Agissilaos Athanassoulis
email: a.athanasoulis@damtp.cam.ac.uk

Thierry Paul
email: thierry.paul@math.polytechnique.fr

STRONG AND WEAK SEMICLASSICAL LIMIT FOR SOME ROUGH HAMILTONIANS

We present several results concerning the semiclassical limit of the time dependent Schrödinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharacteristics can be obtained out of the same initial point is emphasized.

Introduction

The state of a quantum system is described by a positive, trace class operator D ε , which evolves in time under the Heisenberg-von Neumann equation

(1.1) iε ∂ ∂t D ε (t) = -ε 2 2 ∆ + V, D ε (t) , D ε (t = 0) = D ε 0 .
Both the positivity and the trace of D ε are preserved in time [START_REF] Reed | Functional Analysis (Methods of modern Mathematical Physics)[END_REF]; observables, also described by operators, can be measured against the state. For example, the value of observable G ε , associated with operator G ε , at time t is given by the trace G ε (t) = tr(G ε D ε (t) ). The term density matrix is often used for a positive, trace-class operator.

The Heisenberg-von Neumann equation is closely related to the Schrödinger equation. By virtue of the singular value decomposition, the density matrix can be decomposed to a sum of pure states, i.e. rank-1 orthonormal projectors,

D ε = m λ ε m |u ε m u ε m |
(the λ ε m 's are time-independent in the standard context of an external, timeindependent potential, i.e. if V does not depend on D ε , or t. Here we limit our scope to such problems). Then, each of the u ε m 's satisfies a problem of the form

(1.2) iε ∂ ∂t u ε m = -ε 2 2 ∆ + V (x) u ε m , u ε m (t = 0) ∈ L 2 .
The small parameter ε is called Planck's constant, and when it tends to zero one expects the system to behave in a way approaching classical mechanics; i.e. motion under Newton's law in a force field with potential V . This is a singular asymptotic limit, and several techniques have been developed for its study; one is based on the Wigner transform (WT) [START_REF] Gérard | Mesures semi-classiques et ondes de Bloch[END_REF][START_REF] Gérard | Homogenization limits and Wigner transforms[END_REF][START_REF] Lions | Sur les mesures de Wigner[END_REF].

The WT W ε (x, k, t) is defined as

(1.3) W ε (x, k, t) = W ε [D ε (t)](x, k) = y∈R n e -2πiyk K ε (x + ε y 2 , x -ε y 2 , t)dy
where for each t, K ε (t) is the integral kernel of the operator D ε (t). The WT is also closely related to the Weyl symbol of D ε , and the Weyl calculus of pseudodifferential operators in general, see e.g. [START_REF] Folland | Harmonic Analysis in Phase-Space[END_REF] for more details in that direction. It evolves in time under the well-known Wigner equation, (1.4)

∂ t W ε (x, k) + 2πk • ∂ x W ε (x, k)+ + 2 ε Re i e 2πiSx V (S)W ε (x, k -εS 2 , t)dS = 0,
W ε (t = 0) = W ε 0 , an equivalent reformulation of (1.1). In this context, it can be shown that observables can be measured by G ε (t) = G ε W (x, k)W ε (x, k)dxdk, where G ε W (x, k) is the Weyl symbol of the operator G ε . (However this in general is not a Lebesgue integral, but only makes sense as a Cauchy PV one).

In the case of a pure state, D ε = |u ε u ε |, the WT can be written as

(1.5) W ε (x, k, t) = y∈R n e -2πiyk u ε (x + ε y 2 , t)ū ε (x -ε y 2 , t)dy.
As a quadratic transform on an oscillating function u ε , of small typical scale of oscillation ε, it has been widely used in signal processing, see e.g. [START_REF] Flandrin | Time-frequency/Time-scale Analysis[END_REF] and a multitude of references therein for more on that direction. Under appropriate conditions the WT allows for a very natural and compact description of the semiclassical limit. Indeed, it has a physically meaningful limit as ε tends to zero, while in general the operator D ε itself does not. The limit (in an appropriate sense; see below for more precise statement and reference), is called the Wigner measure,

W ε (t) W 0 (t),
and is a probability measure that evolves in time under the Liouville equation of classical statistical mechanics (1.6)

∂ t W 0 + 2πk • ∂ x W 0 -1 2π ∂ x V • ∂ k W 0 = 0, W 0 0 = lim ε 0 W ε 0 .
Observables are now described by bounded continuous functions, which can be measured against the classical state, G 0 (t) = G 0 (x, k)dW 0 (t). The formal probabilistic interpretation e.g. for the expected position of a particle X (t) = xdW 0 (t) should be intuitively clear. Thus the correspondence between classical and quantum physics for the same system is highlighted in a very intuitive way, in terms of models that are, at least formally, similar and directly comparable. Convergence issues are not always painless or completely understood, as we will see in more detail. There is also the well-known caveat that W ε (t) is in general neither a measure of bounded total variation, nor non-negative e.g. as a distribution. Finally asking that the limit problem itself should be well-posed isn't always trivial.

Note at this point that considering initial data with

W ε (0) δ(x -x 0 , k -k 0 )
is particularly natural: they stand after all for classical particles with position x 0 and velocity k 0 .

It was shown in [START_REF] Lions | Sur les mesures de Wigner[END_REF] (Théorème IV.1) that, under appropriate conditions on the initial data (which allow for practically all physically meaningful choices), as long as the potential is smooth enough, V ∈ C 1 , the WT converges (in weak- * sense, and up to extraction of a subsequence) to a weak solution of the limit problem (1.6). It was immediately commented that, for V ∈ C 1 \ C 1,1 , this does not yield an effective computation of the semiclassical limit in general, but only a partial description: even after the extraction of a subsequence (so that there is a unique lim n W εn 0 ), in general there are several weak solutions for the Liouville equation that W εn (t) could converge towards (because the Liouville equation with C 1 \ C 1,1 potential and measure valued initial data has existence, but not uniqueness of weak solutions. This can be seen e.g. using the Peano Theorem). Therefore some kind of selection principle would be needed to supplement the result, if we were to be able to identify the Wigner measure corresponding to our subsequence in ε of initial data. This, in a nutshell, is the context for the present work. Our goal is to understand better the mechanisms that may lead to loss of uniqueness, and develop tools which, at least in some cases, can resolve them.

We are particularly concerned with initial data that concentrate W ε (0) δ(x-x 0 , k-k 0 ), and our findings here apply, under appropriate assumptions, to such data. The main assumption (other than, essentially, C 1,θ potential regularity), is, roughly speaking, a non-concentration condition on the flow associated with the Liouville equation.

Before the statement of the present results, some reference to related works is warranted. The problem of extending semiclassical asymptotics to non-regular potentials (e.g. worse than C 1,1 ) has been studied by many authors -and for good reasons: some of the simplest, most natural systems (such as a gravitational or electrostatic force field) are described by potentials that are not C 1 , 1, or C 1 . In most cases the conclusions make use of a particular form of singularity (e.g. Coulomb/piecewise smooth potential etc) as opposed to a general smoothness class [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF][START_REF] Keraani | Wigner measures dynamics in a Coulomb potential[END_REF], and/or need some additional, non-trivial condition (non-concentration, non-interference etc) [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF]8]. Another type of results is for whole random populations of initial data [START_REF] Ambrosio | Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data[END_REF] (which in particular can be even weaker than a result applying to "almost all" initial data; i.e. there might be no way to just choose "almost any" initial datum at t = 0 and keep track of it; the conclusions apply the population as a whole). Finally, it must be mentioned that in the study of the semiclassical limit in nonlinear problems (e.g. the Schrödinger-Poisson equation, a very natural problem), a lot remains to be understood. One of the main difficulties, comes precisely from the lack of regularity of the potential (which now depends on D ε (t)), and the subsequent lack of wellposedness for the corresponding classical problem for measure-valued initial data; see e.g. [START_REF] Lions | Sur les mesures de Wigner[END_REF][START_REF] Markowich | The classical limit of a self-consistsent quantum Vlasov equation[END_REF][START_REF] Zhang | The limit from the Schrödinger-Poisson to the Vlasov-Poisson equations with general data in one dimension[END_REF].

More specifically, semiclassical limits with rough potentials were considered recently in [START_REF] Ambrosio | Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data[END_REF] where the hypotheses on the potential are, roughly speaking, that it has a generic part with BV loc gradient, plus possibly a repulsive Coulomb part. The setting involves a random population of initial data, and it is shown that the corresponding population of solutions at a later time tends weakly to the push-forward by the Liouville equation of a population of measures. The classical propagation problem for the random family of measures is shown to be well-posed by virtue of the Ambrosio-Di Perna-Lions theory [START_REF] Ambrosio | Transport equation and Cauchy problem for BV vector fields[END_REF][START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]. In [8] similar core ideas were used with deterministic mixed states. In that context, an averaging condition forbids initial data concentrating e.g. to a delta function in phase-space, a fact consistent with the weak limit strategy and the fact that the flow is not defined everywhere. The present work therefore concerns less general potentials than those treated in [START_REF] Ambrosio | Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data[END_REF]8], (still giving rise to ill-posed classical dynamics of course), but can deal with initial data concentrating to point-supported measures.

The possible ill-posedness of the limit equation can originate not only from lack of smoothness of the potential, which we focused on for our discussion so far, but e.g. from working on a torus instead of Euclidean space [START_REF] Macia | High-frequency propagation for the Schroedinger equation on the torus[END_REF].

Finally, let us present a concrete example. Already in [START_REF] Lions | Sur les mesures de Wigner[END_REF] , families of initial data were constructed so that their Wigner measures at later times oscillate between the several solutions of the Liouville equation. The onedimensional case study used for that, was essentially equivalent to the following:

Consider the potential

(1.7) V (x) = -|x| 1+θ • β(x)
with θ ∈ (0, 1), C ∞ β = 1 on [-1, +1], and β(x) = 0 for |x| ≥ 2, and initial data such that W ε 0 δ(x -0, k -0). Then one can easily check that there exist, for t small enough, two families of bicharateristics starting near the origin at t = 0, namely

(1.8) (X ± (t), P ± (t)) = (±c 0 t ν , ± c 0 ν 2π t ν-1 ), with ν = 2 1-θ and c 0 = (1-θ) 2 2 1-θ
. In this case it is possible to find semiclassical limits where one wavepacket (a δ-function in classical phase-space) splits in two wavepackets. (In this particular example it happens immediately, i.e. ∀t > 0, or not at all; different examples can be constructed where it happens after an ε-independent time t * ). The mass of each wavepacket depends on information which can be easily extracted from the quantum problem, but is lost if we take the limit as usual in a straightforward way.

The first of our results deals explicitly with the case study from [START_REF] Lions | Sur les mesures de Wigner[END_REF] mentioned earlier, to provide a clear illustration to the selection principle at work. Our second Theorem gives a general result concerning the approximation, in strong topology, of the Wigner function of the solution of (1.4) with the solution of a "smoothed" Liouville equation. Our third results goes back to the weak approximation, but with explicit remainder estimates.

Main results

We will use a space of test-functions introduced in [START_REF] Lions | Sur les mesures de Wigner[END_REF]: let (2.1)

A = {φ ∈ C 0 (R 2n ) | sup x |F k K [φ(x, k)]|dK < ∞}, equipped with the norm (2.2) ||φ|| A = sup x |F k K [φ(x, k)]|dK.
Our first result will focus on the case study presented in the introduction; a family of initial data which concentrates at the origin at time t = 0 and splits into two separated wave packets at time t > 0. This is only an example among several others presented in section 5 and the more general results of the paper are given by the Theorems 2.2 and 2.3 below.

Theorem 2.1. Let V given by (1.7) and

(2.3) W ε 0 (x, k) = λ 7+3θ 30 w(λ 1+θ 6 x, λ 1-θ 15 k) * 2 ε n e -2π x 2 +k 2 ε with λ = log( 1 ε ), w ∈ H 2 ∩ L ∞ ∩ L 1 , w(x, k) 0, supp w ⊆ {|x| 2 + |k| 2 < 1}, w(x, k)dxdk = 1.
Then ∃T > 0 such that for all t ∈ [0, T ], the solution W ε (t) of (1.4) converges in weak- * sense in A to

(2.4) W 0 (t) = c + δ(X + (t), P + (t)) + c -δ(X -(t), P -(t)),
with (X ± (t), P ± (t)) given by (1.8) and

c ± = ±x>0 w(x, k)dxdk.
Remark: The fact that the initial data contain a smoothing by 2

ε n e -2π x 2 +k 2 ε ensures that W ε 0 corresponds to a density matrix, i.e. a positive trace-class operator (see e.g. Exemple III.7 of [START_REF] Lions | Sur les mesures de Wigner[END_REF], or Lemma 6.7).

In Remarque IV.3 of [START_REF] Lions | Sur les mesures de Wigner[END_REF], one can find a construction of a family of normalized wave functions u + ε concentrating at the origin, and whose Wigner functions, after extraction of a subsequence, converge to a Dirac mass centered on (X + (t), P + (t)). The same construction is obviously possible in order to get a family u - ε whose Wigner functions, after extraction of a subsequence, converge to a Dirac mass centered on (X -(t), P -(t)). Defining

D ε 0 = 1 √ 2 (|u + ε u + ε | + |u - ε u - ε |
) one can check that the Wigner function of D ε 0 will follow the conclusion of Theorem 2.1. The present construction is different, and is not obtained through compactness arguments: there is no need of extraction of subsequence, and the scaling property is explicit (it is given by an implicit diagonal argument in a two scale sequence in [START_REF] Lions | Sur les mesures de Wigner[END_REF]). The construction of [START_REF] Lions | Sur les mesures de Wigner[END_REF] is based on the more-or-less explicit understanding of the flow around the singular point, while our approximation (i.e. Theorems 2.2 and 2.3 below) is built around a non-concentration condition, which is relatively easy to check for other problems, whether the flow can be computed explicitly or not. (This may be increasingly important in high dimensions).

Let us also point out the similarity with the long-time behaviour around a regular separatrix, studied extensively in [START_REF] Paul | Échelles de temps pour l'évolution quantique à petite constante de Planck[END_REF]. In the regular case, the quantum wavepacket leaves the fixed point in "infinite" (i.e. asymptotically long) time, as does the classical flow; here the classical flow leaves the fixed point immediately, and this behaviour (under additional assumptions, of course) is preserved by the quantum system.

We turn now to the more general results of this paper.

In the sequel we will consider potentials V that satisfy the following assumption:

Assumption 1. (2.5) R n | V (S)| S 2 1 + S 2 dS < ∞,
and moreover there are constants C > 0, θ ∈ (0, 1) such that for m ∈ {0, 1, 2} ∀1 a b +∞ :

|S|∈(a,b) | V (S)| |S| m dS C m-1-θ b m-1-θ -a m-1-θ
This is closely modeled after V (x) = C|x| 1+θ ; indeed it is easy to check that the aforementioned potential satisfies this condition. In section 5, we will also see some other relevant types of singularities (also generated from |x| 1+θ in some sense).

Denote

(2.6) ∼γ V (x) = 2 ε γ n 2 e -2π ε γ |x-x | 2 V (x )dx .
Theorem 2.2. Let us suppose that Assumption 1 holds, and there exist T > 0, δ ∈ (0, θ 2+θ ), γ > 2 1+θ and approximate initial data

[W ε 0 ] ∈ H 2 (R 2n ) such that (2.7) ||[W ε 0 ] -W ε 0 || L 2 = o(||W ε 0 || L 2 ) and the solution of (2.8) ∂ t ρ + 2πk • ∂ x ρ - 1 2π ∂ x ∼γ V •∂ k ρ = 0, with initial condition ρ(t = 0) = [W ε 0 ] satisfies (2.9) ||ρ(t)|| H 2 = O(ε -δ ||W ε 0 || L 2 ) uniformly on [0, T ].
Then the Wigner function W ε (t) of the solution of (1.4) satisfies, uniformly on [0, T ],

(2.10) ||W ε (t)-ρ ε 1 (t)|| L 2 = O(ε κ ||W ε 0 || L 2 + ||W ε 0 -[W ε 0 ]|| L 2 ) = o(||W ε 0 || L 2 ). where ρ ε 1 is the solution of (2.8) with initial datum ρ ε 1 (t = 0) = W ε 0 and κ = min{ γ 1+θ 2 -1, θ 2+θ -δ }.
Remark: The non-concentration condition. It is helpful to discuss the motivation behind this type of assumption. Ideally, one would like the solution of the problem

∂ t ρ + 2πk • ∂ x ρ -1 2π ∂ x ∼γ V •∂ k ρ = 0, ρ(t = 0) = W ε 0 to have H 2 regularity in the sense ||ρ(t)|| H 2 e Ct ||W ε
0 || H 2 for some C that is either ε-independent, or grows extremely slowly in ε. Of course, when working with low-regularity potentials, this is in general not possible. So one looks for a weaker version of regularity, which would still be sufficient here; it turns out that the property

∃[W ε 0 ] ∈ H 2 (R 2n ), ||[W ε 0 ] -W ε 0 || L 2 = o(||W ε 0 || L 2 ) such that, if ∂ t ρ + 2πk • ∂ x ρ -1 2π ∂ x ∼γ V •∂ k ρ = 0, ρ(t = 0) = [W ε 0 ] then ||ρ(t)|| H 2 is not too large is an acceptable surrogate.
We call this a non-concentration condition, because of one of the simplest ways to check it. Assume that there is a "small" set S ⊆ R 2n , such that the derivatives of ∼γ V up to order 3 outside S are "not too large" in ε.

Now set B = t∈[0,T ] φ ε -t (S)
, where φ ε t is the flow associated with the regularized Liouville equation (2.8). B contains all the "dangerous" part of phasespace; away from B the original regularity requirement holds automatically.

A "concentrating flow" could pull too big a part of phase-space through the original "small bad neighbourhood" S in time t ∈ [0, T ]. On the other hand, a "repulsive singularity", in general tends to push the flow away, thus keeping the pre-image of S small enough. The approximate initial data [W ε 0 ] can be then constructed by restricting the initial data on B C . Remark: Checking the assumptions for concrete problems. Concrete examples for which the assumptions (in particular the non-concentration condition) can be checked to hold, are examined in section 5. The techniques used can be easily extended to similar problems. It must be noted that the condition for Theorem 2.2 is seen to be satisfied for a generic selection of initial data, as long as the rate of concentration satisfies certain constraints (i.e. the profile that concentrates can be any function with a given regularity; see e.g. Lemmata 5.1, 5.2). In contrast, for the stronger Theorem 2.3 (see next Remark for the difference), in general we need to restrict the shape of the wavepacket with additional assumptions for the same problems; Lemmata 5.3, 5.4.

Remark: L 2 versus measures, and convergence. Theorem 2.2 is an L 2 asymptotic approximation result: two functions, of constant L 2 norm in time (also equal to one another and large in ε), are relatively close in L 2 . It is not, however, a convergence result, since in general neither function has a limit in L 2 . Ideally, we would like to conclude that, since they're (relatively) close, both functions converge (in some appropriate sense) to the same limit, typically a (sum of) delta function(s). However, the (relative) L 2 approximation is not by itself sufficient to ensure that if e.g. ρ e 1 (t) δ(x -x(t), k -k(t)), then also W ε (t) goes to the same limit. Is there some additional information in the problem that we could use to come to that conclusion? Or is there a physical issue here, of L 2 norm really "missing" the solution? For example, is it really possible that in a problem with splitting (as in Theorem 2.1) the conclusion of Theorem 2.2 holds, in the limit ρ ε 1 "goes to the right", but at the same time W ε (or half its mass) "goes to the left"? This is a question that we cannot really answer at the moment. One possible way to go is by strengthening the non-concentration condition, and that is pursued in Theorem 2.3 below. This yields full control over the convergence (in an appropriate sense, see also Corollary 2.4), but at the cost of excluding many interesting, natural problems.

Theorem 2.3. Assume that the hypotheses of Theorem 2.2 hold and ρ ε 1 (t) is as in Theorem 2.2. Moreover, in addition to the condition (2.7) we assume that

(2.11) ||[W ε 0 ] -W ε 0 || L 2 = o(1), ||W ε 0 || L 2 = o(ε -κ ) (κ being the same as in Theorem 2.
2), and the initial data W ε 0 corresponds to a density matrix, i.e. it is the Wigner function of a positive trace-class operator

D ε 0 , tr(D ε 0 ) = 1 ∀ε > 0. For φ ∈ A denote E ε (φ) = W ε (t) -ρ ε 1 (t)
, φ . and, for brevity, by F (ε) the bound of equation (2.10), namely

F (ε) = C(ε κ ||W ε 0 || L 2 + ||[W ε 0 ] -W ε 0 || L 2 ) = o(1)
, for some large enough constant C > 0.

Then, ∀φ ∈ A, t 0, the following estimate holds:

(2.12)

|E ε (φ)| F (ε)||φ|| A + 2 n π n 2 √ Γ(n+1) F (ε)+ +(1 + ||[W ε 0 ]|| L 1 ) || (1 -χ F (ε) 1 2n ) φ || A where χ ∈ C ∞ (R 2n , [0, 1]) is any cutoff function, χ(z) = 1 ∀|z| < 1, χ(z) = 0 ∀|z| > 2, and χ M (z) := χ(M z).
Remark: L 2 versus measures, and convergence. If a generic "nice" function concentrates, f ε δ(x, k), and

||f ε -[f ε ]|| L 1 = o(1), then it follows that [f ε ] δ(x, k) as well. In other words ||[W ε 0 ] -W ε 0 || L 1 = o(1)
(which in general holds in the examples where Theorem 2.2 does) would better suited to control concentration. We are forced however to use the much stronger

||[W ε 0 ] -W ε 0 || L 2 = o(1)
due to technical reasons, and in particular the fact that the Wigner equation doesn't respect the L 1 norm in general (or the total variation of measures, or even the A norm away from the positive cone which also scale like L 1 ). The assumptions of Theorem 2.3 should be considered pessimistic, and probably can be relaxed. In any case, the precise convergence result is summarized in Corollary 2.4 below.

Remark: Size of the error. The last term in rhs of equation (2.12) can be further estimated if there is more information on φ, and in particular its decay in the k variable.

Corollary 2.4. If the assumptions of Theorem 2.3 are satisfied and

||[W ε 0 ]|| L 1 is uniformly bounded in ε, then lim ε 0 E ε (φ) = 0 ∀φ ∈ A, t 0,
and therefore the Wigner function W ε (t) has a weak- * limit in A if and only if ρ ε 1 has a weak- * limit, and the two are equal, i.e.

(2.13) ∃W 0 (t) :

lim ε 0 W ε (t), φ = W 0 (t), φ ∀φ ∈ A, t ∈ [0, T ],
if and only if

(2.14) lim ε 0 ρ ε 1 (t), φ = W 0 (t), φ ∀φ ∈ A, t ∈ [0, T ].
In other words, under the assumptions of Theorem 2.3, one has to solve equation (2.8) in order to compute the propagation in time, ρ ε 1 (t), and then take the limit ε 0. That is, unlike what happens with regular potentials, the semiclassical limit and the propagation in time by (a regularization of) the Liouville flow no longer commute.

ρ ε 1 (0) •φ ε t / / ε 0 ρ ε 1 (t) ε 0 W 0 (0) W 0 (t)
Figure 1. In some cases, there is not enough information in W 0 (0) to determine W 0 (t), i.e. the Cauchy problem (1.6) is ill-posed.

Organization of the paper: Proofs of the main Theorems are in section 4. Checking the assumptions of the Theorems for various concrete problems is done in section 5. Certain auxiliary results we use are collected in section 6.

Definitions and Notations

The Fourier transform is defined as

(3.1) f (k) = F x k [f (x)] = x∈R n e -2πikx f (x)dx.
For compactness, we will use the following notations:

T V ε W = 2 ε Re i e 2πiSx V (S)W (x, k -εS 2 )dS = = 2F -1 X,K x,k V (S) W (X -S, K) sin(πεSK) ε dS , T V 0 W = -1 2π ∂ x ∼γ V •∂ x W = = 2πF -1 X,K x,k V (S) W (X -S, K)S • KdS .
The Sobolev norms of order m on phase-space will be defined as follows:

||f || W m,p (R 2n ) = |a|+|b| m ||∂ a x ∂ b k f || L p (R 2n ) ,
where of course a and b are multi-indices of length n each. Moreover,

H m (R 2n ) = W m,2 (R 2n
f ε 0 (x, k) = λ 7+3θ 30 w(λ 1+θ 6 x, λ 1-θ 15 k),
where of course

λ = log( 1 ε ), i.e. W ε 0 = f ε 0 * 2 ε n e -2π x 2 +k 2 ε . One should note that (4.2) ||W 0 ε -f ε 0 || L 2 √ ε||f ε 0 || H 1 = O(ε 1 2 -η ) ∀η ∈ (0, 1 2 )
(The approximation follows from a standard observation on smoothing operators, see e.g. [START_REF] Athanassoulis | Strong phase-space semiclassical asymptotics[END_REF] for a proof. Moreover, since by construction the constant κ of Theorem 2.2 is κ < 1 3 , the error ||W 0 ε -f ε 0 || L 2 can indeed be dropped without loss of generality, when invoking Theorem 2.3 -and Theorem 2.2 through it).

In Lemma 5.3 it is shown essentially that Theorem 2.3 applies if n = 1, V (x) = -|x| 1+θ in {|x| < 1} (with a smooth cutoff outside of that, the specifics of which are irrelevant) and

W ε 0 = δ -1 x δ -1 k w( x δx , k δ k ), where δ k = R 1+θ 2 , δ x = R 1-θ 5 .
(We avoid duplicating parts of the proof of Lemma 5.3 here. Moreover, it is completely straightforward to see that the assumptions not checked explicitly there also hold). Therefore Corollary 2.4 applies, i.e. it suffices to find the limit in ε of ρ ε 1 . Claim: With [W ε 0 ] defined as in equation (5.15), we have (4.3)

||W ε 0 -[W ε 0 ]|| L 1 = o(1).
Proof of the claim: We will work separately for the two terms

||W ε 0 -[W ε 0 ]|| L 1 ||W ε 0 -f ε 0 || L 1 + ||f ε 0 -[W ε 0 ]|| L 1 . First of all, it is obvious that |(x,k)|>2 |f ε 0 -W ε 0 |dxdk = |(x,k)|>2 |W ε 0 |dxdk = O(ε ∞ ).
Moreover, using the Jensen inequality, one sees that

|(x,k)|<2 |f ε 0 -W ε 0 |dxdk |(x,k)|<2 dxdk |(x,k)|<2 |f ε 0 -W ε 0 | 2 dxdk C||f ε 0 -W ε 0 || L 2 ,
and therefore

||W ε 0 -f ε 0 || L 1 O(ε ∞ ) + C||f ε 0 -W ε 0 || L 2 = o(1). Moreover, (4.4) ||f ε 0 -[W ε 0 ]|| L 1 Cδ -1 x δ -1 k |x|<CR |w( x δx , k δ k )|dxdk = = |x|<C R δx |w(x, k)|dxdk Cδ -1 x R = o(1).
The proof of the claim is complete. This means that ρ ε 2 and ρ ε 1 (the evolution under equation (2.8) of [W ε 0 ], W ε 0 respectively) are interchangeable for our purposes, since

ρ ε 1 , φ = ρ ε 2 , φ + ρ ε 1 -ρ ε 2 , φ and | ρ ε 1 -ρ ε 2 , φ | ||ρ ε 1 -ρ ε 2 || L 1 ||φ|| L ∞ ||W ε 0 -[W ε 0 ]|| L 1 ||φ|| A . (We used the obvious bound ||φ|| L ∞
||φ|| A ; recall that A was defined in eq. (2.1) ).

In particular, this means we can always work with |x| > CR, since ρ ε 2 stays outside of {|x| < CR} by construction. This allows certain ODEs we will use to be well-posed. Without loss of generality we will work for x > 0, k > 0 (working with k < 0 makes no difference other than the opposite sign in the explicit expression for K(0) in equation (4.5), the case x < 0 follows by symmetry). Denote by X(t), K(t) the unique solution of (4.5)

Ẋ(t) = 2πK(t), K(t) = 1+θ 2π (X(t)) θ , X(0) = CR, K(0) = |X(0)| 1+θ 2 ,
i.e. the only branch of the level set of

{2π 2 k 2 + V (x)} = 0 in {x > R ∧ k > 0}.
The interesting property of this trajectory is that, unlike what happens with regular potentials, it leaves zero in finite time (more generally, i.e. without restricting to {x > R ∧ k > 0}, it reaches and leaves zero in finite time). Claim: Any characteristic of the Liouville equation (2.8)

starting in {x > R ∧ k > 0} ∩ supp[W ε 0 ] converges to X(t), K(t) defined in equation (4.

5). Then it readily follows that

χ x>0 (x, k)[W ε 0 ] χ x>0 (x, k)w(x, k)dxdk δ(x -X(t), k -K(t)).
Repeating the argument for x < 0 gives a limit of two delta functions leaving zero along different trajectories.

Proof of the claim: Denote by X 1 (t), K 1 (t) the solution of (4.6)

Ẋ1 (t) = 2πK 1 (t), K1 (t) = -1 2π ∂ x ∼γ V (X 1 (t)), X 1 (0) = x 0 , K 1 (0) = k 0 , where of course (x 0 , k 0 ) ∈ {x > R ∧ k > 0} ∩ supp[W ε 0 ]
are as in the statement of the claim). Then equation (4.6) can be recast as (4.7)

Ẋ1 (t) = 2πK 1 (t), K1 (t) = -1 2π V x (X 1 (t)) -1 2π (∂ x ∼γ V (X 1 (t)) -V x (X 1 (t))), X 1 (0) = x 0 , K 1 (0) = k 0 .
Here we use the estimate |∂ 1) (see Lemma 6.2 with V (x) in the place of f ). It is clear now that any such characteristic converges to (4.8)

x ∼γ V (X 1 (t)) -V x (X 1 (t))| CR θ-1 ε γ 2 = o(
Ẋ2 (t) = 2πK 2 (t), K2 (t) = -1 2π V x (X 2 (t)), X 2 (0) = x 0 , K 2 (0) = k 0 .
To conclude observe that trajectory X 2 (t), K 2 (t) is squeezed between two level sets of 2π 2 k 2 +V (x) converging to each other (namely between {2π 2 k 2 + V (x)} = V (CR) and

{2π 2 k 2 + V (x)} = V (R 1-θ 5
)).

Essentially the same analysis applies to the example of Lemma 5.4 as well.

     ∂ t W ε 1 + 2πk • ∂ x W ε 1 + T ∼γ V ε W ε 1 = 0, W ε 1 (t = 0) = W ε 0 , and ρ ε 2 , (4.10)
     ∂ t ρ ε 2 + 2πk∂ x ρ ε 2 + T ∼γ V 0 ρ ε 2 = 0, ρ ε 2 (t = 0) = [W ε 0 ]
, where [W ε 0 ] for now is simply assumed to exist (and have the properties found in the statement of Theorem 2.2. Such approximate initial data will be constructed, for concrete examples, in Section 5). Finally, recall that ρ ε 1 was defined as the solution of equation (2.8) with initial data ρ 1 ε(t = 0) = W ε 0 . We partition the proof as follows:

(4.11) ||W ε -ρ ε 1 || L 2 ||W ε -W ε 1 || L 2 + +||W ε 1 -ρ ε 2 || L 2 + ||ρ ε 1 -ρ ε 2 || L 2 = o(||W ε 0 || L 2 ).
We will use without further comments the elementary observation (4.12)

|| f (x -s, k)g(s)ds|| L 2 (R 2n ) ||g|| L 1 (R n ) ||f || L 2 (R 2n ) Lemma 4.1 (W ≈ W 1 ). ∀t ∈ [0, T ] ||W ε 1 (t) -W ε (t)|| L 2 = O( ε γ 1+θ 2 -1 ||W ε 0 || L 2 ). Proof: Denote (4.13) h ε = W ε -W ε 1 . Obviously, (4.14) ∂ t h + 2πk • ∂ x h + T ∼γ V h = -T V - ∼γ V W ε 1 , h(x, k, 0) = 0.
Since the Wigner equation has a bounded L 2 propagator, it suffices to bound in L 2 the rhs. Indeed, we have (4.15)

||T V - ∼γ V W ε 1 || L 2 = C|| V (S)(1 -e -π 2 ε γ S 2 ) W ε 1 (X -S, K) sin(πεSK) ε dS|| L 2 O(ε -1 )|| V (S)(1 -e -π 2 ε γ S 2 )|| L 1 ||W ε 1 || L 2 O(ε -1 )||W ε 1 || L 2   ε γ |S| ε -γ 2 | V (S)| |S| 2 dS + |S| ε -γ 2 | V (S)|dS   = = O(ε -1 )||W ε 1 || L 2   ε γ + ε γ ρ∈(1,ε -γ 2 ) ρ -θ dρ + ρ ε -γ 2 ρ -2-θ dρ   = = O(ε -1 )||W ε 0 || L 2 ε γ + ε γ[-1 2 (1-θ)+1] + ε γ 1+θ 2 .
Asking that the remainder is small gives (4.16)

γ > 1, γ[-1-θ 2 + 1] > 1 ⇔ γ > 2 1+θ , γ 1+θ 2 > 1 ⇔ γ > 2 1+θ
. So finally with the calibration γ > 2 1+θ the proof of Lemma 4.1 is complete.

Moreover:

Lemma 4.2 (W 1 ≈ ρ ε 2 ). ∀t ∈ [0, T ] (4.17) ||W ε 1 (t) -ρ ε 2 (t)|| L 2 = O(ε θ 2+θ -δ ||W ε 0 || L 2 ). Proof: It is straightforward to check that (4.18) ∂ t W ε 1 -2πX • ∂ K W ε 1 + 2 ∼γ V (S) W ε 1 (X -S, K) sin(πεS•K) ε dS = 0, ∂ t ρ ε 2 -2πX • ∂ K ρ ε 2 + 2π ∼γ V (S) ρ ε 2 (X -S, K)S • KdS = 0,
and therefore, if

f = W ε 1 -ρ ε 2 , (4.19)
∂ t f -2πX • ∂ K f + 2 ∼γ V (S)f (X -S, K) sin(πεS•K) ε dS = = 2 ∼γ V (S) ρ ε 2 (X -S, K) 1 -sin(πεS•K) πεS•K πS • KdS, f (x, k, 0) = W ε 0 -[W ε 0 ].
Using the Duhamel formula, (4.20)

||f (t)|| L 2 ||W ε 0 -[W ε 0 ]|| L 2 + +2T sup t∈[0,T ] || ∼γ V (S) ρ ε 2 (X -S, K, t) 1 -sin(πεS•K) πεS•K πS • KdS|| L 2 .
Recall that the first term above is (relatively) small by assumption; we will work out the other term: To bound that, first of all observe that (4.21) sin(πεS

• K) πεS • K -1 Cmin {1, |πεS • K|}
and therefore for any b < 0 (4.22)

|| ∼γ V (S) ρ ε 2 (X -S, K) 1 -sin(πεS•K) πεS•K πS • KdS|| L 2 Cε|| 0 |S| 1 | ∼γ V (S)| • |S| 2 | ρ ε 2 (X -S, K)| • |K| 2 dS|| L 2 + +ε 1+2b || 1<|S|<ε b ∼γ V (S) ρ ε 2 (X -S, K)|K| 2 dS|| L 2 + +|| |S|>ε b ∼γ V (S) ρ ε 2 (X -S, K)|SK|dS|| L 2 C ε 1+2b || V (S)min{|S| 2 , 1}|| L 1 ||ρ ε 2 || H 2 + |S|>ε b | ∼γ V (S)| |S|dS||ρ ε 2 || H 1 C ε 1+2b || V S 2 1+S 2 || L 1 ||ρ ε 2 || H 2 + ∞ ρ=ε b ρ -1-θ dρ||ρ ε 2 || H 1 = = C ε 1+2b ||ρ ε 2 || H 2 + ε -θb ||ρ ε 2 || H 1 C(ε 1+2b-δ + ε -θb-δ )||W ε 0 || L 2
Calibrating the parameters is easy; we are given θ ∈ (0, 1), δ as in the statement of the Theorem and we need to find b ∈ (-1 2 , 0) so that

(4.23) 1 + 2b -δ > 0, -θb -δ > 0 ⇔ ⇔ δ-1 2 < b < -δ θ .
It should now be clear that the constraint on δ in statement of the Theorem comes from the self-consistency check -1 2 < δ-1 2 < -δ θ < 0. The proof of Lemma 4.2 is complete. Now the last step is to show that ρ ε 2 ≈ ρ ε 1 ; but this follows by construction, since they satisfy the same equation (which has an L 2 -continuous propagator), and

||ρ ε 2 (t) -ρ ε 1 (t)|| L 2 = ||W ε 0 -[W ε 0 ]|| L 2 .
The proof of Theorem 2.2 is complete.

4.3.

Proof of Theorem 2.3. One observes of course that, by construction,

|E ε (φ)| ||ρ ε 1 (t) -W ε (t)|| L 2 ||φ|| L 2 .
This will be useful in situations when we use a test-function from A ∩ L 2 .

Since in general there are φ ∈ A \ L 2 , we will also use a variation of the space of test-functions A introduced earlier: let

(4.24) B = {φ ∈ C 0 (R 2n ) | sup x |F k K [φ(x, k)]|dK + ||φ|| L 2 < ∞} equipped with the norm (4.25) ||φ|| B = ||φ|| A + ||φ|| L 2 .
In particular, it is clear by construction that B ⊆ A, A ⊆ B , and

||f || L 2 ||f || B , ||f || B ||f || L 2 .

Moreover, one easily sees the following

Claim:

There exists R > 0 such that ||W ε (t)-ρ ε 1 (t)|| A < R for all t ∈ [0, T ]. Proof of the claim: Observe that | f, φ A ,A | = | x,K F k K [f (x, k)]F k K [φ(x, k)]dxdK| K sup x |F k K [φ(x, k)]| x |F k K [f (x, k)]|dxdK K sup x |F k K [φ(x, k)]|dK sup K x |F k K [f (x, k)]|dx, i.e. ||f || A sup K x F k K [f (x, k)]dx = sup K x | k e -2πikK f (x, k)dk|dx ||f || L 1 .
To conclude we use the singular value decomposition of the initial data:

if D ε 0 = m λ m |u m u m |, then W ε 0 = m λ m e -2πiky u ε m (x + εy 2 )u ε m (x - εy 2
)dy, and

W ε (t) = m λ m e -2πiky u ε m (x + εy 2 , t)u ε m (x - εy 2 , t)dy,
where of course each of u m (t), ||u m || L 2 = 1, satisfies the Schrödinger equation (1.2). The point is that the singular values are time-independent and, especially for positive operators, are controlled by the trace:

D ε 0 > 0 ⇒ λ m > 0 ∀m ⇒ tr(D ε 0 ) = m λ m = ||λ m || l 1 .
We readily observe that

sup K x |F k K [W ε (x, k)]|dx = sup K x m |λ m u ε m (x + εK 2 )u ε m (x -εK 2 )|dx tr(D ε 0 ),
and of course

||ρ ε 1 (t)|| L 1 = ||[W ε 0 ]|| L 1 . The proof of the claim is complete; it is obvious that R tr(D ε 0 )+||[W ε 0 ]|| L 1 .
We will also use the following:

Claim: Let χ ∈ C ∞ (R 2n , [0, 1]) be a cutoff function, χ(z) = 1 ∀|z| < 1, χ(z) = 0 ∀|z| > 2, and denote φ M (x, k) = χ( (x,k) M ). Then, ||φ M || L 2 2 n π n 2 Γ( n 2 +1) M n ||φ|| A , and lim M ∞ ||φ M -φ|| A = 0. Proof of the claim: First of all, χ(x, k) ∈ S(R 2n ) ⊆ A(R 2n ). Moreover, keeping in mind that ||φ|| L ∞ ||φ|| A , ||φ M || L 2 ||φ|| L ∞ |(x,k)|<2M dxdk 2 n π n 2 Γ(n + 1) M n ||φ|| A .
The second part, i.e.

lim

M ∞ ||φ M -φ|| A = 0 follows e.g. from the density of S(R 2n ) in A [START_REF] Lions | Sur les mesures de Wigner[END_REF]. Additional information on the decay of φ is needed in order to estimate more precisely the rate of convergence. The proof of the claim is complete.

It is clear now how, given a φ ∈ A, we can select a family {φ M } ⊆ B such that ||φ M -φ|| A 0 as M ∞. Moreover set for brevity

F (ε) = C(ε κ ||W ε 0 || L 2 + ||[W ε 0 ] -W ε 0 || L 2 )
for some large enough C > 0. Now, recalling our remarks in the beginning of the section and the first claim, it follows that (4.26)

| W ε -ρ ε 1 , φ | | W ε -ρ ε 1 , φ M | + | W ε -ρ ε 1 , φ M -φ | F (ε)||φ M || L 2 + (1 + ||W ε 0 || L 1 )||φ M -φ|| A F (ε) 2 n π n 2 √ Γ(n+1) M n + (1 + ||W ε 0 || L 1 )||φ M -φ|| A ∀ε, M.
The proof is completed by selecting, for each value of ε a "small enough" M , so that M = M (ε) ∞, and

lim ε 0 F (ε)||φ M (ε) || L 2 = 0.
In the statement of the Theorem, the calibration

M = F (ε) -1 2n is used; in that case it follows that ||φ M (ε) || L 2 2 n π n 2 √ Γ(n+1) √ F (ε)
.

The proof is complete.

Examples

This section is devoted in understanding better the non-concentration properties that are in the heart of our main results. There are two versions; we isolate them below as "Property 1", a condition to be checked for a given problem that is sufficient for Theorem 2.2 to apply to it, and the stronger "Property 2", which is sufficient for Theorem 2.3 and its Corollary 2.4. Concrete examples of natural problems that satisfy either one or both of them are constructed; it must be noted that it's rather easy to satisfy Property 1 to generic initial data and repulsive singularities. Asking that Property 2 holds is quite restrictive, and it seems that specially prepared initial data are needed for that.

We are only concerned here with examples that give rise to ill-posed classical problems. (Otherwise the existing theory for C 1 potentials has no problem to provide a complete description of the semiclassical limit).

5.1.

Examples for Theorem 2.2. Here we are concerned with Cauchy problems for the Liouville equation,

   ∂ t ρ ε 1 + 2πk • ∂ x ρ ε 1 -1 2π ∂ x ∼γ V •∂ k ρ ε 1 = 0, ρ ε 1 (t = 0) = W ε 0 , which have the following Property 1. There exist T > 0, δ ∈ (0, θ 2+θ ) and [W ε 0 ] ∈ H 2 (R 2n ) such that (5.1) ||[W ε 0 ] -W ε 0 || L 2 = o(||W ε 0 || L 2 )
and the solution of

(5.2)    ∂ t ρ ε 2 + 2πk • ∂ x ρ ε 2 -1 2π ∂ x ∼γ V •∂ k ρ ε 2 = 0, ρ ε 2 (t = 0) = [W ε 0 ], satisfies (5.3) ||ρ ε 2 (t)|| H 2 = O(ε -δ ||W ε 0 || L 2 ), uniformly for t ∈ [0, T ].
We treat κ, γ as given parameters; of course when checking the property in the context of Theorem 2.3 they are controlled by the statement of Theorem 2.2. Lemma 5.1. Let θ ∈ (0, 1), δ ∈ (0, θ 2+θ ), n = 2, V (x) = -|x| 1+θ with an appropriate smooth cutoff outside {|x| < 2}. Assume moreover that there exists a function of compact support w ∈ H 2 (R 4 ) ∩ L ∞ , such that

(5.4) W ε 0 (x, k) = δ -2 x δ -2 k w( x -(z 1 , z 2 ) δ x , k -(z 3 , z 4 ) δ k ), with (z 1 , z 2 ) = -L(z 3 , z 4 ) (i.e. the wavepacket is "shot towards zero"). ||w|| H 2 , ||w|| L ∞ , |z| have to be bounded uniformly in ε, ||w|| H 2 , ||w|| L ∞ , |z| = O (1)
; other than that w and z can be allowed to depend on ε.

The small parameters involved δ x , δ k , R = o(1) are calibrated as follows:

R = log 1 ε -1 3 ,
as in equation ( 6.26), and

δ -1 x (R + δ k ) = o(1), (δ x + δ k ) -2 (δ k + R) -2 = O(ε -δ 2 ). (5.5)
Then Property 1 is satisfied.

Remarks:

• In fact it follows from the Lemma that property 1 holds for any δ ∈ (0, θ 2+θ ). • The difficulties arise from the non-smootheness at zero, and the details of a smooth behaviour away from zero are irrelevant here. So we will work for a small enough time, before any trajectory starting in a neighbourhood of zero reaches the support of the cutoff function. Therefore, to keep the presentation simple, we will not introduce any explicit treatment of the cutoff function. This approach is followed in the sequel as well.

• An example of a scaling satisfying the above constraints is

δ x = √ δ k = √ R.
Proof: Assume without loss of generality that z = (0, -L, 0, 1). Consider φ : R [0, 1] to be a C ∞ function such that

(5.6) φ(x) = 0, |x| < 1 2 , φ(x) = 1, |x| > 1.
The modified initial data [W ε 0 ] is selected as

(5.7) [W ε 0 ](x, k) = W ε 0 (x, k)φ |x 1 | R + 2Lδ k .
The key claim is that, for T small enough but independent from ε,

  t∈[0,T ] suppρ ε 2 (t)   ∩ {|x| < R} = ∅ Indeed [W ε 0 ]
has two disjointly supported components. The calibration of the parameters and the cut-off is such that, if [W ε 0 ] was propagated by a Liouville equation corresponding to V (x) = 0, each of the components would stay on either side of the strip {|x 1 | < R} (see Figure 2). The idea is that since there is a potential driving the flow away from x = 0, the claim follows. (In fact it is slightly more complicated for this case, since the potential depends on x 2 as well: one easily gets an O(1) upper bound for the time it takes for the trajectories to reach x 2 = 0, and then checks that in this longer time, x 1 doesn't have the time to change sign. The difference in an O(1) factor which we absorb in C below). 0 ] has an initial velocity in a small cone around k = (0, 1). If we were in free space, there wouldn't be enough space for sufficient movement in x 1 to reach {|x 1 | < R}. Therefore the solution is the sum of two components, supported on either side of {|x 1 | < R} = 0. The presence of a repulsive potential in many cases can be easily factored in this construction.

Therefore, in this problem we can take ∼γ (5.3), making use of observation 6.6 (enlarging the cut-off area in fact strengthens observation 6.6, i.e. substituting R + 2Lδ k for R is painless) and Lemma 6.4 it follows that (5.8)

V ef f (x) = ∼γ V (x)φ |x| R+CLδ k . Now for condition
||ρ ε 2 (t)|| H 2 ||[W ε 0 ]|| H 2 ε -δ 2 C||W ε 0 || H 2 ε -δ 2 (δ k + R) -2 = = C||W ε 0 || L 2 (δ x + δ k ) -2 (δ k + R) -2 ε -δ 2 ,
and therefore -using the scaling of equation (5.5) -

||ρ ε 2 (t)|| H 2 = O(ε -δ ||W ε 0 || L 2 ). Now for W ε 0 -[W ε 0 ]:
(5.9)

||W ε 0 -[W ε 0 ]|| 2 L 2 δ -2n x δ -2n k |x 1 |<C(R+δ k ) |w( (x 1 ,x 2 ) δx , (k 1 -1,k 2 ) δ k )| 2 dxdk = = δ -n x δ -n k |x 1 |<C R+δ k δx |w(x 1 , x 2 , k 1 -1, k 2 )| 2 dxdk C||W ε 0 || 2 L 2 ||w|| 2 L ∞ δ -1 x (R + δ k )
The same idea can be applied to different configurations:

Lemma 5.2. Consider the setup of Lemma 5.1 with z = (0, -2, 0, 1) and the only difference that (5.10)

V (x) = -|x 1 | 1+θ ψ(x 1 )ψ(x 2 ).
where ψ is a smooth cutoff function, ψ = 1 -φ ∈ S(R) (φ was defined in equation (5.6)).

Then Property 1 holds.

Proof: The geometry is essentially the same as before, and it is clear that the obvious adaptation of observation 6.6, i.e. the one with ∼γ

V ef f (x) = ∼γ V (x)φ( x 1 R + Cδ k ) holds.

5.2.

Examples for Theorem 2.3. In relation to Theorem 2.3, we introduce

Property 2. Given V (x), W e 0 with ||W ε 0 || L 2 = o(ε -κ ), there exist T > 0, δ ∈ (0, θ 2+θ ) and [W ε 0 ] ∈ H 2 (R 2n ) such that ||[W ε 0 ] -W ε 0 || L 2 = o(1) (5.11)
and the solution of (5.12) As was already seen in section 4.1,

   ∂ t ρ ε 2 + 2πk • ∂ x ρ ε 2 -1 2π ∂ x ∼γ V •∂ k ρ ε 2 = 0, ρ ε 2 (t = 0) = [W ε 0 ],
||f ε 0 -f ε 0 * 2 ε n e -2π x 2 +k 2 ε || L 2 = o(1)
. So without loss of generality we can check Property 2 in the context of Theorem 2.1 considering initial data of the form f ε 0 , and not the smoothed version that appears there. (More generally, working with compactly supported initial data -and hence removing the smoothing -might be very helpful whenever we check Properties 1, 2. In any case, we expect that the smoothing can be removed without loss of generality, just like in this case, whenever the initial data decays rapidly and

||W ε 0 || H 1 = o(ε -1 2 )).
Lemma 5.3. Let n = 1, θ ∈ (0, 1), V (x) = -|x| 1+θ with an appropriate smooth cutoff outside {|x| < 2}. Assume moreover that w(x, k) ∈ H 2 ∩L ∞ ∩ L 1 , supp w ⊆ {|x|, |k| < 1}, and

(5.14)

f ε 0 = δ -1 x δ -1 k w( x δ x , k δ k ). Finally, set R = log 1 ε -1 3 .
Then Property 2 is satisfied if

δ k = C R 1+θ 2 , δ x = C R 1-θ 5 .
Proof: We will cutoff a strip of the form {|x| CR}, and show that in fact this suffices.

As is illustrated in Figure 5.2, the preimage under the flow of {|x| <

R ∧ |k| < δ k } is contained between the level sets {2π 2 k 2 - ∼γ V (x) = ∼γ V (0)} and {2π 2 k 2 + ∼γ V (x) = ∼γ V (R)}.
A big help with the algebra will be to approximate

∼γ V (x) = -|x| 1+θ + O(ε γ 2
). (γ is as in the statement of Theorem 2.2). Since all the other small parameters are (negative) powers of log

( 1 ε ), O(ε γ 2
) is negligible everywhere. (The justification is that without loss of generality we can localize the problem on a compact set, and then V (x) ∈ W 1,∞ uniformly in ε, see Lemma 6.5. The conclusion follows by a standard observation on mollifiers, see Lemma 6.2).

It is easily seen that the needed length is x * , defined by

2π 2 δ 2 k + ∼γ V (x * ) = ∼γ V (R) or |x * | 1+θ = 2π 2 δ 2 k + R 1+θ + O(ε γ 2 ).
Here is where

δ k = C R 1+θ 2
comes from; making that scaling we get

x * = ((1 + 2π 2 )R 1+θ + O(ε γ 2 )) 1 1+θ = C R.

So far we have ensured by construction that, if δ

k = R 1+θ 2 , (there is an O(1) constant C so that) if (5.15) [W ε 0 ](x, k) = f ε 0 (x, k)φ( x CR ),
its propagation under equation (5.12) will never enter {|x| < R}. Now for the approximation error:

(5.16)

||f ε 0 -[W ε 0 ]|| 2 L 2 Cδ -2 x δ -2 k |x|<CR |w( x δx , k δ k )| 2 dxdk = = Cδ -1 x δ -1 k |x|<C R δx |w(x, k)| 2 dxdk Cδ -2 x R 1-θ 2 .
This gives the constraint R

1-θ 4

= o(δ x ) for the error to be small, which is satisfied e.g. by our earlier selection δ x = CR 1-θ 5 . Again, since all the small parameters are powers of log(

1 ε ), ||[W ε 0 ]|| H 2 = o(ε -δ
2 ) follows automatically for any δ ∈ (0, θ 2+θ ). The proof is complete.

The following is a somewhat artificial example, but it highlights an interesting behaviour.

(5.17) supp f ⊆ {|x| < 1} × {|k| < 1 ∧ k 1 > 0}.
Denote by f

(5.18) f (x 1 , x 2 , k 1 , k 2 ) = f (-x 1 , x 2 , -k 1 , k 2 )
Now take

(5.19) W ε 0 (x, k) = 1 δ 2 x δ 2 k c 1 f ( x 1 -2R,x 2 +2 δx , k 1 ,k 2 +1 δ k ) + +c 2 f ( x 1 -2R,x 2 +2 δx , k 1 ,k 2 +1 δ k ) . If δ k = C δ x = C √ R,
∂ t f + 2πk • ∂ x f -1 2π ∂ x V (x) • ∂ k f = 0, f (t = 0) = f 0 .
There are constants C 1 , C 2 > 0 depending only on n such that

||f (t)|| H 2 C 1 e tC 2 sup |a| 3 ||∂ a x V (x)|| L ∞ ||f 0 || H 2 .
Proof: The proof follows readily with the method of characteristics. It suffices to observe that if

(6.2) ẋi (t) = 2πk i (t), ki (t) = - 1 2π ∂ x i V (x(t)), (6.3)
z(t) = f (x(t), k(t)), z x i (t) = ∂ x i f (x(t), k(t)), z x i x j (t) = ∂ x i x j f (x(t), k(t)), z k i (t) = ∂ k i f (x(t), k(t)), z k i k j (t) = ∂ k i k j f (x(t), k(t)), z x i k j (t) = ∂ x i k j f (x(t), k(t)),
it follows that ż = 0 (6.4)

żx i (t) = 1 2π m ∂ x i xm V (x(t)) z x i km (t), (6.5) żk i (t) = -2πz x i (t), (6.6) żx i x j (t) = = 1 2π m ∂ x j xm V z x i km (t) + ∂ x i xm V z x j km (t) + ∂ x i x j xm V z km (t) (6.7) żk i k j (t) = -2π(z x j k i + z x i k j ), (6.8) żk i x j (t) = -2πz x i x j (t) + 1 2π m ∂ xmx j V (x(t))z kmk i (t). (6.9)
The result now follows by the Gronwall inequality.

Lemma 6.2. Consider a function f ∈ W 1,∞ , i.e. sup |a| 1 ||∂ a x f || L ∞ < ∞. Then, if f = 2 η n 2 e -2π |x-x | 2 η f (x )dx , we have (6.10) ||f -f || L ∞ = O(||f || W 1,∞ √ η).
Proof: Take any ζ ∈ (0, 1 2 ). Now we have (6.11)

|f (x) -f (x)| = 2 η n 2 e -2π |x | 2 η [f (x -x ) -f (x)]dx = = 2 η n 2 |x |<η 1 2 -ζ e -2π |x | 2 η [f (x -x) -f (x)]dx + O(η ∞ ) C||f || W 1,∞ 2 η n 2 |x |<η 1 2 -ζ e -2π |x | 2 η |x |dx + O(||f || L ∞ η ∞ ) = = O( √ η2 n 2 )||f || W 1,∞ |y|<η -ζ e -2π|y| 2 |y|dy + O(||f || L ∞ η ∞ ) = = O(||f || W 1,∞ √ η).
The proof is complete. Remark: A sharper version is used in subsection 4.1, namely

|f (x) -f (x)| C      sup |a| = 1 |x -x | < η 1 2 -ζ |∂ a x f |      2 η n 2
|x |<η where S is defined as in (6.12), and

(6.14) sup |A| 3 ||∂ A x ∼γ V ef f (x)|| L ∞ = O log ε -δ 2 ,
Then, (6.15)

||ρ ε 2 || H 2 = O( ε -δ 2 ||W ε 0 || H 2 ).
Proof: The proof consists of using ∼γ V ef f in place of ∼γ V , making use of observation 6.3, and then applying directly Theorem 6.1.

We will also use the following Lemma 6.5. It is easy to observe that, if ∇V ∈ L ∞ and suppW ε 0 is compact, it follows that Set x = x R ; then ∂ x = R -1 ∂ x . Now we have (6.25)

||∂ A x |x| 1+θ ψ x 2M -ψ x R || L ∞ = = R θ-2 ||∂ A x |x| 1+θ ψ R 2M x -ψ(x) || L ∞ R θ-2 sup |A| 3 ||∂ A x x 1+θ || L ∞ ([ 1 2 , 2M R ]) sup |A| 3 ||∂ A x ψ R 2M x -ψ(x) || L ∞ (R n ) (2M ) 1+θ ||φ|| W 3,∞ R -3 .
Recall that the constant M is chosen so that any trajectory leaving the support of the initial data doesn't not escape {|x| < M } for t ∈ [0, T ]. That is, to properly quantify it, one needs to consider initial data of compact support associated with the Liouville equation, and a time-scale T . Assuming that the initial data is supported in {|(x, k)| < R 0 }, it is easy to check that one can set M = R 0 + 1 + R 0 T + ||∇V (x)|| L ∞ T 2 . So now for (6.19) to hold it suffices that 2 1+θ (1

+ R 0 + R 0 T + ||∇V (x)|| L ∞ T 2 ) 1+θ 10 C δ 2 R -3 log(ε -1 )
The constant C comes from the O(•) of equation (6.19); it is clear that it can be chosen so that the constraint finally becomes R (log(ε -1 )) -1 3 , which is satisfied by choosing (6.26) R = (log(ε -1 )) -1 3 .

The following is contained in [START_REF] Lions | Sur les mesures de Wigner[END_REF]; we include here a brief mention for completeness: Lemma 6.7 (Density matrices). Let µ be a probability measure on R 2n . Then Proof: We know that tr(D ε ) = W ε dxdk = 1. Let us now look at positivity. To that end, observe that the integral kernel

K ε (x, y) = 2 ε n 2
x 0 ,k 0 e 2πi k 0 ε (x-y) e -π ε [(x-x 0 ) 2 +(y-x 0 ) 2 ] dµ(x 0 , k 0 )

is the kernel of a positive operator. Indeed:

K ε (x, y)u(x)u(y)dxdy = 2 ε n 2 x 0 ,k 0 | e -2πi k 0 ε x+ π ε (x-x 0 ) 2 , u | 2 dµ(x 0 , k 0 ) 0.
The proof is complete by observing that the Wigner function corresponding to the kernel ρ ε is e -2πiky K ε (x + εy 2 , x -εy 2 )dy = W ε 0 .

Figure 2 .

 2 Figure 2. Any trajectory leaving the support of [W ε0 ] has an initial velocity in a small cone around k = (0, 1). If we were in free space, there wouldn't be enough space for sufficient movement in x 1 to reach {|x 1 | < R}. Therefore the solution is the sum of two components, supported on either side of {|x 1 | < R} = 0. The presence of a repulsive potential in many cases can be easily factored in this construction.

Figure 3 .

 3 Figure 3. The construction of Lemma 5.1 holds, with obvious adjustments, to the example of Lemma 5.2.

Figure 4 .

 4 Figure 4. Schematic for Lemma 5.3; cutting off a strip in x, {|x| < x * }, suffices to cut off the preimage of {|x| < R}

2 ηLemma 6 . 4 (

 264 |x |dx + +O(||f || L ∞ η ∞ ).The following observations are used in Section 5: Observation 6.3 (Locality of the Liouville equation). For a Liouville equation with initial data of compact support ρ 0 and a C 1,1 potential, interchanging the potential with any one that coincides with it on the "path" of the solution, An H 2 estimate). Consider equation (2.8) with initial dataρ 0 = [W ε 0 ] ∈ H 2 .Assume that there is a function ∼γ V ef f such that (6.13)∼γ V = ∼γ V ef f on S,

( 6 . 16 )VV 6

 6166 ∃M > 0 : ∀t ∈ [0, T ], ε > 0 :t∈[0,T ] supp ρ ε 1 (t) ⊆ {|(x, k)| < M }. Observation 6.6 ( ∼γ V ef f and R). Let V (x) = -|x| 1+θ .∼γV is a mollified version, according to equation(2.6).Consider φ : R n [0, 1] to be a C ∞ function such that (6.17)φ(x) = 0, |x| < 1 2 , φ(x) = 1, |x| > 1,and ||φ|| W 3,∞ 10. It is clear that such a function exists; this is of course an arbitrary requirement, but one that allows us not to carry the cutoff function φ to other results.Then, setting R log ε -1 -1 3 , R = o(1), it follows that(6.18) ∼γ V ef f (x) := φ( ef f (x)|| L ∞ = O( log ε -ef f = ∼γ V on R n \ {|x| < R}. Proof: Observe that 1 -φ(x) = ψ(x) ∈ S(R n ).Moreover, making use of observations 6.5 and 6.3, we can restrict ∼γ V to {|x| < M } without loss of generality, (6.21) ∼γ V (x) → ∼γ V (x) 1 -φ x 2M . f || L ∞ ||∂ A x f || L ∞ , it suffices to work with (

  is the Wigner function of a density matrix, i.e. the corresponding operator D ε is a positive trace-class operator with tr(D ε ) = 1.

  In this section we will show that Theorem 2.1 follows from Theorems 2.2, 2.3. The latter are proved in the next subsections.Before getting into more detail, let us comment that λ as used in the statement of Theorem 2.1 is an effective upper bound. It is easily checked that if we replace λ by λ λ, 1 λ = o(1), the Theorem still holds. Remark on notation: In this section we use the notation R = λ -1 3 .

	Now denote
	(4.1)

). 4. Proofs 4.1. Proof of Theorem 2.1.

  4.2. Proof of Theorem 2.2. It will be helpful to recall the main objects we are going to use here. The WT for this problem, W = W ε (x, k, t), satisfies the well-known Wigner equation(1.4) with initial data W ε 0 . Moreover, W ε

	and ρ ε 2 are defined as follows	1
	(4.9)

  this problem has Property 2. The scaling of δ x , δ k in this case controls simply the rate of concentration, and therefore ||W ε 0 || H 2 . The result follows.

	6. Auxiliary results
	Lemma 6.1 (2 nd order derivatives equations for the Liouville equation).
	Consider the Cauchy problem for the Liouville equation with potential V (x)
	on R n ,
	(6.1)
	Proof: Two things should be obvious by construction: that W ε 0 (0, 2), k -(0, 1)), and that the propagation of W ε 0 under equation (2.8), i.e. δ(x + ρ ε 1 , is never supported inside {|x 1 | < R}. In other words, for this specifically constructed data, W ε 0 = [W ε 0 ].


Joseph Ben Geloun
email: bengeloun@sun.ac.za

Thomas Krajewski

Jacques Magnen
email: magnen@cpht.polytechnique.fr

Vincent Rivasseau

Linearized Group Field Theory and Power Counting Theorems

Keywords: Pacs numbers: 04.60.-m, 04.60.Pp Key words: Group field theory, renormalization, perturbative study

We introduce a linearized version of group field theory. It can be viewed either as a group field theory over the additive group of a vector space or as an asymptotic expansion of any group field theory around the unit group element. We prove exact power counting theorems for any graph of such models. For linearized colored models the power counting of any amplitude is further computed in term of the homology of the graph. An exact power counting theorem is also established for a particular class of graphs of the nonlinearized models, which satisfy a planarity condition. Examples and connections with previous results are discussed.

Introduction

Group field theories (GFTs) [1] are quantum field theories over group manifolds and can be also viewed as higher rank tensor field theories [START_REF] Freidel | Group field theory: An overview[END_REF][START_REF] Oriti | The group field theory approach to quantum gravity[END_REF]. They provide one of the most promising framework for a background free theory of quantum gravity in which one sums both over topologies and geometries. Indeed, each Feynman graph of a D dimensional GFT can be dually associated with a discrete spacetime (D-simplicial complex) via a specific triangulation and gluing rules given by the covariance and vertices (basic D-simplex) of the theory. The functional integral formalism then defines a weighted sum over triangulations which is a weighted "sum over topologies" with each weight (amplitude) related to a "sum of geometries" (via a spin foam formalism [START_REF] Rovelli | Quantum Gravity[END_REF]). Some early results on power counting and non-perturbative resummation of such models are in [START_REF] Freidel | Non-perturbative summation over 3D discrete topologies[END_REF].

Group field theory is a particular kind of quantum field theory, hence it is natural to ask the question of its renormalizability. A complete renormalization group analysis for GFTs is still lacking and the first steps for such a program have been made only recently [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF][START_REF] Magnen | Scaling behaviour of three-dimensional group field theory[END_REF][START_REF] Ben Geloun | Bosonic Colored Group Field Theory[END_REF]. For the three dimensional Boulatov model a rigorous power counting theorem has been obtained for a particular class of graphs called Type I, which satisfy a contractibility condition [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF]. Given a cutoff Λ in the Peter-Weyl field expansion, Type I graphs diverge as Λ B-1 , where B denotes the number of "bubbles". These bubbles are closed surfaces whose exact definition is clumsy in general [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF]. In [START_REF] Magnen | Scaling behaviour of three-dimensional group field theory[END_REF] scaling bounds for Feynman amplitudes of the Boulatov model have been established together with constructive results. However in that theory generalized tadpoles dominate over other graphs.

This situation improves markedly in the case of colored GFT models [START_REF] Gurau | Colored Group Field Theory[END_REF]. Among the nice features of these models we quote: (1) in the Fermionic case a SU (D + 1) global symmetry in the field indices; (2) a clear definition of "bubbles" of any dimension, which means that the graphs of these colored models define a full D-simplicial complex, not just a 2-complex as in spin foam theory; (3) a computable homology for this complex implying that the corresponding dual triangulated objects are manifolds or pseudo-manifolds but with only point-like singularities; (4) the absence of generalized tadpoles and of tadfaces 1 ;

(5) the existence of an exhaustive sequence of cuts [START_REF] Magnen | Scaling behaviour of three-dimensional group field theory[END_REF] for any graphs of these models. This leads to better scaling bounds in any dimension for the Feynman amplitudes of these models [START_REF] Ben Geloun | Bosonic Colored Group Field Theory[END_REF].

A general power counting theorem is a difficult problem even for the colored GFTs. In this paper, we introduce a further simplification, namely an Abelianized version of GFT that we shortly call "linearized GFT". This model is also interesting as a kind of asymptotic limit of GFT at high Fourier frequencies.

We compute the exact power counting for any graph of linearized GFTs. In the colored case this power counting is reexpressed in terms of the colored homology introduced by Gurau [START_REF] Gurau | Colored Group Field Theory[END_REF]. The connection with the Λ B-1 power counting of Type I graphs in [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF] is clarified; the linearized colored graphs have power counting Λ B-1 if and only if their second homology class vanishes.

Exact power counting is also established for amplitudes of the ordinary nonlinarized models but under a certain planarity condition. This paper is organized as follows. Section 2 fixes our notations, introduces the linearized GFTs and establishes the power counting of their graphs in terms of a certain incidence matrix between lines and faces. The proof relies on evaluation of determinants of quadratic forms through their Pfaffian representations in the manner of [START_REF] Gurau | Parametric representation of noncommutative field theory[END_REF][START_REF] Rivasseau | Parametric representation of 'critical' noncommutative QFT models[END_REF][START_REF] Krajewski | Topological Graph Polynomials and Quantum Field Theory, Part I: Heat Kernel Theories[END_REF][START_REF] Krajewski | Topological graph polynomials and quantum field theory, Part II: Mehler kernel theories[END_REF] and allows not only to find the exact ultraviolet power counting but also an infrared power counting. More generally we define the analog of Symanzik polynomials for the linearized models, which potentially contain much more information. These polynomials are connected to generalized contraction/deletion moves which for colored graphs have been first explored in [START_REF] Gurau | Topological Graph Polynomials in Colored Group Field Theory[END_REF].

Section 3 contains the basic definitions of colored graphs and their homology and reexpresses the power counting of linearized colored graphs in term of their additional structure. We give a computation of the ultraviolet degree of divergence in terms of the homology of the colored graph as defined in [START_REF] Gurau | Colored Group Field Theory[END_REF].

Section 4 introduces an other technique. The jacket of a graph is defined either for restricted Boulatov models or for colored models. It is a matrix-like piece of the theory. An exact non commutative momentum routing is introduced for the dual of the jacket. When the jacket is planar, the power counting is exactly the number of cycles of the complement of the jacket.

Finally, Section 5 provides examples, further discussions and conclusions.

Linearized GFTs and their Feynman amplitudes

An ordinary D-dimensional GFT is a field theory for a field defined over D copies of a group G with a certain gauge invariance and a certain pattern of interaction. For simplicity, we may consider mostly the Euclidean 3D case in which G = SU (2) but indicate which of our results are general for any dimension. We will also assume that the field is not symmetric under any permutation of the D arguments. There will also be no symmetrization operators inserted, neither in the propagators nor in the vertices, so that the category of graphs considered will be the category of stranded graphs without any twists of the strands. The Hilbert space of square integrable functions on GD is H = L 2 ( GD ). The Hilbert space of gauge invariants fields that is of fields invariant under the "diagonal" action of the group G φ(g

1 h, g 2 h, . . . , g D h) = φ(g 1 , g 2 , . . . , g D ) (1) is a subset H 0 ⊂ H.
The interaction is homogenous of degree D + 1 and patterned in the form of a D simplex. For instance in three dimensions this interaction has the form:

S int [φ] := λ 6 i=1
dg i φ(g 1 , g 2 , g 3 )φ(g 3 , g 4 , g 5 )φ(g 5 , g 2 , g 6 )φ(g 6 , g 4 , g 1 ).

(

) 2
The six integration variables are repeated twice, following the pattern of the edges of a tetrahedron.

In the language of Gaussian measures the partition function is

Z(λ) := dµ C [φ] e -λS int [φ] , (3)
where dµ C [φ] is the normalized Gaussian measure whose covariance C can be viewed as the orthogonal projection from H to H 0 implementing (1) through an average over the diagonal action of the group (Cφ)(g 1 , g 2 , g 3 ) := dh φ(g 1 h, g 2 h, g 3 h).

Graphs are therefore stranded with D strands per propagator. The ordinary Feynman rules to build graphs out of propagators and vertices give for a stranded graph G of this model an amplitude which is the following formal expression

A G = e dg e dh s∈ δ(g es h g -1 e s ) (5)
where each line propagates D strands s from edge variable e s to e s . Suppose that we choose both an arbitrary orientation of each line of the graph and of each face (closed set of strands). The integration over all edge variables can be explicitly performed, leading to an other (still formal) expression for the Feynman amplitude, but solely in terms of an integral over line variables:

A G = ∈L G dh f ∈F G δ ∈f h , (6)
where L G , F G are the set of lines and faces of G, respectively. The oriented product l∈∂f h l means that the variables h of the delta functions (which are the SU (2) holonomies along each face) has to be taken in the following way: h if the orientations of the line and of the face f coincide and h -1 if they do not. Surprisingly the formal amplitude (6) neither depends of the arbitrary orientation of the lines, nor of those of the faces! The deep reason is that the two formulas ( 5) and ( 6) are equal, but a more pedestrian way to see this is to exploit carefully the parity of the δ function and of the Haar measure under h → h -1 . However a full proof of this fact should address the fact that both ( 5) and ( 6) are infinite in general.

Indeed multiplication of distributions is not always allowed. Hence formulas ( 5) and ( 6) suffer from what one should consider from the usual field theoretic point of view as an ultraviolet2 divergence. However since groups like SU (2) are compact there is no infrared or large volume divergence associated to the group integrations over the h variables in these Euclidean GFTs.

A linearized GFT is obtained by just changing the group G to the additive commutative group of a vector space R d and repeating all the arguments above. It can also be considered as an expansion around the origin of unity of the group, in which each group element h is replaced by e ih and a Taylor expansion is performed which only retains the linear forms in the new h variables in all the δ functions of [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF]. In that case, d would just be the dimension of the group or of its Lie algebra. But in the rest of this paper we shall consider d = 1 for simplicity.

Let us introduce now some terminology. Non-regular faces, also called tadfaces are faces which goes several times through the same propagator. A regular graph is a graph without any tadface. Unfortunately, ordinary GFTs typically generate tadfaces and nonregular graphs. Non-regular faces play a bit the same annoying role than tadpoles play in ordinary graphs and ordinary field theory. They are difficult to capture through an incidence matrix. Although they are somewhat trivial they may perturb naive versions of contraction-deletion rules.

The linearized GFT amplitude is conveniently defined in terms of a rectangular F by

L incidence matrix ε f, , where F = |F G | is the number of faces of G and L = |L G | is its number of lines.
Definition 2.1 ε f, is defined as the algebraic number of passages of the face f through the line , counting as +1 each passage with coinciding orientation of f and and as -1 each passage with opposite orientation of f and . Hence ε f, ∈ Z.

For a regular graph ε f, ∈ {-1, 0, 1} with

ε f, =    +1, if or(l) = or(f ), ∈ ∂f -1, if or(l) = -or(f ), ∈ ∂f 0, if / ∈ ∂f (7)
where or(•) stands for the orientation.

The linearized amplitiude of a regular graph is then given by

A G = R L ∈L G dh f ∈F G δ ∈∂f ε f, h . (8)
Again although the incidence matrix does depend on the chosen orientations, the amplitude (8) does not depend on them, as we can arbitrarily multiply each line or column of ε f, and compensate for that using either parity of the δ function or changing h to -h . Formula (8) remains ill-defined and the corresponding amplitude is formally infinite but now for two distinct reasons:

-the set of L linear forms ∈∂f ε f, h may typically not be made of independent forms, i.e. the rank r of the rectangular matrix ε f, may be smaller than F , in which case the corresponding δ functions cannot be multiplied. This impossibility of multiplication of distributions should be considered as an ultraviolet divergence.

-even if the corresponding multiplication of distributions is regularized, e.g. by replacing δ distributions by Gaussian integrals, the rank r of the rectangular matrix ε f, could be strictly smaller than L, the number of internal lines. In that case, the corresponding quadratic form in the h variables would not be definite, so that the h integrations do not fully converge at infinity because there is no decay in some directions in R L . This kind of divergence should be considered of infrared type in field theory. It has no analog in the non-linearized Euclidean group field theories because the group used there is compact; but it has some analogs in Lorentzian group field theories.

Hence we expect:

Proposition 2.1 The ultraviolet degree of divergence of a linearized regular GFT graph is F -r and its infrared degree of divergence is L -r, where r is the rank of the ε f, matrix3 .

To give a precise mathematical content to this statement the simplest an dmost natural proposal, which we follow in this paper, is to regularize the amplitudes (8) through Gaussian regulation both of the δ functions and of the h integrals.

Hence we are lead to consider infrared regulators m which break gauge invariance but allow the amplitude to be well defined even for a noncompact group as is R, and to replace the δ function of each face f by a regularized normalized Gaussian function of the form:

δ Λ f (h) = Λ f √ 2π e -Λ 2 f h 2 /2 . ( 9
)
Therefore the ill-defined amplitude of a regular vacuum graph ( 8) is re-expressed using the regularized delta function (9) as4

A G (m , Λ f ) = dh e -m 2 h 2 /2 f Λ f e -(Λ 2 f /2)( ε f, h ) 2 = dh e -m 2 h 2 /2 f dk f e -Λ -2 f k 2 f /2 e i f, ε f, h k f ( 10
)
where we forgot inessential normalizations and we introduced the momentum variables k f labeled by faces (see Appendix 1 for useful identities pertaining to that transformation). The expression [START_REF] Gurau | Parametric representation of noncommutative field theory[END_REF] corresponds to a kind of phase space representation of the amplitude. The limit ( 8) is formally recovered as all Λ f → ∞ and all m → 0.

We start now the derivations of our main results on exact power counting theorems. At first we will discuss the case of vacuum graphs and then we will focus on graphs with external legs. The proof of the second theorem will be shortened following similar steps as for the situation without external legs.

Theorem 2.1 For any connected vacuum graph G of the linearized GFT, we have

A G (1, Λ) Λ→∞ KΛ F -r . ( 11
)
where K is an inessential factor. Similarly

A G (m, 1) m→0 Km -(L-r) . (12)
Proof. We shall deduce all these results from a more general formula which computes the Feynman amplitudes A G (m , Λ f ) in terms of a topological "Symanzik polynomial". We introduce the vector h = (h 1 , h 2 , . . . , h L , k 1 , k 2 , . . . , k F ) t and the matrix defined by its block sub-matrices

M =   m 2 δ i t ε ,f i ε f, Λ -2 f δ f f   (13)
Gaussian integration gives, up to inessential normalizations

A G = dh f dk f e -h t M h/2 = 1 √ Det M . ( 14
)
There exists many ways to determine the behaviour of A G at large Λ f 's. For instance, the Schur complement formula gives [START_REF] Rivasseau | Parametric representation of 'critical' noncommutative QFT models[END_REF], for m = 1 ∀ , Λ f = Λ ∀f . However this method does not give the detailed dependence of A G (m , Λ f ) in all joint variables. More information is obtained by the method of [START_REF] Gurau | Parametric representation of noncommutative field theory[END_REF] which provides a Pfaffian expansion of a determinant det(D + R), where D is diagonal and R is antisymmetric. It expands the determinant as:

Det (D + R) = I⊂1,2,...,N i∈I D ii Pf 2 R I , (15)
where R I is the matrix obtained from R after removing lines and columns of indices in I.

We first remark that the determinant of M coincides with the determinant of

M =   m 2 δ -t ε ,f ε f, Λ -2 f δ f f   (16)
obtained from M after multiplying the block iε t by i and the block iε by -i. This decomposes M as the sum of a diagonal and an antisymmetric matrix. Hence

Det M = I⊂{1,2,...,L} J⊂{1,2,...F } ∈I m 2 f ∈J Λ -2 f Pf 2 ε Î, Ĵ (17)
where Pfε Î, Ĵ means the Pfaffian of ε with lines and columns in I and J erased.

Remark that this is a polynomial in m 2 and Λ -2 f with positive integer coefficients which we could call the Symanzik topological polynomial of the graph. It should obey deletion-contraction relations.

We are now in position to find the scaling behaviour of A G (Det M) -1/2 . This can be done by putting m = 1 ∀ , Λ f = Λ ∀f and identifying the non-zero monomial with the lowest power p in Λ -2 in (17). One notices that because equation ( 17) is a sum of squares this is equivalent to find the subset J with the minimal order such that Pfε Î, Ĵ = 0, in other words to find the minor of maximal rank in ε. Then p = |J| = F -rank ε, since the rank of ε is the cardinal of Ĵ, the complement of J in {1, 2, . . . F }. We are led to A G (Λ -2p ) -1/2 = Λ p = Λ F -rank ε . This achieves the proof of the ultraviolet power counting. Similarly to get the infrared power counting one should identifying the non-zero monomial with the lowest power q in m 2 in (17), hence find the subset I with the minimal order such that Pfε Î, Ĵ = 0. This means to find again the minor of maximal rank in ε. Therefore q = |I| = L -rank ε, since the rank of ε is the cardinal of Î, the complement of I in {1, 2, . . . L}.

We consider now graphs with external legs. Such a graph has L internal lines of index , N external legs of indices j, F internal (closed) faces of indices f and F = DN/2 open strands of indices s. We define a rectangular L + N by F + F incidence matrix by choosing again an arbitrary orientation of each line whether it is internal or external and each face or open strand.

An interesting quantity which corresponds naturally to Feynman amplitudes with fixed external momenta is (up to inessential normalization constants)

A G (k 1 , . . . , k s , . . . , k F ) = j m -1 j e -m -2 j ( s ε j,s ks) 2 s e -Λ -2 s k 2 s ÃG (k 1 , . . . , k s , . . . , k F ) ÃG (k 1 , . . . , k s , . . . , k F ) = dh e -m h 2 /2 f Λ f e -Λ 2 f ( ε f, h ) 2 s e i( ε ,s h ks) = l dh l e -m 2 h 2 /2 f dk f e -Λ -2 f k 2 f /2 e i ,f ε f, h k f +i ,s ε s, h ks = (DetM ) -1/2 exp [-t K t S M -1 SK] (18)
where M = D + R, R being the antisymmetric L + F by L + F matrix whose only non zero upper triangular part is the L by F rectangular matrix ε ,f and S being the L + F by DN/2 matrix with only non trivial part ε s, . K is the vector with components k s .

The quadratic form t SM -1 S is a rational fraction in the Λ -2 f and m 2 variables whose denominator is an analog of the first Symanzik polynomial and whose numerator is an analog of the second Symanzik polynomial.

At zero external momenta the scaling properties are the same as those of Theorem 2.1 but for the graph with the reduced set of internal faces and internal lines. This graph has typically no longer D strands per line but less, since the external strands have been deleted.

Colored GFTs and their linearization 3.1 Colored GFTs

Colored GFTs have many improvements over ordinary GFTs. All their graphs are regular and equipped with canonical orientations of lines, faces, and higher bubbles. The graphs have a well defined homology [START_REF] Gurau | Colored Group Field Theory[END_REF] and admit an exhaustive sequence of cuts [START_REF] Magnen | Scaling behaviour of three-dimensional group field theory[END_REF]. We show in this section that the power counting of any corresponding linearized colored amplitudes, still expressed through Theorem 2.1, can be also computed in terms of their homology.

Let us first review the definition of the colored models along the lines of [START_REF] Gurau | Colored Group Field Theory[END_REF]. Colored GFT models in dimension D are defined in terms of D + 1 complex Bosonic or Fermionic fields denoted by φ , where = 1, 2, . . . , D + 1 is referred to as the color index or simply color. These fields are also defined on GD , where e.g. G = SU (2). The dynamics is described by an action expressed in term of left-invariant fields as δ(g i j (g j i ) -1 ) +λ 2 dg i j e -g 2 i j /2 φ1

S D [φ] := D i=1 dg i e -g 2 i /2 D+1 =1 φ 1,2,...,D φ D,...,2,1 +λ 1 dg i j e -g 2 i j /2 φ 1 1 2 ,1 3 ,...,1 (D+1) φ (
1 2 ,1 3 ,...,1 (D+1) φD+1 (D+1) 1 ,(D+1) 2 ,(D+1) 3 ,...,(D+1) D φD D D+1 ,D 1 ,D 2 ,...,D D-1 . . . . . . . . . φ3 3 4 ,3 5 ,...,3 D+1 ,3 1 ,3 2 φ2 2 3 ,2 4 ,...,2 D+1 ,2 1 D+1 j =i δ(g i j (g j i ) -1 ) (19)
where φ φ are quadratic mass terms and integrations are performed over copies of G using products Haar measures, and λ 1,2 are the coupling constants. We have introduced the shorthand notation φ (g i , g j , . . . , g k ) = φ i , j ,..., k , where the label of group element g k reminds us the link of two colors and k in the vertex. The left-invariant constraint is again given by (1).

Again the linearized version of this model is simply expressed by replacing the group G by R.

The propagator and vertex φ D+1 are drawn in Fig. 1; the vertex for φD+1 can be easily found by reversing the order of all labels in Fig. 1.

An interesting property of colored GFTs is that the coloring can be used to define canonical orientations and a full homology for any colored graph, including external legs [START_REF] Gurau | Colored Group Field Theory[END_REF][START_REF] Gurau | Topological Graph Polynomials in Colored Group Field Theory[END_REF]. More precisely it is shown in [START_REF] Gurau | Colored Group Field Theory[END_REF][START_REF] Gurau | Topological Graph Polynomials in Colored Group Field Theory[END_REF]

d p : U p → U p-1 , d ∂ p : U p ∂ → U p-1 ∂ . ( 20
)
These cellular complexes are made of k-bubbles, which are connected components of strands of subsets of k-colors. Note that the sets of such 0-bubbles B 0 , of 1-bubbles B 1 and of 2-bubbles B 2 naturally coincide with the sets of vertices V, of lines L and of faces F, respectively.

In order to define properly the simplicial complex of the boundary, one needs some further ingredients by making as a connected object the boundary which can be a priori disconnected. This is realized by the "Gurau pinching" [START_REF] Gurau | Topological Graph Polynomials in Colored Group Field Theory[END_REF], introducing new vertices (any boundary D-simplex itself can be dually associated with a vertex φ D for a D dimensional group field model) in the theory which mainly renders connected the dual boundary graph.

Then one can show that indeed

d p • d p+1 = 0 and d ∂ p • d ∂ p+1 = 0.
We also define the extreme differential operators d D+1 := 0 and d ∂ D := 0. From this stage, the homology groups can be defined either for (U * , d * ) or for (U * ∂ , d ∂ * ). We have for the "vacuum" complex (U * , d * ) the p-th homology group defined by H p = Ker d p /Im d p+1 and for the boundary complex

(U * ∂ , d ∂ * ), H ∂ p = Ker d ∂ p /Im d ∂ p+1
which can be called the p-th boundary homology group.

For a general graph G with external legs, one identifies easily the boundary graph G ∂ generated by external strands pinched and a remaining part denoted by Ḡ = G \G ∂ which is generally a sum of vacuum graphs. Using the definitions of cellular complexes for vacuum graphs (U * , d * ) and for a boundary graph (U * ∂ , d ∂ * ), one can define the complex (V * , δ * ) for a graph G with external legs such that

V p = U p ⊕ U p ∂ , ∀p, (21)
δ p = d p ⊕ d ∂ p , ∀p. (22)
A quick checking proves that

δ p • δ p+1 = (d p • d p+1 ) ⊕ (d ∂ p • d ∂ p+1 ) = 0,
since there is no cross terms. The definition of homology group of graphs follows naturally H p = Ker δ p /Im δ p+1 .

As a consequence of the direct sum, we have Ker

δ * = Ker d * ⊕ Ker d ∂ * , Im δ * = Im d * ⊕ Im d ∂ * and H * ≡ H * ⊕ H ∂ * . Finally it can be proved that Ker d D = Z, Im d D = ⊕ B D -1 k=1 Z, H D ≡ Ker d D ; (23) Ker d ∂ D-1 = Z, Im d ∂ D-1 = ⊕ B D-1 ∂ -1 k=1 Z, H ∂ D-1 ≡ Ker d ∂ D-1 . (24)

Homology formula for linearized color power counting

We can now define canonically the incidence matrix ε f, , using the ordering of the colors.

Definition 3.1 ε f,l =    +1, if or(l) = or(f ), l ∈ ∂f -1, if or(l) = or(f ), l ∈ ∂f 0, if l / ∈ ∂f (25)
where or(•) stands for the orientation, and -any line is oriented from φ to φ, -any face is oriented [START_REF] Gurau | Colored Group Field Theory[END_REF] according to its lines with highest color index. Hence each face being bicolored, for such a face f made of the two alternating colors i 1 < i 2 we have

ε f,i 2 = +1 and ε f,i 1 = -1.
Then the amplitude of a colored graph G in the linearized model is formally given by

A G = ∈L G dh l f ∈F G δ ∈∂f ε f, h . (26)
Let us now relate the power counting of Theorem 2.1 to a homology formula Theorem 3.1 For any connected colored vacuum graph G of the linearized D dimensional colored group field model, the amplitude of G behaves as

A G KΛ D+1 k=3 (-1) k-1 B k + D-1 k=2 (-1) k h k , B D+1 := 1 ( 27
)
where K is an inessential factor, B k is the number of k-bubbles and h k stand for Betti numbers for G namely dimH k .

Corollary 3.1 In three dimensions, any connected colored vacuum linearized amplitude G behaves as

A G KΛ B 3 -1+h 2 (28)
where h 2 is the second Betti number.

Hence we recover the same formula than for Type I graphs in [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF], except for the correcting factor h 2 , which must therefore vanish for Type I graphs. The proof of Theorem 3.1 simply follows from Theorem 2.1 and the following lemma Lemma 3.1 Let G be a colored vacuum graph in dimension D ≥ 3 with F faces and an incidence matrix ε between its faces and lines, then

F -rank ε = D+1 k=3 (-1) k-1 B k + D-1 k=2 (-1) k h k , B D+1 := 1 ( 29
)
where B k is the number of k-bubbles and h k the k-th Betti number of G.

Proof. This is a straightforward derivation after noting the fact that, by our definition (3.1), the matrix ε coincides with the matrix of the differential operator d 2 defined in [START_REF] Gurau | Colored Group Field Theory[END_REF].

Introducing the differential complex defined for a fixed graph G

0 d 0 ← U 0 d 1 ← U 1 d 2 ← U 2 d 3 ← U 3 ← . . . . . . d D-1 ← U D-1 d D ← U D d D+1 ← 0 (30)
we have by a simple recurrence

F -rank ε = B 2 -dim Im d 2 = dim Ker d 2 = dim Im d 3 + dim H 2
for G = R, so that we recover the Dirac distribution on G in the limit Λ → ∞. The heat kernel regularization has the obvious advantage of preserving the positivity but any other regularization fulfilling δ Λ (gh) = δ Λ (hg) could also be used in the following argument.

Definition 4.1 For a GFT graph G, we define its jacket G as the ribbon graph made of the 2 external strands.

Note that the jacket can be defined only because we considered only models generating untwisted stranded graphs without any symmetrization on the strands. The jacket then fully determines the underlying GFT graph. For a colored model the jacket can be uniquely defined in terms of the colors only instead of the external strands. This definition is motivated by the following result, which exhibits a class of graphs for which the amplitude do not depend on the nature of the group G.

Theorem 4.1 For a connected GFT vacuum graph G and whose jacket is planar, the amplitude reads

A G = δ Λ (1) N +1 , ( 36
)
where N is the number of circuits followed by the D -2 strands that do not define its jacket. 33) can be written as a product of these δ-functions on the lines of T times a δ-function expressing the momentum conservation around T . More precisely, let us choose a vertex on T , the root, and orient all lines T from the root to the leaves. The lines not in T join two vertices in T and we orient them towards the root, that is, in such a way that they form a consistently oriented cycle with the lines of T between these two vertices. Finally, all faces in G inherit the orientation of the surface it is embedded in. We denote by T | l the tree formed by the lines of T between the root and l, l not included. Then,

f faces of G δ Λ l∈∂f h l = δ Λ (h T ) l∈T δ Λ (h T | l ) (37)
where h T is product of the cyclically ordered momenta incoming to the tree T .

If we further assume that G is planar, then the lines l / ∈ T incident to T are either nested or disjoint, so that each line appears at most once in each δ functions in (37). Accordingly, δ Λ (h T ) = δ Λ (1). Consider now any closed circuit followed by the D -2 strands that do not form the jacket and draw it on G * using perpendicular lines. This circuit encircles a certain number of vertices of G * and momentum conservation implies that the product of the h l along the circuit is 1. For instance, on the example of figure [START_REF] Rovelli | Quantum Gravity[END_REF]. the tree momenta h 2 , h 5 , h 6 , h 7 , h 8 are determined in terms of the loop momenta h

1 , h 3 , h 4 h-5 h-6 h-8 h-7 h-3 h-4 h-2 h-1
Figure 2: A graph, its dual and a spanning tree on the dual.

2

In the general case of a non planar jacket, (37) still holds. However, the lines not in T may have crossings, so that we have factors of the type δ Λ (h

• • • h -1 • • • ),
with the dots denoting group elements other than h. In particular δ Λ (h T ) encodes the defining relation of the fundamental group of the surface G is drawn on. Then, the amplitude cannot be computed without using specific group theoretical relations, unless the δ-functions of the closed circuits yield some extra simplifications. Nevertheless, troublesome factors of the type δ Λ (h • • • h -1 • • • ) are trivial for the commutative group R. Since the divergence of the amplitude is a consequence of the redundancy of the δ functions, it may be reasonably asserted that if a diagram is divergent in the non-linear theory, so it is in the linearized one. 2 Example 4.1 (The octahedron) Consider a graph whose structure is an octahedron (6 vertices and 12 lines). Its jacket is planar and the strands not in the jacket form 3 circuits. This immediately yields A octahedron = δ Λ (1) 4 .

(38)

Conclusion and Discussions

We have defined the linearized group field theories and established their power counting.

The result sheds more light on a result of [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF], where the power counting for a particular class of graphsin three dimensions is given in terms of the number of their 3-bubbles.

Our results show how this formula has to be completed in the linear and colored situation by the alternating sum of the number of higher dimensional bubbles and Betti numbers for the graphs. This study also shows the simplicity of colored models with their more regular structures.

This work has introduced also other power counting results under planarity conditions.

Apart from these properties, we remark that for linearized colored graphs the power counting is B 3 -1, like for Type I graphs, if and only if h 2 = 0. This proves that Theorem 5.1 Every Type I graph in the sense of [START_REF] Freidel | Group field theory renormalization -the 3d case: power counting of divergences[END_REF] must have h 2 = 0.

It is however not so easy to build colored graphs with h 2 = 0. Remark that the three colored graphs considered in [START_REF] Gurau | Colored Group Field Theory[END_REF] all have h 2 = 0. However such graphs should exist. By Poincaré duality, any compact manifold (such as the 3-torus T 3 ) for which one has h 1 = 0 has also h 2 = h 1 = 0. From a theorem by Kaufmann, any piecewise-linear manifold admits a colored triangulations [START_REF] Lins | Gems, Computers and Attractors for 3-Manifolds[END_REF], so there should exist such graphs but we have not found any explicit example yet.

We should explore elsewhere the properties of the Symanzik polynomials for tensor theories introduced in this paper under deletion-contraction of lines, faces or higher "bubbles" and their link to the polynomials introduced in [START_REF] Gurau | Topological Graph Polynomials in Colored Group Field Theory[END_REF].

We shall also study in a future publication the behaviour of leading graphs such as the chain of Figures 3 and4 These are maximal vacuum graphs in the sense that at order n their linearized power counting is 2 + n/2. They can be colored easily.

Figure 1 :

 1 Figure 1: Propagator and vertex φ D+1 in D dimensional GFT.

1 ,3 2 φ 2 2 3

 123 D+1) (D+1) 1 ,(D+1) 2 ,(D+1) 3 ,...,(D+1) D φ D D D+1 ,D 1 ,D 2 ,...,D D-1 . . . . . . . . . φ 3 3 4 ,3 5 ,...,3 D+1 ,3 ,2 4 ,...,2 D+1 ,2 1 D+1 j =i

Proof:

  It is convenient to pass to the dual G * . In this case the faces of G become vertices of G * and the Dirac distribution around faces become more familiar momentum conservation around vertices. Lines of G are in bijection with lines of G * , which may be drawn perpendicularly to those of G. Note however that momenta are group elements and do not necessarily commute. This is taken into account by the cyclic ordering at the vertices of G * . Next, choose a spanning tree T in G * . Because of momentum conservation, momenta on the lines of T are entirely determined by the momenta in G * -T . Then, the product around the faces of the δ-function pertaining to the faces of G in (

Figure 3 : 2 Figure 4 :

 324 Figure 3: A chain with power counting 2 + n/2

  that the p-bubbles and boundary p-bubbles define two cellular complexes (U * , d * ) and (U * ∂ , d ∂ * ) and therefore two cellular homologies induced by two boundary operators d * and d ∂

* such that

A tadpole is a line which goes twice through the same vertex. A "tadface" is a face that goes at least twice through a line. A generalized tadpole is a subgraph with a single external vertex.

In the loop quantum gravity context this divergence is often interpreted as an infrared one for the effective space-time theory, so to underline this ambiguity we might also call it an ultraspin divergence.

Of course this rank is independent of the chosen orientations to define this matrix.

Since there is no ambiguity, we henceforth omit the full notations l ∈ L G and f ∈ F G . We also forget inessential multiplicative factors.

Acknowledgments

The work of J.B.G. is supported by the Laboratoire de Physique Théorique d'Orsay (LPTO, Université Paris Sud XI) and the Association pour la Promotion Scientifique de l'Afrique (APSA). J.M. thanks the Laboratoire de Physique Théorique d'Orsay for their hospitality.

The result follows after noting that from (23), dim Im D = B D -1. Similarly Theorem 3.2 For any connected colored graph G with N external legs of the linearized D dimensional colored group field model, the amplitude of G at fixed bounded external momenta behaves as

where K is an inessential factor, B k is the number of k-bubbles and h k stand for the Betti numbers for Ḡ, the graph obtained by deleting the external open strands in G.

Power counting for graphs with a planarity condition

Here we present a comparison of the power counting for a particular class of graphs in the linear and non-linear model. Most of our results in this section are valid for the graphs of the BF theory with group G in dimensions D = 2, 3, 4, be it colored or not. It turns out that the amplitude is largely independent of the group G for a specified category of graphs. First, recall that the lines GFT graphs in these dimensions are made of D strands. Their vertices of valence D + 1 involve a specific routing of the incoming strands. Closed circuits followed by the strands define the faces and for a vacuum diagram, the amplitude reads

The integration measure on the group is assumed to be normalized, for instance by taking the normalized Haar measure on a compact Lie group or the Lebesgue measure with a Gaussian weight for R. The delta functions are regularized either using the heat-kernel

for G a simple Lie group and