Quantum mean field asymptotics and multiscale analysis

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Quantum mean field asymptotics and multiscale analysis

Zied Ammari, Sébastien Breteaux, Francis Nier

To cite this version:

Zied Ammari, Sébastien Breteaux, Francis Nier. Quantum mean field asymptotics and multiscale analysis. Tunisian Journal of Mathematics, Mathematical Science Publishers, 2019, 1 (2), pp.221-272.

�10.2140/tunis.2019.1.221�. �hal-01442714�

Quantum mean-field asymptotics and multiscale analysis

Z. Ammari

^∗

, S. Breteaux

^†

and F. Nier

^‡

January 20, 2017

Abstract

We study, via multiscale analysis, some defect of compactness phenomena which occur in bosonic and fermionic quantum mean-field problems. The approach relies on a combination of mean-field asymptotics and second microlocalized semiclassical measures. The phase space geometric description is illustrated by various examples.

Keywords and phrases: Semiclassical and multiscale measures, reduced density matrices, second quantization, microlocal analysis.

2010 Mathematics subject classification: 81S30, 81Q20, 81V70, 81S05.

1 Introduction

Motivations: Over the past two decades, it becomes clear that microlocal analysis provides inter- esting mathematical tools for the study of quantum field theories and quantum many-body theory, see for instance [AmNi1, Rad]. In particular, in the analysis of general bosonic mean-field problems, as done in [AmNi1, AmNi2, AmNi3, AmNi4], the following defect of compactness problem arises.

If γ

_ε^(p)

denotes the p-particles reduced density matrix, one may have

ε

lim

→0

Tr[γ

_ε^(p)

˜ b] = Tr[γ

^(p)₀

˜ b] (1) for any p-particle compact observable ˜ b , while it is not true for a general bounded ˜ b , e.g.

ε

lim

→0

Tr[γ

_ε^(p)

] > Tr[γ

₀^(p)

] .

In the fermionic case, it is even worse, because mean-field asymptotics cannot be described in terms of finitely many quantum states and the right-hand side of (1) is usually 0 while lim

_ε→0

Tr[γ

ε^(p)

] > 0 . From the analysis of finite dimensional partial differential equations, it is known that such defect of compactness can be localized geometrically with accurate quantitative information by introducing

∗[email protected], IRMAR, Universit´e de Rennes I, UMR-CNRS 6625, campus de Beaulieu, 35042 Rennes Cedex, France.

†[email protected], BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Spain.

‡[email protected], LAGA, UMR-CNRS 9345, Universit´e de Paris 13, av. J.B. Cl´ement, 93430 Villetaneuse, France.

scales and small parameters within semiclassical techniques (e.g. [Ger, GMMP]). We are thus led to introduce two small parameters ε > 0 for the mean-field asymptotics and h > 0 for the semiclassical quantization of finite dimensional p-particles phase space. The small parameter ε stands for 1/n , where n → ∞ is the typical number of particles, while h is the rescaled Planck constant measuring the proximity of quantum mechanics to classical mechanics. The combined analysis of this article is concerned with the general situation when ε = ε(h) with lim

_h→0

ε(h) = 0 . In order to keep track of the information at the quantum level especially in the bosonic case we also introduce finite dimensional multiscale observables in spirit of [Bon, FeGe, Fer, Nie].

Framework: The one particle space Z is a separable complex Hilbert space endowed with the scalar product h , i (anti-linear in the left-hand side). For a Hilbert space h the set of bounded operator is denoted by L (h) , while the Schatten class is denoted by L

^p

(h) , 1 ≤ p ≤ ∞ , the case p = ∞ corresponding to the space of compact operators. Let Γ

_±

( Z ) be the bosonic (+ sign) or fermionic ( − sign) Fock-space built on the separable Hilbert space Z :

Γ

_±

( Z ) = M

⊥

n∈N

S

_±ⁿ

Z

⊗n

,

where tensor products and direct sum are Hilbert completed. The operator S

_±ⁿ

is the orthogonal projection given by

S

_±ⁿ

(f

₁

⊗ · · · ⊗ f

_n

) = 1 n!

X

σ∈Sn

s

_±

(σ)f

_σ(1)

⊗ · · · ⊗ f

_σ(n)

,

where s

₊

(σ) equals 1 while s

₋

(σ) denotes the signature of the permutation σ and S

_n

is the n- symmetric group.

The dense set of many-body states with a finite number of particles is Γ

^{f in}_±

( Z ) =

⊥,alg

M

n∈N

S

_±ⁿ

Z

^⊗n

,

where the

^⊥,alg

superscript stands for the algebraic orthogonal direct sum.

We shall use the notations [A, B]

₊

= [A, B] = ad

_A

B = AB − BA for the commutator of two oper- ators and the notation [A, B]

₋

= AB + BA for the anticommutator.

One way to investigate the mean-field asymptotics relies on parameter-dependent CCR (resp.

CAR) . The small parameter ε > 0 has to be thought of as the inverse of the typical number of particles and the Canonical Commutation (resp. Anticommutation) Relations are given by

[a

_±

(g), a

_±

(f )]

_±

= [a

^∗_±

(g), a

^∗_±

(f )]

_±

= 0 , [a

_±

(g), a

^∗_±

(f )]

_±

= ε h g , f i .

Let (̺

_ε

)

_ε>0

be a family of normal states (i.e., non-negative and normalized trace-class operators)

on the Fock space Γ

_±

( Z ) , depending on ε > 0 , we want to investigate the asymptotic behaviour

of reduced density matrices, defined below, as ε → 0 , by possibly introducing another scale h > 0

on the p-particles phase-space, with ε = ε(h) and lim

_h→0

ε(h) = 0 .

Outline: In Section 2, we recall how Wick observables are used to define the reduced density matrices γ

_ε^(p)

. Note that it is much more convenient here, in the general grand canonical framework, to work with non normalized reduced density matrices. Some symmetrization formulas are also recalled in this section. In Section 3, we present the geometry of the classical p-particles phase space and introduce the formalism of double scale semiclassical measures, after [Fer, FeGe]. In Section 4, we combine the mean-field asymptotics with semiclassical analysis, the two parameters ε and h being related through ε = ε(h) with lim

_h→0

ε(h) = 0 . Instead of studying the collection of non normalized reduced density matrices (γ

_ε(h)^(p)

)

_p∈N

, it is more convenient to associate generating functions z 7→ Tr

̺

_ε(h)

e

^{z dΓ±(aQ,h)}

, and to use holomorphy arguments presented there. In Section 5, some classical examples with various asymptotics illustrate the general framework: coherent states in the bosonic setting; simple Gibbs states in the fermionic case; more involved Gibbs states in the bosonic case, which make explicit the separation of condensate and non condensate phases for rather general non interacting steady Bose gases. The Appendices collect or revisit known things about multiscale semiclassical measures, the (PI)-condition of bosonic mean-field problems, Wick composition formulas, and traces of non self-adjoint second quantized contractions.

2 Wick observables and reduced density matrices

2.1 Wick observables

Notation: For n ∈ N , the operator S

_±ⁿ

is an orthogonal projection in Z

⊗n

so that ( S

_±ⁿ

)

^∗

= S

_±ⁿ

. However, we consider S

_±ⁿ

as a bounded operator from Z

⊗n

onto S

_±ⁿ

Z

⊗n

and its adjoint, denoted by S

_±^n,∗

: S

_±ⁿ

Z

⊗n

→ Z

⊗n

, is nothing but the natural embedding.

Let ˜ b ∈ L ( S

_±^p

Z

⊗p

; S

_±^q

Z

⊗q

) , the Wick quantization of ˜ b is the operator on Γ

^{f in}_±

( Z ) defined by

˜ b

^{W ick}

S±^n+pZ^⊗(n+p)

= ε

^p+q2

p (n + p)!(n + q)!

n! S

_±^n+q

(˜ b ⊗ Id

Z⊗n

) S

_±^n+p,∗

.

In the bosonic case, an element ˜ b ∈ L ( S

+^p

Z

^⊗p

; S

+^q

Z

^⊗q

) is determined by the symbol Z ∋ z 7→

b(z) = h z

^⊗q

, ˜ bz

^⊗p

i owing to the relation ˜ b =

_q!p!¹

∂

_z^q

∂

_z^p

b . Observe that b(z) admits higher Fr´echet derivatives with the natural identification of ∂

^k_z

b(z) as a continuous form on S

+^k

Z

^⊗k

and ∂

_z^k_¯

b(z) as a vector in S

+^k

Z

^⊗k

. We shall use also the notation b

^{W ick}

= ˜ b

^{W ick}

.

Examples:

a) The annihilation operator a

_±

(f ) , f ∈ Z , is the Wick quantization of ˜ b = h f | : Z

⊗1

= Z ∋ ϕ 7→ h f , ϕ i ∈ Z

⊗0

= C .

b) The creation operator a

^∗_±

(f ) , f ∈ Z , is the Wick quantization of ˜ b = | f i : Z

^⊗0

= C ∋ λ 7→

λf ∈ Z

⊗1

= Z .

c) For ˜ b ∈ L ( Z ) its Wick quantization ˜ b

^{W ick}

is nothing but dΓ

_±

(˜ b)

S±ⁿZ^⊗n

= ε h

˜ b ⊗ Id

Z

⊗ · · · ⊗ Id

Z

+ · · · + Id

Z

⊗ · · · ⊗ Id

Z

⊗ ˜ b i .

When ˜ b is self-adjoint one has

dΓ

_±

(˜ b) = i∂

_t

e

^{−itdΓ±(˜b)}

t=0

= i∂

_t

Γ

_±

(e

^−iεt˜b

)

t=0

,

while for a contraction C ∈ L ( Z ; Z ) , Γ

_±

(C)

S±ⁿZ^⊗n

= C ⊗ · · · ⊗ C .

A particular case is ˜ b = Id

Z

associated with the scaled number operator (N

_±,ε=1

stands for the usual ε-independent number operator):

˜ b

^{W ick}

= dΓ

_±

(Id

Z

) = N

_±

= εN

_±,ε=1

.

From the definition of the Wick quantization one easily checks the following properties.

Proposition 2.1. For ˜ b ∈ L ( S

_±^p

Z

⊗p

; S

_±^q

Z

⊗q

) :

• h

˜ b

^{W ick}

i

_∗

= [˜ b

^∗

]

^{W ick}

.

• The operator (1 + N

_±

)

^−m/2

˜ b

^{W ick}

(1 + N

_±

)

^−m′/2

extends to a bounded operator on Γ

_±

( Z ) as soon as m + m

^′

≥ p + q with

k (1 + N

_±

)

^−m/2

˜ b

^{W ick}

(1 + N

_±

)

^−m′/2

k

_L(Γ±(Z))

≤ C

_m,m^′

k ˜ b k

_L(S^p±Z;S±^qZ)

, (2) with C

_m,m^′

independent of ˜ b and of ε ∈ (0, ε

0

) .

• (˜ b ≥ 0) ⇔ (˜ b

^{W ick}

≥ 0) , while this makes sense only for q = p .

The Wick quantized operators generally are unbounded operators on Γ

_±

( Z ) (e.g. N

_±

) but they are well defined on the dense set Γ

^{f in}_±

( Z ) which is preserved by their action. Hence ˜ b

^{W ick}₁

◦ ˜ b

^{W ick}₂

makes sense at least on Γ

^{f in}_±

( Z ) and the following composition law holds true.

Proposition 2.2 (Composition of Wick operators). Let ˜ b

_j

∈ L ( S

_±^pj

Z

⊗pj

; S

_±^qj

Z

⊗qj

), j = 1, 2, then

˜ b

^{W ick}₁

◦ ˜ b

^{W ick}₂

=

min{p1,q2}

X

k=0

( ± 1)

^{(p1−k)(p2+q2)}

ε

^k

k! (˜ b

₁

♯

^k

˜ b

₂

)

^{W ick}

, (3) where ˜ b

₁

♯

^k

˜ b

₂

:=

_(p^p1!

1−k)!

q2!

(q2−k)!

S

_±^q1+q2−k

(˜ b

₁

⊗ Id

^⊗q2−k

) (Id

^⊗p1−k

⊗ ˜ b

₂

) S

_±^{p1+p2−k,∗}

. For reader’s convenience, the proof of Prop. 2.2 is provided in Appendix C.

In the bosonic case the symbols b(z) = h z

^⊗q

, ˜ bz

^⊗p

i are convenient for writing the composition of Wick quantized operators. If b

₁

♯

^{W ick}

b

₂

denotes the symbol of ˜ b

^{W ick}₁

◦ ˜ b

^{W ick}₂

, the composition law is summarized below (see [AmNi1]).

Proposition 2.3 (Composition of Wick symbols in the bosonic case).

b

1

♯

^{W ick}

b

2

(z) = e

^{ε∂z1·∂z2}

b

1

(z

1

)b

2

(z

2

)

z1=z2=z

=

min{p1,q2}

X

k=0

ε

^k

k! ∂

_z^k

b

1

(z) · ∂

_z^k

b

2

(z) . The commutator of Wick operators in the bosonic case:

h b

^{W ick}₁

, b

^{W ick}₂

i

=





max{min{p1,q2},min{p2,q1}}

X

k=1

ε

^k

k! { b

₁

, b

₂

}

^(k)





W ick

,

where the k-th order Poisson bracket is given by { b

₁

, b

₂

}

^(k)

= ∂

_z^k

b

₁

(z) · ∂

_z^k

b

₂

(z) − ∂

_z^k

b

₂

(z) · ∂

_z^k

b

₁

(z) .

Proposition 2.4. Let p, m, m

^′

∈ N , such that m + m

^′

≥ 2p − 2. Then, there exist coefficients C

_j1,...,jk

≥ 0 such that, for any ˜ b ∈ L ( Z ; Z ) ,

dΓ

_±

(˜ b)

^p

− (˜ b

^⊗p

)

^{W ick}

=

p−1

X

k=1

ε

^p−k

X

0≤j1≤···≤j_k j1+···+j_k=p

C

_j1,...,jk

S

_±^k

˜ b

^j1

⊗ · · · ⊗ ˜ b

^jk

S

_±^k,∗

W ick

(4)

and the estimate

k (1 + N

_±

)

^−m/2

dΓ

_±

(˜ b)

^p

− (˜ b

^⊗p

)

^{W ick}

(1 + N

_±

)

^−m′/2

k

_L(Γ±(Z))

≤ ε B

p

k ˜ b k

^p_L(Z)

holds in both the bosonic case and the fermionic case, with B

_p

the p-th Bell number.

Remark 2.5. The p-th Bell number B

_p

can be defined as the number of partitions of a set with p elements and satisfies B

_p

<

0.792p ln(p+1)

p

(see [BeTa]), and hence it grows much slower than p! . Proof. We first prove Formula (4) by induction on p ∈ N

^∗

.

For p = 1, Formula (4) holds because dΓ

_±

(˜ b) = (˜ b)

^{W ick}

.

We then set r

p

(˜ b) := dΓ

_±

(˜ b)

^p

− (˜ b

^⊗p

)

^{W ick}

. Assuming the result holds for some p ∈ N

∗

, one can compute

dΓ

_±

(˜ b)

^p+1

= (˜ b

^⊗p

)

^{W ick}

(˜ b)

^{W ick}

+ r

_p

(˜ b)

^{W ick}

(˜ b)

^{W ick}

using the composition formula (3) for

(˜ b

^⊗p

)

^{W ick}

(˜ b)

^{W ick}

= (˜ b

^⊗p+1

)

^{W ick}

+ pε S

_±^p

˜ b

^⊗p−1

⊗ ˜ b

²

S

_±^p,∗

W ick

and for

ε

^p−k

S

_±^k

˜ b

^j1

⊗ · · · ⊗ ˜ b

^jk

S

_±^k,∗

W ick

(˜ b)

^{W ick}

= ε

^p+1−(k+1)

S

_±^k+1

˜ b ⊗ ˜ b

^j1

⊗ · · · ⊗ ˜ b

^jk

S

_±^k+1,∗

W ick

+ kε

^p+1−k

S

_±^k

˜ b

^j1

⊗ · · · ⊗ ˜ b

^jk

S

_±^k,∗

S

_±^k

˜ b ⊗ Id

^⊗_Z^{j1+···+jk−1}

S

_±^k,∗

W ick

, which yields the expected form for r

_p+1

(˜ b), and achieves the induction.

We then remark that the sum of coefficients of order k, S

₂

(p, k) = X

0≤j1≤···≤j_k j1+···+j_k=p

C

_j1,...,jk

,

satisfies the recurrence relation S

₂

(p, k) = kS

₂

(p − 1, k)+S

₂

(p − 1, k − 1), with S

₂

(p, 1) = 1 = S

₂

(1, k) for all p, k ∈ N

∗

, where the S

₂

(p, k) are the Stirling numbers of the second kind. Observe that, for M/2 ≥ k, and for any ˜ c ∈ L ( S

_±^k

Z

^⊗k

),

k ˜ c

^{W ick}

(1 + N

_±

)

^−M/2

k

_L(Γ±(Z))

≤ k c ˜ k

_L(S±^kZ^⊗k;S^k±Z^⊗k)

. We thus get,

k (1 + N

_±

)

^−m/2

dΓ

_±

(˜ b)

^p

− (˜ b

^⊗p

)

^{W ick}

(1 + N

_±

)

^−m′/2

k

_L(Γ±(Z))

≤

p−1

X

k=1

ε

^p−k

S

₂

(p, k) k ˜ b k

^p_L(Z)

and the estimate then follows from P

p−1

k=1

ε

^p−k

S

₂

(p, k) ≤ ε P

p

k=1

S

₂

(p, k) = εB

_p

with B

_p

the p-th

Bell number.

2.2 Reduced density matrices

Reduced density matrices emerge naturally in the study of correlation functions of quantum gases and in particular in the quantum mean-field theory they are the main quantities to be analysed.

We shall work with non normalized reduced density matrices, which are easier to handle. Going back to the more natural reduced density matrices with trace equal to 1 , requires attention when normalizing and taking the limits.

Definition 2.6. Let ̺

_ε

∈ L

¹

(Γ

_±

( Z )) (ε > 0 is fixed here) be such that ̺

_ε

≥ 0 , Tr [̺

_ε

] = 1 and Tr ̺

_ε

e

^cN±

< ∞ for some c > 0 . The non normalized reduced density matrix of order p ∈ N , γ

ε^(p)

∈ L

¹

( S

_±^p

Z

⊗p

) , is defined by duality according to

∀ ˜ b ∈ L ( S

_±^p

Z

^⊗p

; S

_±^p

Z

^⊗p

) , Tr h γ

_ε^(p)

˜ b i

= Tr h

̺

_ε

˜ b

^{W ick}

i .

The definition makes sense owing to the number estimate (2) and to (1 + N

_±

)

^k

e

^−cN±

∈ L (Γ

_±

( Z )) . When Tr h

γ

ε^(p)

i 6 = 0 , the normalized density matrix ¯ γ

ε^(p)

is defined by ¯ γ

ε^(p)

=

^γε(p)

Trh γ^(p)ε

i

, that is

∀ ˜ b ∈ L ( S

_±^p

Z

^⊗p

), Tr h

¯ γ

_ε^(p)

˜ b i

=

Tr h

̺

_ε

˜ b

^{W ick}

i Tr h

̺

_ε

(Id

_S^p

±Z⊗p

)

^{W ick}

i

=

Tr h

̺

_ε

˜ b

^{W ick}

i

Tr [̺

_ε

N

_±

(N

_±

− ε) . . . (N

_±

− ε(p − 1))] .

These normalized reduced density matrices ¯ γ

ε^(p)

are commonly used, especially when ̺

_ε

∈ L

¹

( S

±

Z

⊗n

), with nε ∼ 1 (see [BGM, BEGMY, BPS, KnPi, LNR]), for the following reason: When ̺

_ε

∈ L

¹

( S

_±ⁿ

Z

⊗n

) ⊂ L

¹

( Z

⊗n

) lies in the n-particles sector (with nε → 1 for the mean-field regime) it is simply given by the partial trace of ̺

_ε

when n > p . Actually

˜ b

^{W ick}

S±ⁿZ⊗n

= ε

^p

n!

(n − p)! S

_±ⁿ

(˜ b ⊗ Id

Z⊗(n−p)

) S

_±^n,∗

and, as ε

^p

n(n − 1) · · · (n − p + 1) → 1 if nε → 1, it follows that

nε

lim

∼1 ε→0

Tr h γ

_ε^(p)

˜ b i

= lim

nε∼1 ε→0

Tr h

̺

_ε

(˜ b ⊗ Id

Z⊗(n−p))

) i .

When Z = L

²

(M; dv) one thus often considers:

˜

γ

_ε^(p)

(x

₁

, . . . , x

_p

; x

^′₁

, . . . , x

^′_p

) = Z

M^n−p

̺

_ε

(x

₁

, . . . , x

_p

, x; x

^′₁

, . . . , x

^′_p

, x) dv

^⊗(n−p)

(x) .

But, if the states ̺

_ε

are not localized on the n-particles sector, such an alternative definition does not coincide with γ

ε^(p)

, even asymptotically in the mean-field regime.

As well there is no general relation between the non normalized density matrices γ

ε^(p+1)

and γ

_ε^(p)

. Actually we have

S

_±^p+1

(˜ b ⊗ Id

Z

) S

_±^p+1,∗

W ick

S±^n+p+1Z

= ε

^p+1

(n + p + 1)!

n! S

_±^n+p+1

(˜ b ⊗ Id

Z⊗n+1

) S

_±^n+p+1,∗

= ε(n + 1)˜ b

^{W ick}

S±ⁿZ^n+p+1

,

from which we deduce

Tr h

γ

_ε^(p+1)

(˜ b ⊗ Id

Z

) i

= Tr h

̺

_ε

N

_±

˜ b

^{W ick}

i

+ O (ε) , while Tr h

γ

_ε^(p)

˜ b i

= Tr h

̺

_ε

˜ b

^{W ick}

i ,

where we have again identified γ

ε^(p+1)

as an element of L

¹

( Z

⊗(p+1)

) . We thus conclude with the following important remark.

Remark 2.7. Simple asymptotic relation between the γ

_ε^(p)

and γ

_ε^(p′)

(or the normalized version γ ¯

_ε^(p)

and γ ¯

_ε^(p′)

) can be expected when ̺

_ε

= ̺

_ε

1

_[ν−δ(ε),ν+δ(ε)]

(N

_±

) with ν > 0 and lim

_ε→0

δ(ε) = 0 but not otherwise (of course the condition above is sufficient but not necessary).

We shall use recurrently with variations the following lemma, with the following notations

˜ b

1

⊙ · · · ⊙ ˜ b

p

= 1 p!

X

σ∈Sp

˜ b

_σ(1)

⊗ · · · ⊗ ˜ b

_σ(p)

,

for ˜ b

₁

, . . . , ˜ b

_p

∈ L ( Z ) .

We also write shortly (˜ b

1

⊙ . . . ⊙ ˜ b

p

)

^{W ick}

and (˜ b

^⊗p

)

^{W ick}

instead of

S

_±^p

(˜ b

1

⊙ . . . ⊙ ˜ b

p

) S

_±^p,∗

W ick

and S

_±^p

(˜ b

^⊗p

) S

_±^p,∗

W ick

.

Lemma 2.8. Quantum symmetrization lemma: In the bosonic and fermionic cases for any p ∈ N , the equality

S

_±^p

(˜ b

₁

⊗ · · · ⊗ ˜ b

_p

) S

_±^p,∗

= S

_±^p

(˜ b

₁

⊙ · · · ⊙ ˜ b

_σ(p)

) S

_±^p,∗

, (5) holds in L ( S

_±^p

Z

⊗p

; S

_±^p

Z

⊗p

) for all ˜ b

1

, . . . , ˜ b

p

∈ L ( Z ; Z ) .

As a consequence, under the assumptions of Definition 2.6, the non normalized (resp. normalized if possible) reduced density matrix γ

_ε^(p)

(resp. ¯ γ

_ε^(p)

) , p ∈ N , is completely determined by the set of quantities n

Tr h

̺

_ε

(˜ b

^⊗p

)

^{W ick}

i

, ˜ b ∈ B o

when B is any dense subset of L

^∞

( Z ; Z ) . Remark 2.9. While computing Tr h

γ

ε^(p)

i or studying γ ¯

^(p)ε

one can simply add to B the element Id

Z

owing to S

_±^p

Id

^⊗_Z^p

S

_±^p,∗

= Id

_S^p

±Z⊗p

. For ε > 0 fixed it is not necessary because compact observables are sufficient to determine the total trace owing to Tr h

γ

ε^(p)

i

= sup

_B∈L∞(S±^pZ⊗p),0≤B≤Id

Tr h γ

ε^(p)

B i

. However, while considering weak

^∗

-limits as ε → 0 , adding the identity operator Id

_S^p

±Z⊗p

to the set of compact observables, or possibly replacing B by the Calkin algebra C Id( Z ) ⊕ L

^∞

( Z ), is useful in order to control the asymptotic total mass.

Proof. For ˜ b

₁

, . . . , ˜ b

_p

∈ L ( Z ) , we decompose S

_±^p

˜ b

₁

⊗ · · · ⊗ ˜ b

_p

S

_±^p,∗

S

_±^p

(ψ

₁

⊗ · · · ⊗ ψ

_p

) as

S

_±^p



 1 p!

X

σ^′∈Sp

s

_±

(σ

^′

)(˜ b

₁

ψ

_σ^′₍₁₎

) ⊗ · · · ⊗ (˜ b

_p

ψ

_σ^′_(p)

)





= 1 p!p!



 X

σ∈Sp

X

σ^′∈Sp

s

_±

(σ)s

_±

(σ

^′

)(˜ b

_σ(1)

ψ

_σ◦σ^′₍₁₎

) ⊗ · · · ⊗ (˜ b

_σ(p)

ψ

_σ◦σ^′_(p)

)



 .

Setting σ

^′′

= σ ◦ σ

^′

, with s

_±

(σ

^′′

) = s

_±

(σ)s

_±

(σ

^′

) yields the Eq. (5), after noting that ˜ b

₁

⊙ · · · ⊙ ˜ b

_p

=

1 p!

P

σ∈Sp

˜ b

_σ(1)

⊗ · · · ⊗ ˜ b

_σ(p)

commutes with S

_±^p

in both the bosonic case and the fermionic case.

Now the non normalized reduced density matrix is determined by Tr h

γ

_ε^(p)

B ˜ i

= Tr h

̺

_ε

B ˜

^{W ick}

i

for ˜ B ∈ L

^∞

( S

_±^p

Z

⊗p

) as L

¹

( S

_±^p

Z

⊗p

) is the dual of L

^∞

( S

_±^p

Z

⊗p

) . But ˜ B ∈ L

^∞

( S

_±^p

Z

⊗p

) means B ˜ = S

_±^p

B ˜

^′

S

_±^p,∗

with ˜ B

^′

∈ L

^∞

( Z

⊗p

) , while the algebraic tensor product L

^∞

( Z )

^⊗algp

is dense in L

^∞

( Z

⊗p

) .

With the estimate Tr h

̺

_ε

B ˜

^{W ick}

i =

Tr h

e

^c/2N

̺

_ε

e

^c/2N

e

^−c/2N

B ˜

^{W ick}

e

^−c/2N

i

≤ CTr

̺

ε

e

^cN

k B ˜ k

_L(S±^pZ⊗p;S±^pZ⊗p)

,

it suffices to consider ˜ B = S

_±^p

B ˜

^′

S

_±^p,∗

with ˜ B

^′

∈ L

^∞

( Z )

^⊗algp

. By linearity and density, γ

_ε^(p)

is determined by the quantities Tr[̺

_ε

B ˜

^{W ick}

] with ˜ B

^′

= ˜ b

₁

⊗ · · · ⊗ ˜ b

_p

, ˜ b

_i

∈ B . We conclude with

S

_±^p

˜ b

₁

⊗ · · · ⊗ ˜ b

_p

S

_±^p,∗

= S

_±^p

˜ b

₁

⊙ · · · ⊙ ˜ b

_p

S

_±^p,∗

, and the polarization identity

˜ b

₁

⊙ · · · ⊙ ˜ b

_p

= 1 2

^p

p!

X

εi=±1

ε

₁

· · · ε

_p

( X

p

i=1

ε

_i

b ˜

_i

)

^⊗p

.

Remark 2.10. In the bosonic case, the non normalized reduced density matrices γ

ε^(p)

are also characterized by the values of Tr[γ

_ε^(p)

B] for B in B = {| ψ

^⊗p

ih ψ

^⊗p

| , ψ ∈ Z } ∪ n

Id

^⊗_Z^p

o

. This does not hold in the fermionic case.

The rest of the article is devoted to the asymptotic analysis of γ

ε^(p)

as ε → 0 . In particular we shall study their concentration at the quantum level while testing with fixed observable ˜ b (with ˜ b compact) and their semiclassical behaviour after taking semiclassically quantized observables, e.g.

a(x, hD

_x

) with some relation ε = ε(h) between ε and h .

3 Classical phase-space and h -quantizations

When Z = L

²

(M

¹

, dx) , with M

¹

= M a manifold with volume measure dx , the classical one particle phase-space is X

¹

= X = T

^∗

M

¹

and we will focus on the h-dependent quantizations which associates with a symbol a(x, ξ) = a(X) , X ∈ X

¹

an operator a

^Q,h

= a(x, hD

_x

) with the standard semiclassical quantization or when M

¹

= R

^d

, a

^Q,h

= a

^W,h

= a

^W

(h

^t

x, h

^1−t

D

x

) by using the Weyl quantization, t ∈ R being fixed.

Note that in later sections the parameters ε and h will be linked through ε = ε(h) with

lim

_h→0

ε(h) = 0 . In relation with the symmetrization Lemma 2.8, we introduce the adapted

p-particles phase-space which was also considered in [Der1], and the corresponding semiclassical

observables.

3.1 Classical p -particles phase-space

A fundamental principle of quantum mechanics is that identical particles are indistinguishable.

The classical description is thus concerned with indistiguishable classical particles. If one classical particle is characterized by its position-momentum (x, ξ) ∈ X

¹

= T

^∗

M

¹

, x ∈ M being the position coordinate and ξ the momentum coordinates, p indistinguishable particles will be described by their position-momentum coordinates (X

₁

, . . . , X

_p

) = (x

₁

, ξ

₁

, . . . , x

_p

, ξ

_p

) ∈ X

^p

/S

_p

= (T

^∗

M )

^p

/S

_p

= T

^∗

(M

^p

)/S

_p

, where the quotient by S

_p

simply implements the identification

∀ σ ∈ S

_p

, (X

_σ(1)

, . . . , X

_σ(p)

) ≡ (X

₁

, . . . , X

_p

) .

The grand canonical description of a classical particles system then takes place in the disjoint union

p

⊔

∈N

X

^p

/S

_p

= ⊔

p∈N

(T

^∗

M)

^p

/S

_p

.

A p-particles classical observable will be a function on X

^p

/S

_p

and when the number of particles is not fixed a collection of functions (a

^(p)

)

p∈N

each a

^(p)

being a function on X

^p

/S

p

. The situation is presented in this way in [Der1]. A p-particles observable is a function a

^(p)

on X

^p

/S

_p

and a p-particles classical state is a probability measure (and when the normalization is forgotten a non negative measure) on X

^p

/S

p

.

However while quantizing a classical observable, it is better to work in X

^p

which equals T

^∗

(M

^p

) , a function a

^(p)

on X

^p

/S

_p

being nothing but a function on X

^p

which satisfies

∀ σ ∈ S

_p

, σ

^∗

a

^(p)

= a

^(p)

= 1 p!

X

˜ σ∈Sp

˜ σ

^∗

a

^(p)

,

with ∀ X

₁

. . . X

_p

∈ X

^p

, σ

^∗

a

^(p)

(X

₁

, . . . , X

_p

) = a

^(p)

(X

_σ(1)

, . . . , X

_σ(p)

) .

In the same way, we define for a Borel measure ν on X

^p

and σ ∈ S

_p

, the measure σ

_∗

ν by R

X

σ

^∗

a

^(p)

dν = R

X^p

a

^(p)

d(σ

_∗

ν) for all a

^(p)

∈ C

c⁰

( X

^p

) , or alternatively σ

_∗

ν(E) = ν(σ

⁻¹

E) for all Borel subset E of X

^p

. A non-negative measure on X

^p

/S

_p

is identified with a non-negative measure ν on X

^p

such that

∀ σ ∈ S

_p

, σ

_∗

ν = ν = 1 p!

X

˜ σ∈Sp

˜

σ

_∗

ν . (6)

Lemma 3.1 (Classical symmetrization lemma). Any Borel measure µ

^(p)

on X

^p

/S

_p

is characterized by the quantities R

X^p

a

^⊗p

dµ

^(p)

, a ∈ C where the tensor power a

^⊗p

means a

^⊗p

(X

₁

, . . . , X

_p

) = Q

p

i=1

a(X

i

) and C is any dense set in C

_∞⁰

( X

¹

) =

f ∈ C

⁰

( X

¹

) , lim

X→∞

f (X) = 0 .

Proof. By Stone-Weierstrass Theorem the subalgebra generated by the algebraic tensor product C

^⊗algp

is dense in C

_∞⁰

( X

^p

) . Hence it suffices to consider

a

₁

⊙ · · · ⊙ a

_p

= 1 p!

X

σ∈Sp

a

_σ(1)

⊗ · · · ⊗ a

_σ(p)

, a

_i

∈ C .

We conclude again with the polarization identity a

₁

⊙ · · · ⊙ a

_p

= 1

2

^p

p!

X

εi=±1

ε

₁

. . . ε

_p

X

p

i=1

ε

_i

a

_i

!

_⊗p

.

We will work essentially with M = R

^d

and X = T

^∗

R

^d

and therefore on X

^p

= T

^∗

R

^dp

∼ R

^2dp

and recall the invariance properties, if possible by a change of variable in order to extend it to the general case. Remember that on R

^dp

, the standard and Weyl semiclassical quantization are asymptotically equivalent a(x, hD

_x

) − a

^W

(x, hD

_x

) = O(h) when a ∈ S(1, dX

²

) (sup

_X∈T∗R^dp

| ∂

^α_X

a(X) | < ∞ for all α ∈ N

^2d

) . Moreover on R

^dp

, a

^W

(x, hD

_x

) is unitary equivalent to a

^W

(h

^t

x, h

^1−t

D

_x

) for any fixed t ∈ R so that result can be adapted to different scalings.

3.2 Semiclassical and multiscale measures

We recall the notions of semiclassical (or Wigner) measures and multiscale measures in the finite dimensional case. We start with the results on M = R

^D

(think of D = d p) and review the invariance properties for applications to some more general manifolds M .

3.2.1 In the Euclidean Space

On R

^D

the semiclassical Weyl quantization of a symbol a ∈ S

^′

( R

^2D

) will be written a

^W,h

= a

^W

(h

^t

x, h

^1−t

D

_x

) with t > 0 fixed and a kernel given by

[a

^W

(x, D

_x

)](x, y) = Z

R^d

e

^iξ·(x−y)

a( x + y

2 , ξ) dξ (2π)

^d

.

Definition 3.2. Let (γ

_h

)

_h∈E

with 0 ∈ E , E ⊂ (0, + ∞ ) , be a family of trace-class non-negative operators on L

²

( R

^D

) such that lim

_h→0

Tr [γ

_h

] < + ∞ . The semiclassical quantization a 7→ a

^W,h

= a

^W

(h

^t

x, h

^1−t

D

x

) is said adapted to the family (γ

_h

)

_h∈E

if

δ

lim

→0⁺

lim sup

h∈E, h→0

Re Tr h

(1 − χ(δ · )

^W,h

)γ

_h

i

= 0 for some χ ∈ C

₀^∞

(T

^∗

R

^D

) such that χ ≡ 1 in a neighborhood of 0 .

The set of Wigner measures M (γ

_h

, h ∈ E ) is the set of non-negative measures ν on T

^∗

R

^D

such that there exists E

^′

⊂ E , 0 ∈ E

^′

such that

∀ a ∈ C

0^∞

(T

^∗

R

^D

) , lim

h∈E^′ h→0

Tr h

γ

_h

a

^W,h

i

= Z

T^∗R^D

a(X) dν(X) .

The following well known statement (see [CdV, HMR, Ger, GMMP, LiPa, Sch]) results from the asymptotic positivity of the semiclassical quantization and it is actually the finite dimensional version of bosonic mean-field Wigner measures (with the change of parameter ε = 2h) (see [AmNi1]–

Section 3.1).

Proposition 3.3. Let (γ

_h

)

_h∈E

with 0 ∈ E , E ⊂ (0, + ∞ ) , such that γ

_h

≥ 0 and lim

_h→0

Tr [γ

_h

] <

+ ∞ . The set of semiclassical measures M (γ

_h

, h ∈ E ) is non-empty. The semiclassical quantization a

^W,h

is adapted to the family (γ

_h

)

_h∈E

, if, and only if, any ν ∈ M (γ

_h

, h ∈ E ) satisfies ν( R

^2D

) = lim

_h→0

Tr [γ

_h

] .

Remark 3.4. The manifold version, with a

^Q,h

= a(x, hD

_x

) instead of a

^W,h

, results from the semiclassical Egorov theorem.

By reducing E to some subset E

^′

(think of subsequence extraction), one can always assume that

there is a unique semiclassical measure. While considering a time evolution problem, or adding

another non countable parameter, (γ

_t,h

)

_h∈E,t∈R

finding simultaneously the subset E

^′

for all t ∈ R requires some compactness argument w.r.t the parameter t ∈ R , usually obtained by equicontinuity properties.

We now review the multiscale measures introduced in [FeGe, Fer]. For reader’s convenience, details are given in Appendix A, concerning the relationship between Prop. 3.5 below and the more general statement of [Fer].

The class of symbols S

⁽²⁾

is defined as the set of a ∈ C

^∞

( R

^2D

× R

^2D

) , such that

• there exists C > 0 such that ∀ Y ∈ R

^2D

, a( · , Y ) ∈ C

0^∞

(B (0, C)) ;

• there exists a function a

_∞

∈ C

0^∞

( R

^2D

× S

^2D−1

) such that a(X, Rω)

^R→∞

→ a

_∞

(X, ω) in C

^∞

( R

^2D

× S

^2D−1

) .

Those symbols are quantized according to

a

^(2),h

= a

^W,h_h

, a

_h

(X) = a(X, X h

^1/2

) .

A geometrical interpretation of those double scale symbols can be given by matching the compact- ified quantum phase space with the blow-up at r = 0 of the macroscopic phase space, see Figure 1.

Figure 1: On the left-hand side, the macroscopic phase space with its sphere at infinity. On the right-hand side the matched quantum and macroscopic phase spaces for which the quantum sphere at infinity and the r = 0 macroscopic sphere coincide.

Proposition 3.5. Let (γ

_h

)

_h∈E

be a bounded family of non-negative trace-class operators on L

²

( R

^D

) with lim

_h→0

Tr [γ

_h

] < + ∞ . There exist E

^′

⊂ E , 0 ∈ E

^′

, non-negative measures ν and ν

_(I)

on R

^2D

and S

^2D−1

, and a γ

₀

∈ L

¹

(L

²

( R

^D

)) , such that M (γ

_h

, h ∈ E

^′

) = { ν } and, for all a ∈ S

⁽²⁾

,

h

lim

∈E^′ h→0

Tr h

γ

_h

a

^(2),h

i

= Z

R^2D\{0}

a

_∞

(X, X

| X | ) dν(X) +

Z

S2D−1

a

_∞

(0, ω) dν

_(I)

(ω) + Tr [a(0, x, D

x

)γ

0

] .

Definition 3.6. M

⁽²⁾

(γ

_h

, h ∈ E ) denotes the set of all triples (ν, ν

_(I)

, γ

₀

) which can be obtained in Prop. 3.5 for suitable choices of E

^′

⊂ E , 0 ∈ E

^′

.

Remark 3.7. Actually when a

^W,h

= a

^W

( √ hx, √

hD

_x

) , this trace class operator γ

₀

is nothing but the weak

^∗

-limit of γ

_h

. Take simply a(X, Y ˜ ) = χ(X)α(Y ) with χ , α ∈ C

0^∞

( R

^2D

) , χ ≡ 1 in a neighborhood of 0 , for which lim

_h→0

k a ˜

^(2),h

− α

^W

(x, D

_x

) k

_L(L²)

= 0 . The above results says lim

_h→0

Tr

γ

_h

α

^W

(x, D

_x

)

= Tr

γ

₀

α

^W

(x, D

_x

)

for all α ∈ C

0^∞

( R

^2D

) ⊂ L

²

( R

^2D

, dX) , and by the density of the embeddings C

0^∞

( R

^2D

) ⊂ L

²

( R

^2D

, dx) ∼ L

²

(L

²

( R

^D

)) ⊂ L

^∞

(L

²

( R

^D

)) , the test ob- servable α

^W

(x, D

_x

) , can be replaced by any compact operator K ∈ L

^∞

(L

²

( R

^D

, dx)) . Moreover the relationship between ν and the triple (1

_(0,+∞)

( | X | )ν, ν

_(I)

, γ

₀

) can be completed in this case by

ν( { 0 } ) = Z

S^2D−1

dν

_(I)

(ω) + Tr [γ

₀

] , (7)

and ν

_(I)

≡ 0 is equivalent to ν ( { 0 } ) = Tr [γ

₀

] .

Because products of spheres are not spheres, handling the par ν

_(I)

in the p-particles space, D = dp , is not straitghtforward within a tensorization procedure, see Figure 2. Actually we expect

Figure 2: Tensor product of two blow-ups. The product of the two matching spheres is not a sphere:

the corners of the grey square correspond to the case when the quantum variables | X

1

| and | X

2

| go to infinity without any proportionality rule.

in the applications that a well chosen quantization leads to ν

_(I)

= 0 . This leads to the following definition.

Definition 3.8. Assume that the quantization a

^W,h

= a

^W

( √ hx, √

hD

x

) is adapted to the family (γ

_h

)

_h∈E

, γ

_h

≥ 0 , Tr [γ

_h

] = 1 . We say that the quantization a

^W,h

= a

^W

( √

hx, √

hD

_x

) is separating for the family (γ

_h

)

_h∈E

if one of the three following (equivalent) conditions is satisfied:

1. For any triple (ν, ν

_(I)

, γ

₀

) ∈ M

⁽²⁾

(γ

_h

, h ∈ E ), ν

_(I)

= 0 . 2. M (γ

_h

, h ∈ E

^′

) = { ν } ,

w

^∗

− lim

h∈E^′,h→0

γ

_h

= γ

₀

in L

¹

(L

²

( R

^D

)) )

⇒ ν ( { 0 } ) = Tr [γ

0

] .

3. For any triple (ν, ν

_(I)

, γ

₀

) ∈ M

⁽²⁾

(γ

_h

, h ∈ E ), ν( { 0 } ) = Tr [γ

₀

] .

Remark 3.9. This terminology expresses the fact that the mass localized at any intermediate scale vanishes asymptotically when ν

_(I)

≡ 0 . Accordingly, the microscopic quantum scale and the macro- scopic scale are well identified and separated.

Hence we can get all the information by computing the weak

^∗

-limit of γ

_h

and the semiclassical measure ν and then by checking a posteriori the equality ν( { 0 } ) = Tr [γ

₀

] .

This will suffice when the quantum part corresponds within a macroscopic scale, to a point in the phase-space. When M = R

^d

, we have enough flexibility by choosing the small parameter h > 0 and using some dilation in R

^D

in order to reduce many problems to such a case. On a manifold M if we can first localize the analysis around a point x

₀

∈ M , the problem can be transferred to R

^D

and then analyzed with the suitable scaling.

3.2.2 On a Compact Manifold

We now consider another interesting case of a compact manifold M with the semiclassical calculus a

^Q,h

= a(x, hD

_x

). This case is not completely treated in [Fer] because the geometric invariance properties do not follow only from the microlocal equivariance of semiclassical calculus. We assume Z = L

²

(M, dx) to be defined globally on the compact manifold M (e.g. by introducing a metric, dx being the associated volume measure).

Remark 3.10. When M is a general manifold, replace a

^W,h

in Def. 3.2 by a

^Q,h

= a(x, hD

x

) , and χ(δ · ) with δ → 0 by some increasing sequence of comptacly supported cut-off functions (χ

_n

)

_n∈N

, such that ∪

n∈N

χ

⁻_n¹

( { 1 } ) = T

^∗

M .

To adapt Prop. 3.5 to the case of a compact manifold, we consider another notion instead of the symbols S

⁽²⁾

. For the observables we shall consider the pair (K, a) where K ∈ L

^∞

(L

²

(M, dx)) and a ∈ C

0^∞

(S

^∗

M ⊔ (T

^∗

M \ M)) with S

^∗

M ⊔ (T

^∗

M \ M) being described in local coordinates through the identification

M × [0, ∞ ) × S

^D−1

∋ (x, r, ω) 7→

( (x, ω) ∈ S

^∗

M if r = 0 , (x, ξ = rω) ∈ T

^∗

M \ M otherwise .

We have identified the 0-section of the cotangent bundle T

^∗

M with M . After introducing an additional parameter δ > 0 , δ ≥ h , and a C

^∞

partition of unity (1 − χ) + χ ≡ 1 on T

^∗

M with 1 − χ ∈ C

0^∞

(T

^∗

M ) , 1 − χ ≡ 1 in a neighborhood of M , we can quantize a as

a

^(2)Q,δ,h

= [χ(x, ξ)a(x, hδ

⁻¹

ξ)]

^Q,δ

.

Note that K and the quantization of a are geometrically defined modulo O (δ) when h ≤ δ in L (L

²

(M, dx)): Use local charts for the semiclassical calculus with parameter δ while L

^∞

(L

²

(M, dx)) is globally defined like all natural spaces associated with L

²

(M, dx) . Actually in local coordinates the seminorms of the symbol χ(x, ξ)a(x, hδ

⁻¹

ξ) in S(1, dx

²

+ dξ

²

) are uniformly bounded w.r.t. h ∈ (0, δ] by seminorms of a in C

0^∞

((T

^∗

M \ M) ⊔ S

^∗

M) . When a ≥ 0 one also has

k (χa( · , hδ

⁻¹

· )

^Q,δ

− Re h

(χa( · , hδ

⁻¹

· )

^Q,δ

i

k ≤ C

_a

δ (8)

k a k

L^∞

+ C

_a

δ ≥ Re h

(χa( · , hδ

⁻¹

· )

^Q,δ

i

≥ − C

_a

δ (9)

uniformly w.r.t to h ∈ (0, δ] .

Proposition 3.11. Let (γ

_h

)

_h∈E

be a family of non-negative trace class operators on L

²

(M, dx) , such that lim

_h→0

Tr [γ

_h

] < + ∞ . Then there exist E

^′

⊂ E , 0 ∈ E

^′

, with M (γ

_h

, h ∈ E

^′

) = { ν } , a non-negative measure ν

_(I)

on S

^∗

M and a non-negative γ

₀

∈ L

¹

(L

²

(M, dx)) such that, for any K ∈ L

^∞

(L

²

(M, dx)),

h

lim

∈E^′ h→0

Tr [γ

_h

K] = Tr [γ

₀

K]

and, for any a ∈ C

0^∞

(S

^∗

M ⊔ (T

^∗

M \ M )), and any partition of unity (1 − χ) + χ ≡ 1 with 1 − χ ∈ C

0^∞

(T

^∗

M) , 1 − χ ≡ 1 in a neighborhood of M ,

δ

lim

→0

lim

h∈E^′ h→0

Tr h

γ

_h

a

^(2)Q,δ,h

i

= Z

T^∗M\M

a(X) dν(X) + Z

S^∗M

a(X)dν

_(I)

(X) . Additionally (ν

_(I)

, γ

₀

) is related to ν by

ν (E) = ν

_(I)

(π

⁻¹

(E)) + ν

₀

(E) ,

for any Borel set E ⊂ M identified with E × { 0 } , when π : S

^∗

M → M is the natural projection and ν

₀

is defined by R

M

ϕ(x)dν

₀

(x) = Tr [γ

₀

ϕ] , where ϕ ∈ C

^∞

(M ) is identified with the multiplication operator by the function ϕ.

Proof. When γ

_h

is bounded in L

¹

(L

²

(M, dx)) , after extraction of a sequence h

_n

→ 0 from E , M ((γ

_hn

)

_n∈N

) = { ν } , and the weak

^∗

limit γ

₀

of (γ

_hn

), and the associated measure ν

₀

are well- defined objects on the manifold M .

Let us construct a measure ˜ ν on (T

^∗

M \ M) ⊔ S

^∗

M = n

(x, rω) , x ∈ M, ω ∈ S

^d−1

, r ∈ [0, ∞ ) o and a subset E

^′

⊂ E , 0 ∈ E

^′

, such that

δ

lim

→0

lim

h∈E^′ h→0

Tr h

γ

_h

(χa( · , hδ

⁻¹

· ))

^Q,δ

i

= Z

(T^∗M\M)⊔S^∗M

a d˜ ν (10)

holds for all a ∈ C

0^∞

((T

^∗

M \ M ) ⊔ S

^∗

M) .

Fix first the partition of unity (1 − χ) + χ ≡ 1, 1 − χ ∈ C

0^∞

(T

^∗

M), 1 − χ ≡ 1 in a neighborhood of M , and δ = δ

₀

> 0 . For a given a ∈ C

0^∞

((T

^∗

M \ M ) ⊔ S

^∗

M) , the inequalities (8) and (9) imply that one can find a subsequence (h

_k,χ,δ0,a

)

_k∈N

of (h

_n

)

_n∈N

, such that

k

lim

→∞

Tr h γ

_hk,χ,δ

0,a

(χa( · , h

_k,χ,δ0,a

δ

₀⁻¹

· )

^Q,δ0

i

= ℓ

_χ,δ0,a

∈ C . (11) For a different partition of unity (1 − χ) + ˜ ˜ χ ≡ 1 the symbol [χ − χ]a(x, hδ ˜

₀⁻¹

ξ) is supported in C

_χ,⁻¹_χ,δ˜

0

≤ | ξ | ≤ C

_{χ,χ,δ˜ 0}

and equals

[χ − χ]a(x, hδ ˜

₀⁻¹

ξ) = [χ − χ]a ˜

0

(x, ξ

| ξ | ) + hr

_{χ,χ,δ˜ 0,h}

(x, ξ) , where a

₀

= a

S^∗M

and with r

_{χ,χ,δ˜ 0,h}

uniformly bounded in S(1, dx

²

+ dξ

²

) . For δ

₀

> 0 fixed, the operator [(χ − χ)a ˜

0

]

^Q,δ0

is a compact operator and we obtain

h

lim

→0

Tr h

γ

_h

(χa( · , hδ

₀⁻¹

· ))

^Q,δ0

i

− Tr h

γ

_h

( ˜ χa( · , hδ

₀⁻¹

· ))

^Q,δ0

i

= Tr h

γ

₀

((χ − χ)a ˜

₀

)

^Q,δ0

i

.

Therefore the subsequence extraction, which ensures the convergence (11) can be done independently of the choice of ˜ χ and by taking ˜ χ(x, ξ) = χ(x, δδ

₀⁻¹

ξ) independently of δ > 0 . For E

a

= (h

_k,a

)

_k∈N

such a sequence of parameters, the limits can be compared by ℓ

_χ,δ,a˜

− ℓ

_χ,δ0,a

= lim

h∈Ea

h→0

Tr h

(γ

_h

χa(., hδ ˜

⁻¹

.))

^Q,δ

i

− Tr h

(γ

_h

χa( · , hδ

⁻₀¹

· ))

^Q,δ0

i

= Tr h

(( ˜ χ(δδ

₀⁻¹

) − χ)a

₀

)

^Q,δ0

γ

₀

i

. (12)

By choosing ˜ χ = χ above, the inequality 0 ≤ (χ − χ(δδ

⁻₀¹

))a

₀

≤ χa

₀

, for a

₀

≥ 0 and δ ≤ δ

₀

, and the δ

₀

-Garding inequality implies

| Tr h

(( ˜ χ(δδ

⁻₀¹

) − χ)a

0

)

^Q,δ0

γ

0

i | ≤ Tr[(χa

0

)

^Q,δ0

γ

0

] + O (δ

0

)

uniformly with respect to δ ≤ δ

₀

. Thus the quantity ℓ

_χ,δ,a

thus satisfies the Cauchy criterion as δ → 0 because s − lim

_δ0→0

(χa

0

)

^Q,δ0

= 0 and γ

0

is fixed in L

¹

(L

²

(M, dx)) . Hence the limit

ℓ

_χ,a

= lim

δ→0

ℓ

_χ,δ,a

= lim

δ→0

lim

h∈E^a h→0

Tr h

(γ

_h

χa( · , hδ

⁻¹

· ))

^Q,δ

i

exists for any fixed a ∈ C

0^∞

((T

^∗

M \ M ) ⊔ S

^∗

M ) . Using (12) with δ = δ

₀

but a general pair (χ, χ) and ˜ taking the limit as δ → 0 shows ℓ

_χ,a˜

= ℓ

_χ,a

= ℓ

_a

. The inequalities (8) and (9) give 0 ≤ ℓ

_a

≤ k a k

L^∞

. By the usual diagonal extraction process according to a countable set N ⊂ C

0^∞

((T

^∗

M \ M ) ⊔ S

^∗

M) dense in the set of continuous functions with limit 0 at infinity , we have found a subset E

^′

⊂ E , 0 ∈ E

^′

, and a non-negative measure ˜ ν such that (10) holds. Note that we have also proved

Z

(T^∗M\M)⊔S^∗M

a d˜ ν = lim

δ→0

lim

h∈E^′ h→0

Tr h

γ

_h

(χa(., hδ

⁻¹

.))

^Q,δ

i

= lim

δ→0

lim

h∈E^′ h→0

Tr h

(γ

_h

− γ

0

)(χa(., hδ

⁻¹

.))

^Q,δ

i

where both limits do not depend on the partition of unity (1 − χ) + χ ≡ 1 with 1 − χ ∈ C

0^∞

(T

^∗

M) equal to 1 in a neighborhood of M .

We still have to compare ˜ ν and ν . For this take a ∈ C

0^∞

(T

^∗

M) and set a

₀

(x, ω) = ϕ(x) = a(x, 0) . The symbol identity

a(x, hδ

⁻¹

ξ) = a(x, hδ

⁻¹

ξ)(1 − χ) + a(x, hδ

⁻¹

ξ)χ = ϕ(x)(1 − χ) + a(x, hδ

⁻¹

ξ)χ + hr

_a,χ,δ,h

with r

_a,δ,χ,h

uniformly bounded in S(1, dx

²

+ dξ

²

) w.r.t. h , leads after δ-quantization to

Z

T^∗M

a dν = lim

h∈E^′ h→0

Tr h

γ

_h

a

^Q,h

i

= lim

h∈E^′ h→0

Tr h

γ

_h

(ϕ(x)(1 − χ))

^Q,δ

i + lim

h∈E^′ h→0

Tr h

γ

_h

(χa( · , hδ

⁻¹

· ))

^Q,δ

i .

For δ > 0 fixed (ϕ(x)(1 − χ))

^Q,δ

is a fixed compact operator so that the first limit is

h

lim

∈E^′ h→0

Tr h

γ

_h

(ϕ(x)(1 − χ))

^Q,δ

i

= Tr h

γ

₀

(ϕ(x)(1 − χ))

^Q,δ

i

,