FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOLTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK F. CASTELLA 1

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1 FROM THE VON-NEUMANN EQUATION TO THE QUANTUM BOTZMANN EQUATION IN A DETERMINISTIC FRAMEWORK F. CASTEA 1 J. Stat. Phys., Vol. 104, N. 1/2, pp (2001). Abstract In this paper, we investigate the rigorous convergence of the Density Matrix Equation (or Quantum iouville Equation) towars the Quantum Boltzmann Equation (or Pauli Master Equation). We start from the Density Matrix Equation pose on a cubic box of size with perioic bounary conitions, escribing the quantum motion of a particle in the box subject to an external potential V. The physics motivates the introuction of a amping term acting on the off-iagonal part of the ensity matrix, with a characteristic amping time α 1. Then, the convergence can be prove by letting successively ten to infinity an α to zero. The proof relies heavily on a lemma which allows to control some oscillatory integrals pose in large imensional spaces. The present paper improves a previous announcement [CD]. Key wors: Density matrix, iouville equation, Pauli Master Equation, time-epenent scattering theory, Fermi s Golen Rule, oscillatory integrals in large imensions. AMS subject classification: 81H05, 81F20, 82A15, 82A70, 34E20, 46F10 Acknowlegements. The author woul like to express his eepest gratitue to P. Degon an B. Perthame for initiating the present project, an for so many passionating iscussions on this work. He woul like to thank A. Arnol, E. Grenier, G. Métivier, an F. Nier for fruitfull iscussions as well. Part of this work was one while the author was visiting the Erwin Schröinger Institute in Vienna (Austria). The author woul like to thank this institution for its hospitality, an he particularly thanks P. Markowich for his invitation. This work was supporte by the TMR-Network Asymptotic Methos in Kinetic Theory number ERB FMBX CT an by the Erwin Schröinger Institute in Vienna (Austria). 1 CNRS et IRMAR - Université e Rennes 1 - Campus e Beaulieu Rennes Ceex - France - castella@maths.univ-rennes1.fr 1

2 Contents 1 Introuction Introuction Motivation of the present work Presentation of the results The moel an its formal asymptotics The mathematical moel uner consieration Elimination of the non-iagonal part A formal investigation of the infinite volume limit in the case of zero amping Statement of the Theorems The rigorous convergence results An oscillatory integral estimate Proofs Proof of emma 2.1: Positivity of the ensity matrix Proof of Theorem 2.1: A close equation on the iagonal part of the ensity matrix Proof of emma 3.1: Controlling oscillatory integrals Proof of Theorem 3.1: getting apriori bouns Proof of Theorem 3.2: convergence as Proof of Theorem 3.3: convergence as α Conclusion: possible extensions of the present results Appenix: The physisicts view of Fermi s Golen Rule 45 1 Introuction 1.1 Introuction In this paper we are intereste in the quantum ynamics of an electron in a given perioic istribution of obstacles in imensions of space ( 3, we will actually restrict ourselves with the case = 3). More exactly the electron is assume to evolve on a Torus, an the present ynamics is naturally escribe by a perioic Von-Neumann equation. In our moel, the size of the perio is measure by the large scaling parameter, an each elementary cell contains one obstacle occupying a volume of the orer O(1). We consier the asymptotic ynamics as. In orer to obtain a non-trivial limiting ynamics, one has to rescale time as well, an to look at the evolution of the electron on long time scales of the orer T, with T, unless the electron essentially performs a free flight in the limit. The present paper is concerne with the so-calle low-ensity regime (or: Boltzmann-Gra regime) where the ratio T is prescribe. In this scaling inee, the obstacles occupy a proportion 1/ of the total volume, so that the probability for the electron to hit an obstacle once per unit time on this time scale is unity. 2

3 The issue in consiering such a moel is the following: it is physically expecte (see e.g. [Pa], [VH1,2,3], [K1,2], [Ku], [Pr], [Vk], [Zw] or also [Ck], see [Fi] for recent evelopments) that the present system tens to be escribe by a linear Boltzmann equation in the lowensity asymptotics, an precise convergence results in this irection have been actually prove in various situations where the obstacles are ranomly istribute (see e.g. [Sp1], [HW], [a], [EY]). In particular, the initially time-reversible moel is expecte to be asymptotically escribe by a time-irreversible equation. Contrary to the ranom situation, the present paper eals at variance with a moel which is both eterministic an perioic, which is a very strong constraint as well as a non-generic case. While the stochastic approach somehow ensures that the esire convergence towars a linear Boltzmann equation hols for almost every istribution of obstacles, the present paper wishes to investigate the same convergence in one particular configuration, namely the perioic one. Obviously, the perioic situation leas to specific coherence, or resonance effects, which are somehow smoothe out in the ranom situation. These effects turn out to be extremely strong, an to abnormally ominate the limiting proceure: it is actually prove in [CD1,2] that the irect limit, T in the original perioic Von-Neumann equation oes not lea to the expecte linear Boltzmann equation. The two papers [CD1,2] show in fact that the above mentionne coherence effects can be precisely quantifie making use of number theoretical consierations, an the limiting ynamics is prove to remain time-reversible in this case. For this reason, the present paper introuces some noise in the moel, in that we let the system { electron + potential } aitionaly interact with another system, typically a bath of photons or phonons. This leas to the aing of an aitional amping term in the original Von-Neumann equation, which moelizes at a phenomenologic level ([NM], [SS]) the above mentionne interaction. It is measure by the relaxation time 1/α > 0. It makes the moifie Von-Neuman equation (2.8) in which we pass to the limit reaily time-irreversible. This amping term, which at first sight can be viewe as an artificial trick, is actually present in most physically realistic cases [Boy], [o], [NM], see also [Hu] for the necessity of consiering such aitional interactions. Our main result is the following: in the low-ensity limit, T, followe by the limit α 0 where the interaction with the external system vanishes, the moifie ynamics of the electron is asymptotically escribe by a linear Boltzmann equation (Theorem 3.3). Moreover, the cross-section involve in the limiting Boltzmann equation is given by a full power series expansion in the potential (3.32), (3.34), an the lower orer term satisfies Fermi s Golen Rule (3.35). engthy calculations which are eferre to another work [Ca2] actually prove that this expansion coincies with the usual Born series expansion of conventional scattering theory (3.38), (3.39). In view of the negative results prove in [CP1,2], the two limits in an α o not commute. The results presente here heavily rely on the control of some oscillatory integrals in large imensions (emma 3.1). We close this introuction with some references. The first formal erivations of the Pauli Master Equation are ue to [Pa], [K1,2], [Ku], [VH1,2,3]. Other works in this irection are ue to [Zw], [Pr], [Ja], [Mei] an recently, to [Fi]. ater, many authors i inee rigorously justify this erivation [Sp1], [a], [HW], an more recently, [EY], but all these references eal with ranom potentials. The present paper is, to our knowlege, the first to present a erivation of 3

4 the Quantum Boltzmann Equation for eterministic potentials an to follow the formal approach base on time-epenent scattering theory as outline in the appenix. Similar questions are ealt with in the framework of classical mechanics in [Sp2,3] an [VbS]. We also mention [Ni1,2], where the square of the scattering matrix is prove to be the right object allowing to escribe some semi-classical limits, but only finitely many fixe scatterers are consiere. Finally, we mention [Ca1], in which the size of the box is kept fixe an the limit equation is a reversible retare time integral equation. A review about the convergence result presente here an the non-convergence results prove in [CP1,2] can be foun in [Ca3]. 1.2 Motivation of the present work We give here two remarks motivating the present work. At first, we wish to unerline a strong parallel between the results iscusse here at the quantum level, an other, somehow similar, results vali at the classical level, since it is a major motivation for this paper as well as [CP1,2]. The present paper investigates at the quantum level the convergence of the ynamics of an electron in a perioic istribution of obstacles in the low-ensity limit. Aing a amping term which moelizes the interaction of the electron with an external reservoir (a stanar proceure in many contexts of statistical physics, see [Sp2,3], see also [CEFM]), we prove the expecte convergence towars a linear Boltzmann equation in the appropriate limit. Though the aition of a amping term makes our moel time-irreversible while the stochastic approach proves the convergence (in expectation) of a true time-reversible ynamics towars an irreversible ynamics, the present result is somehow analogous, in a eterministic framework, to the convergence results establishe in the stochastic framework. On the other han, the results prove in [CP1,2] inicates that the true perioic situation in the case of zero amping leas to specific coherence effects, measure in terms of number theoretical arguments, which in turn prevent the convergence towars the esire Boltzmann equation. These convergence, respectively non-convergence results at the quantum level have the following important analogon at the classical level: it has been prove in [BBS] that the classical ynamics of an electron in a ranom istribution of obstacles converges, in the appropriate low-ensity limit, towars a linear Boltzmann equation. However, the work [BGW] proves that, when the istribution is perioic, the convergence towars a Boltzmann equation cannot hol, thus showing that the perioic situation is egenerate in the sense that it is in the zero-measure set where the expecte convergence fails. On the more, the non-convergence result of [BGW] relies as in [CP1,2] on coherence effects specific to the perioic case which can be precisely quantifie thanks to number theoretical consierations. Roughly speaking, we may say that too many trajectories never hit any obstacle in the perioic situation, which makes this situation specific both at the classical an at the quantum level.. A secon strong motivation for the present work, an more precisely for the limit in we consier here, is the following. This paper aims at giving a rigorous erivation of the Pauli Master Equation (or Quantum Boltzmann Equation) starting from the Quantum iouville Equation (or Von-Neumann Equation). This problem has receive consierable interest since the early formal erivation of Pauli [Pa]. Our approach actually tries to mimic the elementary erivation of Fermi s Golen Rule base on time-epenent scattering theory which can be foun in many 4

5 textbooks e.g. [Boh], [CTDRG], [CTD], [Mes], [SS], an which has been recently revive in [Co]. (We recall that Fermi s Golen Rule gives the expression of the transition rate involve in the Pauli Master Equation). This is where our limit in originates. Elementary time-epenent scattering theory [Boh] consiers inee a Hamiltonian of the form H 0 + V in a finite box of size, where H 0 is the free kinetic energy. Initially, particles are suppose concentrate on an eigenstate state ψ n of H 0 associate with the energy ε n n 2 / 2. The goal is to compute the time asymptotics of the population C p (t) 2 = (ψ(t), ψ p ) 2 of the initially voi states of energy ε p, p n, where ψ(t) enotes the wave-function of the particle at time t. It is well known that, if the energy levels remain iscrete (i.e. if the size of the box is left unchange), the behaviour of C p (t) 2 is either an oscillatory function of time if the energies are ifferent (ε p ε n ) or a quaratic function of time if the energies are equal (ε p = ε n ), but can by no means be a linear function of time. However, scattering theory aims at proucing a rate of change of the populations, i.e. a probability per unit of time. Therefore, one looks for a linear in time behaviour of C p (t) 2. To remove this apparent paraox, one must therefore let the size of the box go to infinity. Inee, when the level spacings are set to zero, i.e. as, the oscillatory function of time C p (t) 2, where now p is a continuous variable ranging in the wave-vector space R 3, formally is, in the sense of istributions of the p variable, asymptotic as time goes to infinity to a linear function of time multiplie by a elta function of the ifferences of the energies of the initial an final states δ(ε n ε p ). In this sense, one recovers the usual Fermi Golen Rule as a succession of two limits then t. Note that we recall these stanar an formal computations below in the appenix. In particular, the process of taking the size of the box to infinity is of primary importance an not just a technicality, because one cannot eal from the onset with an infinite meium ue to the above mentionne reasons: in an infinite box, the eigenstates of the unperturbe Hamiltonian H 0 are not normalize, an therefore unphysical, an it cannot be given any meaning to the quantity C p (t) 2. This point of view thus emphasizes the contrast between the case where the spectrum is iscrete, an the case where it becomes continuous. On this particular point, we wish to quote the work [Co], where the effect of iscretizing the spectrum is stuie from a physical point of view, [CP1,2] where this effect is quantitatively stuie in the perioic case, as well as [Ca1] for a mathematical work showing that the large-time/small potential limit (analogous to the weak-coupling limit) cannot give the physically expecte linear Boltzmann equation when stuie in a perioic box of fixe finite size. 1.3 Presentation of the results The present paper follows the subsequent lines: 1- (section 2) 1a- As a starting point (subsection 2.1), we write own the equation escribing the ynamics of an electron in a perioic box, an we introuce the phenomenological amping term neee in our analysis. We also introuce the appropriate scalings an initial ata uner consieration. We are thus le to consiering a (ampe) Von-Neumann equation pose on the so-calle ensity matrix of the electron. We recall that the iagonal part of the ensity matrix (in the momentum representation ψ n ), enote by ρ,α (t, n) (see below for the notations), precisely 5

6 gives the populations C n (t) 2. Note that the amping term acts on the off-iagonal part of the ensity matrix only. 1b- Then (subsection 2.2), we follow the computations in [Ca1]: the iscreteness of the energy levels an the fact that the evolution of the iagonal part only epens on the action of the perturbe Hamiltonian on the off-iagonal part (a universal phenomenon ue to the commutator structure of the equation) allow us to write a close equation for the populations, which is equivalent to the original quantum iouville equation (Theorem 2.1). This equation has the form of a retare time integral equation (non-markovian ynamics), t ρ,α (t, n) = k Z 3 0 σ,α (s, n, k)ρ,α (s, k) s 0 σ,α (s, n)ρ,α (s, n) s, (1.1) for some cross-sections σ,α an σ,α epening on the scaling parameters an α, whose explicit value is compute as a power series in the potential V. We refer to the sequel for the precise meaning of the notations. 1c- Then (subsection 2.3), we present a formal approach in the case of zero amping. It inicates that the low-ensity limit, T 3 (which, in the language of conventional scattering theory use in subsection 1.2, provies at the same time the two limits i.e. the size of the box to infinity an the time to infinity), turns out to formally give the esire convergence towars a pointwise in time (Markovian) Boltzmann equation, with a cross-section satisfying the Fermi-Golen-Rule at the lower orer. However, we mention that this formal limit cannot be rigorously justifie because it involves a limit in an oscillatory series which becomes an oscillatory integral, a situation which is not escribe by the classical stationary phase theorem. Moreover, the formal result is actually wrong in one imension as well as in large imensions 3, as it is prove in [CP1,2]. 2- (section 3) To rigorously obtain the true Quantum Boltzmann equation, one nees therefore to make use of the amping term: from a technical point of view, it provies us with a regularizing parameter α > 0, aitional to the scaling parameter escribing the low-ensity process. In the language of subsection 1.2, this parameter allows to somehow ecouple the limits an t. Now we perform the various limiting proceures: 2a- (Theorem 3.1) Firstly, the convergence proof relies on establishing various apriori estimates which are inepenent of (but not on α). 2b- (Theorem 3.2) The limit is then taken first an takes the original ampe retare time integral equation to a ampe local-in-time equation of the form (see below for the notations), t ρα (t, n) = σ α (n, k)ρ α R 3 (t, k) k σ α (n)ρ α (t, n), (1.2) for some cross-sections σ α an σ α which we explicitely compute as power series in the potential V. In particular, we see that the ynamics of the electron becomes Markovian along the asymptotic process, while being by no means Markovian initially, see also [EY] on this point. 2c- (Theorem 3.3) The limit α 0 now allows to remove the amping an to obtain (see below for the notations), t ρ (t, n) = R 3 σ(n, k)ρ (t, k) k σ(n)ρ (t, n), (1.3) 6

7 where the cross sections σ an σ are given as power series of the potential V, an they satisfy the Fermi Golen Rule at lower orer in V (Theorem 3.3), i.e., σ(n, k) = 2πδ(n 2 k 2 ) V (n k) 2 + O( V 3 ). (1.4) The full series efining σ can actually be prove to be ientical with the celebrate Born series - see [Ca2] - a fact which far from obvious in view of formulae given in Theorem (emma 3.1) The last limit α 0 relies in an absolutely essential way on the control of the singularity an the growth with the imension of certain oscillatory integrals with quaratic phases in large imension. This emma lies at the core of the convergence proof presente here. It gives the estimates inepenent of α neee along the asymptotic process. Also, since the cross-section involve in (1.3) appears as a power-series in the potential, this emma allows to control the convergence of these power series. 3- (Section 4) The full convergence proofs of the above statements are given. Our main results are Theorem 2.1 (getting a close equation for the iagonal part only), Theorem 3.1 (a priori bouns on the solution to the Von-Neumann equation), Theorem 3.2 (the limit, where the ynamics becomes Markovian), Theorem 3.3 (the limit α 0, where we recover the Quantum Boltzmann equation with the correct cross-section), as well as the key emma 3.1 (control of oscillatory integrals in higher imensions). 2 The moel an its formal asymptotics 2.1 The mathematical moel uner consieration Before writing own the equation in which we actually perform the asymptotic process an introucing the amping term, we write, as a first step, the equation escribing the evolution of an electron evolving on a Torus [0, 2π] 3 uner the mere influence of a potential V. As is well known, it is actually escribe by the perioic Von-Neumann equation in the box [0, 2π] 3, with perioic bounary conitions, i t ρ(t, x, y) = ( x + y ) ρ(t, x, y) + (V (x) V (y)) ρ(t, x, y), (2.1) where ρ(t, x, y) is the particle ensity matrix, which is the mathematical object escribing the state of the electron at time t R (see [CTD]), epening on two space variables x an y both belonging to [0, 2π] 3. We mention that the bounary conitions can easily be moifie into Dirichlet or Neumann bounary conitions, See section 5. Here an throughout the paper, we make the following important assumption on the given potential V (x), { V (x) is real-value an compactly supporte, Supp(V (x)) ]0, 2π[ 3, V (x) is a smooth function of x R 3, (2.2) where the precise smoothness assumption on V is written in the next section (See efinition 3.1 an assumption (3.5)). The compactness assumption ensures that the potential (or: obstacle) 7

8 occupies a volume of the orer O(1) in a cell of the orer O( 3 ), which is the natural lowensity situation. Note that this assumption can somewhat be relaxe, an the results that we present are also vali if the potential V = V epens on an converges to a fixe profile V belonging to the same spaces as in efinition 3.1 an (3.5). Now, the various asymptotic processes are naturally performe in the Fourier space rather than irectly on (2.1). For this reason, we efine, for any n an p Z 3, the following Fourier transforms, ρ(t, n, p) := (2.3) ( 1 ρ(t, x, y) exp +i n y ) ( 1 exp i p x ) x y, [0,2π] 6 (2π) 3 2 (2π) 3 2 as well as the more stanar, V (n) := V (x) exp( in x) x [0,2π] 3 ( ) = V (x) exp( in x) x, (2.4) R 3 for any n R 3. The last equality comes from the assumption on the support of V an V ( ) is by assumption a fixe smooth profile. Here, bol letters n, p,... enote continuous variables belonging typically to R 3, whereas plain letters n, p,... enote iscrete variables belonging typically to Z 3, a convention use throughout the paper. With these notations, the original Von-Neumann equation (2.1) becomes, t ρ(t, n, p) = n 2 ip2 + i (2π) 3 2 ρ(t, n, p) (2.5) { ( ) ( ) k n V ρ(t, k, p) V p k } ρ(t, n, k). k Z 3 Note that the efinition (2.3) is the natural one since the functions, ψ n (x) := (2π) /2 exp( in x/), (n Z 3 ) are the eigenfunctions of the operator x on the space of perioic functions in the box [0, 2π] 3, with egenerate eigenvalues, E n := n 2 / 2 (n Z 3 ). Before proceeing further, we now efine the iagonal an off-iagonal parts of ρ respectively as, ρ (t, n) := ρ(t, n, n), ρ n (t, n, p) := ρ(t, n, p)1(n p), (2.6) where 1(n p) enotes the characteristic function of the set {n p}. The iagonal part ρ (t, n) represents the occupation number of the n-th eigenstate of the aplacian, whereas the non-iagonal part represents the correlations between the occupation numbers of the n-th an p-th eigenstates. We now come to writing the moel in which we shall actually perform the asymptotic process. This nees the introuction a phenomenological amping term on the off-iagonal elements in the original equation (2.5). Also, this leas to rescaling the various quantities, namely time, 8

9 potential an absorption. More precisely, let T enote the time-scale, λ the typical amplitue of the potential, an α the amping parameter. Our starting point reas, with these notations, 1 T t ρ,α (t, n, p) = i p2 n 2 + iλ (2π) 3 2 ρ,α (t, n, p) αρ,α (t, n, p)1(n p) (2.7) { ( ) ( ) k n V ρ,α (t, k, p) V p k } ρ,α (t, n, k), k Z 3 (compare with (2.5)), where we now explicitely inex the epenence of the ensity matrix upon the scaling parameters an α (an write ρ,α an ρ,α n as well for the iagonal an off-iagonal parts of ρ,α respectively). Equation (2.7) also reas, upon splitting ρ,α into its iagonal an off-iagonal parts, T 1 t ρ,α n (t, n, p) = p 2 +in2 ρ,α n (t, n, p) αρ,α n (t, n, p) ) { 2 ρ,α ) ( λ +i (2π) V p n 3 λ ( k n +i V (2π) 3 k p T 1 t ρ,α (t, n) = + iλ (2π) 3 (t, p) ρ,α (t, n) ρ,α n (t, k, p) iλ k n { V ( ) k n } (2π) 3 k n ( ) p k V ρ,α n (t, k, n) V ( n k ρ,α n (t, n, k), ) ρ,α n (t, n, k) }, To be complete, we now precise the exact asymptotic regime an the specific initial ata uner consieration. Firstly, we are intereste in the following low ensity regime, (2.8) T = (2π) 3, λ λ 0, (2.9) where λ 0 is some (small) constant inepenent of α an. Note that equation (2.7) introuces a amping scale by α in the new time scale T, which is an extremely strong amping. Seconly, an as it is stanar in this fiel (see e.g. [Ku], [K1,2], [Ck], [Zw]) we are only intereste in performing the asymptotics, α 0 in (2.8) for particular initial ata which are stationary states of the free Von-Neuman equation it 1 ρ/ t = ( x + y ) ρ. In other wors, we wish to quantify the large time influence of the potential for initial states which are equilibrium states of the unperturbe hamiltonian x. The initial ata of interest in the present paper are thus taken of the form, ρ,α n (t, n, p) t=0 = 0, ρ,α (t, n) t=0 = (2π) 3 ρ 0 ( n ) ρ 0 (n) 0 is a given smooth function of n R3,, (2.10) where again the precise smoothness assumption neee on ρ 0 is written in the next section (See efinition 3.1 an assumption (3.5) ). The assumption (2.10) generalizes both the case 9

10 of initial thermoynamical equilibrium where ρ,α (t, n, p) t=0 3 exp( βn 2 / 2 ) 1[n = p] an β is the inverse temperature, an the more general case where ρ,α (t = 0) is an arbitrary function of the energy ρ,α (t, n, p) t=0 3 f(n 2 / 2 )1[n = p] for some reasonable function f. Summarizing, we are intereste in obtaining a linear Boltzmann equation on the (noncommuting) limit ρ = lim α 0 lim ρ,α, in (2.8), (2.10), (2.9). We en the presentation of the moel with two comments. Physically, the amping term accounts for elastic interactions of the particles with an external bath of, typically, ions, atoms, phonons, photons, etc. (see e.g. [SS], Chap. 7.3 for applications to light-matter interaction). Uner such interactions inee, the off-iagonal terms of the ensity matrix (i.e. the correlations between the energy levels) are exponentially ampe ue to ranom changes of phases of the particle wave-functions, while, simultaneously, the iagonal part is left unchange ue to the elastic character of the collisions (see also [o], [Boy], [NM]). In the most general case, when both elastic an inelastic interactions are present, the latter act on much longer time scales than the former. Consequently, the ecay of the off-iagonal terms of the ensity matrix by elastic collisions is much quicker than that of the iagonal terms by inelastic collisions [NM]. The amping of the off-iagonal terms bears similarities with the so-calle Ranom Phase Approximation [VH1,2,3], [Zw], [Kr], [Pr]. Also, the effect of this amping term is similar to that of an average over ranom variables like in [Sp1], [a], [HW], [EY]. Finally, for the sake of completeness, me mention the following emma, the proof of which is given in section 4 below. It states that the amping term in (2.8) preserves the positivity of the ensity matrix. It actually satisfies the stronger inbla property [i]. emma 2.1 Assume (2.2). Then, the solution ρ,α (t, n, p) to (2.8) with initial ata ρ,α (t, n, p) t=0 satisfying (2.10) is unique in C 0 (R t ; l 2 (Z 6 n,p)). Besies, for all non-negative values of time t 0, ρ,α (t, n, p) is a hermitian, positive, an trace-class operator acting on l 2 (Z 3 ). In particular, we have, for all t 0, ( ) ρ,α (t, n) 0, an, ρ,α n (t, n, p) = ρ,α n (t, p, n). (2.11) 2.2 Elimination of the non-iagonal part We wish to erive the Pauli master equation for the occupation number ρ,α (t, n) in the asymptotic limits, α 0, starting from the couple system (2.8) on ρ,α an ρ,α n. As formally observe in [Zw], [Pr], it is actually possible to write a close equation on ρ,α before any scaling limit. Following [Ca1], the next theorem gives the explicit form of the equation on ρ,α. Its proof is eferre to section 4. Theorem 2.1 Assume (2.2). et ρ,α (t, n), ρ,α n (t, n, p) be the unique solution to the rescale system (2.8) with initial ata given by (2.10). Then, for positive values of time, t 0, ρ,α (t, n) satisfies the following close equation of Boltzmann-type, t ρ,α + (t, n) = l=1 λ l+1 (Q,α l ρ,α )(t, n), (2.12) 10

11 where the ampe collision operators Q,α l are given by, (Q,α l ρ,α )(t, n) = T (2π) 3(l+1) ( 2R) ε 1,,ε l k 1,,k l u 1,,u l ( 1) ε 1+ + ε l exp ( i (n + ε 1k 1 ) 2 (n ε 1 k 1 ) 2 2 u 1 αu 1 ) (2.13) exp ( i (n + ε 1k 1 + ε 2 k 2 ) 2 (n ε 1 k 1 ε 2 k 2 ) 2 2 u 2 αu 2 ) exp ( i (n + ε 1k ε l k l ) 2 (n ε 1 k 1 ε l k l ) 2 2 u l αu l ) [i V ( k 1 )] [i V ( k 2 )] [i V ( k l )] [i V ( k 1 + k 2 + k l )] ρ,α (t T 1 (u 1 + u u l ), n + ε 1 k 1 + ε 2 k ε l k l ). Here an in the sequel, R enotes the real part of a complex number. Also, the sums ε 1,,ε l carry over the variables (ε 1,, ε l ) {0, 1} l an (k 1,, k l ) (Z 3 ) l such that, an we have mae the convention, k 1 0, k 1 + k 2 0,, k 1 + k 2 + k l 0, (2.14) j = 1,, l ε j = (1 ε j ) = 1 or 0. (2.15) Finally, the integrals carry over the set, u 1,,u l 0 u 1 T t, 0 u 2 T t u 1,, 0 u l T t u 1 u l 1. (2.16) Moreover, the first collision operator Q,α 1 (case l = 1 above) has the more explicit value, λ 2 (Q,α 1 ρ,α )(t, n) = 2 λ 2 T t (2π) 6 V 2 ( n k exp( αu) cos ( n2 k 2 2 u) (2.17) k 0 ) [ρ,α (t T 1 u, k) ρ,α (t T 1 u, n)] u. Remarks. 1- The system (2.12)-(2.13) is a linear Boltzmann-like equation with memory in time, of the form (1.1). The typical gain-loss structure of the right-han-sie of (2.12)-(2.13) is particularly transparent on the first collision operator λ 2 Q,α 1, in spite of the fact that the (time epenent) scattering rate, k 1,,k l 2 λ 2 (2π) 6 exp( αu) cos k 2 (n2 2 u) V 2 ( n k ), (2.18) is not positive. More generally, each term λ l+1 Q,α l ρ,α escribes how the occupation numbers (t, n) are affecte by (l + 1) consecutive interactions with the potential V, so that the ρ,α 11

12 expansion l λl+1 Q,α l is also a power series in V. Obviously, its structure coul be illustrate by means of a Feynman iagram. 2- We can reaily observe that, for fixe t, we have the following boun on the right-han-sie of (2.12), λ l+1 Q,α T l ρ,α l 2 n [ 2λ ] l+1 T l t l (2π) 3 l! ( use that [ C λ T t u 1 =0 [ (2π) 3 V ( k ) ] l lk 1 (2π) 3 V ( k ) lk ρ,α u1 u l 1 1 u 1 u l = t l /l! ) u l =0 ] l l! ρ,α (t, n, p) l 2 n,p ( for some constant C, thanks to (2.2) ), [ C λ T t 3 ] l+1 1 l! C (t), (t, n) l 2 n, for some time epenent function C (t), because ρ,α (t) C 0 (l 2 ) (emma 2.1). Obviously, this is enough to give a rigorous meaning to the series involve in Theorem 2.1, at least for fixe values of the scaling parameters T an. 2.3 A formal investigation of the infinite volume limit in the case of zero amping We are intereste in the infinite volume limit ( + ) in (2.12)-(2.13) when the amping is set to zero. On the one han, we escribe the main scaling properties of equation (2.12)-(2.13), which lea to thinking that the irect an formal limit in (2.12)-(2.13) with α 0 coul give the Quantum Boltzmann equation. Simultaneously, we inicate that this rough an formal limit cannot be mae rigorous. To simplify the presentation we restrict ourselves to the leaing orer term namely, t ρ = λ 2 Q 1 ρ + O(λ3 ), with vanishing amping, i.e., t ρ (t, n) = 2 T λ2 (2π) 6 k 0 cos ( n2 k 2 2 u) V 2 ( n k ) [ρ (t u, k) ρ (t u, n)] u + O(λ 3 ), ρ (t = 0, n) = 1 (2π) 3 ρ0 ( n ). (2.19) We first observe that formula (2.19) ressembles the following basic formula, vali for all smooth functions ψ, N ψ(k/) ψ(k) k, but the factor (2π) 6 in front of the k Z N k R N sum k is too strong, since the vector k belongs to the 3-imensional space only. Therefore, the low-ensity scaling T = (2π) 3 is natural in (2.19). With this rescaling in time we obtain 12

13 inee, (2π) 3 t ρ (t, n) = 2λ2 (2π) 3 t k 0 [ρ (t u (2π) 3, k) ρ (t ρ t=0(n) = (2π) 3 ρ 0 (n/). cos ( n2 k 2 2 u) V 2 ( n k ) u (2π) 3, n)] u + O(λ3 ), (2.20) If we now try to perform the infinite volume limit on (2.20), three istinct phenomena take place, at least formally: 1- The moel shoul converge towars a moel where k ( )(k/, n/) is replace by k ( )(k, n) as, an with the initial ata in (2.20) simply replace by ρ t=0 = (2π) 3 ρ 0 (n). 2- The time integration u [0, t] in the original moel (2.12)-(2.13) has become, after rescaling in time, u [0, (2π) 3 t]. Therefore, for positive values of time t (this is where time irreversibility appears), the limit + in (2.20) shoul give rise to an integral over the positive real line, The factor ρ (t T 1 u, n) in the original equation has become ρ (t u, n) an shoul (2π) 3 converge towars ρ (t, n) in the limit +. In particular, the infinite volume limit seems to transform an equation with memory in time into a Boltzmann equation without memory in time. For all these reasons, the infinite volume limit in (2.20) formally gives the asymptotic behaviour, + t ρ (t, n) 2λ 2 0 R 3 cos ([n 2 k 2 ]u) (2.21) V 2 (n k) [ ρ (t, k) ρ (t, n) ] k (2π) 3 u + O(λ3 ). In particular, the time integral + in (2.21) has a meaning as an oscillatory integral, an (2.21) gives (See emma 3.1 an the subsequent remarks), t ρ (t, n) 2πλ 2 R 3 δ(n 2 k 2 ) V 2 (n k) [ ρ (t, k) ρ (t, n) ] k (2π) 3 + O(λ3 ), (2.22) which is precisely the Quantum Boltzmann equation that we are looking for. The first two steps cannot be justifie because they involve the tranformation of a iscrete series into an oscillatory integral. Inee, the notion of oscillatory integral heavily relies on the fact that the k, n variables are continuous. Moreover, it can be prove [CP1,2] that, at least in imension one as well as in large imensions ( 3), this formal analysis is actually false, an the limiting equation remains reversible. This is the reason of our introuction of the amping variable α. This is also the reason why, in previous works, expectation values over ranom phases were consiere [Sp1], [a], [HW], [EY]. k (2π) 3 13

14 3 Statement of the Theorems 3.1 The rigorous convergence results We are now able to escribe the asymptotics +, α 0 in Theorem 2.1. We begin with some notations. Definition 3.1 (i) For any D 0, we efine the following spaces of test functions, T D (R N ) := {Ψ(n) C 0 (R N ) s.t. n D Ψ(n) (R N ) }, (3.1) where we use the usual notation, n := (1 + n 2 ) 1/2. In the sequel, T D (R N ) will often be written in short T D, without refering to the actual imension of the unerlying space R N. The space T D is a Banach space, an its norm is given by, (ii) For any D 0, we efine, Ψ TD := n D Ψ(n). (3.2) S D (R N ) := {Ψ(n) C 0 (R N ) s.t. n a n b Ψ(n) C 0 (R N ), 0 a, b D },(3.3) where as usual, n := (1 n ) 1/2. The space S D is a Banach space, an its norm is given by, Ψ SD := D a,b=0 n a n b Ψ(n). (3.4) Using these notations, we shall assume throughout the paper that the profiles ρ 0 an V (the ata of the problem) satisfy, ρ 0 (n) T D, V (n) S 2D, for some large (but fixe) D, say D 4. (3.5) Also, we pick up a coupling parameter λ satisfying, λ λ 0, where λ 0 is some small constant, whose actual value only epens on the norms ρ 0 (n) T D an V (n) S2D, but it oes not epen on nor on α. We are now in position to state the three main Theorems of the present paper. Theorem 3.1 (A priori bouns an existence of weak limits). et ρ,α (t, n) an ρ,α n (t, n, p) be the unique solutions to (2.8) in C0 t (ln) 2 an Ct 0 (ln,p) 2 respectively. Assume that the initial atum satisfies (2.10), an the potential satisfies (2.2), with profiles ρ 0 an V satisfying (3.5). Then the following hols. (i) For any t 0, the following a-priori estimates hol for the iagonal part, ρ,α (t, n) l 1 n C (preservation of trace), (3.6) ρ,α (t, n) l n C 3 (maximum principle), (3.7) ρ,α (t, n) l 2 n C 3/2 (interpolation), (3.8) 14

15 where the constant C is of the form, C := C( ρ 0 (n) TD, V (n) S2D ), (3.9) but C is inepenent of t an α, a convention that we shall use throughout the paper. Also, the non-iagonal part satisfies, for any t 0, ρ,α n (t, n, p) l 2 n,p C α 3. (3.10) (ii) Uner the same circumstances, the following estimate on the erivative in time t ρ,α n hols, for any t 0, t ρ,α n (s, n, p) 2 l 2 n,p s C α 3. (3.11) (iii) Finally, the sequence ρ,α (t) is uniformly ifferentiable in time, in the sense that for any t 0, t ρ,α (t, n) l 2 n C. (3.12) α3/2 (iv) Define now the istributions ρ,α (t) an ρ,α n (t) (enote by the same name for convenience) acting respectively on functions φ(n) Cc (R 3 ) an Φ(n, p) Cc (R 3 R 3 ) through formulae, ρ,α (t), φ := n ρ,α (t, n)φ( n ), (3.13) ρ,α n (t), Φ := ρ,α n (t, n, p)φ( n n,p, p ). (3.14) Then ρ,α (t) an ρ,α n (t) exten continuously to the spaces T D(R 3 ) an T D (R 6 ) respectively, an, for any t 0, the following estimates hol, ρ,α (t), φ C φ(n) TD (R 3 ), (3.15) t ρ,α (t), φ C α φ(n) T D (R 3 ), (3.16) ρ,α n (t), Φ C α Φ(n, p) T D (R 6 ). (3.17) In particular, up to extracting subsequences, the istributions ρ,α (t) an ρ,α n (t) both possess weak limits as, enote by ρ α (t) an ρα n (t), so that, ρ,α (t) ρ α (t) in C 0 (R + t ; [ T D (R 3 ) ] weak ), (3.18) t ρ,α (t) t ρ α (t) in [ 1 (R + t ; T D(R 3 )) ] weak, (3.19) ρ,α n (t) ρ α n(t) in [ 1 (R + t ; T D(R 6 )) ] weak. (3.20) Furthermore, ρ α (t) possesses in turn a weak limit as α 0, say ρ (t), so that, ρ α (t) α 0 ρ (t) in [ 1 (R + t ; T D(R 3 )) ] weak. (3.21) 15

16 Remarks 1- The estimates (i) shoul be seen as scaling estimates in terms of the parameter. They tell us, in some sense, that for each t 0, the sequences ρ,α (t, n) an ρ,α n (t, n, p) behave in the same way as sequences of the form, 1 3 u(t, n ) an 1 6 v(t, n, p ), (3.22) respectively, where u(t, n) an v(t, n, p) shoul be, say, smooth profiles efine on the whole space. This fact is naturally obvious at time 0. On the more, if one explicitely solves the Von- Neumann equation (2.8) using Neumann-series an expressing ρ,α n (t) as well as ρ,α (t) in terms of the initial atum, this is also formally the case at time t if we o not ask questions of convergence. However the series built up in this way is very ifficult to boun in a non-trivial way, for we are not able to take avantage of the many oscillations present at the iscrete level. We may inee raw the following parallel: when the continuous limit has been taken (See Theorem 3.2), we are le to manipulate series of collision operators (See (3.26) an (3.27)) which involve iterate oscillating integrals. The oscillations then allow to make the series converge in a nice way, thanks to emma 3.1 below. Now obviously this kin of argument is completely forbien while performing the continuous limit, where we still have to eal with iscrete sums. The part (i) of the Theorem makes sure in this perspective that the natural scaling properties in of the solution to the Von-Neumann equation is actually propagate through time-evolution, without writing own the above-mentionne Neumann series. This leas however to worse estimates (ii) than expecte, but fortunately these will be enough to take care of the continuous limit. Note finally that Theorem 3.1 is a-priori false for negative values of time. 2- As mentionne above, the estimate (ii) on the erivative in time t ρ,α n is somehow weaker than what one woul expect, in two respects: firstly, the estimate (ii) hols for averages in time, an we o not know whether it hols true pointwise ; seconly, the scaling parameter 3 on the right-han-sie of (3.11) is much weaker than the 6 one woul expect for profiles of the form (3.22). However, it is surprising that estimate (ii) hol uniformly in t. 3- This Theorem relies in an absolutely essential way on the following features of the Von- Neumann equation uner consieration: the Von-Neumann equation (2.8) has the so-calle inbla property (it preserves the positivity of the ensity matrix - see emma 2.1) ; the initial ata is purely iagonal (See (2.10)) ; the equation on the iagonal part involves the noniagonal part only on the right-han-sie. We emphasize in this respect that the l boun (3.7) is very specific (it is actually a irect consequence of the inbla property), an it is not at all a general feature of the Schröinger-like equations. Arme with the apriori bouns of Theorem 3.1, we are now able to characterize the consecutive asymptotics an α 0, as follows. Theorem 3.2 (Convergence as : the equation becomes Markovian) et ρ,α (t) an ρ,α n (t) be as in Theorem 3.1 an efine their weak limits ρα (t) an ρα n (t) as 16

17 state in Theorem 3.1. Then the following hols, (i) The weak limits satisfy the following system of equations, t ρ α [ (t, n) = iλ V (n k)ρ α n(t, k, n) V (k n)ρ α n(t, n, k) ] R 3 k (2π) 3, (3.23) ρ α n(t, n, p) = iλ + (2π) 3 e [i(n2 p 2 ) α]u V (p n) [ρ α (t, p) ρ α (t, n)] u (3.24) + +iλ e [i(n2 p2 ) α]u [ V (k n)ρ α n(t, k, p) V (p k)ρ α n(t, n, k) ] k R 3 (2π) 3 u, with initial atum, ρ α (t, n) t=0 = ρ 0 (n). (3.25) The equations (3.23) an (3.24) hol between istributions belonging to [ 1 (R + t ; T D) ]. (ii) The system (3.23)-(3.24) implies that the following Boltzmann equation is satisfie, t ρ α (t, n) = l 1 λ l+1 [Q α l ρ α ](t, n), (3.26) with initial atum (3.25), where the collision operators Q α l are efine as, [Q α l ρα ](t, n) := (2π) 3l ( 2R) ( 1) ε 1+ + ε l ε 1,,ε l R 3l u 1 =0 u 2 =0 exp ( i[(n + ε 1 k 1 ) 2 (n ε 1 k 1 ) 2 ]u 1 αu 1 ) exp ( i[(n + ε 1 k ε l k l ) 2 (n ε 1 k 1 ε l k l ) 2 ]u l αu l ) [i V (k 1 )] [i V (k l )] [i V (k k l )]ρ α (t, n + ε 1k ε l k l ) k 1 k l. u l =0 (3.27) Here, variables ε 1,, ε l are as in Theorem 2.1. Also, the leaing term in the series (3.26) is, + ( ) λ 2 [Q α 1 ρ α ](t, n) = 2 exp( αu) cos [n 2 k 2 ]u (3.28) R 3 V (n k) 2 [ρ α (t, k) ρ α k (t, n)] (2π) 3. (iii) The equation (3.26) makes sense weakly, when teste against test functions φ(n) S 2D (R 3 ). More precisely, for any such φ, we have, t ρ α (t), φ = l 0 λ l+1 ρ α (t), t Q α l φ, (3.29) where t Q α l φ enotes the formal ajoint of Qα l, efine as, [ t Q α l φ](t, n) := + + (2π) 3l (+2R) ε 1,,ε l R 3l u 1 =0 u 2 =0 + u l =0 ( 1) ε 1+ +ε l exp ( i[(n + ε 2 k ε l k l ) 2 (n + k 1 + ε 2 k ε l k l ) 2 ]u 1 αu 1 ) exp ( i[(n + ε 3 k ε l k l ) 2 (n + k 1 + k 2 + ε 3 k ε l k l ) 2 ]u 1 αu 1 ) exp ( i[n 2 (n + k k l ) 2 ]u l αu l ) [i V (k 1 )] [i V (k l )] [i V (k k l )] φ(t, n + ε 1 k ε l k l ) k 1 k l. (3.30) 17

18 Also, the following estimate hols, λ l+1 ρ α (t), t Q α l φ λ l+1 C C l 0 V (n) l+1 S 2D φ S2D, (3.31) for some universal constant C 0, together with some constant C of the form (3.9). The estimate (3.31) implies the convergence of the series (3.29), at least for λ small enough. Also, the estimate (3.31) together with the equation (3.29) imply that ρ α (t) is actually uniformly (in α) boune in the space, C 1 (R + t ; [ S 2D ] weak ), so that the weak convergence ρ α (t) ρ (t) also hols in C 0 (R + t ; [ S 2D ] weak ). Theorem 3.3 (Convergence as α 0: obtaining the Quantum Boltzmann equation) et ρ α (t) be as in Theorem 3.2 an efine its weak limit ρ (t) as state in Theorem 3.1. Then the following hols, (i) The following Boltzmann equation is satisfie, t ρ (t, n) = l 1 λ l+1 [Q l ρ ](t), (3.32) with initial atum, where the collision operators Q l are efine as, [Q l ρ ](t, n) := (2π) 3l ( 2R) ε 1,,ε l ρ (t, n) t=0 = ρ 0 (n), (3.33) R 3l + + u 1 =0 u 2 =0 + ( 1) ε 1+ + ε l u l =0 exp ( i[(n + ε 1 k 1 ) 2 (n ε 1 k 1 ) 2 ]u 1 ) exp ( i[(n + ε 1 k ε l k l ) 2 (n ε 1 k 1 ε l k l ) 2 ]u l ) [i V (k 1 )] [i V (k l )] [i V (k k l )]ρ (t, n + ε 1 k ε l k l ) k 1 k l. (3.34) Here, variables ε 1,, ε l are as in Theorem 2.1. The equations (3.32), (3.33) an (3.34) hol between istributions belonging to, say, (R + t ; [ S 2D ] weak ). (ii) Also, the leaing term in the series (3.32) is, λ 2 [Q 1 ρ ](t, n) = 2πλ R 2 δ(k 2 n 2 ) V (n k) 2 k [ρ (t, k) ρ (t, n)] 3 (2π) 3, (3.35) as preicte by the Fermi Golen Rule. (iii) The equation (3.26) makes sense weakly, when teste against test functions φ(n) S 2D (R 3 ). More precisely, for any such φ, we have, t ρ (t), φ = l 0 λ l+1 ρ (t), t Q l φ, (3.36) where t Q l φ enotes the formal ajoint of Q l (See (3.30)), an where the following estimate hols, λ l+1 ρ (t), t Q l φ λ l+1 C C l 0 V (n) l+1 S 2D φ S2D, (3.37) for some universal constant C 0, together with some constant C of the form (3.9). 18

19 Remark Firstly, equations (3.26) an (3.32) are inee of the form (1.2) an (1.3) respectively. Seconly, it can be prove that the full series in the potential which efines the cross-section in (3.32) is nothing else than the celebrate Born series. By this, we mean that the equation (3.32) can be put uner the form, t ρ (t, n) = 2π δ(k 2 n 2 )b(k, n)[ρ (t, k) ρ (t, n)]k, (3.38) R 3 where the coefficient b(k, n) is given by the Born series, b(k, n) = λ 2 V 2 (n k) 2λ 3 I V (n k) V (k k ) V (k n) R 3 k 2 n 2 k +... (3.39) + i0 This point is not obvious from the expression (3.34). The proof will be etaile in a future work [Ca2]. 3.2 An oscillatory integral estimate Before turning to the proofs of the above theorems, we state the key estimate which allows to pass to the limit an α 0 in (2.8). This estimate allows to control the oscillations prouce by the free Hamiltonian so as to recover the oscillatory integrals in Theorem 3.3 an in particular the Dirac mass in energy in formula (3.35) as α 0. Also, since our metho relies on the iteration of Duhamel s formula an leas therefore to oscillatory integrals in large imensions, it allows to control the growth of the oscillatory integrals with the imension. Both aspects are of key importance (See remarks 2 an 3 below). This emma heavily relies on the commutation structure i t ρ = ( x + y ) ρ + of the original Von-Neumann equation. emma 3.1 (Oscillatory Integrals with quaratic phases in large imensions) et ψ(n) satisfy ψ (R 3 ). Then, the collision kernel Q l (ψ)(n) as efine in Theorem 3.3 is well efine an acts continuously on the Sobolev space W 4, (R 3 ). More precisely, for any choice of the test function φ W 4, (R 3 ), an for any choice of ε := (ε 1,, ε l ) as in Theorem 2.1, we have, (i) The following oscillatory integral makes sense, + + ε Q l (ψ), φ := exp ( i [(n + ε 1 k 1 ) 2 (n ε 1 k 1 ) 2 ) ] u 1 u 1 =0 u l =0 R 3(l+1) exp ( i [(n + ε 1 k 1 + ε 2 k 2 ) 2 (n ε 1 k 1 ε 2 k 2 ) 2 ) ] u 2 exp ( i [(n + ε 1 k ε l k l ) 2 (n ε 1 k 1 ε l k l ) 2 ]u l ) V (k 1 ) V (k 2 ) V (k l ) V (k k l ) ψ(n + ε 1 k ε l k l ) φ(n) n k 1 k l u 1 u l. (ii) Moreover, the integral efining ε Q l (ψ), φ converges absolutely in the variables u 1,, u l. 19

20 In particular, lim Q α α 0 + ε l (ψ), φ := = lim α u 1 =0 u l =0 exp ( i [(n + ε 1 k ε l k l ) 2 (n ε 1 k 1 ε l k l ) 2 ) ]u l αu l R 3(l+1) exp ( i [(n + ε 1 k 1 ) 2 (n ε 1 k 1 ) 2 ] u 1 αu 1 ) V (k 1 ) V (k l ) V (k k l ) ψ(n + ε 1 k ε l k l ) φ(n) = ε Q l (ψ), φ. (iii) Finally, the following exponential boun hols, for some universal constant C 0, ε Q l (ψ), φ C l 0 V l+1 W 8,1 (R 3 ) ψ (R 3 ) φ W 4, (R 3 ), (3.40) an the same boun (with a ifferent absolute constant) obviously hols for Q l (ψ), φ as well, since Q l = ε ε Q l, where the sum carries over all possible sequences ε. Remarks 1- The metho of proof given below allows to establish various estimates of the same kin than (3.40). For instance, we may mention without proof the estimate, ε Q l (ψ), φ C l 0 V l+1 W 8,1 (R 3 ) W 8, (R 3 ) ψ M b (R 3 ) φ W 4, (R 3 ), (3.41) where M b stans for the space of boune measures. 2- The iea of the proof is the following: all the above quantities are of the form, R 3(l+1) + u 1,,u l =0 exp[iq u1,,u l (x)]φ(x) x u 1 u l, (3.42) where q u1,,u l (x) is a quaratic form in x, whose coefficients epen on u 1,, u l, an the ifficulty stems from the nee to integrate (3.42) up to u 1 = = u l =. By the stationnary phase formula, we know that the quantity exp[iq u1,,u l (x)]φ(x)x ecays like et(q u1,,u l ) 1/2 at infinity as a function of the u-variables, at least if the function Φ is sufficiently smooth. Hence the control of quantities like (3.42) reuces to controlling that et(q u1,,u l ) grows fast enough at infinity, as well as to controlling the number of erivatives neee by the use of the stationnary phase formula. 3- The boun (3.40) is important in two respects. Firstly, the left-han-sie is uniformly boune using a fixe (inepenent of l) number of erivatives in V, φ, although the singularity create by the oscillatory exponentials apparently grows with l. In other wors, the oscillatory integrals above o not act as iterate erivatives on φ, but rather as istributions of fixe orer. Seconly, the constant involve in (3.40) grows geometrically with l. Both points are crucial since the case of iterate erivatives coul lea to a growth of ε Q l (ψ), φ like l! (hence to iverging series). 4- Up to some change of variables, one can see this emma as a efinition of istributions like, (n 2 k i0) 1 (n 2 k i0) 1 (n 2 k 2 l + i0) 1 (3.43) 20

21 in D (R 3(l+1) ). Inee, emma 3.1 shoul be compare with the well-known formula, vali in D (R), + exp (isx) s = +πδ(x) + iv.p.(1/x) = i(x + i0) 1. However, the efinition of istributions like (3.43) is not in general a consequence of the usual theorems about the prolongation an composition of homogeneous istributions (See [Hö]), nor it is a consequence of the usual theorems about the composition an proucts of istribution with well-behave wave-fronts. This is ue to the fact that the singularity near the origin n 2 = k 2 1 = = k2 l = 0 an near all the axes is too severe. In particular, the efinition of such istributions a priori involves proucts of Dirac masses, a forbien operation. The above emma shows that this is fortunately not the case, an this relies on the fact that the +i0 an i0 occur at the right places ue to the specific commutator structure of the original equation (See (4.20) an the last step of the proof of this emma). This is formally in analogy with the well-known fact that the istribution (x + i0) 2 is well-efine, whereas the prouct (x + i0) 1 (x i0) 1 oes not make sense. 5- In the particular case l = 1, we can ientify the oscillatory integrals as, R + R 6 exp (i[n 2 k 2 ]u)ψ(n, k) n k u = π R 6 δ(n 2 k 2 ) ψ(n, k) n k, (3.44) where the istribution δ(n 2 k 2 ) has the usual meaning (See [Hö]). 5- The proof below goes through when the variables n, k 1, etc. in emma 3.1 belong to the -imensional space R, for any imension 3 (See (4.25)), up to replacing V W 8,1 (R 3 ) by V W 2(+1),1 (R ), an φ W 4, (R 3 ) by φ W +1, (R ). We mention in passing the obvious imbeing S 8 (R 3 ) W 8,1 (R 3 ). 4 Proofs 4.1 Proof of emma 2.1: Positivity of the ensity matrix Equation (2.5) is linear. For fixe time, we can obviously boun the l 2 (Z 6 n,p)-norm of the righthan-sie of (2.5) by, l 2 2 (2π) 3 V (k/) l 1 (Z 3 k ) ρ(t, n, p) l 2 (Z 6 n,p), The quantity (2π) 3 V (k/) l 1 (Z 3 ) is obviously boune for fixe if V satisfies (2.2). Also k the l 2 -norm of ρ(t, n, p) t=0 is boune initially. Inee, ρ(t, n, p) t=0 l 2 n,p (2π) 3 ρ 0 (n/) l 2 n, an this last quantity is boune for fixe since ρ 0 ecays nicely (assumption (2.10)). All this clearly proves the first part of the emma (existence an uniqueness in C 0 (l 2 )). 21