Page 1

arXiv:1006.3924v1 [math-ph] 20 Jun 2010

On the nature of Bose-Einstein condensation

enhanced by localization

Thomas Jaeck1, Joseph V. Pul´ e2

School of Mathematical Sciences, University College Dublin

Belfield, Dublin 4, Ireland

Valentin A. Zagrebnov3

Universit´ e de la M´ editerran´ ee (Aix-Marseille II),

Centre de Physique Th´ eorique - UMR 6207, Luminy - Case 907

13288 Marseille, Cedex 09, France

Abstract

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.

Keywords: Generalized Bose-Einstein Condensation, Random Potentials, Integrated Den-

sity of States, Multiscale Analysis, Diagonal Particle Interactions.

PACS: 05.30.Jp, 03.75.Fi, 67.40.-w

AMS: 82B10, 82B23, 81V70

1PhD student at UCD and Universit´ e de la M´ editerran´ ee (Aix-Marseille II, France),

e-mail: Thomas.Jaeck@ucdconnect.ie, phone: +353 1 7162571

2e-mail: joe.pule@ucd.ie, phone: +353 1 7162568

3e-mail: Valentin.Zagrebnov@cpt.univ-mrs.fr, phone: +33 491 26 95 04

1

Page 2

1 Introduction

It can be easily seen from the explicit formula for the occupation numbers in the non-

interacting (perfect) Bose gas that, the condition for Bose-Einstein Condensation (BEC) to

occur, is that the density of states of the one particle Schr¨ odinger 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 intro-

duces 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

[4], are able to produce BEC. In both cases, it is possible to obtain condensation even in

dimension 1 or 2.

While BEC has historically been associated with the macroscopic occupation of the ground

state only, it was pointed out in [5] 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 condensa-

tion 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 Pul´ e,

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 macro-

scopic 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 Schr¨ odinger 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-

2

Page 3

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 Schr¨ odinger operator can be correctly described

as of “Bose-Einstein” type in the traditional sense. This opens up the possibility of formu-

lating 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 ex-

ternal 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).

2 Notation and models

Let {Λl:= (−l/2,l/2)d}l?1be a sequence of hypercubes of side l in Rd, centered at the origin

of coordinates with volumes Vl= ld. We consider a system of identical bosons, of mass m,

contained in Λl. 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:

h0

l:= −1

2∆D,(2.1)

3

Page 4

acting in the Hilbert space Hl:= L2(Λl), where ∆ is the usual Laplacian. The subscript D

stands for Dirichlet boundary conditions. We denote by {ψl

eigenfunctions and eigenvalues corresponding to h0

(counting multiplicity) as 0 < εl

following bound

k,εl

k}k?1the set of normalized

l. By convention, we order the eigenvalues

3... . Note that all kinetic states satisfy the

1? εl

2? εl

|ψl

k(x)| ? l−d/2. (2.2)

Next we define the Hamiltonian with an external potential

hl:= h0

l+ vl, (2.3)

also acting in Hl, where vl: Λl?→ [0,∞) is positive and bounded. Let {φl

be respectively the sets of normalized eigenfunctions and corresponding eigenvalues of hl.

Again, we order the eigenvalues (counting multiplicity) so that El

that the non-negativity of the potential implies that El

lower end of the spectrums of h0

This assumption will be proved for the models considered in this paper.

i}i?1and {El

i}i?1

1? El

2? El

3... . Note

1> 0. We shall also assume that the

land hlcoincide in the limit l → ∞, that is, liml→∞El

1= 0.

Now, we turn to the many-body problem. Let Fl:= Fl(Hl) be the symmetric Fock space

constructed over Hl. Then Hl:= dΓ(hl) denotes the second quantization of the one-particle

Schr¨ odinger operator hlin Fl. Note that the operator Hlacting in Flhas the form:

Hl=

?

i?1

El

ia∗(φl

i)a(φl

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 ∈ Hl. Then, the grand-canonical

Hamiltonian of the perfect Bose gas in an external potential is given by:

H0

l(µ) := Hl− µNl =

?

i?1

(El

i− µ) Nl(φl

i) (2.5)

where Nl(φ) := a∗(φ)a(φ) is the operator for the number of particles in the normalized state

φ, Nl :=

?

chemical potential.

iNl(φl

i) is the operator for the total number of particles in Λl and µ is the

As well as for the perfect gas, the results in this paper hold also when a mean-field

interaction is introduced, that is, for a Bose gas with the following many particle Hamiltonian

Hl(µ) := H0

l(µ) +

λ

2VlN2

l, (2.6)

where λ is a non-negative parameter.

We recall that the thermodynamic equilibrium Gibbs state ?−?lassociated with the Hamil-

tonian Hl(µ) is defined by:

?A?l:=TrFl{exp(−βHl(µl))A}

TrFlexp(−βHl(µl))

where the value of µ, µlis determined by fixing the mean density ρ > 0:

1

Vl?Nl?l= ρ.(2.7)

4

Page 5

When referring specifically to the perfect Bose gas state, we shall use the notation ?−?0

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.

l.

A normalized single particle state ϕ is macroscopically occupied if

lim

l→∞

1

Vl?Nl(ϕ)?l> 0

and in particular there is condensation in the ground state if

lim

l→∞

1

Vl?Nl(φl

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 φl

iif

lim

δ↓0lim

l→∞

1

Vl

?

i:El

i?δ

?Nl(φl

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 com-

monly known notion of BEC as condensation in the ground state only), type II condensation,

when condensation occurs in a 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 φl

kinetic energy states ψl

that for models which are diagonal in the occupation numbers of the eigenstates of the

Hamiltonian (2.3), the generalized BEC in the kinetic states is never less then that in the

eigenstates of the single particle Hamiltonian coincide. To be more precise, we proved that

ibut in the

k, and for both random and weak positive potentials we established

lim

δ↓0lim

l→∞

1

Vl

?

k:εl

k<δ

?Nl(ψl

k)?l? lim

δ↓0lim

l→∞

1

Vl

?

i:El

i<δ

?Nl(φl

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 φl

densation in the kinetic energy states ψl

macroscopically occupied.

iso that the con-

kis of type III, that is no kinetic energy state is

It is easy to see that both the perfect and mean-field gases satisfy the following commu-

tation relation

[Hl(µ),Nl(φl

j)] = 0, for all j. (2.8)

5

Page 6

This property implies that ?a∗(φl

relation between the mean occupation numbers for the ψl

i)a(φl

j)?l = 0 if i ?= j and allows us to obtain a simple

k’s and φl

k’s:

1

Vl?Nl(ψl

k)?l=1

Vl?a∗(ψl

k)a(ψl

k)?l

=

1

Vl

?

i,j

(φl

i,ψl

k)(φl

j,ψl

k)?a∗(φl

i)a(φl

j)?l

(2.9)

=

1

Vl

?

i

|(φl

i,ψl

k)|2?Nl(φl

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

Ul :=

λ

Vl

?

i,j

ai,jNl(φi)Nl(φj)

with suitable assumptions on the coefficients ai,j in order to make the associated many-

particle 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 ai,j = δi,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 ?Nl(φl

which so far we are unable to prove beyond the mean-field case.

i)?l(see Lemma 3.1),

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.

3 Localization and kinetic single-state BEC

First we shall prove the following lemma which is trivial for the perfect gas. For the mean-

field 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 either the perfect or mean-field Bose gases, i.e. for bosonic systems with

Hamiltonians (2.5) or (2.6), the function i → ?Nl(φi)?lis non-increasing.

Proof:

This proof is straightforward in the perfect Bose gas, since the occupation numbers are

known explicitly

?Nl(φl

i)?0

l=

?

eβ(Ei−µl)− 1

?−1,

and the lemma follows. In the mean-field case, let us define f : R+?→ R by

f(t) := ln Tre−βHl(µ;t),

Hl(µ;t) := Hl(µ) + t(Nl(φl

where

m) − Nl(φl

n)),

6

Page 7

for some 1 ? m < n. It follows that

f′(0) = β−1?Nl(φl

n) − Nl(φl

m)?l

and since the function f is convex, we have the following inequality

β−1?Nl(φl

n) − Nl(φl

m)?l ? f′(t), (3.1)

for any t ? 0. Now we set t =1

assumed that m < n. From the explicit expression (2.6) for Hl(µ), we have

2(El

n− El

m). Note that with this choice t ? 0, since we have

Hl(µ;t) =

?

i?=m,n

(El

i− µ)Nl(φl

i) +

λ

2VlN2

l+ (El

m+ El

2

n

− µ)Nl(φl

m) + (El

m+ El

2

n

− µ)Nl(φl

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 Hl(µ;t) is symmetric with respect to the exchange of

m and n. Hence

f′(t) =

Tr?Nl(φl

n) − Nl(φl

Tre−βHl(µ;t)

m)?e−βHl(µ;t)

= 0,

which in view of (3.1) gives

?Nl(φl

n) − Nl(φl

m)?l ? 0,

and the lemma follows since m < n are arbitrary.

?

Let us introduce the notation

ρl

i:=

1

Vl?Nl(φl

i)?l.

With this notation we can write the standard fixed density condition (2.7) for a given density

ρ, as

?

i

ρl

i= ρ,

and so for any N ∈ N,

N

?

i=1

ρl

i? ρ.

Letting

ρi := limsup

l→∞

ρl

i,

and taking the infinite volume limit, we then get

N

?

i=1

ρi = limsup

l→∞

N

?

i=1

ρl

i? ρ.

7

Page 8

Letting N tend to infinity, this gives?∞

i0 < ∞ such that ρi0 < 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 ψl

normalized, we obtain

i=1ρi? ρ, and hence, for any ε > 0, there exists

kare

1

Vl?Nl(ψl

k)?l

=

?

i?i0

?

i?i0

|(φl

i,ψl

k)|2ρl

i+

?

i>i0

|(φl

i,ψl

k)|2ρl

i

?

|(φl

i,ψl

k)|2ρl

i+ ρl

i0

?

i>i0

|(φl

i,ψl

k)|2

? ρ

?

i?i0

?

i?i0

|(φl

i,ψl

k)|2+ ρl

i0

? ρ

l−d/2||φl

i||1+ ρl

i0.

Therefore if l−d/2||φl

i||1→ 0 as l → ∞ for each i, then

limsup

l→∞

1

Vl?Nl(ψl

k)?l ? ε,

and since ε is arbitrary

lim

l→∞

1

Vl?Nl(ψl

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 φl

ilocalized if it satisfies the following condition

lim

l→∞

1

ld/2

?

Λl

dx|φl

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 Schr¨ odinger 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 φl

i. Then, no kinetic state ψl

iare localized in the sense of (3.2) for all

kcan be macroscopically occupied, that is

1

Vl?Nl(ψl

lim

l→∞

k)?l = 0, (3.3)

which implies in particular that any possible kinetic generalized BEC in these models is of

type III.

8

Page 9

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

4.1 Random potentials

Before we specify the random model under consideration, let us emphasize again that the

localization property (3.2) is very different from what is usually called “exponential local-

ization” in the literature about random Schr¨ odinger operators (see for example [12]). In the

standard literature localization refers to the eigenfunctions of the infinite volume Hamilto-

nian 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 L1norm not to diverge too fast, because

our eigenfunction depends 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 [12]. It consists of impurities located

at points of the lattice Zd, 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 Λl(x) the cubic box of side l centered at x. The single-site potential f,

Λ1(0) → R has the following properties:

1. f ∈ Lp(Λ1(0)) where p = 2 if d ? 3, and p > d/2 if d > 3.;

2. f is bounded, and f(x) ? σ > 0 for all x ∈ Λ1(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 H¨ older-continuous, that is for some α > 0,

sup

{s,t}

?µ([s,t]) : 0 ? t − s ? η?

? ηα,∀ 0 ? η ? 1(4.1)

The random potential is then defined by

vω(x) :=

?

k∈Zd

qω(k)f(x − k), (4.2)

9

Page 10

where the qω(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 non-

random M < ∞ such that vω(x) < M for any x and all ω.

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

hω

l = h0

l+ vω

l, (4.3)

where vω

by φω,l

−1

lis the restriction of vωto Λl. The eigenfunctions and eigenvalues of hω

and Eω,l

i

respectively. We denote by hω

2∆ + vωto the region Λl(x), with Dirichlet boundary conditions.

lare denoted

i

l(x) the restriction of the Schr¨ odinger operator

Before we establish the localization criterion (3.2) we prove our assumption that the eigen-

values of hω

ltend to zero as l tends to ∞:

Lemma 4.1 With probability one, for each i

lim

l→∞Eω,l

i

= 0. (4.4)

Proof: Let ν denote the limiting integrated density of states for the Hamiltonians hω

is for any A ⊂ R+,

1

Vl♯{i : Eω,l

Since by ergodicity ν is nonrandom [12], 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 [16],

l, that

ν(A) := lim

l→∞

i

∈ A}.(4.5)

ν([0,E]) ?

1

VLE?♯{i : Eω,L

i

? E}??

1

VLP{ω : Eω,L

1

? E}. (4.6)

which is satisfied for any L > 0. From the min-max principle, we obtain

Eω,L

1

? εL

1+

?

ΛL

dx|ψL

1|2vω(x) (4.7)

where εL

|ψL

1is the first kinetic eigenvalue and ψL

1(x)|2? 1/VL, we have

1the corresponding eigenfunction. Since

Eω,L

1

? εL

1+

1

VL

?

ΛL

dxvω(x) ? εL

1+

A

VL

?

k∈Zd∩ΛL

qω(k). (4.8)

where A :=

?

Λ1dxf(x). Letting L := π(E/2)−1/2so that εL

1

VLP{ω :

1= E/2, (4.6) and (4.8) give

ν([0,E]) ?

?

k∈Zd∩ΛL

qω(k) ? EVL/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 [12]). For convenience here we follow the

version in [12].

Adhering to the terminology of [12], we first define so-called “good boxes”:

10

Page 11

Definition 4.1 Given x ∈ Zd, a scale l ∈ 2N + 1, an energy E, a rate of decay γ > 0, we

call the box Λl(x) (γ,E)-good for a particular realization ω of the random potential (4.2) if

E / ∈ σ(hω

l(x)) and

||χout

l

(hω

l(x) − E)−1χint

l|| ? e−γl. (4.10)

Here σ(hω

in L2(Λl(x)), and χint

respectively, which we define as follows

l(x)) denotes the spectrum of hω

l,χout

l(x), the norm in (4.10) refers to the operator norm

are the characteristic functions of the regions Λint

ll(x),Λout

l (x)

Λint

l(x) := Λl/3(x),Λout

l (x) := Λl(x) \ Λl−2(x).

Our proof depends crucially on the following important multiscale analysis result extracted

from [12]. 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.

Proposition 4.1 Assume that hω

for any ζ > 0 and any α ∈

satisfying l1? 2 and lα

that if I := [0,r],

lis as above with random potential given by (4.2). Then

?1,2 − (4d/(4d + ζ))?, there exist a sequence {lk},k ? 1,

k−1+ 6 for k ? 2, and constants r > 0 and γ > 0 such

k−1? lk? lα

P

?

ω : for all E ∈ I, either Λlk(x) or Λlk(y) is (γ,E)-good

?

? 1 − (lk)−2ζ, (4.11)

for all k ? 1 and for all x,y ∈ Zd, satisfying |x − y| > lk.

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 [12] (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 hω

Eω,l

then the following inequality holds

lbe defined as above, and φω,l

i

to be one eigenfunction with eigenvalue

does not belong to the spectrum of hω

i. Let x ∈ Λl, such that Λlk(x) ⊂ Λl. If Eω,l

ilk(x),

?χint

lk(x)φω,l

i? ? κ?χint

lk(x)(hω

lk(x) − Eω,l

i)−1χout

lk(x)?, (4.12)

where the norms are L2(Λl)-norm, and κ is a constant depending only on M.

We are now ready to prove that for our model the localization condition (3.1) is satisfied.

Lemma 4.2 Assume that hω

almost surely, for all i,

lis as in (4.3) with random potential given by (4.2). Then

lim

l→∞

1

V1/2

l

?

Λl

dx|φω,l

i(x)| = 0.(4.13)

11

Page 12

Proof: We first choose 0 < δ < 1/4 and ζ > (2d + 1)/2δ and then we take the constants

α, γ and r and the sequence {lk} 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

δ

αlnl1

lnl < k <

1 − δ

αln(l1+ 6)lnl.

The fact that δ < 1/4 ensures that there exists such an integer k. Then, by Proposition 4.1,

we have

lδ< lk< l1−δ. (4.14)

Now let us define A(ω,l) to be the event in which, for all E ∈ I, for any x,y ∈ Λl∩ Zdsuch

that |x − y| > lk, either Λlk(x) or Λlk(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

Xl :=

?

ω : A(ω,l)is not true at scalel

?

.

Then we can write

Xl :=

?

such that bothΛlk(x) and Λlk(y)are not(γ,E)-good

ω : ∃E ∈ I, ∃x,y ∈ Λl∩ Z with |x − y| > lk,

?

=

?

x,y∈Λl∩Z

|x−y|>lk

?

ω : ∃E ∈ I, such that bothΛlk(x) and Λlk(y)are not(γ,E)-good

?

,

and by Proposition 4.1 we obtain

P(Xl) ? l2d(lk)−2ζ? l−2(δζ−d),

where the last step follows from (4.14). Since 2(δζ − d) > 1, it follows that

?

l

P(Xl) < ∞.

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, Eω,l

large enough almost surely. Therefore there exists˜Ω ⊂ Ω with P(˜Ω) = 1 such that for each

ω ∈˜Ω there is L1(ω) < ∞ such that for all l > L1(ω) and for any x,y ∈ Λl∩ Zdsatisfying

|x − y| > lk, either Λlk(x) or Λlk(y) are (γ,Eω,l

i

tends to 0 as l tends to ∞, Eω,l

i

∈ I for l

i)-good.

Now we take ω ∈˜Ω and l > L1(ω) and partition the box Λl(0) into Λ1

Λ2

corridor Λ2

l

l:= Λl−lk(0) and

land the

l:= Λl(0) \ Λ1

l. We then split up the integral in (4.13) into the interior cube Λ1

?

Λl

dx|φω,l

i(x)| =

?

Λ1

l

dx|φω,l

i(x)| +

?

Λ2

l

dx|φω,l

i(x)|.(4.15)

12

Page 13

In the second term, we can use the Schwarz inequality and the fact that the eigenfunctions

are L2(Λl)-normalized to obtain

?

Λ2

l

dx|φω,l

i(x)| ? |Λ2

l|1/2? l(d−1)/2l1/2

k

? l(d−δ)/2. (4.16)

For the first term in (4.15), we shall use the eigenfunction decay inequality (4.12) of Propo-

sition 4.2. We cover the “interior cube” Λ1

call {xj} their respective centers. Then for each j the cube Λlk(xj) is included in Λland Λj

coincides with Λint

Using the Schwarz inequality and Proposition 4.2, we obtain for any j the estimate:

lby disjoints subcubes Λj of side lk/3. Let us

lk(xj).

?

Λj

dx|φω,l

i(x)| ? ld/2??

Λl

dx|χint

lk(xj)φω,l

i(x)|2?1/2

? ld/2?

κ?χint

lk(x)(hω

lk(xj) − Eω,l

i)−1χout

lk(x)?

?1/2.

Hence, for any j such that Λjis (γ,Eω,l

i)-good, one has the following upper bound

?

Λj

dx|φω,l

i(x)| ? ld/2e−1

2γlk? ld/2e−1

2γlδ. (4.17)

Now, we distinguish two cases.

The first one corresponds to the situation where all cubes Λlk(xj) are (γ,Eω,l

then follows directly from (4.16) and (4.17) that

i)-good. It

l−d/2

?

Λl

dx|φω,l

i(x)| ?

l(d−δ)/2

ld/2

+ l−d/2?

xj∈Λ1

ld/2e−1

2γlδ

? l−δ/2+ 3d(l − lδ)d

ld(1−δ)e−1

2γlδ. (4.18)

The second case corresponds to the situation when there exists at least one subcube Λlk(xj)

which is not (γ,Eω,l

and l > L1(ω), for x,y ∈ Λl∩Zdsatisfying |x−y| > lk, either Λlk(x) or Λlk(y) are (γ,Eω,l

good. It therefore follows that, outside of a box of side 2lkcentered at ˜ x, all other Λlk(xj)

are (γ,Eω,l

Schwarz inequality as we did for Λ2

i)-good. Let us denote by ˜ x the center of one such bad cube. Since ω ∈˜Ω

i)-

i)-good. We treat the good boxes as above, and deal with Λ2lk(˜ x) by using the ,

l, to obtain:

?

Λ1

l

dx|φω,l

i(x)| =

?

Λ1

l\Λ2lk(˜ x)

?

xj∈Λ1

l\Λ2lk(˜ x)

? ld/23d(l − lδ)d

dx|φω,l

i(x)| +

?

Λ2lk(˜ x)

dx|φω,l

i(x)|

?

ld/2e−1

2γδ+ |Λ2lk(˜ x)|d/2

ld(1−δ)e−1

2γlδ+ (2l)d(1−δ)/2.

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

l−d/2

?

Λl

dx|φω,l

i(x)| ? l−δ/2+ 3d(l − lδ)d

ld(1−δ)e−1

2γlδ+ 2d(1−δ)/2l−dδ/2.(4.19)

13

Page 14

Therefore for any ω ∈˜Ω either (4.18) or (4.19) is satisfied for all l large enough and (4.13)

follows.

?

4.2Weak 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 Λ1⊂ Rdwhich satisfies the following

two conditions:

1. There is a finite, nonempty subset of Λ1, D := {yj}n

if x ∈ D.

j=1such that v(x) = 0 if and only

2. For each yj∈ D there are strictly positive numbers {αj}, {cj} such that

lim

x→yj

v(x)

|x − yj|αj= cj. (4.20)

We order the yj’s in such a way that 0 < α1 ? α2 ? ... ? αn.

The one-particle Schr¨ odinger operator with a weak (scaled) external potential in a box Λl

is defined by:

hl = −1

2∆D+ v(x1/l,...,xd/l) . (4.21)

We recall that the eigenfunctions and eigenvalues of hlare denoted by φl

The aim of this section is to prove that our localization condition (3.2) holds for this class

of weak potentials.

iand El

irespectively.

Lemma 4.3 Let hlbe as in (4.21). Then, for all i

lim

l→∞

1

ld/2

?

Λl

dx|φl

i(x)| = 0. (4.22)

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

(cj− ε)|x − yj|αj? v(x) ? (cj+ ε)|x − yj|αj, (4.23)

for all x ∈ Ij(δ), the ball of radius δ centered at yj. Note also that by continuity there

exists a constant κ > 0 such that v(x) ? κ, for all x ∈ Λ1\

min(κ,c1− ε,...,cn− ε) and C := max(c1+ ε,...,cn+ ε).

??n

j=1Ij(δ)

?

. We let K :=

The first step in our proof is to obtain an estimate for the eigenvalue El

let us denote by h(n)

l

the restriction of the Schr¨ odinger operator to the regions In(δl), with

Dirichlet boundary conditions. Then we have

i. To this end,

hl ? h(n)

l

(4.24)

14

Page 15

in quadratic form sense (c.f. [14], Chapter VIII, Proposition 4). From the inequality (4.23),

we obtain

h(n)

l

?˜h(n)

l

:=

1

2∆D+ C

???x

l

???

αn, (4.25)

where the last operator acts on L2?In(δl)?. Let U : L2?In(δl)?

unitary transformation defined by

?→ L2?In(δl1−γn)?

be the

(Uϕ)(x) :=lγn/2ϕ(lγnx),

where γn:= αn/(2+αn). By direct computation, one can check that˜h(n)

where

ˆh(n)

l

:= (−1

l

= l−2γnUˆh(n)

l

U−1

2∆ + C|x|αn),

acting on L2?In(δl1−γn)?. Let 0 < Dl

D2? ... the eigenvalues ofˆh(n)where

1? Dl

2? ... be the eigenvalues ofˆh(n)

l

and 0 < D1?

ˆh(n):= (−1

2∆ + C|x|αj),

acting on L2(Rd). Since for each i, Dl

Dl

i→ Dias l → ∞, there are constants˜Disuch that

i?˜Dj

ifor all l. Using this and the operator inequalities (4.24) and (4.25) we finally get

El

i? Dl

il−2γn?˜Dil−2γn. (4.26)

The rest of our proof relies on the methods developed in [15]. We start with some defini-

tions. Let Ωt, for some t > 0, to be the set of all continuous trajectories (paths) {ξ(s)}t

in Rdwith ξ(0) = 0, and let wtdenote the normalized Wiener measure on this set. For a

given x ∈ Rd, we define the following characteristic function

s=0

χx,l(ξ) := 1?ξ : ξ(s) ∈ Λl− x, for all0 ? s ? t?.

We now use the following identity (c.f. [13]),

(e−thlφl

i)(x) =

?

Ωtwt(dξ)e

−

?t

0

dsv((x + ξ(s))/l)

φl

i(x + ξ(t))χx,l(ξ),

from which, since El

iis the eigenvalue of hlcorresponding to φl

?t

0

i, we get

|φl

i(x)| ? etEl

i

?

Ωtwt(dξ)e

−

dsv((x + ξ(s))/l)

|φl

i(x + ξ(t))|χx,l(ξ). (4.27)

Now, we insert into the right-hand side of (4.27) the following bound proved in [11],

|φl

i(x)| ? cd(El

i)d/4,

where cd:= (e/π)d/4and we obtain from (4.27) the following estimate

|φl

i(x)| ? cdetEl

i(El

i)d/4

?

Ωtwt(dξ)e

−

?t

0

dsv((x + ξ(s))/l)

χx,l(ξ)

= cdetEl

i(El

i)d/4

?

?

Ωtwt(dξ)e

Ωtwt(dξ)1

−1

t

?t

0

dstv((x + ξ(s))/l)

χx,l(ξ)

? cdetEl

i(El

i)d/4

t

?t

0

dse−tv((x + ξ(s))/l)χx,l(ξ),

15

Page 16

where the last step follows from Jensen’s inequality. Therefore, integrating over Λl with

respect to x, and then changing the order of integration, yield

l−d/2

?

Λl

dx|φl

i(x)| ? cdl−d/2etEl

i(El

i)d/4

?

Λl

?

dx

?

Ωtwt(dξ)1

?t

t

?t

0

?

{x∈?

dse−tv((x + ξ(s))/l)χx,l(ξ)

? cdl−d/2etEl

i(El

i)d/4

Ωtwt(dξ)1

t

0

ds

s′(Λl−ξ(s′))}

dxe−tv((x + ξ(s))/l).

Letting y = x + ξ(s) in the second integral we get

l−d/2

?

Λl

dx|φl

i(x)| ? cdl−d/2etEl

i(El

i)d/4

?

Ωtwt(dξ)1

dy e−tv(y/l).

t

?t

0

ds

.

?

{y−ξ(s)∈?

s′(Λl−ξ(s′))}

Since?

over y to Λland use the fact that the Wiener measure wtis normalized to obtain

?

Λl

s′(Λl− ξ(s′) + ξ(s)) ⊂ Λlfor all s, we can now extend the domain of integration

l−d/2

dx|φl

i(x)| ? cdetEl

i(El

i)d/4l−d/21

t

?t

0

ds

?

Λl

dy e−tv((y)/l)

(4.28)

= cdetEl

i(El

i)d/4ld/2

?

Λ1

dz e−tv(z).

Next, we obtain an upper bound for the last integral in (4.28). We have

?

Λ1

dz e−tv(z)

?

n

?

j=1

?

Ij

dz e−tv(z)+

?

Λ1\??n

i=1Ij

?dz e−tv(z)

(4.29)

? e−tK+

n

?

j=1

?

Ij

dz e−tK|x−yj|αj.

For each j,

?

Ij

dz e−tK|x−yj|αj? t−d/αjKd

?

Rdd˜ z e−|˜ z|αj?˜K t−d/αj,

where˜K := Kdmaxj

?

Rdd˜ z e−|˜ z|αj, which, in view of (4.29), gives the following bound

?

Λ1

dz e−tv(z)? e−tK+˜K

n

?

j=1

t−d/αj.

Now, fixing t = (El

i)−1, we get from the last inequality and (4.28)

l−d/2

?

Λl

dx|φl

i(x)| ? cde(El

i)d/4ld/2?

e−K(El

i)−1+˜K

n

?

j=1

(El

i)d/αj?

.

Since by (4.26), El

··· < αn, there exist new constants Aisuch that the following bound holds for l large enough

?

Λl

Inserting the bound (4.26), we finally obtain for l large enough

?

Λl

and the lemma follows since γn< 1.

i→ 0 as l → ∞, and since we have ordered the αi’s such that α1< α2<

l−d/2

dx|φl

i(x)| ? Aild?El

i

?d(1/4+1/αn)= Aild/2?El

i

?d(2−γn)/(4γn). (4.30)

l−d/2

dx|φl

i(x)| ? Ai˜Dd(2−γn)/(4γn)

i

l−d(1−γn)/2

?

16

Page 17

Acknowledgments

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

Scholarship.

References

[1] J.V. Pul´ e, J. Math. Phys. 24 (1983), 138–142

[2] M. Van den Berg, J.T. Lewis, Commun. Math. Phys. 81 (1981), 475-494

[3] O. Lenoble, L.A. Pastur, V.A. Zagrebnov, Comptes-Rendus de l’Acad´ emie des Sciences

(Paris), Physique 5 (2004), 129-142

[4] L.A. Pastur, A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer-

Verlag, Berlin, 1992

[5] M. Girardeau, J. Math. Phys. 1 (1960), 516-523

[6] M. Van den Berg, J.T. Lewis, J.V. Pul´ e, Helv. Phys. Acta 59 (1986), 1271-1288

[7] O. Lenoble, V.A. Zagrebnov, Markov Processes and related fields 13 (2007), 441-468

[8] M. Fannes, A. Verbeure, J. Math. Phys. 21 (1980), 1809-1818

[9] T. Jaeck, J.V. Pul´ e, V.A. Zagrebnov, J. Stat. Phys. 137 (2009), 19-55

[10] T. Jaeck, V.A. Zagrebnov, in preparation (2010)

[11] E.B. Davies, J. Lond. Math. Soc. 7 (1973), 483

[12] P. Stollmann, Caught by disorder. Bound states in random media, Progress in Mathe-

matical Physics, 20. Birkh¨ auser Boston, Inc., Boston, MA, 2001

[13] D. Ray, Trans. Amer. Math. Soc. 77 (1954), 299-321

[14] M. Reed, B. Simon, Methods of Mathematical Physics, IV: Analysis of Operators,

Academic Press, London, 1978

[15] P. Mac Aonghusa, J.V. Pul´ e, Lett. Math. Phys. 14 (1987), 117-121

[16] W. Kirsch, F. Martinelli, Commun. Math. Phys. 89 (1983), 27-40

17