den Berg-Lewis-Pulé [6].

After Ginibre [4] it was tempting to conjecture that the limiting value of the solution to the finite volume condensate equation yields the correct condensate density under all circumstances. By means of a counter example it was shown in [7] that this is not so. Although the Bogoliubov c-number approximation still gives the right pressure in these systems, it has been shown for the mean-field boson gas [7], that the solution α max of the condensate equation does not provide the ground-state condensate density ρ 0 , but the total amount of the generalized condensate, i.e., ρ 0 ≤ |α max | 2 . This was in a striking contrast with a general conviction that the gauge-invariant, translation invariant boson systems always manifest the total amount condensation in the ground state, and hence that the Bogoliubov c-number approximation in the zero-mode coincides with this amount.

We return to this point at the end of the present paper where we discuss our main result (Theorem 3.1) in relation to the results of [4], [7], [2], and [5].

The aim of our paper is to prove that the Bogoliubov c-number approximation to interacting Bose-gas can be extended to the case of homogeneous random external potentials (random media). Note that the arguments in [5] are valid for inhomogeneous systems and allows to treat condensation in many modes, as long as its number is much smaller than the volume V . In the case of random media this problem is more complicated. First, we have to investigate a random inhomogeneous system, albeit with non-random properties (self-averaging) in the thermodynamic limit [START_REF] Lifshitz | Introduction to the Theory of Disordered Systems[END_REF]. Secondly, since the randomness seems to force the generalised condensation to be of type III, as we disucussed in [15], the number of modes occupied by the condensation in the random potentials is a priori of the order of V .

Recall that for non-interacting (perfect) bosons systems embedded into a bona fide random potential, the generalized condensation occurs even in low-dimensional systems (see [START_REF] Kac | [END_REF]10,11] and [12], [15] for a review) even though it does not occur for the corresponding translation-invariant systems. This is caused by the fact that the one-particle density of states has Lifshitz tails, that is an extremely low density of quantum states near the bottom of the spectrum, a well-known feature of random systems widely believed to be associated with the existence of localised eigenstates. Kac and Luttinger [START_REF] Kac | [END_REF]10] conjectured that in a homogeneous random potential condensation occurs in the state with the lowest energy (ground state). Indeed, one can check this conjecture [13] for the particular case of the Luttinger-Sy model [11]. Recently in [15], it was shown that whenever condensation occurs in the random perfect Bose-gas, then there is a generalized condensation in the kinetic-energy momentum states, and the both densities have the same value. This result can be partially extended to some simple models of interacting Bose gases (mean-field models). It was proved in [16] that under a fairly weak assumption about localization this condensation is of the type III, i.e. there is no macroscopic accumulation of particles in any single momentum state. We conjecture that this holds also for general interacting systems in the presence of homogeneous random potentials.

For the above reasons, we would like to follow the general philosophy of the Bogoliubov c-number substitution ansatz and extend it to a generalized Bogoliubov approximation, in order to cover the possible case of type III generalized condensation. By this, we mean a replacement of all creation/annihilation operators corresponding to momentum states with kinetic energy ε k in the energy band 0 ≤ ε k < δ by complex numbers { √ V α ♯ k } {k:ε k <δ} . First we show that this extension of the Bogoliubov approximation applied to interacting Bose-gas in homogeneous random potentials is valid as far as the pressure is concerned (Section 3). In this case as in the case one-mode condensation the corresponding trial pressure is maximized with respect to these complex numbers and then one lets the parameter δ → 0 after the thermodynamic limit. Note that for each realization of the random potential, the system is not translation invariant and the minimizer has a random value before the thermodynamic limit. For this reason, the proof use the stationarity and ergodicity of the random potential with respect to the space translations. Finally, we discuss the variational problem established for the pressure (Section 4). In particular, we highlight the fact that the link between the c-numbers that maximize the trial pressure and the structure of the condensate is highly non-trivial. By the mean of a simple example, we show that the Bogoliubov quasi-average technique of adding external sources [2] is not "satisfactory" in this case, since we suspect that the generalised condensate should be of type III, while this procedure is able to drastically alter the fine structure of the generalised condensate. This brings us back to the discussion of the gauge symmetry breaking and the physical reliability of the Bogoliubov quasi-averages in the case of condensation [2], [5].

Model and definitions

Let {Λ l := (-l/2, l/2) d } l 1 be a sequence of hypercubes of side l in R d , d 1, centered at the origin of coordinates with volumes V l = l d . We consider a system of identical bosons, of mass m, contained in Λ l . For simplicity, we use a system of units such that = m = 1. First we define the self-adjoint one-particle kinetic-energy operator of our system by:

h 0 l := -1 2 ∆ P , (2.1)
acting in the Hilbert space H l := L 2 (Λ l ). The subscript P stands for the periodic boundary condition. We denote by {ψ l k , ε l k } k∈Λ * the set of normalized eigenfunctions and eigenvalues corresponding to operator h 0

l ψ l k (x) = 1 √ V l e ik•x , ε l k = 1 2 k 2 , (2.2)
and Λ * l is the usual dual space to Λ l , that is Λ *

l := {k ∈ R d : k 2 = (2πn) 2 /l 2 , n ∈ Z d }.
Finally, we denote by ν 0 l the finite-volume integrated density of states (IDS), that is,

ν 0 l (E) := 1 V l ♯{k ∈ Λ * l : ε l k E}, (2.3)
and we let ν 0 (E) := lim l→∞ ν 0 l (E). Note that the limiting IDS, ν 0 (E), has support on [0, ∞) and that it is known explicitly:

ν 0 (E) = C d E d/2 (the Weyl formula). Definition 2.1 We define an external random potential v (•) (•) : Ω×R d → R, x → v ω (x)
as a measurable random field on a probability space (Ω, F , P), satisfying the following conditions:

(i) v ω , ω ∈ Ω, is non-negative; (ii) p := P{ω : v ω (0) = 0} < 1.
As usual, we assume also (see e.g. Appendix B in [15]) that this field is: (iii) homogeneous (stationary) and ergodic with respect to the group {τ x } x∈R d of probability preserving translations on (Ω, F , P) ; (iv) ϕ-mixing for Σ Λ -measurable functions, where Σ Λ is the σ-algebra generated by the field

{v ω (x)} x∈Λ for Λ ⊂ R d .
Then the corresponding self-adjoint random Schrödinger operator acting in H := L 2 (R d ) is a perturbation of the kinetic-energy operator:

h ω := -1 2 ∆ ∔ v ω , (2.4)
defined as a sum in the quadratic-forms sense. The restriction to the box Λ l , is specified by the periodic boundary conditions and for regular potentials one gets the self-adjoint operator:

h ω l := -1 2 ∆ + v ω P = h 0 l ∔ v ω l , (2.5)
acting in H l , where v ω l is the restriction of v ω to Λ l . We denote by {φ ω,l i , E ω,l i } i 1 the set of normalized eigenfunctions and the corresponding eigenvalues of the random operator h ω l . We order the eigenvalues (counting the multiplicity) in such a way that E ω,l 1 E ω,l 2 E ω,l 3 . . . . Note that the non-negativity of the random potential implies that E ω,l 1 > 0. So, for convenience we assume also that in the thermodynamic limit the lowest edge of this random one-particle spectrum σ(h ω l ) satisfies the fifth condition: (v) lim l→∞ E ω,l 1 = 0 , almost surely (a.s.) with respect to the probability P.

Remark 2.1 Note that (v) is in fact an implicit condition on the random potential saying that a.s., one can find a sequence of regions (gaps) with v ω (x) = 0, with volumes tending to infinity in the van Hove sense. Now, we turn to the many-body problem. Let F l := F l (H l ) be the symmetric Fock space constructed over H l . Then H 0 l := dΓ(h ω l ) denotes the second quantization of the oneparticle Schrödinger operator h ω l in F l . For simplicity, we omit the explicit mention of the randomness of the Hamiltonians and all related quantities, unless this is necessary for the sake of clarity. Since for any ω ∈ Ω the one-particle eigenstates {φ i := φ ω,l i } i ≥1 of h ω l form a basis in H l , the operator H 0 l acting in F l can be expressed as:

H 0 l = i 1 E ω,l i a * (φ i )a(φ i ) = i 1 E ω,l i N l (φ i ) . (2.6)
Here a * (φ i ), a(φ i ) are the creation and annihilation operators, satisfying the boson Canonical Commutation Relations, and N l (φ i ) is the particle-number operator in the state φ ω,l i . Note that, since [h 0 l , h ω l ] = 0, the Hamiltonian (2.6) cannot be expressed as a function of the operators N l (ψ l k ) in the kinetic-energy eigenstates (2.2).

(vi) Below we assume that the particles interact through a suitable non-negative two-body translation-invariant potential Φ(x, y) := u(|x -y|). More precisely, we assume that the function u has a continuous, bounded Fourier transformation û(q), and there is γ < ∞ such that |û(q)| < γ for all q ∈ Λ * l and for all l. For example, one can choose u ∈ L 1 (R d ).

Then the second quantization of interaction: U l := dΓ(Φ) has a simple form in the translation-invariant basis {ψ k := ψ l k } k∈Λ * l of the kinetic-energy operator h 0 l :

U l = 1 2V l q,k,k ′ ∈Λ * l û(q) a * (ψ k+q )a * (ψ k ′ -q )a(ψ k ′ )a(ψ k ) . (2.7)
The full Hamiltonian with the chemical potential µ included has the form:

H l (µ) := H 0 l -µN l + U l (2.8)
Note that the creation and annihilation operators in the interaction term (2.7) are in the momentum eigenstates {ψ k } k∈Λ * l , although the perfect Bose-gas Hamiltonian (2.6) is not diagonal when expressed in this basis.

Byl we denote below the grand-canonical equilibrium state defined by the Hamiltonian H l (µ):

A l (β, µ) := 1 Ξ l (β, µ) Tr F l exp(-βH l (µ)),
and by p l (β, µ) its associated grand-canonical pressure

p l (β, µ) := 1 βV l ln Ξ l (β, µ), (2.9)
where

Ξ l (β, µ) := Tr F l exp(-βH l (µ))
is the corresponding grand-canonical partition function.

It is known that the pressure of the corresponding non-random model (i.e. for v ω (x) = 0) with a bona fide interaction exists and is independent of the boundaries condition for a large class of them, including the periodic case, see e.g. [START_REF] Robinson | The thermodynamic pressure in quantum statistical mechanics[END_REF]. The proof of this statement consists essentially in showing the existence of the Dirichlet pressure using sub-additivity

p D Λ (β, µ) p D Λ ′ (β, µ) + p D τxΛ ′′ (β, µ)
where Λ ′ , Λ ′′ are disjoints subsets of Λ, and τ x denotes translation by x. The exact value of x is chosen according to the usual tempering condition required of the two-body interaction potential (vi). Then, using translation invariance of the non-random model, one obtains

p D Λ (β, µ) p D Λ ′ (β, µ) + p D Λ ′′ (β, µ) , (2.10)
and the boundeness of the pressure, which is provided by the superstability of the interaction (u ≥ 0, (vi)), leads to the existence and finiteness of the limiting pressure for any µ. Then, one can show using functional integration techniques, see [20], that the others boundary conditions converge to the same limit.

The last part of this prove can be carried through verbatim in the presence of the external random potential. However, because of the lack of translation invariance in the random case, the inequality (2.10) for the Dirichlet pressure is modified as follows:

p D,ω Λ (β, µ) p D,ω Λ ′ (β, µ) + p D,ω τxΛ ′′ (β, µ) = p D,ω Λ ′ (β, µ) + p D,τxω Λ ′′ (β, µ) . (2.11)
We used the stationarity of the random potential in the last identity. To prove the the existence of the thermodynamic limit one can use the Kingman sub-additive ergodic theorem, see [21]:

Proposition 2.1 Let τ be measure preserving transformation of the probability space (Ω, F , P) and {g n } n≥1 be a sequence of functions g n ∈ L 1 (Ω, F , P) satisfying the condition:

g n+m (ω) ≤ g n (ω) + g m (τ n ω) (2.12)
Then one gets that a.s.lim

n→∞ g n (ω)/n = g(ω) , (2.13)
where the function g(ω) is τ -invariant: g(τ s ω) = g(ω). If in addtion, the functions g n are ergodic, it follows that the limit g(ω) is a.s. non random.

3 Generalized Bogoliubov c-numbers approximation

Existence of the approximating pressure

Following the Bogoliubov approximation philosophy, we want to replace all creation/annihilation operators in the momentum states ψ k with kinetic energy less than some δ > 0 by c-numbers.

To this end let I δ ⊂ Λ * l be the set of all replaceable modes, that is,

I δ := k ∈ Λ * l : k 2 /
2 δ , and we denote n δ := ♯{k : k ∈ I δ }. The number of quantum states n δ is of the order V l , since by definition of the IDS (2.3): n δ = V l ν 0 l (δ). Let H δ l to be the subspace of H l spanned by the set of ψ l k with k ∈ I δ , and P δ be orthogonal projector onto this subspace. Hence, we have a natural decomposition of the total space H l and the corresponding representation for the associated symmetrised Fock space:

H l = H δ l ⊕ H ⊥ l , F l ≈ F δ l ⊗ F ⊥ l . (3.1)
Then we proceed to the Bogoliubov substitution a k → c k and a * k → c k for all k ∈ I δ , which provides an approximating Hamiltonian that we denote by H low l (µ, {c k }). The meaning of the superscript low will become clear in the next section. We postpone to Appendix A a description of the explicit form of this operator. The partition function and the corresponding pressure for this approximating Hamiltonian have the form:

Ξ low l (µ, {c k }) := Tr F ⊥ l e -βH low l (µ,{c k }) , (3.2)
p low l,δ (µ, {c k }) := 1 V l ln Ξ low l (µ, {c k }) . (3.3)
The principal result of the present paper is the following main theorem:

Theorem 3.1 The c-numbers substitution for all operators in the energy-band I δ does not affect the original pressure (2.9) in the following sense:

a.s.-lim l→∞ p l (β, µ) = lim δ↓0 lim inf l→∞ max {c k } p low l,δ (µ, {c k }) = lim δ↓0 lim sup l→∞ max {c k } p low l,δ (µ, {c k }) . (3.4)
Note that the number of the substituted modes is of order V , since we let δ ↓ 0 after the thermodynamic limit.

Proof of the main Theorem

Our method is a generalisation of the one invented in [5]. For the convenience of the reader, we postpone the proof of some technical lemmas to the next section.

First we define a family of normalized coherent vectors in the Fock space F l , with the vacuum state |0 :

|c = k∈I δ e -|c k | 2 /2+c k a * k |0 , (3.5)
labeled by the set of complex numbers {c k } k∈I δ . With the help of vectors (3.5), we define the lower symbol A low for any operator A in F l by the partial inner product:

A low ({c k }) := c|A|c ,
which is an operator in F ⊥ l , see (3.1). Similarly, using the tensor structure of (3.1), we can define on F ⊥ l the upper symbol A up of the operator A in F l through the integral representation A =:

C n δ d 2 c 1 . . . d 2 c n δ A up ({c k }) |c c| .
Here d 2 c j := d Re(c j )d Im(c j )/π and |c c| := k∈I δ |c k c k | is the projector on the coherent vectors (3.5). Similar to the one-mode case one has the completeness property

C n δ d 2 c 1 . . . d 2 c n δ |c c| = I.
Note that, contrary to the lower symbols, the upper symbol does not necessarily exists, and it may not be unique. Although, this poses no problem in our case, since the existence (though not the unicity) of the upper symbols follows from the fact that the Hamiltonian (2.8) is polynomial in creation/annihilation operators. We postpone the explicit expressions of these upper symbols to Appendix A.

We then define two approximating Hamiltonians, that we denote by H low l (µ, {c k }) and H up l (µ, {c k }). They are obtained by replacing all creations and annihilations operators a ♯ k by their lower, respectively upper, symbols. We refer the reader to Appendix A for the details and the explicit expressions. Note that H low l (µ, {c k }) is obtained simply by replacing all operators {a ♯ k } k∈I δ by the corresponding complex numbers {c ♯ k } k∈I δ . Formally it corresponds to the Hamiltonian obtained by the standard Bogoliubov approximation, that is why it appears in Theorem 3.1.

In a way similar to (3.2), (3.3), one can define by Ξ up l (µ, {c k }) the partition function for the Hamiltonian H up l (µ, {c k }), and by p up l,δ (µ, {c k }) the corresponding pressure.

Finally, we denote bylow andup the grand-canonical equilibrium states related to the following (integrated) partition functions:

Ξ low l (µ) := C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) , (3.6)
Ξ up l (µ) :=

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH up l (µ,{c k }) , (3.7)
and we denote the associated pressures by p low l,δ (µ), p up l,δ (µ).

Now one can mimic the arguments of [5] and extend them to the multi-mode projections |c c| case to produce the Bogoliubov-Peierls and the Berezin-Lieb inequalities for (3.6) and (3.7). These inequalities yield respectively lower and upper estimates for the grand partition function:

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) Ξ l (µ) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH up l (µ,{c k }) . (3.8)
By a straightforward generalization of the arguments in [5] to the case of multi-mode coherent projection |c c|, we obtain for any { c k } k∈I δ the bound

Tr F ⊥ l e -βH low l (µ,{ c k }) Ξ l (µ)
on the integrand in the left-hand side of (3.8). This in particular implies max

{c k }∈C n δ Tr F ⊥ l e -βH low l (µ,{c k }) Ξ l (µ) , (3.9)
i.e. the estimate of the grand partition function from below.

To find a similar bound for the right-hand side of (3.8) from above, we note that H low l (µ, {c k }) and H up l (µ, {c k }) are related by

H up l (µ, {c k }) = H low l (µ, {c k }) + κ(µ, {c k }) ,
see Appendix A, equation (A.22) for an explicit expression of κ(µ, {c k }). If this is combined with the Bogoliubov convexity inequality: ln ln Tr F l e -βH l (µ) ln

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH up l (µ,{c k }) -ln C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l -κ(µ, {c k })e -βH up l (µ,{c k }) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH up l (µ,{c k }) , ( 3
C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) -κ(µ, {c k }) up . (3.11)
Using the orthogonal projection P δ : H l → H δ l , and in view of (A.22), one can estimate the last term in (3.11) explicitly:

-κ(µ, {c k })
Tr(h ω l -µ)P δ (3.12)

-γ ν 0 l (δ) + V l 2 ν 0 l (δ) 2 + V l ν 0 l (δ)ν 0 l (2δ) + γ 2 4 V l + 2ν 0 l (δ) + 2ν 0 l (2δ) k∈I δ |c k | 2 + γ 2 2ν 0 l (δ) + 2ν 0 l (δ) k∈I c δ a * k a k Tr (h ω l -µ)P δ -γν 0 l (δ) 1 -4V l ν 0 l (2δ) + V l 2 ν 0 l (δ) + V l ν 0 l (2δ) + 4γν 0 l (2δ) k∈I δ |c k | 2 -1 + k∈I c δ a * k a k .
Here I c δ := Λ * l \ I δ . Then taking into account the upper symbol of the total number operator, we find that

H up l (µ, {c k }) + a k∈I δ (|c k | 2 -1) + k∈I c δ a * k a k = H up l (µ -a, {c k }) , (3.13)
which together with equation (3.11) provides the following estimate ln Tr F l e -βH l (µ) ln

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) (3.14) + Tr (h ω l -µ)P δ -γν 0 l (δ) 1 -4V l ν 0 l (2δ) + V l 2 ν 0 l (δ) + V l ν 0 l (2δ) + 4γν 0 l (2δ)∂ µ ln C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) (3.15)
To finish the proof, we need three lemmas. We postpone their proofs to the next section.

Lemma 3.1 Suppose that the system of interacting bosons (2.8) has a bounded limiting particle density ρ(µ) for any fixed µ ∈ R:

ρ(µ) := ∂ µ p(µ) := ∂ µ lim l→∞ p l (β, µ) < ∞ (3.16)
see (2.9). Then one gets the estimates:

lim sup δ↓0 lim sup l→∞ 1 βV l ∂ µ p low l,δ (µ) ρ(µ), (3.17)
lim sup

δ↓0 lim sup l→∞ 1 βV l ∂ µ p up l,δ (µ) ρ(µ) . (3.18)
Next, we relate the integrated pressure p low l,δ (µ) defined by (3.6) to the maximum of the corresponding integrand. Lemma 3.2 For any α > 1, one has the estimate:

1 βV l ln C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) (3.19) 1 βV l ln max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) - 1 βV l ln(1 -1/α) + ν 0 l (δ) β ln(α∂ µ p low l,δ (µ)) + 1 β ν 0 l (δ) - 1 2β ln V l V l - ν 0 l (δ) β ln ν 0 l (δ) - 1 2βV l ln ν 0 l (δ) .
Notice that above statements are independent of the possible presence of random potentials. The next Lemma serves to control (ergodic) random external potentials.

Lemma 3.3 Taking into account our assumptions on the random potentials, see Section 2, one gets the following inequality:

lim sup l→∞ 1 V l Tr(h ω l -µ)P δ ν 0 (δ) (δ -µ) + E v ω (0) .
Here E(•) denotes the expectations in the probability space (Ω, F , P).

Returning back to the proof of Theorem 3.1, we get from (3.9) and (3.14) the estimates:

max {c k } p low l,δ (µ, {c k }) p l (µ) p low l,δ (µ) + 1 βV l Tr (h ω l -µ)P δ -γν 0 l (δ) 1 -4V l ν 0 l (2δ) + V l 2 ν 0 l (δ) + V l ν 0 l (2δ) + 4γν 0 l (2δ) 1 βV l ∂ µ ln C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) .
By Lemma 3.2, this implies for any configuration ω the estimates:

max {c k } p low l,δ (µ, {c k }) p l (µ) max {c k } p low l,δ (µ, {c k }) + K ω (l, δ) , (3.20)
where the random parameter K ω (l, δ) is given by

K ω (l, δ) : = 1 βV l Tr (h ω l -µ)P δ -γν 0 l (δ) 1 -4V l ν 0 l (2δ) + V l 2 ν 0 l (δ) + V l ν 0 l (2δ) + 4γν 0 l (2δ) 1 βV l ∂ µ ln C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) .
Note that, by Lemmas 3.1 and 3.3, we can control this error term since for any configuration one gets:

lim δ↓0 lim inf l→∞ K ω (l, δ) = lim δ↓0 lim sup l→∞ K ω (l, δ) = 0 .
Hence, (3.20) yields lim sup

l→∞ max {c k } p low l,δ (µ, {c k }) p(µ) lim sup l→∞ max {c k } p low l,δ (µ, {c k }) + lim inf l→∞ K ω (l, δ) lim sup δ↓0 lim sup l→∞ max {c k } p low l,δ (µ, {c k }) p(µ) lim inf δ↓0 lim sup l→∞ max {c k } p low l,δ (µ, {c k }) ,
which proves the first equality in (3.4) in Theorem 3.1. The second one can be proven in a similar way.

Proofs of technical results

Recall that the finite-volume IDS (2.3): ν 0 l (δ) ≥ 0, converges in the limit l → ∞ to the Weyl formula ν 0 (δ) = C d δ d/2 , uniformly in δ on any finite interval 0 ≤ δ ≤ E. Below we use this uniformity in a systematic way.

Proof of Lemma 3.1 Since

H low l (µ, {c k }) + a k∈I δ |c k | 2 + k∈I c δ a * k a k = H low l (µ -a, {c k }) , (3.21)
the estimate (3.12) yields:

H up l (µ, {c k }) H low l (µ + 4γν 0 l (2δ), {c k }) -Tr (h ω l -µ)P δ -γν 0 l (δ) 1 + V l 2 ν 0 l (δ) + ν 0 l (2δ) .
Applying now the Bogoliubov-Peierls and the Berezin-Lieb inequalities, we obtain:

p low l,δ (µ) p l (µ) p up l,δ (µ) p low l,δ (µ + 4γν 0 l (2δ)) + M ω (l, δ) V l ,
where

M ω (l, δ) := Tr (h ω l -µ)P δ + γν 0 l (δ) 1 + V l 2 ν 0 l (δ) + ν 0 l (2δ) . (3.22)
Consequently, for any configuration ω one gets in the limit the estimates:

lim sup l→∞ p low l,δ (µ) p(µ) lim inf l→∞ p up l,δ (µ) lim sup l→∞ p up l,δ (µ) lim inf l→∞ p low l,δ (µ + 4γν 0 l (2δ)) + lim inf l→∞ M ω (l, δ) V l . ( 3
(2δ)) + lim inf l→∞ 1 V l M ω (l, δ, µ) -p(µ) 1 t p(µ + t + 4γν 0 (2δ)) + lim inf l→∞ 1 V l M ω (l, δ, µ) -p(µ) .
Since by (3.22)

p up l,δ (µ -4γν 0 l (2δ)) -lim inf l→∞ 1 V l M ω (l, δ, µ -4γν 0 l (2δ)) .
Applying the estimate (3.23) to the last inequality once more, we get lim sup l→∞ ∂ µ p low l,δ (µ)

1 t p(µ + t) -p(µ -4γν 0 (2δ)) -lim inf l→∞ 1 V l M ω (l, δ, µ -4γν 0 l (2δ))
and in view of the limit (3.22) and Lemma 3.3, the result follows by letting δ ↓ 0 and then, letting t ↓ 0.

Proof of Lemma 3.2 Let

C n δ ξ := {z ∈ C n δ : |z| 2 ξ}.
and we denote the volume of this ball by Vol(C

n δ ξ ) = π n δ ξ n δ /n δ Γ(n δ ).
We then obtain the following bound

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) = C n δ ξ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) + C n δ \C n δ ξ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) Vol(C n δ ξ ) π n δ max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) + 1 ξπ n δ C n δ d 2 c 1 . . . d 2 c n δ k∈I δ |c k | 2 Tr F ⊥ l e -βH low l (µ,{c k }) Vol(C n δ ξ ) π n δ max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) + 1 ξπ n δ k∈I δ |c k | 2 low C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) .
Here we used the expectation • low defined by the integrated partition function Ξ low l (µ), see (3.6). Notice that, using the explicit form of the lower symbol for the total particle number operator, see (3.21), we can represent the last integral in the form:

C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) Vol(C n δ ξ ) π n δ max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) + 1 ξπ n δ V l ∂ µ p low l,δ (µ) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) = ξ n δ n δ Γ(n δ ) max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) + 1 ξπ n δ V l ∂ µ p low l,δ (µ) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) .
Thus, we get that

1 - V l ξ ∂ µ p low l,δ (µ) C n δ d 2 c 1 . . . d 2 c n δ Tr F ⊥ l e -βH low l (µ,{c k }) ξ n δ n δ Γ(n δ ) max {c k } Tr F ⊥ l e -βH low l (µ,{c k }) .

Concluding remarks

We discuss here the meaning of our main result and in particular the way to interpret the solutions of the variational problem established in Theorem 3.1. Note that the theorem is established for a random system in the thermodynamic limit. Its aim is to take into account the possibility of type III generalized condensation, which we believe is favored by the randomness even in interacting systems.

First we recall a result established in [5], cf. Section 1.1. For a homogeneous system and the single-mode substitution at k = 0, the solution of the variational problem gives the total condensate density in the mode k = 0, if one adds to the Hamiltonian the zero-mode gauge-breaking term (quasi-average sources):

H l (µ; η) := H l (µ) + V l η a 0 + η a * 0 .
This means that after the Bogoliubov c-number substitution the solution α max,l (β, µ; η) of the (finite-volume) variational problem not only provides the right pressure in the thermodynamic limit, but it also coincides with quasi-averages that give the total amount of the condensate in the zero mode:

lim |η|↓0 lim l→∞ α max,l (β, µ; η) = lim |η|↓0 lim l→∞ a * 0 a 0 /V l l (β, µ; η) .
Herel (β, µ; η) is the equilibrium state defined by H l (µ; η).

Using a simple example, we discuss the relevance of this quasi-average approach to more subtle cases of the condensation of type II and III. We show that the Bogoliubov quasiaverage sources breaking the gauge invariance [2] are able to alter the fine structure of the condensate reducing it to one-mode (type I).

To see this, consider the perfect Bose-gas in a cubic three-dimensional anisotropic parallelepiped Λ l := V αx l × V αy l × V αz l , with periodic boundary condition and α x ≥ α y ≥ α z , α x + α y + α z = 1. The Hamiltonian is:

H 0 l (µ) := k∈Λ * l (ε k -µ)a * k a k , (4.27)
with quadratic spectrum and ε k=0 = 0.

It is well known that this system exhibits a generalised condensation of type II for α x = 1/2 and of type III for α x > 1/2 for a standard critical density ρ c , whereas for α x < 1/2, the whole condensate is sitting in the mode k = 0 , i.e, in the ground state (type I) [6]. Consider the system (4.27) with the quasi-average source in a single mode k: V l (e β(ε k -µ) -1)

-1 + 1 V l k = k 1 e β(ε k -µ) -1 + |η| 2 (ε k -µ) 2 .
+ 1 2V l k∈I c δ k ′ ∈I δ q: k+q∈I c δ k ′ -q∈I δ û(q) c k ′ -q c k ′ a * k+q a k (A.16) + 1 2V l k∈I c δ k ′ ∈I δ q: k+q∈I δ k ′ -q∈I δ û(q) c k+q c k ′ -q c k ′ a k (A.17) + 1 2V l k∈I c δ k ′ ∈I c δ q: k+q∈I c δ k ′ -q∈I c δ û(q) a * k+q a * k ′ -q a k ′ a k (A.18)

+ 1 2V l k∈I c δ k ′ ∈I c δ q: k+q∈I δ k ′ -q∈I c δ û(q) c k+q a * k ′ -q a k ′ a k (A.19) + 1 2V l k∈I c δ k ′ ∈I c δ q: k+q∈I c δ k ′ -q∈I δ û(q) c k ′ -q a * k+q a k ′ a k (A.20) + 1 2V l k∈I c δ k ′ ∈I c δ q: k+q∈I δ k ′ -q∈I δ û(q) c k+q c k ′ -q a k ′ a k (A.21)
Now, we give an explicit form of the upper symbols. We recall the general form of this symbols for polynomials in the creation/annihilation operators of some mode k ∈ I δ :

(a k ) up = c k , (a * k ) up = c k , (a k a k ) up = (c k ) 2 , (a * k a k ) up = (c k ) 2 (a * k a k ) up = |c k | 2 -1, (a * k a * k a k a k ) up = |c k | 4 -4|c k | 2 + 2
Note that, since the interaction term of the Hamiltonian term considered on its own does have a momentum conservation law, it is not possible to get exactly three out of four operators in the same mode k. In view of this, it can be seen that the lower and upper symbols of the Hamiltonian will differ only when two or four operators in the same mode appears, that is only terms (A.1), (A.9), (A.12), (A.13), (A.15), (A. 16) and (A.17) differs in both approximating Hamiltonians. Splitting further the sums in these terms leads finally to the final upper symbol of the Hamiltonian H up l (µ, {c k }) = H low l (µ, {c k }) + κ(µ, {c k }), where

κ(µ, {c k }) = k∈I δ i 1 |(φ i , ψ k )| 2 (E i -µ) (A.22) + - 1 2V l 2 k∈I δ û(0) + k,k ′ ∈I δ k =k ′ û(0) + q =0 k∈I δ ∩I-q û(q) + 1 2V l 4 k∈I δ û(0)|c k | 2 + k,k ′ ∈I δ k =k ′ û(0)(|c k | 2 + |c k ′ | 2 ) + q =0 k∈I δ ∩I-q k ′ ∈I δ ∩I-q k ′ -k=q û(q)(|c k | 2 + |c k ′ | 2 ) + 1 2V l 2 k∈I δ û(0) k ′ ∈I c δ a * k ′ a k ′ + q =0 k∈I δ k ′ ∈I c δ ∩I+q k ′ -k=q û(q)a * k a k + q =0 k∈I c δ ∩I-q k ′ ∈I δ k ′ -k=q û(q)a * k a k .

. 10 )

 10 then (3.8) and (3.10) provide the upper bound:

  H 0 l (µ; η) := H 0 l (µ) + V l η a k + η a * k . (4.28)Then for a fixed density ρ, the finite-volume equation which defines the corresponding chemical potential µ = µ l (ρ, η) takes the form:ρ = ρ l (β, µ, η) := 1 V l k∈Λ * l a * k a k 0 l (β, µ, η) (4.29) = 1

  and Lemma 3.3 for any configuration ω one gets:

	Similarly one proves the first estimate (3.17). Again using the convexity and (3.23) we
	obtain
	lim sup l→∞	∂ µ p low l,δ (µ)	1 t	lim sup l→∞	p low l,δ (µ + t) -lim inf l→∞	p low l,δ (µ)
			1 t	l→∞ lim sup	p l (µ + t)
			-lim sup l→∞
				lim δ↓0	lim inf l→∞	1 V l	M ω (l, δ) = 0 ,
	we obtain from (3.24)
		lim sup δ↓0	lim sup l→∞	∂ µ p up l,δ (µ)	1 t	p(µ + t) -p(µ)	(3.25)
	which is valid for any t > 0. Letting t ↓ 0 leads to the second estimate (3.18).

Acknowledgments

One of the authors (Th.Jaeck) is supported by funding from the UCD Ad Astra Research Scholarship. We are thankful to J.V.Pulé and J-B Bru for very useful remarks and suggestions.

If we put ξ := αV l ∂ µ p low l,δ (µ) for some α > 1, and use the Stirling formula for large n δ , then the coefficient in the right-hand side can be estimated as:

α∂ µ p low l,δ (µ)

V l ν 0 l (δ)+1/2 e -n δ .

Hence, one finally obtains the estimate:

which leads to the desired result

Proof of Lemma 3.3 First we note that since the projection P δ is constructed with respect to the basis of eigenvectors of h 0 l , one gets:

and consequently:

Thus, the ergodic theorem yields:

To investigate a possible condensation, we must take the thermodynamic limit in the righthand side of (4.29), and then switch off the source, that is let |η| → 0. Let us denote by I(β, µ) the limit of ρ l (β, µ, η = 0), that is the limiting density function of the gauge-invariant system,

with critical density ρ c := sup µ<0 I(µ).

Now we have to distinguish two cases: (i) For any k such that lim l→∞ ε k > 0, we obtain from (4.29)

i.e. the quasi-average coincides with the average and we return to the analysis of the condensate equation (4.29) for η = 0. This gives again all possible types of condensation as a function of α x . (ii) On the other hand, if k is such that lim l→∞ ε k = 0, then the condensate equation (4.29) yields for the quasi-average of the total particle density:

(4.30)

If ρ ≤ ρ c , then the asymptotic solution of (4.30) is µ ∞ (ρ) = lim η→0 lim l→∞ µ l (ρ, η) < 0 and there is no condensation in any mode.

. By explicit calculation one also gets that only k-mode quasi-average is non-zero:

i.e. for any α x the condensation is type I. Recall that the only condition on k is that the corresponding eigenvalue ε k vanishes in the infinite volume limit. Since

by virtue of (4.31) and (4.32) we see that the Bogoliubov quasi-average procedure not only transforms the generalised condensates of type II or III into a one-mode condensate (i.e., type I), but this mode does not even need to be the ground-state. Therefore, using the quasi-average approach [2], one can force the condensate to be in any given mode k, as long as its energy ε k vanishes in the limit l → ∞.

Appendix A The approximating Hamiltonians

In this section, we provide an explicit form of the upper and lower symbols for the Hamiltonian (2.8). For a short-hand we put E ω,l i = E i . We note that, as the coherent vector (3.5) is defined as a tensor product of one-mode coherent states, its effect on each creation/annihilation operator a ♯ k is independent of all the others modes operators. First, we give an explicit form of the lower symbol of the full Hamiltonian (2.8):