Julien Audiffren

Liva Ralaivola

Ralaivola Liva

Mixing Stationary

2014 Bandits

Mixing Hal-01011112 Stationary

Stationary Mixing Bandits

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Introduction

Bandit with mixing arms. The bandit problem consists in an agent who has to choose at each step between K arms. A stochastic process is associated to each arm, and pulling an arm produces a reward which is the realization of the corresponding stochastic process. The objective of the agent is to maximize its long term reward. It is classically assumed that the stochastic process associated to each arm is a sequence of independently and identically distributed (i.i.d) random variables (see, e.g. [START_REF] Lai | Asymptotically efficient adaptive allocation rules[END_REF]). In that case, the challenge the agent has to face is the well-known exploration/exploitation problem: she has to simultaneously make sure that she collects information from all arms to try to identify the most rewarding ones -this is exploration-and to maximize the rewards along the sequence of pulls she performs -this is exploitation. Many algorithms have been proposed to solve this trade-off between exploration and exploitation [START_REF] Auer | Finite-time analysis of the multi-armed bandit problem[END_REF][START_REF] Auer | Ucb revisited: Improved regret bounds for the stochastic multi-armed bandit problem[END_REF][START_REF] Bubeck | Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems[END_REF][START_REF] Lai | Asymptotically efficient adaptive allocation rules[END_REF]. We propose to go a step further than the i.i.d setting and to work in the situation where the process associated with each arm is a stationary ϕ-mixing process and the rewards are thus dependent from one another, but with a strength of dependence that weakens over time. From an application point of view, this is a reasonable dependence structure: if a user clicks on some ad (a typical use of bandit algorithms) at some point in time, it is very unlikely that she will click again on this ad in the near future. As it shall appear in the sequel, working with such dependent observations poses the question of how informative are some of the rewards with respect to the value of an arm since, because of the dependencies and the high correlation between close-by (in time) rewards, they might not reflect the true 'value' of the arms. However, as the dependencies weaken over time, some kind of independence might be recovered if rewards are ignored. This actually requires to deal with a new trade-off exploration/exploitation/independence that need be precisely handled.

Rested and Restless case. A closely related setup that addresses the bandit problem with dependent rewards is when they are distributed according to Markov processes, such as Markov chains and Markov decision process (MDP) [START_REF] Ortner | Regret bounds for restless markov bandits[END_REF][START_REF] Tekin | Online learning of rested and restless bandits[END_REF], where the dependences between rewards are of bounded range, which is what distinguishes those works with ours. Contributions in this area study two settings, that we will analyze as well: the rested case, where the process attached to an arm evolves only when the arm is pulled, and the restless case, where all processes simultaneously evolve at each time step.

Contributions and structure of the paper. We define the notion of a ϕ-mixing bandit and its regret (Section 2), we provide a general analysis and an algorithm to solve the rested case where the size of independence blocks is fixed (Section 3). We provide another approach where these sizes are computed from the data by introducing another algorithm (Section 4). Finally, in Section 5, we provide an algorithm and a regret analysis to deal with the restless case.

Overview of the problem

Let (ω, F , P) be a probability space. We recall the definitions of stationary and of ϕ-mixing processes:

Definition 1 (Stationarity). A sequence of random variables X = {X t } +∞
t=-∞ is stationary if, for any t and nonnegative integer m and s, the random subsequences (X t , . . . , X t+m ) and (X t+s , . . . , X t+m+s ) are identically distributed.

Definition 2 (ϕ-mixing process). Let X = {X t } +∞ t=-∞ be a stationary sequence of random variables. For any i, j ∈ Z∪{-∞, +∞}, let σ j i denote the σ-algebra generated by the random variables X t , i ≤ t ≤ j. Then, for any positive integer n, the ϕ-mixing coefficient ϕ(n) of the stochastic process X is defined as

ϕ(n) = sup t,A∈σ +∞ t+k ,B∈σ t -∞ |P [A|B] -P [A]| . (1)
X is said to be ϕ-mixing if ϕ(n) → 0 as n → ∞.

We are interested in the problem of sampling from a K-armed ϕ-mixing bandit. In our setting, pulling arm k at time t provides the agent with a realization of the random variable X k τ k (t) , where τ k (t) = t in the restless case and τ k (t) is the number of times arm k was pulled in the rested case, and where the family (X k t ) t≥1 satisfies the following hypotheses : 1. ∀k, (X k t ) t∈Z is stationary; 2. the sequences (X k t ) t∈Z are ϕ-mixing; 3. each X k 1 takes values in a discrete finite set.

This setting assumes the possibility of long-term dependencies between the rewards output by the arms. It is important to note that by definition of the ϕ-mixing processes, the amount of dependence decreases with time. Hence, as evoked earlier, in order to choose which arm to pull, the agent is forced to address the exploration/exploitation/independence trade-off where independence may be partially recovered by ignoring some rewards so as to make computations on data that are distant in time, i.e. data that are not too correlated (thanks to the mixing property).

It is critical to note that unlike in the i.i.d. framework, Hoeffding inequality cannot be applied in this setting, thus the widely used upper confidence bound (UCB) algorithms cannot be used here. In the case of stationary ϕ-mixing distributions, we have the following theorem from [START_REF] Kontorovich | Concentration inequalities for dependent random variables via the martingale method[END_REF].

Theorem 1 ( [START_REF] Kontorovich | Concentration inequalities for dependent random variables via the martingale method[END_REF][START_REF] Mohri | Stability bounds for stationary -mixing and -mixing processes[END_REF]). Let ψ : U m → R be a function defined over a countable space U, and X be a stationary ϕ mixing process. If ψ is l-Lipschitz with respect to the Hamming metric for some l > 0, then the following holds for all t > 0:

P X [|ψ(X) -Eψ(X)| > t] ≤ 2 exp - t 2 2ml 2 Λ m 2 ∞ , (2)
where Λ m ∞ ≤ 1 + 2 m k=1 ϕ(k). In the following, we consider a more general framework than the one usually encountered in the bandit literature. Instead of looking at cumulative gains, we look at rewards computed according to Lipschitz functions meeting the requirements of the concentration inequality stated in Theorem 1.

More precisely, we suppose that we have K known families (ψ k t ) t≥1 k = 1, . . . , K of functions such that for every non-negative integer m:

1. ψ k m : U m → R accounts for the reward associated with m consecutive outcomes of arm k; 2. ψ k m is 1-Lipchitz with respect to the Hamming metric. In the following sections, we use m to identify the reward functions (ψ k m ) 1≤k≤K we want to optimize, b to refer to the size of the independence blocks and we consider blocs of s trials. We study three different scenarios. In the next section, we will present a general algorithm and regret analysis in the rested case where m is fixed and s . = m + b. In Section 4, with an additional hypothesis on the ϕ-mixing processes, we will take another approach to the rested case when m + b | s (i.e. s is a multiple of m + b) by including the notion of dependency into the regret. Finally, in Section 5, we present a general algorithm and regret analysis for the restless case.

Mixing Bandits in the Rested Case

Regret analysis. Here, we are going to analyze the situation where m and b are fixed. Our goal is to show that a simple algorithm derived from UCB that works by making blocks s . = m + b consecutive trials on each arm has low regret, with a notion of regret that we define in the sequel. The running time index is therefore of the form of st, with t the number of times arm selection has been performed, and the sequence accessed to are such as (X k st , . . . , X k (s+1)t-1 ). The reward that is accessed to at time t (with a slight abuse of notation that makes us use t as the time index) when pulling arm k does not make use of the full information provided by this sequence but instead is

ψ k m (X k t,s,b ) where X k t,s,b . = (X k st , . . . , X k (s+1)t-b-1 ); (3)
this means that only the first m points from (X st , . . . , X (s+1)t-1 ) are taken advantage of. Given τ , the total number of trials of s-blocks, the regret that we are going to work with is

R . = τ µ * ψ,m - K k=1 Eτ k (τ )µ k ψ,m (4)
where:

µ * ψ,m . = max k=1,...,K µ k ψ,m , µ k ψ,m . = E X k 1 ,...,X k m ψ k m (X k 1 , . . . , X k m ) and τ k (t)
is the number of times arm k has been chosen given that a total (i.e. over all arms) of t pulls of s-blocks have been performed.

The arm selection procedure of the algorithm that we propose and dub Block-UCB, is depicted in Algorithm 1, where the function Λ k is defined as

Λ k (t) . = 1 + 2 t r=1 ϕ k (rb + (r -1)m). (5)
It is possible to show that Block-UCB has the following regret.

Theorem 2 (Regret of Block-UCB). The regret of Block-UCB is bounded by

k:µ k ψ,m <µ * ψ,m u k ∆ k + 1 α -2 ,
where the u k 's are the solutions of the problems

u k ∆ 2 k -8αΛ 2 k (u k ) log τ = 0, k = 1, . . . , K. (6)
The result of Theorem 2 hinges on the derivation of a concentration inequality for each arm k that relates the random variable

1 τ τ -1 r=0 ψ k m X k r,s,b to µ k ψ,m .
To establish this concentration inequality, we study the random variables

Γ k b (X k 0,s,b , . . . , X k τ -1,s,b ), defined for b ≥ 0 as Γ k b (X k 0,s,b , . . . , X k τ -1,s,b ) . = 1 τ τ -1 r=0 ψ k m (X k rs , . . . , X k (r+1)s-b-1 ) -µ k ψ,m . (7)
The concentration inequality that we are going to use to prove our regret bound is the following:

Algorithm 1 Main iteration of Block-UCB Choose arm I t as

I t ∈ argmax k 1 t t-1 r=0 ψ k m X k r,s,b + Λ k (τ k (t -1)) 2α log t τ k (t -1) .
Theorem 3. For all τ, k, b, and assuming that ψ k m takes value in [0; 1]:

P Γ k b (X k 0,s,b , . . . , X k τ -1,s,b ) ≥ ε ≤ exp - τ ε 2 2Λ 2 k (τ ) , (8)
where Λ k is defined in [START_REF] Ortner | Regret bounds for restless markov bandits[END_REF].

Proof. We make the proof for some arm k. We also assume that s and b are fixed and to lighten the notation, we drop the dependence on these variables when no confusion is possible: we use X r (resp. Γ) for X k r,s,b (resp. Γ b ). The proof hinges on the fact that since (X t ) t≥0 is a stationary mixing sequence with mixing coefficients (ϕ(t)) t≥1 , (X r, ) r≥0 is a stationary mixing sequence with mixing coefficients (ϕ(q)) q≥1 such that ϕ(q) . = ϕ(qb + (q -1)m) (see Proposition 5, Appendix). To obtain the targeted result, we make use of the concentration inequality of Theorem 1 with the function Γ to get (8) and we exploit the observation that EΓ k b = 0, which results from the stationarity of (X t ) t≥0 .

Let q be an integer in {0, . . . , r -1}, x 0 , . . . , x r-1 and x ′ q blocks from U s-b . Then:

Γ(x 0 , . . . , x q , . . . , x r-1 ) -Γ(x 0 , . . . , x ′ q , . . . , x r-1 ) = 1 τ (ψ s-b (x q ) -ψ s-b (x ′ q )) ≤ 1 τ ,
which comes from the range of ψ s-b being [0; 1]. Γ is therefore 1/τ -Lipschitz with respect to the Hamming metric, which, combined with (X r ) r≥0 being a ϕ-mixing sequence, gives [START_REF] Mohri | Stability bounds for stationary -mixing and -mixing processes[END_REF].

The proof of the previous theorem uses the following more general result, that is of independent interest.

Theorem 4 (General Regret). Suppose that the arms we work with are such that

∀k, P( μk τ -µ k ≥ ε) exp (-θ k (τ )γ k (ε)) , ( 9
)
where τ is the number of data the empirical mean μk τ is computed on, and θ k and ε k are increasing functions defined on (0; +∞].

Consider the regret defined by

R . = τ µ * - K k=1 Eτ k (τ )µ k
where τ k (t) is the number of times a (suboptimal) arm k has been chosen up to time t. The (α, θ, γ)-UCB that chooses at iteration t an arm I t according to

I t ∈ argmax μk τ k (t-1) + γ -1 k α θ k (τ k (t -1)) log t
has regret bounded by:

k:µi<µ * θ -1 k α log τ γ k (∆ k /2) ∆ k + 1 α -2 . ( 10
)
Proof. Note that Theorem 2 is a consequence of this theorem with

θ k (s) = s/Λ 2 k (s) and γ k (ε) = ε 2 /2.
The proof use the standard technique to prove the regret of UCB-like algorithms. Namely, at iteration t, if I t = i for i not optimal, then one of the following events E * 1 (t), E 2 (i, t), E 3 (i, t) must occur

E * 1 (t) . = μ * < µ * -γ -1 * α θ * (τ * (t -1)) log t , (11)
E 2 (i, t) . = µ i ≤ μi -γ -1 i α θ i (τ i (t -1)) log t , (12)
E 3 (i, t) . = τ i (t -1) ≤ θ -1 i α log τ γ i (∆ i /2) . (13)
Indeed, if none of the events occurs then (using

∆ i = µ * -µ i ) μ * +γ -1 * α θ * (τ * (t -1)) log t (11)
≥ µi+∆i

(12) > µi+2γ -1 i α θi(τi(t -1)) log τ (13)
> μi+γ

-1 i α θi(τi(t -1)) log t
where we have used that t → γ -1 i (a log t) is an increasing function of t on [1; ∞) whenever a > 0. This implies that I t = i, which contradicts our working hypothesis.

If we let u be defined as:

u . = θ -1 i α log τ γ i (∆ i /2) ,
then, for i suboptimal, we have the following

Eτ i (τ ) = τ t=1 E½ [It=i] ≤ u + τ t=u+1 E½ [It=i ∧ ¬E3(i,t)] ≤ u + τ t=u+1 E½ [E * 1 (t) ∨ E2(i,t)] ≤ u + τ t=u+1 [P(∃t : E * 1 (t)) + P(∃t : E 2 (i, t))]
Using the union bound and Equation (9), both probabilities P(∃t : E * 1 (t)), P(∃t : E 2 (i, t)) can be bounded from above by 1/t α . Standard calculations allow us to get desired result (10).

Discussion. Some observations must be made regarding the result of Theorem 2. First, as we used Theorem 4 to prove our regret bound, it is necessary for the result to hold for the functions θ k . = τ /Λ 2 k (τ ) to be increasing. In addition, the regret only makes sense if it is bounded, i.e. if the u k ∆ k are bounded. Finally, if these conditions hold, it might be interesting to find, for some fixed horizon τ , the value of b that minimizes the regret. We now depicts common settings for the ϕ k that yield instructive results and that build upon the previous remarks. For the sake of conciseness, we will assume that all ϕ 1 = . . . = ϕ K and we use ϕ to refer to the mixing coefficients. Independent case. if ϕ = 0, i.e. we are in the independent case, and the θ k 's are naturally increasing. In addition, it is straightforward to observe that the best use of the data is achieved for b = 0, i.e. each and every reward is used to estimate the quality of an arm.

Case Λ < +∞. In that case, we are again back to a situation almost similar to the usual independent case. The θ k 's are increasing, the u k are well-defined and the regret as the usual

O( k log τ /∆ k ) form.
Algebraically mixing case. Here, ϕ(t) = ϕ 0 t -p for p > 1, and a few calculations give

Λ(τ ) = 1 + 2ϕ0 τ r=1 1 (rs -m) p ≤ 1 + 2ϕ0 1 + τ 1 1 (rs -m) p dr = 1 + 2ϕ0 + 2ϕ0 s(p -1) 1 b p-1 - 1 (τ s -m) p-1
Using this upper bound to find the u k s as in [START_REF] Tekin | Online learning of rested and restless bandits[END_REF] and to solve for b so that this bound is minimized provides a way to find a data-dependent b. Another (coarser) way to look at the algebraically mixing situation is not to optimize for b and to consider that it is a particular case of the previous case, since t ϕ(t) < ∞, i.e. Λ < +∞. This assumption is made in the rest of the paper.

m + b | s: Expressing the Independence Trade-Off in the Regret

This section introduces another way of encoding the trade-off between exploration, exploitation and independence. As before, we consider sequences of s trials, but among the s results, we seek to optimize the number m of results we use to update the empirical value of the arms and the number b of results we ignore in order to improve the independence between the considered realization of our random variables.

In addition to the case s = m + b, we also consider the situation where β(m + b) = s with β ∈ N and β > 1. The sequence of s trials can then be interpreted as β successive sequences of m + b trials, and the value of this particular (m, b) distribution is thus multiplied by β.

Hypotheses and Regrets

In this section we make the following additional assumption on the ϕ-mixing processes (X k t ) t≥0 :

∀1 ≤ k ≤ K, ∀b ∈ N, M k (b) . = 1 + i≥1 ϕ k (b(i + 1)) < +∞.
Note that this is equivalent to the widely used assumption that the ϕ k (i) are summable over i (see for instance the case of algebraically mixing sequence mentioned before). Also, note that M k (1) is an upper bound of (Λ k m ) m which appears in Theorem 1, and

M k (•) is a decreasing function such that M k (b) ≥ 1.
The setting is the following: at each step, the agent pulls an arm s times, and has to choose how to split those s elements between a meaningful part of m elements, used to update empirical values of the arm, and the non-significant part of b element, used to strengthen the independence between the variables. For each such combination m + b | s and for each arm k, we define the value of the combination (m, b, k) as:

ν k m,b . = β m,b M k (b) µ k m , (14)
where With this in mind, we define the regret R at time t as :

µ k m . = E X k 1 ,...,X k m ψ k m (X k 1 , . . . , X k m )
R . = K k=1 E(T (k)) ν k * m * ,b * -ν k m * k ,b * k (15) where (k * , m * , b * ) = arg max (k,m,b) ν k m k ,b k , (m * k , b * k ) = arg max (m,b) ν k m k ,b k ,
and T (k) denotes the number of times arm k was pulled.

It is important to note that one of the main difference between (15) and the classical formulation of regret from a multi arm bandit in the i.i.d. case is that in our setting, we are comparing the value of the best combination of the best arm with the value of the best combination of the pulled arm.

Concentration inequality and algorithm

We now introduce a concentration inequality tailored for the ν k m,b of (14).

Proposition 1. Let 1 ≤ k ≤ K, 1 ≤ m ≤ s, b = m -s, n ∈ N * , ψ k m : U m →
R be a 1-Lipschitz with respect to the Hamming metric function defined over a countable space U. Suppose that the ϕ k (i) are summable over i, and let us define M k (b) = 1 + i∈N * ϕ k (i(b + 1)), and ζ s m : N → N, ζ s m (t) = t + b⌊(t -1)/m⌋. Then the following holds for all t > 0:

P 1 M k (b) 1 n n-1 i=0 ψ k m X k ζ s m (im+1) , • • • , X k ζ s m ((i+1)m) -µ k m > t ≤ 2 exp - nt 2 2 . ( 16
)
Algorithm 2 Block-UCB with parameters s, α fixed

t ← 0, ψ k m,b,0 ← 0, k = 1, . . . , K, m + b | s, τ k,0 ← 0, k = 1, . . . , K for t = 1, . . . , τ do Select arm k with k ∈ argmax k max m+b|s s m + b ψ k m,b,t-1 + 2α(m + b) log(t) sτ k,t-1 Update the block counts τ k,t ← τ k,t-1 + δ k k , ∀k Compute the values of the ψ k m,b,t , ∀m + b | s ψ k m,b,t ← 1 M k (b)βτ k,t β(τ k,t -1) r=0 ψ k m X k r(m+b) , . . . , X k (m+b)r+m-1 ψ k m,t ← ψ k m,t-1 , for k = k end for
Proof. This proposition naturally follows from the proof of Theorem 1 with the function

φ k n = 1 n n-1 i=0 ψ k m X ζ s m (im+1) , • • • , X ζ s m ((i+1)m)
which is 1/n-Lipschitz with respect to the Hamming metric, and Proposition 5 ( see Appendix).

Note that the upper bound on the probability that appears in ( 16) is uniform over (k, m, b). This is crucial to define our algorithm and to analyze its regret (more details in the next subsection). We now introduce Algorithm 2 that is designed for the particular setting of ϕ-mixing bandit problem just described. First, note that since the pair (m, b) (with m + b | s) which gives the best result for each arm is unknown, the algorithm needs to compute an empirical estimator for each of these combinations for each arm. In other words, the algorithm needs to efficiently and simultaneously learn both the best combination and the best arm.

Regret Analysis

In this subsection we provide an upper bound for the regret of Algorithm 2.

Proposition 2. Let R be the regret as defined in (15) and let η :

N → N, η(s) . = s i=1 i1 i|s Then, with ∆ i = ν k * m * ,b * -ν i m * i ,b * i R ≤ 1≤i≤K (1 + η(s))∆ i + 8αs log(t) ∆ i .
Proof. The main difference with the standard technique to proving regret bounds comes from the fact that the value of each arm is the maximum of its coordinate: as suchs we have to consider the following event

ψ k * t ≤ ν k * m * ,b * ∃m + b | s, ν k m * k ,b * k ≤ s m + b ψ k m,b,t-1 - 2α(m + b) log(t) sτ k,t-1 ∃m + b | s, 2 2α(m + b) log(t) sτ k,t-1 ≥ ν k * m * ,b * -ν k m * k ,b * k
By carefully using the property of the maximum, the result can be recovered. All the details of the proof can be found in the supplementary material.

We have just seen another approach to the rested mixing bandit. By assuming the summability of the ϕ k , and by properly defining the value of an arm, we were able using Algorithm 2 to address the case where m and b are no longer fixed, but must be computed from the data by the agent.

It is interesting to note that the upper bound in Proposition 2 differs from the usual bound in the classical dependence-free setting by two multiplicative constants: η(s), which encodes the total number of combination of pair (m, b) such that m + b | s, and s, which is in fact used as an upper bound for s/(m + b).

Restless ϕ mixing bandits

In this section, we provide an analysis for the restless ϕ-mixing bandit. Recall that contrarily to the rested case studied previously, the stochastic processes associated to each arm evolves regardless of the actions of the agent. This difference is of paramount importance in the ϕ-mixing setting. Indeed, in the rested case, the agent was bound to ignore some realizations obtained from an arm to enforce the independence and therefore the accuracy of its predictor. In the restless case, instead of pulling an arm to no avail, an agent willing to increase the independence of the realization of an arm k can pull another arm k ′ = k to gather information about k ′ while enforcing the independence of k. This idea is central to this section.

Like in Section 4, we assume that ∀k = 1, • • • , K, the stochastic process X k is a ϕ-mixing process, and its mixing coefficients ϕ k (i) are summable, and we define the following upper bound function M k , which differs for the one defined in the previous section:

∀1 ≤ k ≤ K, ∀b ∈ N, M k (b) . = 1 + i≥b ϕ k (i) < +∞.
In the restless ϕ-mixing bandit, the agent pulls the arm k in sequences of m k trials, where the m k are fixed parameters and may differ for each arm. The mean value of this sequence is defined as follows:

µ k . = E X k 1 ,...,X k m k ψ k m (X k 1 , . . . , X k m k )
and we use the same definition of regret as defined in Section 3. In the restless setting, an interesting way of dealing with the trade-off exploration/exploitation/independence appears: in addition to the usual exploration, it might be interesting for the agent to pull an apparently sub-optimal arm to get an increased independence on the result of the other arms -since the time between two consecutive sequences of pull decrease their dependency. In order to study this trade-off, we introduce a suitable concentration inequality.

Proposition 3. Let 1 ≤ k ≤ K, 1 ≤ m ≤ s, b = m -s, n ∈ N * , ψ k m : U m →
R be a 1-Lipschitz with respect to the Hamming metric function defined over a countable space U. Let ζ s m : N → N, be such that ζ s m (t) = t + b1 t≥m+1 . Then the following holds for all t > 0:

P 1 n n-1 i=0 ψ k m X k ζ s m (im+1) , • • • , X k ζ s m ((i+1)m) -µ k > t ≤ 2 exp - nt 2 2M 2 k (b) . ( 17
)
Proof. This proposition follows from the proof of Theorem 1 using the same technique as Proposition 1.

It is interesting to note that the independence trade-off naturally appears in the right term of inequality 17 within the M k , and will modify the upper confidence bound. We introduce algorithm 3 to solve this particular setting of ϕ mixing bandit problem, and we provide a regret analysis for this algorithm.

Proposition 4 (Regret analysis). Let R(t) be the regret at time t for Algorithm 3. Then, with

∆ i = µ k * -µ i , R t ≤ 1≤i≤K ∆ i + 8α log(t) ∆ i . Algorithm 3 Restless Block-UCB with parameters m k , α fixed t ← 0, ψ k 0 ← 0, τ k,0 ← 0, , η k,0 ← 0, k = 1, . . . , K for t = 1, . . . , τ do Select arm k ∈ argmax k ψ k t-1 + 2α log(t) τ k,t-1 M 2 k (η k,t
) Update counters and timers :

η k,t ← 0, τ k,t ← τ k,t-1 + 1, ∀k = k, τ k,t ← τ k,t-1 , and η k,t ← η k,t-1 + m k Compute the values of the ψ k t , m + b | s ψ k t ← 1 τ k,t τ k,t -1 r=0 ψ k m X k rm k , . . . , X k (r+1)m k -1 ψ k t ← ψ k t-1 , for k = k end for
Proof. The proof uses the same ideas as the previous regret analysis from Section 3 and 4, and naturally follows from Proposition 3. The proof can be found in the supplementary materials.

In this section, we have provided an algorithm for the restless ϕ-mixing framework. We have seen that the restless setting is a natural framework to use with ϕ-mixing bandit, as it naturally makes it possible to decrease the dependence of the variables for the arms not chosen. Our algorithm takes advantage of this observation and we were able to show that is has low regret. In order to do so, in addition to the usual τ k , the number of time a given arm has been pulled, the Algorithm 3 computes the η k , the time spent since the last time the arm k was sampled. Indeed, as seen in (17), as η k increases, M 2 k (η k ) decreases and the optimistic value of the arm increases.

Conclusion

We have studied an extension of the multi-armed bandit problem to the stationary ϕ-mixing framework, both in the rested and in the restless case. We have provided both a theoretical analysis in a general framework, and a more practical study of the problem in the case of fast mixing sequences (with ϕ(i) < +∞). For each of theses cases, we provided algorithms and accompanying regret analyses, which are strict extensions of the methods that exist for the i.i.d situation, as usual results might be recovered from our bounds when the mixing coefficients are all 0. Future works might include a study of the restless case where the m k has to be computed from the data, as well as a study in the more difficult case of β-mixing processes.

A Appendix

Proposition 5. Let m, b ∈ N and s = m + b, X t be a ϕ-mixing process on Ω taking value in R, with mixing coefficient ϕ X (•) , and and ψ : R m → R be a measurable function. Then the stochastic process Z t defined by Z t = ψ(X st+1 , . . . , X st+m ) is also a ϕ-mixing process with mixing coefficient ϕ Z = ϕ X • κ, where κ(t) = bt + m(t -1).

Proof. In the following we use σ(A) to denote the σ-algebra generated by A. First note that since ψ is measurable σ(ψ -1 (R)) ⊂ σ(R m ), and as a consequence σ(Z t ) ⊂ σ(X st+1 , . . . , X st+m ) (since σ-algebra are closed under countable intersection) . Now for any i, j ∈ Z ∪ {-∞, +∞}, let σ j i (Z) denote the σ-algebra generated by the random variables Z k , i ≤ k ≤ j. Then, for any positive integer k, the ϕ-mixing coefficient ϕ Z (t) of the stochastic process Z is defined as Since the last line is true ∀m + b | s,, we deduce that ψ k * t ≥ ψ k t hence T (t) = k, which is absurd. Then, we need to bound the probability of the events defined by (18), ( 19) and (20). Because ψ k * t is defined as a maximum, we have Proof. If at time t, the arm chosen is k instead of k * , then one of the following must be true :

ψ k * t ≤ µ k * (21) µ k ≤ ψ k t-1 - √ αJ(k, t -1) (22) 2 2α log(t) M 2 k (η k,t )τ k,t-1 ≥ µ k * -µ k (23)
From the last inequality, we deduce that :

τ k,t-1 ≤ 8α log(t) M 2 k (η k,t )∆ 2 k ≤ 8α log(t) ∆ 2 k ( 24
)
since M 2 k (η k,t ) ≤ 1.

  and β m,b = s/(m + b). This value explicitly shows the trade-off between independence (through M k (b)) and exploitation (through the µ k m,b ). The value of an arm k is then defined as the maximum value of the possible combination (m, b, k), for m + b|s.

1 ≥

 1 ϕ Z (t) = sup n,A∈σ +∞ n+t (Z),B∈σ n -∞ (Z) |P [A|B] -P [A]| ≤ sup n,A∈σ +∞ s(n+t) (X),B∈σ ns+m -∞ (X) |P [A|B] -P [A]| = ϕ X (s(n + t) -ns -m) = ϕ X (tb + (t -1)m)B Proof of Proposition 2Proof : If at time t, the arm chosen is k instead of k * , then one of the following must be true :ψ k * t ≤ ν k * m * ,b * (18) ∃m + b | s, ν k m * k ,b * k ≤ β m,b ψ k m,b,t-1 -√ αJ m,b (k, t -1) (19) ∃m + b | s, 2 2αβ m,b log(t) τ k,t-1 ≥ ν k * m * ,b * -ν k m * k ,b * k (20) Otherwise, ∀m + b | s, ψ k * t ≥ ν k * m * ,b * ≥ ν β m,b ψ k m,b,t-1 + √ αJ m,b (k, t -1)

P,P) 2

 2 (ψ k * t ≤ ν k * m * ,b * ) ≤ P β m * ,b * ( ψ k m,b,t-1 -√ αJ m * ,b * (k * , t -1)) ≤ ν k * m * ,b * ≤ 1 t α , using Proposition 1. Now, by definition of ν k m * k ,b * k ∃m + b | s, ν k m * k ,b * k ≤ β m,b ( ψ k m,b,t-1 -√ αJ m,b (k, t -1)) ≤ P ∃m + b | s, ν k m,b ≤ β m,b ( ψ k m,b,t-1 -√ αJ m,b (k, t -1)) ≤ σ(s) max m+b|s P ν k m,b ≤ β m,b ( ψ k m,b,t-1 -√ αJ m,b (k, t -1)) ≤ σ(s) t αwhere we used Proposition 1 again at the last line. Finally,∃m + b | s, 2 2αβ m,b log(t) τ k,t-1 ≥ ν k * m * ,b * -ν k m * k ,b * k ⊂ 2 2αs log(t) τ k,t-1 ≥ ν k * m * ,b * -ν k m * k ,b * k = 8αs log(t) (ν k * m * ,b * -ν k m * k ,b * k ) 2 ≥ τ k,t-1i.e. the event defined by (20) happens at most u = 8αs log(t) (ν k * m * ,b * -ν k m * k ,b * k times. hence the conclusion C Proof of Proposition 5


Alain Bretto

Yannick Silvestre

Thierry Vallée

Factorization of products of hypergraphs: Structure and algorithms

come


Paolo Antonelli
email: paolo.antonelli@sns.it

AND Émi Carles
email: remi.carles@math.cnrs.fr

Jorge Drumond Silva
email: jsilva@math.ist.utl.pt

SCATTERING FOR NONLINEAR SCHR ÖDINGER EQUATION UNDER PARTIAL HARMONIC CONFINEMENT

Keywords: 2000 Mathematics Subject Classification. Primary: 35Q55. Secondary: 35B40, 35P25, 81U30

We consider the nonlinear Schrödinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

i∂ t u + 1 2 ∆u = 0, (t, x) ∈ R × R d ,
with u |t=0 ∈ S(R d ), has large time dispersive properties which allow, for instance, to develop scattering theories both in the presence of a linear perturbation and in a nonlinear setting (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Érard | Scattering theory of quantum and classical N-particle systems[END_REF][START_REF] Ginibre | An introduction to nonlinear Schrödinger equations[END_REF][START_REF] Yafaev | Scattering theory: some old and new problems[END_REF]). On the other hand, in the presence of an (isotropic) harmonic potential, (1.2)

i∂ t u + 1 2 ∆u = |x| 2 2 u, (t, x) ∈ R × R d ,
the solution is periodic in time. This is a consequence of the existence of a Hilbert basis for the harmonic oscillator H 0 = -1 2 ∆ + |x| 2 2 (see e.g. [START_REF] Lifshitz | Quantum mechanics: non-relativistic theory[END_REF]): the pure point spectrum is

σ p (H 0 ) = d 2 + k =: λ k ; k ∈ N ,
and the associated eigenfunctions are given by (tensor products of) Hermite functions (denoted by ψ jk , associated to λ k ), which form a basis of L 2 (R d ). So if

u(0, x) = j,k∈N α jk ψ jk (x),
then for all t ∈ R, u(t, x) = j,k∈N α jk ψ jk (x)e -i( d 2 +k)t .

Perturbations of (1.2) rather lead to KAM type results, in the linear or in the nonlinear setting (see e.g. [START_REF] Thomann | KAM for the quantum harmonic oscillator[END_REF][START_REF] Kuksin | Nearly integrable infinite-dimensional Hamiltonian systems[END_REF]). In this paper, we study the large time behavior of solutions to a nonlinear perturbation of a mixture of the above two linear dynamics:

(1.3) i∂ t u = Hu + λ|u| 2σ u, (x, y) ∈ R n × R d-n , H = -

1 2 ∆ x + |x| 2 2 - 1 2 ∆ y ,
with d 2, 1 n d -1, λ ∈ C and σ > 0. We show that if the power of the nonlinearity is sufficiently large, then a scattering theory is available: if f (z) = |z| 2σ z is short range in space dimension d -n, then the nonlinearity is negligible for large time. In the rest of the introduction, we make this statement more precise.

We emphasize also the fact that the harmonic potential in (1.3) corresponds to the standard modelling for magnetic traps in the contexts of Bose-Einstein condensation (see e.g. [START_REF] Pomeau | Nonlinear aspects of the theory of Bose-Einstein condensates[END_REF][START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF]). So for instance, our result shows that turning off the confinement in some of the directions may suffice for the condensate to evolve asymptotically freely, and that turning it off in all directions is not necessary: it is possible to keep some properties of the linear confinement, associated to the linear Hamiltonian H.

1.2.

A quick review on scattering for NLS. Consider (1.4)

i∂ t u + 1 2 ∆u = λ|u| 2σ u, x ∈ R d ,
and introduce the space

Σ = f ∈ L 2 (R d ) ; f Σ := xf L 2 (R d ) + ∇f L 2 (R d ) < ∞ .
The following statement summarizes several results which can be found in e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | An introduction to nonlinear Schrödinger equations[END_REF]. 1. Existence of wave operators in H 1 . Suppose that σ 2/d. Let u -∈ H 1 (R d ). There exist T > 0, depending on |λ|, d, σ and u -H 1 , and a unique solution u ∈ C((-∞, -T ]; H 1 ) to (1.4) such that u(t) -e i t 2 ∆ u - 5. Asymptotic completeness in Σ for small data. Let λ ∈ R. Suppose that

σ > 1 if d = 1, σ > 2 d + 2 if d 2,
and λ 0 if σ 2/d. If u 0 Σ is sufficiently small, there exist a unique u ∈ C(R; Σ) solution to (1.4) with u |t=0 = u 0 , and a unique u + ∈ Σ such that e -i

u(t) -e i t 2 ∆ u + L 2 = e -i t 2 ∆ u(t) -u + L 2 -→ t→+∞ 0,
then necessarily, u = u + = 0.

Remark 1.2. Uniqueness is actually granted in smaller spaces than in the above statement, involving a mixed time-space norm which we have omitted to simplify the presentation.

Remark 1.3. As pointed out in the above statement, the free group e i t 2 ∆ is unitary on H 1 (R d ), while it is not on Σ.

The above (positive) results all rely on (global in time) Strichartz estimates. It is essentially the only tool to prove the first point, while the second point uses the operator x + it∇, which provides an explicit decay rate in time, to improve the assumption on σ. The third point relies on Morawetz estimates (see also [START_REF] Ginibre | Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations[END_REF][START_REF] Vega | Bilinear virial identities and applications[END_REF]). The fourth point relaxes the assumption of the third point on σ, since

2 d > 2 -d + √ d 2 + 12d + 4 4d > 1 d .
It relies on the pseudo-conformal evolution law, which provides a rather simple relation concerning the time evolution of the L 2 norm of (x + it∇)u. The fifth point is essentially the same as the second one, plus a bootstrap argument. The final point means that if σ 1/d, linear and nonlinear dynamics can no longer be (easily) compared, due to long range effects.

To be complete, we recall the above mentioned pseudo-conformal evolution law: if u solves (1.4), then d dt

1 2 (x + it∇ x )u 2 L 2 + λt 2 σ + 1 u 2σ+2 L 2σ+2 = λt σ + 1 (2 -dσ) u 2σ+2 L 2σ+2 .
This law was discovered in [START_REF] Ginibre | On a class of nonlinear Schrödinger equations. II Scattering theory, general case[END_REF]. A generalization to the case of an external potential can be found in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]. Note however that in the presence of an external potential, it seems that the corresponding evolution law can be exploited only in the case where the potential is exactly quadratic and isotropic ([5]). In particular, in the case of a partial confinement as in (1.3), the law presented in [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF] seems helpless as far as the description of large time behavior is concerned.

1.3. Heuristics. Leaving out all the technical aspects, scattering for (1.4) stems from the following remark. As t → +∞, we have

e i t 2 ∆ f (x) = 1 (it) d/2 f x t e i |x| 2 2t + o(1) in L 2 (R d ),
where the Fourier transform is normalized as

F f (ξ) = f (ξ) = 1 (2π) d/2 R d f (x)e -ix•ξ dx.
Assuming that the solution to (1.4) behaves as the free evolution of a scattering state u + as t → +∞, the nonlinear potential in (1.4) satisfies

|u(t, x)| 2σ ≈ 1 t dσ u + x t 2σ .
The function on the right hand side belongs to L 1 t L ∞ x provided that (u + is sufficiently localized and) σ > 1/d. From the linear scattering point of view [START_REF] Érard | Scattering theory of quantum and classical N-particle systems[END_REF], this is consistent with the fact that u behaves asymptotically like a solution to the free equation (1.1). In the case σ 1/d, the (nonlinear) potential remains relevant at leading order for all time, no matter how large, as proved in [START_REF] Barab | Nonexistence of asymptotically free solutions for nonlinear Schrödinger equation[END_REF]; long range effects were described for the first time in [START_REF] Ozawa | Long range scattering for nonlinear Schrödinger equations in one space dimension[END_REF] in the case of (1.4). Note however that except in the cases d = 1 and d = 2, Theorem 1.1 introduces restrictions on the lower bound for σ which are not intuitive, and are likely to be "only" technical.

In the case of (1.3), the x-part is not expected to yield large time dispersion, while the ypart should provide large time dispersion. From this perspective, it seems natural to expect a scattering theory for (1.3) with the same numerology as in Theorem 1.1, with d replaced by d -n as far as lower bounds on σ are concerned.

Such a problem is to be compared with the approach in [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF]. The authors study (1.3) where, instead of considering the harmonic oscillator on R n , they examine the Laplacian on a compact manifold without boundary. Also, in [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF], Z. Hani and B. Pausader consider a similar problem in the particular case where λ = 1, σ = 2 (quintic, defocusing nonlinearity), x ∈ T 2 (n = 2) and d = 3. This problem is in addition both L 2 and energycritical, an aspect which introduces new difficulties compared to (1.3) as it is treated in the present paper. In both [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF] and [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF], the nonlinearity in supposed to be short range in the y variable so that scattering in H 1 can be expected, σ 2/(d -n), and small data scattering is established: for small initial data (in spaces that we do not describe here), the solution to the nonlinear equation is global in time, and behaves for large time like a solution to the linear equation.

More recently, in [START_REF] Visciglia | Private communication on a joint work in progress[END_REF], the authors have considered the issue of asymptotic completeness for the nonlinear Schrödinger equation posed on R N × T, a case which corresponds to n = 1 for (1.3). They prove that a scattering theory is available with the same numerology as on R N , provided that the nonlinearity in energy-subcritical in total space dimension N + 1. Such a result is qualitatively very similar to our Theorem 1.10.

Note that like with H in (1.3), the dynamics in the x variable prevents the free dynamics from being as trivial as in the case of (1.1). We will return to a comparison with these papers on a more technical level in Sections 2 and 3.

1.4. Main results. Note that the definition of Σ has to be adapted to the present notations:

Σ = f ∈ L 2 (R d ) ; f Σ := xf L 2 (R d ) + yf L 2 (R d ) + ∇ x,y f L 2 (R d ) < ∞ .
In view of [START_REF]Nonlinear Schrödinger equation with time dependent potential[END_REF]Theorem 1.8], we have:

Proposition 1.4. Let d 2, 1 n d -1, λ ∈ R, and σ > 0 with σ < 2/(d -2) if d 3. If σ 2/d, assume in addition that λ 0. For all u 0 ∈ Σ, (1.3) has a unique solution u ∈ C(R; Σ) ∩ L (4σ+4)/(dσ) loc R; W 1,2σ+2 (R d )
such that u |t=0 = u 0 . The following conservations hold:

d dt u(t) 2 L 2 (R d ) = d dt u(t), Hu(t) + λ σ + 1 u(t) 2σ+2 L 2σ+2 (R d ) = 0.
Our first result regarding scattering theory is the existence of wave operators, along with asymptotic completeness for small data:

Theorem 1.5. Let d 2, 1 n d -1, λ ∈ C, and σ > 0 with σ < 2/(d -2) if d 3.
Suppose in addition that

(1.5) σ > 2d d + 2 1 d -n .
• Existence of wave operators. Let u -∈ Σ. There exist T > 0, depending on |λ|, d, n, σ and u -Σ , and a unique solution u ∈ C((-∞, -T ]; Σ) to (1.3) such that

e itH u(t) -u -Σ -→ t→-∞ 0.
• Asymptotic completeness for small data. Let u 0 ∈ Σ and u be the solution provided by Proposition 1.4. If u 0 Σ is sufficiently small, there exists a unique u + ∈ Σ such that

e itH u(t) -u + Σ -→ t→+∞ 0.
Remark 1.6 (Uniqueness). Uniqueness is stated rather loosely in the first point, for an extra property will be needed, which is too involved to be made precise here. See Section 4 for full details.

Remark 1.7 (Lower bound on σ). The assumption (1.5) is the expected one, except possibly for the factor 2d/(d + 2). If d = 2 (hence n = 1), (1.5) corresponds to the heuristics in the previous subsection. On the other hand if d 3, we have 2d/(d + 2) > 1, and the expected lower bound for σ is not reached. Note that this factor is the same as in the second point in Theorem 1.1. It actually appears for the same technical reason as in the context of Theorem 1.1, as will be clear in Section 4.

Remark 1.8 (Values allowed for n). The assumptions of Theorem 1.5 always allow the value n = 1 (one direction of confinement). The value n = 2 is allowed for d 3, since the range

2d d + 2 1 d -2 < σ < 2 d -2
is non-empty. The value n = 3 is allowed only if d 7, and no value n 4 can be considered, since for d 5,

2d d + 2 1 d -4 > 2 d -2 .
So in the physically relevant cases d = 2 or 3, any n such that 1 n d -1 is allowed.

Regarding asymptotic completeness for large data, the first remark is that due to the anisotropy of the operator H, no analogue of the pseudo-conformal evolution law, which is used in the proof of the fourth point in Theorem 1.1, must be expected. Therefore, it is more sensible to rely on Morawetz type estimates, which we prove in a more general context than the framework of (1.3) with a quadratic potential. Proposition 1.9 (Anisotropic Morawetz estimates). Consider H = -1 2 ∆ x +V (x)-1 2 ∆ y , where V is a real-valued potential depending only on x ∈ R n . Let λ > 0 and assume that the solution u to (1.3) with u |t=0 = u 0 ∈ L 2 (R d ) exists globally, and satisfies

sup t∈R ∇ y u(t) L 2 (R d ) < ∞.
Then we have

(1.6) |∇ y | 3-(d-n) 2 R 2 L 2 t,y (R×R d-n ) ≤ C u 0 3 L 2 (R d ) sup t∈R ∇ y u(t) L 2 (R d ) ,
where

R(t, y) := R n |u(t, x, y)| 2 dx,
is the marginal of the mass density.

Using these estimates, we can prove asymptotic completeness results. Note however that since we consider energy-subcritical problems on R d , we assume

σ < 2 (d -2) + .
On the other hand, proving asymptotic completeness like in R d-n via Morawetz estimates (d -n is the "scattering dimension") is sensible, in view of the third point of Theorem 1.1, provided that

σ > 2 d -n .
The above two conditions are compatible if and only if n = 1. This restriction is expected to be only technical, since scattering should occur under the weaker assumption

σ > 1 d -n ,
but we lack of tools to decrease the value of σ down to this threshold. Note that in the small data case, asymptotic completeness is established for n = 2 provided that d 3, for instance, from Theorem 1.5. On the other hand, it is our choice to state the result only for d 4, since the case d 5 would bring new technicalities in the presentation, but no new real difficulty (see also [START_REF] Visciglia | Private communication on a joint work in progress[END_REF]).

Theorem 1.10 (Asymptotic completeness). Let n = 1, 2 d 4, and λ > 0. Assume moreover:

• If d = 2: σ > 2. • If d = 3: 1 < σ < 2. • If d = 4: 2/3 < σ < 1.
For all u 0 ∈ Σ, there exists a unique u + ∈ Σ such that the solution u provided by Proposition 1.4 satisfies

e itH u(t) -u + Σ -→ t→+∞ 0.
1.5. Structure of the paper. In Section 2, we recall some of the aspects in [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF][START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF], with emphasis on aspects related to Strichartz estimates. In Section 3, we propose a slight generalization of a result from [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF], and establish global in time Strichartz estimates associated to e -itH which are isotropic in space, in the same fashion as in [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF], and as opposed to [START_REF] Visciglia | Private communication on a joint work in progress[END_REF][START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF]. These estimates allow us to prove Theorem 1.5 in Section 4, thanks to suitable vector-fields. Proposition 1.9 is established in Section 5, and Theorem 1.10, in Section 6. Finally in an appendix, we present formal arguments suggesting the existence of long range effects if σ is not sufficiently large.

NONLINEAR SCHR ÖDINGER EQUATION ON PRODUCT SPACES

In this section, we recall some results which are highly related to the framework of the present paper, at least qualitatively.

A key step both in [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF] and in [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF] consists in proving global in time Strichartz estimates, even though the free dynamics is not fully dispersive due to the boundedness of the compact manifold (T 2 in the case of [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF]). For M n an n-dimensional compact manifold without boundary, consider

(2.1) i∂ t u + 1 2 ∆ M n ×R d-n u = F.
We introduce once and for all the notion of admissible pairs inspired by Strichartz estimates on R k . The definition includes a notion of "dispersive dimension", and will be of constant use in this paper.

Definition 2.1. Let k be a positive integer. A pair (q, r) is k-admissible if 2 r < 2k k-2 (2 r ∞ if k = 1, 2 r < ∞ if k = 2) and 2 q = k 1 2 - 1 r . Proposition 2.2 (From [31]). Let d 2, 1 n d -1 and M n be an n-dimensional compact manifold without boundary. For (x, y) ∈ M n × R d-n , the following estimate holds: there exists C r1,r2 such that e it∆x,y f L p 1 t L r 1 y L 2 x + t 0 e i(t-s)∆x,y F (s, x, y)ds L p 1 t L r 1 y L 2 x C r1,r2 f L 2 x,y + F L p ′ 2 t L r ′ 2 y L 2 x ,
where the pairs (p 1 , r 1 ) and (p 2 , r 2 ) are (d -n)-admissible.

The proof in [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF] relies on the fact that ∆ M n possesses an eigenbasis in L 2 (M n ): by decomposing any solution to (2.1) on this eigenbasis, each coefficient solves a Schrödinger equation on R d-n , and satisfies global Strichartz estimates. Proposition 2.2 follows by summing these inequalities in L 2 (M n ) and invoking Minkowski inequality. In Section 3.1, we present a generalization of this result, which does not require any spectral analysis.

In the case where d = 3, n = 2 and M 2 = T 2 , another family of estimates has been established in [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF]. For γ ∈ Z, set I γ = 2π[γ, γ + 1). Proposition 2.3 (From [START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF]). Let N 1 be dyadic, then

e it∆ R×T 2 P N u 0 ℓ q γ L p t,x,y (Iγ ×R×T 2 ) N ( 3 2 -5 p ) u 0 L 2 (R×T 2 ) , whenever p > 4 and 2 q + 1 p = 1 2 ,
and where P N stands for a frequency cut-off.

An important difference with the approach in [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF] is that in the case M n = T n , the eigenbasis of the Laplacian possesses a group structure, which makes it possible to proceed with more explicit computations. Extensive use of Strichartz estimates on T 2 [START_REF] Bourgain | Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I, Schrödinger equations[END_REF] (see also [START_REF] Ginibre | Le problème de Cauchy pour des EDP semi-linéaires périodiques en variable d'espace[END_REF]) is also made to prove Proposition 2.3. We emphasize however that the proof of Proposition 2.3 is rather involved technically. We prove an analogous result in the framework of (1.3) in Section 3.2. Even though the group structure of the eigenfunctions of H is lost (the eigenfunctions are given by Hermite functions), computations are less involved, and one does not have to face a loss of regularity for local Strichartz estimates associated to the harmonic oscillator.

GLOBAL IN TIME STRICHARTZ ESTIMATES

3.1. A slight generalization. In this subsection, we generalize Proposition 2.2 in a case where the existence of an eigenbasis for the operator in the x variable is not assumed.

Proposition 3.1. Let d 2, 1 n d -1,
X n be an n-dimensional space, and y ∈ R d-n . Let P be an operator acting on the x variable, but not on the y variable. In particular,

(3.1) [P, ∆ y ] = 0.
Assume that the flow generated by P is uniformly bounded on L 2 (X n ), at least in the future,

∃C > 0, e itP v 0 L 2 (X n ) C v 0 L 2 (X n ) , ∀t 0.
Then for all u 0 ∈ L 2 (R d ), all (d -n)-admissible pairs (p 1 , r 1 ) and (p 2 , r 2 ), there exists C r1,r2 such that the solution to

i∂ t u + P u + ∆ y u = F, u |t=0 = u 0 , satisfies: u L p 1 t (R+;L r 1 y L 2 x ) C r1,r2 u 0 L 2 x,y + F L p ′ 2 t (R+;L r ′ 2 y L 2 x )
.

For instance, X n may be any manifold without boundary, and P a Schrödinger operator, including a real-valued external potential or magnetic field, with decay or sign assumptions to ensure that the flow is well-defined on L 2 (X n ). In this cases, it may happen that no spectral theory like in [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF] is available. In general, the X n part can be viewed as a black box, and Proposition 3.1 is a way to take advantage of the Schrödinger dispersion on R d-n . Example 3.2. In the case of Schrödinger operators on R n , P [30, p. 199]) V 1 , V 2 real-valued and measurable, and

= -∆ x + V (x), it suffices to consider V = V 1 + V 2 , with (see
• V 1 (x) -a|x| 2 -b for some constants a and b. • V 2 ∈ L p (R n ) with p 2 if n 3, p > 2 if n = 4, and p n/2 if n 5.
This includes in particular the case studied in this paper,

V (x) = |x| 2 /2.
Example 3.3. Still on R n , P may be any polynomial in D x = -i∂ x , not necessarily elliptic, or more generally a real-valued Fourier multiplier.

Proof. The commutation assumption (3.1) implies e it(P +∆y) = e itP e it∆y = e it∆y e itP .

Using this remark, the assumption that e itP is bounded on L 2 (X n ) for positive time, and the standard properties of the Schrödinger group on R d-n , we have:

e it(P +∆y) L 2 x,y →L 2 x,y
1, e it(P +∆y)

L 1 y L 2 x →L ∞ y L 2 x 1 t (d-n)/2 , t > 0.
Invoking [START_REF] Keel | Endpoint Strichartz estimates[END_REF]Theorem 10.1], with

B 0 = H = L 2 x,y , B 1 = L 1 y L 2 x ,
we have

e it(P +∆y) u 0 L p t (R+;B * θ )
u 0 H , t 0 e i(t-s)(P +∆y) F (s)ds

L p 1 t (R+;B * θ 1 ) F L p ′ 2 t (R+;B θ 2 )
, where for 0 θ 1, B θ denotes the real interpolation space (B 0 , B 1 ) θ,2 , and

2/p = (d -n)θ/2, p 2, (p, θ, d -n) = (2, 1, 1)
, hence the proposition.

3.2. The case of partial harmonic confinement. We split the Hamiltonian H into two parts, the confining one and the fully dispersive one, by denoting

H 1 = - 1 2 ∆ x + |x| 2 2 ; H 2 = - 1 2 ∆ y .
As in the previous subsection, we note that both operators commute, [H 1 , H 2 ] = 0, therefore their corresponding propagators also commute

e -itH = e -it(H1+H2) = e -itH1 e -itH2 = e -itH2 e -itH1 .
The classical formula

e -itH2 g(y) = 1 (2iπt) (d-n)/2 R d-n e i |y-y ′ | 2 2t f (y ′ )dy ′ yields the standard global dispersive estimate (3.2) e -itH2 L 1 (R d-n )→L ∞ (R d-n ) 1 (2π|t|) (d-n)/2 , ∀t = 0.
Similarly, the fundamental solution associated to H 1 is given by Mehler's formula [START_REF] Mehler | Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceseschen Functionen höherer Ordnung[END_REF]:

e -itH1 f (x) = 1 (2iπ sin t) n/2 R n e i 2 sin t ((|x| 2 +|x ′ | 2 ) cos t-2x•x ′ ) f (x ′ )dx ′ ,
where the square root of sin t means implicitly that the singularities of the fundamental solution are taken into account: every time sin t = 0, a phase shift appears (Maslov index; see e.g. [START_REF] Kapitanski | On the fundamental solution of a perturbed harmonic oscillator[END_REF][START_REF] Zelditch | Reconstruction of singularities for solutions of Schrödinger's equation[END_REF]). The only information that we shall actually use is that local in time dispersive estimates are available, periodically in time:

(3.3) e -itH1 L 1 (R n )→L ∞ (R n ) 1 (2π| sin t|) n/2 , ∀t ∈ πZ.
A property that we will use crucially to establish global Strichartz estimates for e -itH is the fact that the above right hand side is periodic (see Remark 3.5 below). Accordingly, for γ ∈ Z, we set

I γ = π[γ -1, γ + 1). Theorem 3.4 (Global Strichartz estimates). Let d 2 and 1 n d -1. If 2 r < 2d d-2
and the pairs (q, r) and (p, r) are, respectively, d-admissible and (d -n)-admissible, then the following two inequalities hold,

(3.4) e -itH u 0 ℓ p γ L q (Iγ ;L r (R d )) u 0 L 2 (R d ) , and
(3.5) t∈R e itH F (t)dt L 2 (R d ) F ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d )) .
Also, if (p 1 , q 1 , r 1 ) and (p 2 , q 2 , r 2 ) are two such triplets, then

(3.6) t 0 e -i(t-s)H F (s)ds ℓ p 1 γ L q 1 (Iγ ;L r 1 (R d )) F ℓ p ′ 2 γ L q ′ 2 (Iγ ;L r ′ 2 (R d ))
.

The above statement can be understood as follows. Locally in time, we have the same Strichartz estimates as for e it∆ , a fact that is standard in the presence of a smooth potential growing at most quadratically in space (see e.g. [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]). In the present case, these estimates are made global in time thanks to the dispersion in the y directions, corresponding to the free Hamiltonian H 2 : there are d -n such directions, so the assumption for (p, r) to be (d -n)-admissible appears very natural.

Proof. The scheme of this proof follows the same general method as for standard Strichartz inequalities, by using duality and the T T * argument of Tomas-Stein. The added novelty here is strongly inspired by the arguments in [START_REF] Cuccagna | On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D[END_REF][START_REF] Hani | On scattering for the quintic defocusing nonlinear Schrödinger equation on R × T 2[END_REF].

The homogeneous inequalities (3.4) and (3.5) are dual to each other and therefore equivalent.

We now prove (3.5

). Let F, G ∈ C ∞ (R × R d ).
Then, (3.5) is equivalent to the bilinear inequality

(3.7) t∈R e itH F (t)dt, s∈R e isH G(s)ds L 2 F ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d )) G ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d ))
, which, again by duality, is in turn equivalent to (3.8) s∈R e -i(t-s)H G(s)ds

ℓ p γ L q (Iγ ;L r (R d )) G ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d )) .
To prove (3.7) we use a partition of unity in time

γ∈Z χ(t -πγ) = 1, ∀t ∈ R with suppχ ⊂ [-π, π].
We thus have t∈R e itH F (t)dt, s∈R e isH G(s)ds

L 2 = α,β∈Z t,s∈R χ(t -πα)e itH F (t), χ(t -πβ)e isH G(s) L 2 dt ds.
This sum is now bounded by splitting it into the diagonal terms, |α -β| 1, and the non-diagonal ones, |α -β| > 1.

The sum of the diagonal terms is bounded as follows

|α-β| 1 t,s∈R χ(t -πα)e itH F (t), χ(t -πβ)e isH G(s) L 2 dt ds |α-β| 1 π -π χ(t)e i(t+πα)H F (t + πα)dt L 2 π -π χ(s)e i(s+πβ)H G(s + πβ)ds L 2 = |α-β| 1 π -π χ(t)e itH F (t + πα)dt L 2 π -π χ(s)e isH G(s + πβ)ds L 2
, where, in this last step, we have used the fact that the propagator e -itH is unitary in L 2 . At this point, we use Strichartz estimates in their dual form (analogous to (3.5)), which are available locally in time, for the partially confined operator H, with d-admissible pairs (q, r),

|α-β| 1

F (• + πα) L q ′ ([-π,π];L r ′ (R d )) G(• + πβ) L q ′ ([-π,π];L r ′ (R d )) F ℓ 2 γ L q ′ (Iγ ;L r ′ (R d )) G ℓ 2 γ L q ′ (Iγ ;L r ′ (R d ))
. For the non-diagonal part of the sum -the crucial and deeper one -we proceed with a different approach, involving a localized T T * argument.

|α-β|>1

t,s∈R

χ(t -πα)e itH F (t), χ(t -πβ)e isH G(s) L 2 dt ds = |α-β|>1 t,s∈R χ(t)χ(s) F (t + πα), e -i(t-s+π(α-β))H G(s + πβ) L 2 dt ds β∈Z |γ|>1 (t,s)∈[-π,π] 2 χ(t)χ(s) F (t + π(β + γ)), e -i(t-s+πγ)H G(s + πβ) L 2 dt ds β∈Z |γ|>1 (t,s)∈[-π,π] 2 F (t + π(β + γ)) L r ′ e -i(t-s+πγ)H G(s + πβ) L r .
We have now reached the stage where we use the time decay estimate of the dispersive propagator. For that, we exploit the commutativity property of the partial propagators e -itH1 and e -itH2 , as well as the periodicity in time of e -itH1 . Thus, following the remarks at the beginning of this section, we know that e -i(t-s+πγ)H = e -i(t-s+πγ)H1 e -i(t-s+πγ)H2 .

The propagator for H 2 corresponds to the free Schrödinger case in the y variables, thus has a global time decay with rate t -(d-n)/2 , as seen in (3.2). Whereas the propagator for H 1 corresponds to the time periodic harmonic oscillator in the x variables, with local in time decay with rate t -n/2 from (3.3). So, doing the full

L 1 (R d ) → L ∞ (R d ) estimate
for the propagator of H by first splitting the variables x and y, then applying the partial propagators one at a time for each of the corresponding variables and using the periodicity of H 1 , we obtain the combined time decay rate

e -i(t-s+πγ)H f L ∞ x,y (R d ) 1 |t -s| n/2 1 |γ| (d-n)/2 f L 1 x,y (R d ) ,
which is local in the t, s variables (from the partially confining operator H 1 acting on the x variables) and global in γ (from the free operator H 2 acting on the y variables). Finally, interpolating the above L 1 → L ∞ estimate with the L 2 conservation, we obtain the intermediate L r ′ → L r decay to be used within the T T * proof of the Strichartz estimates,

e -i(t-s+πγ)H f L r x,y (R d ) 1 |t -s| n(1/2-1/r) 1 |γ| (d-n)(1/2-1/r) f L r ′ x,y (R d ) ,
for 2 r ∞.

So, plugging this estimate back into the non-diagonal sum inequality, we thus obtain

β∈Z |γ|>1 (t,s)∈[-π,π] 2 F (t + π(β + γ)) L r ′ 1 |t -s| n( 1 2 -1 r ) 1 |γ| (d-n)( 1 2 -1 r ) G(s + πβ) L r ′ β∈Z |γ|>1 1 |γ| (d-n)( 1 2 -1 r ) (t,s)∈[-π,π] 2 1 |t -s| n( 1 2 -1 r ) F (t + π(β + γ)) L r ′ G(s + πβ) L r ′ β∈Z |γ|>1 1 |γ| (d-n)( 1 2 -1 r ) F (• + π(β + γ)) L q ′ ([-π,π];L r ′ (R d )) G(• + πβ) L q ′ ([-π,π];L r ′ (R d )) ,
where we have used the Hardy-Littlewood-Sobolev inequality in the t, s variables, in the last step, for (q, r) now an n-admissible pair. To conclude this estimate, we again apply the Hardy-Littlewood-Sobolev inequality, this time in its discrete version for the γ variable, to obtain

(3.10) F ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d )) G ℓ p ′ γ L q ′ (Iγ ;L r ′ (R d )) , whenever 2 p (d -n) 1 2 - 1 r ,
in particular for (p, r) a (d -n)-admissible pair.

To finish the proof of (3.7) we gather the diagonal and non-diagonal estimates, (3.9) and (3.10), to obtain t∈R e itH F (t)dt, s∈R e isH G(s)ds

L 2 F ℓ 2 γ L q ′ 1 (Iγ ;L r ′ (R d )) G ℓ 2 γ L q ′ 1 (Iγ ;L r ′ (R d )) + F ℓ p ′ γ L q ′ 2 (Iγ ;L r ′ (R d )) G ℓ p ′ γ L q ′ 2 (Iγ ;L r ′ (R d ))
, where the pair (q 1 , r) is d-admissible, while (q 2 , r) is n-admissible. Therefore q ′ 2 < q ′ 1 and using the inclusion L q ′ 1 (I γ ) ⊂ L q ′ 2 (I γ ), while ℓ p ′ ⊂ ℓ 2 , we obtain (3.7). Thus, we have established (3.4) and (3.5), as well as (3.8).

There still remains establishing (3.6) to conclude the proof of the theorem. But this finite interval version follows from the similar global integral estimate (3.8) by using the now standard Christ-Kiselev lemma, together with interpolation between admissible triplets, just as in the analogous final stage of a classical Strichartz estimate proof. Remark 3.5 (More general potentials). The two properties of e -itH1 that we have used in the proof of Theorem 3.4 are the existence of local dispersive estimates, and the fact that such estimates are available periodically in time. The first point remains for smooth potentials which grow at most quadratically, from [START_REF] Fujiwara | Remarks on the convergence of the Feynman path integrals[END_REF]. The fact that these dispersive estimates are available periodically in time remains for instance when the harmonic potential is suitably perturbed with a smooth one (Schwartz class), so the new operator has the same spectrum as the harmonic oscillator; this has been established in the case n = 1 in [START_REF] Mckean | The spectral class of the quantum-mechanical harmonic oscillator[END_REF]. Also, we may consider more general quadratic potentials, when n 2,

n j=1 ω 2 j x 2 j 2
, ω j > 0.

If the ω j 's are rationally dependent, then the above proof can easily be repeated, since local in time dispersive estimates remain, as well as time periodicity of the propagator (each component leads to dispersive estimates which are π/ω j -periodic in time). On the other hand, if the ω j 's are rationally independent, either by using Hermite functions (eigen decomposition) or by invoking Mehler's formula, we see that the time periodicity is lost.

The argument that we have followed to obtain global Strichartz estimates no longer works; the dynamics might be rather involved, typically due to the presence of small divisors. On the other hand, since Proposition 3.1 only assumes that the flow associated to the operator P (= 1 2 ∆ x + V (x) here), is bounded on L 2 , its conclusion remains valid for all quadratic potentials.

EXISTENCE OF WAVE OPERATORS AND SMALL DATA SCATTERING

In this section, we prove Theorem 1.5. The existence of wave operators is obtained by a fixed point argument in a suitable space, constructed in view of global Strichartz estimates provided by Theorem 3.4. Asymptotic completeness for small data follows essentially from the same estimates, plus a standard bootstrap argument. 4.1. Vector fields. As pointed out in the introduction, proving the existence of wave operator in Σ in the absence of external potential, that is for (1.4), relies on the use of the vector field x + it∇. In the case of (1.3), this means that we consider

J = y + it∇ y = it e i|y| 2 /(2t) ∇ y e -i|y| 2 /(2t) • .
In view of the second identity, J acts on the gauge invariant nonlinearity |u| 2σ u like ∇ y , and Gagliardo-Nirenberg inequalities involving ∇ y export to similar inequalities in terms of J, with an extra time dependent factor:

(4.1) f L r (R d ) C r |t| (d-n)(1/2-1/r) f 1-d(1/2-1/r) L 2 (R d ) (∇ x , J)f d(1/2-1/r) L 2 (R d ) , for all 2 r 2d d-2 (2 r < ∞ if d = 2)
. Finally, J commutes with the linear part of (1.3), [i∂ t -H, J] = 0.

We now turn to the x-dependent part of H. In view of the above weighted Gagliardo-Nirenberg inequality, we will get the same time decay as on R d-n provided that we can estimate ∇ x u (in L 2 ). However, unlike J, ∇ x does not commute with the linear part of (1.3):

[i∂ t -H, ∇ x ] = -x.
We remark that

[i∂ t -H, x] = ∇ x ,
so it is possible to obtain a closed system of estimates for (u, Ju, ∇ x u, xu). This remark can be used to construct a solution to (1.3) locally in time, thanks to the Gronwall lemma; see [START_REF] Oh | Cauchy problem and Ehrenfest's law of nonlinear Schrödinger equations with potentials[END_REF] (see also [START_REF] Carles | Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential[END_REF]). However, the above commutators generate an exponentially growing factor after the use of the Gronwall lemma, an aspect which ruins the algebraic time decay provided by (4.1) when large time is considered, like in scattering theory. Therefore, we need to replace the operators ∇ x and x by adapted operators which enjoy properties similar to those of J. In the case of the harmonic potential, such operators are available: see e.g. [START_REF] Carles | Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential[END_REF]. We introduce

A 1 (t) = x sin t -i cos t∇ x , A 2 (t) = x cos t + i sin t∇ x A 3 (t) = -i∇ y , A 4 (t) = y + it∇ y .
To ease the forthcoming formulas, we also set

A 0 = Id.
The operators A 3 and A 4 are the standard ones in the absence of external potential. The operators A 1 and A 2 are adapted to the harmonic potential, and satisfy

A 1 (t) -A 2 (t) = sin t cos t -cos t sin t x -i∇ x .
Indeed, the operators A 1 and A 2 account for the fact that the harmonic oscillator rotates the phase space at a speed which is uniform in (x, ξ). In particular, we have the pointwise identity for, say, f ∈ S(R d ),

(4.2) |A 1 (t)f (x, y)| 2 + |A 2 (t)f (x, y)| 2 = |xf (x, y)| 2 + |∇ x f (x, y)| 2 , ∀t ∈ R.
Lemma 4.1. For 1 j 4, the operators A j satisfy the following properties.

• They correspond to the conjugation of gradient and momentum by the free flow,

A 1 (t) = e -itH (-i∇ x )e itH , A 2 (t) = e -itH x e itH , A 3 (t) = e -itH (-i∇ y )e itH , A 4 (t) = e -itH y e itH .
Therefore, they commute with the linear part of (1.3): [i∂ t -H, A j (t)] = 0. • They act on gauge invariant nonlinearities like derivatives. In particular, we have the pointwise estimate

A j (t) |u| 2σ u |u| 2σ |A j (t)u|. • Weighted Gagliardo-Nirenberg inequalities: for all 2 r 2d d-2 (2 r < ∞ if d = 2), there exists K r such that for all f ∈ Σ, t = 0, f L r (R d ) K r t (d-n)(1/2-1/r) f 1-δ L 2 (R d )   4 j=1 A j (t)f L 2 (R d )   δ with δ = d(1/2 -1/r).
The last two points follow directly from the "nonlinear" counterpart of the "linear" factorization of the first point:

A 1 (t)f = -i cos t e -i 2 |x| 2 tan t ∇ x e i 2 |x| 2 tan t f , A 2 (t)f = i sin t e i|x| 2 /(2 tan t) ∇ x e -i|x| 2 /(2 tan t) f , A 4 (t)f = it e i|y| 2 /(2t) ∇ y e -i|y| 2 /(2t) f .
We emphasize that analogous operators are available for anisotropic quadratic potentials.

On the other hand, the existence of operators enjoying the above three properties seems to be bound to potentials which are polynomials of degree at most two ( [START_REF] Miller | Semiclassical nonlinear Schrödinger equations with potential and focusing initial data[END_REF]).

In view of the first point in Lemma 4.1, we have the equivalence, in the sense of equivalence of norms:

e itH u(t) -u -Σ ∼ 4 j=0 A j (t)u(t) -A j (t)e -itH u -L 2 ∼ 4 j=0 A j (t)u(t) -e -itH A j (0)u -L 2 .

4.2.

The fixed point argument. Let u -∈ Σ. For γ 0 -1, we denote

J γ0 = (-∞, π(γ 0 + 1)) = γ γ0 I γ ,
and we set

X γ0 = u ∈ L ∞ (J γ0 ; H 1 (R d ))
; for all j ∈ {0, . . . , 4},

A j u L ∞ (Jγ 0 ;L 2 ) 2 A j (0)u -L 2 , A j u ℓ p γ γ 0 L q (Iγ ;L r ) 2C r A j (0)u -L 2 ,
where C r stems from Theorem 3.4. The indices p, q, r are precised below: in view of Theorem 3.4, they are such that (q, r) is d-admissible, and (p, r) is (d -n)-admissible. Theorem 1.5 follows from a fixed point argument on Duhamel's formula

u(t) = e -itH u --iλ t -∞ e -i(t-s)H |u| 2σ u (s)ds =: Φ(u)(t),
applied in X γ0 for γ 0 ≪ -1 and (p, q, r) well chosen. Thanks to Theorem 3.4 and Lemma 4.1, the argument becomes rather close to the proof the second point in Theorem 1.1, for which the reader is referred to, e.g., [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF][START_REF] Ginibre | An introduction to nonlinear Schrödinger equations[END_REF].

Proof of Theorem 1.5. The key step, for the stability of X γ0 under the action of the map Φ as well as for the contraction of Φ on X γ0 with respect to the weaker norm ℓ p γ γ0 L q (I γ ; L r ) (Kato's method, see [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]: X γ0 equipped with that norm is complete, so it is not necessary to prove contraction for norms involving derivatives of Φ), consists in controlling B(|u| 2σ u) in, say, ℓ p ′ γ γ0 L q ′ (I γ ; L r ′ ), for all B ∈ {Id, A 1 , . . . , A 4 }. This is so thanks to Strichartz estimates, and to the first point in Lemma 4.1. In view of the second point in Lemma 4.1 and Hölder inequality,

B(|u| 2σ u) ℓ p ′ γ γ 0 L q ′ (Iγ ;L r ′ ) u 2σ ℓ θ γ γ 0 L ω (Iγ ;L k ) Bu ℓ p γ γ 0 L q (Iγ ;L r ) ,
with a constant independent of γ 0 , provided that (4.3)

1 p ′ = 1 p + 2σ θ , 1 q ′ = 1 q + 2σ ω , 1 r ′ = 1 r + 2σ k .
If we can find (p, q, r) with the above algebraic requirements, such that u ∈ X γ0 =⇒ u ∈ ℓ θ γ γ0 L ω (I γ ; L k ), with (θ, ω, k) given like above, for some finite θ, then the result follows, up to taking |γ 0 | sufficiently large.

Let u ∈ X γ0 . In view of the last point in Lemma 4.1,

u(t) L k (R d ) C |t| (d-n)(1/2-1/k) , t < π(γ 0 + 1),
where C depends only on d, n, k and u -Σ . Since |I γ | = 2π, we infer for any ω, γ γ 0 ,

u(t) L ω (Iγ ;L k (R d )) (2π) 1/ω u(t) L ∞ (Iγ ;L k (R d )) 1 |γ| (d-n)(1/2-1/k) .
Therefore,

X γ0 ⊂ ℓ θ γ γ0 L ω (I γ ; L k ) as soon as (d -n) 1 2 - 1 k θ > 1.
We distinguish two cases: d = 2 and d 3.

Case d = 2. Then necessarily, n = 1. We take r = 2 + ε, with 0 < ε ≪ 1. In view of (4.3), we compute:

k = 2σ(2 + ε) ε , p 4(2 + ε) ε , q = 2(2 + ε) ε , θ = 2σ 4 + 2ε 4 + ε , ω = (2 + ε)σ, hence (d -n) 1 2 - 1 k θ = 4σ + 2(σ -1)ε 4 + ε .
This is larger than 1 for ε > 0 sufficiently small as soon as σ > 1.

Case d 3. We consider the largest possible value for k, which corresponds to the endpoint of Sobolev embedding,

k = 2d d -2 .
In view of (4.3), we compute

r = 2d d -(d -2)σ .
Since the nonlinearity is energy-subcritical, σ < 2/(d -2), we have 2 r < 2d/(d -2), so this value is acceptable. We also have

p = 4d (d -n)σ(d -2) , q = 4 σ(d -2) , θ = 4dσ 2d -(d -n)(d -2)σ ,
where the last formula defines a positive θ, since σ < 2/(d -2) and n > 0. We infer

(d -n) 1 2 - 1 k θ = d -n d 4dσ 2d -(d -n)(d -2)σ .
This quantity is larger than 1 if and only if

σ > 2d d + 2 1 d -n ,
which is the condition stated in Theorem 1.5.

4.3.

Asymptotic completeness for small data. We now turn to the proof of the second part of Theorem 1.5. Resume exactly the same computations as the previous subsection. For B ∈ {Id, A 1 , . . . , A 4 },

Bu ℓ p γ 1 L q (Iγ ;L r ) B(0)u 0 L 2 + u 2σ ℓ θ γ 1 L ω (Iγ ;L k ) Bu ℓ p γ 1 L q (Iγ ;L r
) . For t 0, we estimate, in view of the last point in Lemma 4.1,

u(t) L k (R d ) C t (d-n)(1/2-1/k) u 0 1-δ   4 j=1 A j (t)u L 2   δ , with δ = d(1/2 -1/k). Now let M (t) = 4 j=0 A j u ℓ p 0 γ t L q (Iγ ;L r ) + A j u L ∞ ([0,t];L 2 ) .
It satisfies

M (t) u 0 Σ + M (t) 1+2σδ ,
with δ as above. Recall that standard bootstrap argument: Lemma 4.2 (Bootstrap argument). Let M = M (t) be a nonnegative continuous function on [0, T ] such that, for every t ∈ [0, T ],

M (t) ε 1 + ε 2 M (t) κ ,
where ε 1 , ε 2 > 0 and κ > 1 are constants such that

ε 1 < 1 - 1 κ 1 (κε 2 ) 1/(κ-1) , M (0) 1 (κε 2 ) 1/(κ-1) .
Then, for every t ∈ [0, T ], we have

M (t) κ κ -1 ε 1 .
We infer that M is bounded for all t 0 provided that u 0 Σ is sufficiently small. Once we know that A j u ∈ L ∞ ([0, ∞); L 2 ) for all j ∈ {0, . . . , 4}, we readily infer, by using Strichartz estimates again, that e itH A j (t)u t 0 is a Cauchy sequence as t → ∞, hence the result.

ANISOTROPIC MORAWETZ ESTIMATES

In this section, we prove anisotropic Morawetz estimates, as stated in Proposition 1.9.

Proof. We proceed as in [START_REF] Ginibre | Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations[END_REF], by working on quadratic quantities which appear naturally in the hydrodynamical reformulation of (1.3). We deduce a monotonicity formula for an appropriate virial quantity and then we infer the a priori bounds for the desired quantities. To shorten the notations, we set z = (x, y). In [START_REF] Ginibre | Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations[END_REF] the calculations rely on the conservation laws for the first two momenta related to the wave function: if u is a solution to (1.3), then we have (5.1)

   ∂ t ρ + div J = 0 ∂ t J + div (Re(∇ū ⊗ ∇u)) + λ σ σ + 1 ∇ρ σ+1 + ρ∇V = 1 4 ∇∆ρ,
where ρ(t, z) := |u(t, z)| 2 and J(t, z) := Im(ū∇u)(t, z). Let us define the virial potential

I(t) := 1 2 R d ×R d ρ(t, z)a(|z -z ′ |)ρ(t, z ′ ) dzdz ′ = 1 2 ρ, a * ρ ,
where a is a sufficiently smooth weight function which will be chosen later. Here •, • denotes the scalar product in L 2 (R d ). By using (5.1), we see that the time derivative of

I(t) reads (5.2) d dt I(t) = -ρ, ∇a * J = ρ(t, z ′ )∇a(|z -z ′ |) • J(t, z) dz ′ dz =: M (t),
where M (t) is the Morawetz action. By using again the balance laws (5.1) we have

d dt M (t) = -J, ∇ 2 a * J + ρ, ∇ 2 a * Re(∇ū ⊗ ∇u) + λσ σ + 1 ρ, ∆a * ρ σ+1 -ρ, ∇a * (ρ∇V ) - 1 4 ρ, ∆a * ∆ρ = -Im(ū∇u), ∇ 2 a * Im(ū∇u) + ρ, ∇ 2 a * (∇ū ⊗ ∇u) + λσ σ + 1 ρ, ∆a * ρ σ+1 -ρ, ∇a * (ρ∇V ) - 1 4 ρ, ∆a * ∆ρ ,
where in the second term we dropped the real part because of the symmetry of ∇ 2 a. For the first term we have

Im(ū∇u), ∇ 2 a * Im(ū∇u) = ū∇u, ∇ 2 a * (ū∇u) -Re(ū∇u), ∇ 2 a * Re(ū∇u) = ū∇u, ∇ 2 a * (ū∇u) - 1 4 ∇ρ, ∇ 2 a * ∇ρ ,
hence we may write (5.3)

d dt M (t) = -ū∇u, ∇ 2 a * (ū∇u) + ρ, ∇ 2 a * (∇ū ⊗ ∇u) + 1 4 ∇ρ, ∇ 2 a * ∇ρ - 1 4 ρ, ∆a * ∆ρ + λσ σ + 1 ρ, ∆a * ρ σ+1 - ρ(t, z ′ )ρ(t, z)∇a(|z -z ′ |) • ∇V (z)dzdz ′ .
Let us consider the second line in (5.3). It is equal to

1 4 ∇ρ, ∇ 2 a * ∇ρ - 1 4 ρ, ∆a * ∆ρ = 1 2 ∇ρ, ∇ 2 a * ∇ρ = 1 2 ∇ρ, ∆a * ∇ρ .
Now, we notice the first two terms on the right hand side of (5.3) may be rewritten as

-ū∇u, ∇ 2 a * (ū∇u) + ρ, ∇ 2 a * (∇ū ⊗ ∇u) = 1 2 R d ×R d ∇ 2 a(|z -z ′ |) • (ū(t, z ′ )∇ū(t, z) -ū(t, z)∇ū(t, z ′ )) • (u(t, z ′ )∇u(t, z) -u(t, z)∇u(t, z ′ )) dz ′ dz,
which is non-negative if the Hessian matrix ∇ 2 a is non-negative definite. On the other hand, by choosing the weight a(|z|) depending only on y ∈ R d-n the last term in (5.3) involving the potential V = V (x), vanishes. Therefore, the natural choice for the weight is a(|z|) = |y|. In this way we have (5.4)

d dt M (t) 1 2 ∇ y ρ, ∆ y a * ∇ y ρ + λσ σ + 1 ρ, ∆ y a * ρ σ+1 . Notice that ∆ y a(|y|) = d-n-1
|y| is, up to a multiplicative constant, the integral kernel of the operator (-∆ y ) -d-n-1

2

, that is,

(-∆ y ) -d-n-1 2 f (y) = R d-n c |y -y ′ | f (y ′ ) dy ′ .
Thus, by recalling z = (x, y), we obtain

R d ×R d 1 |y -y ′ | ∇ y ρ(t, z ′ ) • ∇ y ρ(t, z) dz ′ dz = R n ×R n ×R d-n ∇ y ρ(t, x, y) • ∇ y (-∆ y ) -d-n-1 2 ρ(t, x ′ , y) dxdx ′ dy.
Hence, if we define the marginal of the mass density R(t, y) := R n ρ(t, x, y) dx, the last integral also reads

R d-n |∇ y | 3-(d-n) 2 R(t, y) 2 dy.
We now plug this expression into (5.4) and we integrate in time. Furthermore, since λ > 0 the second term in the right hand side in (5.4) is positive. We then infer (5.5)

T 0 R d-n |∇ y | 3-(d-n) 2 R(t, y) 2 dydt C sup t∈[0,T ] M (t).
Furthermore, with our choice of the weight a(|z|), we have

M (t) = ρ(t, z ′ ) y -y ′ |y -y ′ | • Im(ū∇ y u)(t, z) dz ′ dz u 0 3 L 2 (R d ) ∇ y u(t) L 2 (R d ) ,
where we have used the conservation of the L 2 -norm of u (V is real-valued). We infer that the right hand side in (5.5) is uniformly bounded,

T 0 R d-n |∇ y | 3-(d-n) 2 R(t, y) 2 dydt C u 0 3 L 2 (R d ) sup t∈R ∇ y u(t) L 2 (R d )
.

By letting T go to infinity, we prove (1.6).

Remark 5.1. Let us come back to (5.4) in the proof of Proposition 1.9. We see that the second term in the right hand side is nonnegative, and is equal to

λσ σ + 1 ρ(t, z ′ ) d -n -1 |y -y ′ | ρ σ+1 (t, z) dz ′ dz λσ σ + 1 d -n -1 |y -y ′ | ρ σ+2 2 (t, z ′ )ρ σ+2 2 (t, z) dz ′ dz.
As before, 1 |y-y ′ | is the integral kernel of (-∆ y ) -d-n-1

2

, hence the last integral may be written in the following way,

C R d-n |∇ y | -d-n-1 2 R n ρ σ+2 2 (t, x, y) dx 2 dy.
Consequently, from Morawetz estimates we also infer the following a priori bound

R×R d-n |∇ y | -d-n-1 2 ρ (σ+2)/2 (t, x, y) dx 2 dydt C u 0 3 L 2 sup t∈R ∇ y u(t) L 2 .
This term does not appear in the statement of Proposition 1.9, since (1.6) is the only estimate we are going to use to prove asymptotic completeness.

6. ASYMPTOTIC COMPLETENESS 6.1. Outline of the proof and technical remarks. The strategy of the proof of Theorem 1.10 is the same as in e.g. [START_REF] Vega | Bilinear virial identities and applications[END_REF] (Section 5.1, for the one-dimensional example), or [START_REF] Banica | On scattering for NLS: from Euclidean to hyperbolic space[END_REF]. Global in time estimates for the solution u are obtained by an inductive bootstrap argument: Lemma 6.1. Let t 0 ∈ R, h ∈ L q loc ([t 0 , ∞)) for some 1 q ∞. Suppose that there exist C 0 , η, κ > 0, and f ∈ L p ([t 0 , ∞)) with 1 p < ∞, such that for all t ′ t t 0 , (6.1)

h L q ([t,t ′ ]) C 0 + f η L p ([t,t ′ ]) h κ L q ([t,t ′ ]) .
Then h is globally integrable, h ∈ L q ([t 0 , ∞)).

Proof. If κ > 1, split [t 0 , ∞) into finitely many sub-intervals I j , on which f η L p (Ij ) ε 2 , with ε 2 = ε 2 (η, f, C 0 , κ) so small that Lemma 4.2 implies

h L q (Ij ) κ κ -1 C 0 ,
and the result follows. If κ 1, we may invoke Young inequality ab a q + b q ′ , 1 < q < ∞, to fall back into the first case.

Remark 6.2. When κ > 1, it is crucial in the above lemma that C 0 is independent of t, t ′ . If we suppose for instance that we have inequalities of the form

M (t ′ ) M (t) + f η L p ([t,t ′ ]) M (t ′ )
κ , then M need not be bounded. Consider for instance M (t) = t: for t ′ t 1,

t ′ = t + t ′ -t = t + t ′ -t tt ′ tt ′ t + t ′ -t tt ′ (t ′ ) 2 , that is, M (t ′ ) M (t) + t ′ t dt t 2 M (t ′ ) 2 .
We also need a Gronwall type argument: Lemma 6.3. Let t 0 ∈ R, M ∈ C([t 0 , ∞)). Suppose that there exist η > 0, and f ∈ L p ([t 0 , ∞)) with 1 p < ∞, such that for all t ′ t t 0 , (6.2)

M (t ′ ) M (t) + f η L p ([t,t ′ ]) M (t ′ ). Then M is bounded, M ∈ L ∞ ([t 0 , ∞)). Proof. Split [t 0 , ∞) into finitely many sub-intervals I j , [t 0 , ∞) = N j=0 I j , I j = [t j , t j+1 ), t N +1 = ∞, on which f η L p (Ij ) 1/2. Then sup t∈Ij M (t) 2M (t j ), hence sup t t0 M (t) 2 N +1 M (t 0 ).
Now the goal is to use Strichartz inequalities and Hölder inequality in order to obtain (6.1) with h(t) = u(t) Z for some suitable Z, so that the existence of such a function f follows from the conservation of energy and Morawetz estimates. Lemma 6.3 is then applied to M (t) = Bu L p ([0,t];Z) , where B ∈ {A 1 , . . . , A 4 }. Essentially, the nonlinear estimates of Lemma 6.1 become linear estimates as in Lemma 6.3 because all the vectorfields A j act like first order derivatives on gauge invariant nonlinearities. Note that in view of the conservation of energy (see Proposition 1.4) and the identity (4.2), we may replace h at the first step with

h(t) = 3 j=0 A j (t)u Z ,
since the conservations of mass and energy yield an a priori bound in

Σ anis = {f ∈ L 2 (R d ), xf, ∇ x f, ∇ y f ∈ L 2 (R d )}.
In that case, only the operator A 4 , which contains the large time decay information, is left out. We will follow both approaches below.

At this stage, the strategy may appear relatively standard. The less standard aspect is that the numerology associated to Strichartz estimates in the present framework does not allow to follow directly the above road map by working in ℓ p γ L q (I γ , L r ), with (p, q, r) as in Theorem 3.4. Indeed, apply Hölder inequality to the quantity

|u| 2σ u ℓ p ′ 1 γ L q ′ 1 L r ′ 1 ,
which appears after the use of inhomogeneous Strichartz estimates, to bound it by

u (2σ+1)(1-η) ℓ θ γ L ω L s u (2σ+1)η ℓ p γ L q L r
, where η ∈ (0, 1), and (p 1 , q 1 , r 1 ), (p, q, r) are triplets as in Theorem 3.4, with

1 = 1 p 1 + (2σ + 1)(1 -η) θ + (2σ + 1)η p , 1 = 1 q 1 + (2σ + 1)(1 -η) ω + (2σ + 1)η q , 1 = 1 r 1 + (2σ + 1)(1 -η) s + (2σ + 1)η r .
Then since d -n < d, we necessarily have θ < ω, unless r 1 = r = 2, a case where Strichartz estimates are not even needed (energy estimate). Unless we proceed as in Section 4, which means unless we use Gagliardo-Nirenberg inequality, we cannot claim that u ∈ ℓ θ γ L ω L s with θ < ω, by invoking only Morawetz estimates and the conservation of energy. To overcome this issue, we shall use the Strichartz estimates from Proposition 3.1 (which, in our case, follow essentially from [START_REF] Tzvetkov | Small data scattering for the nonlinear Schrödinger equation on product spaces[END_REF], since the harmonic oscillator possesses an eigenbasis): the above space Z will be of the form L r y L 2 x , or a more sophisticated version of it. We prove Theorem 1.10 from the easiest case to the most involved technically: if d = 3 or 4, a rather straightforward estimate allows to apply Lemmas 6.1 and 6.3, successively. When d = 2, the same strategy works for 2 < σ < 4: since this does not cover the whole range of values for σ, we directly present another approach, more in the spirit of [START_REF] Visciglia | Private communication on a joint work in progress[END_REF].

∈ L 4 t L 4 y L 2 x . Let B ∈ {Id, A 1 , A 2 , A 3 }. As recalled above, Bu ∈ L ∞ t L 2
x,y from the conservation of the energy (see also (4.2)). In view of Proposition 3.1 and Hölder inequality, we have, since ( 10 3 , 10 3 ) is 3-admissible, for t ∈ I any interval

Bu L 10/3 t L 10/3 y L 2 x u 0 Σ + |u| 2σ Bu L 10/7 t L 10/7 y L 2 x u 0 Σ + u 2σ L 5σ t L 5σ y L ∞ x Bu L 10/3 t L 10/3 y L 2
x .

Since n = 1, we invoke the standard estimate

f L ∞ (R) √ 2 f 1/2 L 2 (R) ∂ x f 1/2 L 2 (R) , to infer u L 5σ t L 5σ y L ∞ x u 1/2 L p t L p y L 2 x u 1/2 L 10/3 t L 10/3 y H 1 x , where
1 p = 4 -3σ 10σ .
For σ ∈ ( 2 3 , 1), we have

1 10 < 1 p < 3 10 ,
thus we can always interpolate

u L p t L p y L 2 x u α L 4 t L 4 y L 2 x u 1-α L q t L q y L 2
x , with 0 < α < 1, and where q is either 10 or 10 3 . The value 10 is motivated by the fact that the pair (10, 10) in 3-admissible at the level of Ḣ1 (instead of L 2 so far), and so

u L 10 t L 10 y L 2 x ∇ y u L 10 t L 30/13 y L 2 x ,
where the pair (10, 30 13 ) is 3-admissible. Letting

h(t) = sup (p,r) 3-admissible 3 j=0 A j (t)u p L r y L 2 x ,
we conclude that this function satisfies, for all interval I,

h L 1 (I) 1 + u β L 4 (I;L 4 y L 2
x ) h κ L 1 (I) , for some β, κ > 0. Lemma 6.1 yields

u, A 1 u, A 2 u, A 3 u ∈ L p (R; L r y L 2
x ) for all 3-admissible pairs (p, r). Now taking B = A 4 in the above estimate, Lemma 6. 6.4. The case d = 2. In this case, instead of working mainly with an L 2 regularity in x, we consider a Banach algebra. Since x is associated to a harmonic oscillator, it is not sensible to require only a Sobolev type regularity of the form H s x , since the harmonic oscillator rotates the phase space (see e.g. [START_REF]On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential[END_REF]). Instead, we consider the domain of the fractional harmonic oscillator: for s 0 a real number, let

Σ s x = f ∈ L 2 (R), f Σ s x := f L 2 (R) + -∂ 2 x + x 2 s/2 f L 2 (R) < ∞ .
We know from [3, Theorem 2.1] that (for any nonnegative real number s) the above norm enjoys the following equivalence,

f Σ s x ∼ f L 2 (R) + f Ḣs (R) + |x| s f L 2 (R) .
In particular, Σ s

x is a Banach algebra as soon as s > 1/2. Lemma 6. .

Proof. For s = 0, the result is a particular case of Proposition 2.2 (or Proposition 3.1). For s = 1, apply the vector-fields A 1 and A 2 to obtain the result from the case s = 0 and the identity (4.2). The case 0 < s < 1 follows by interpolation.

Morawetz estimate yields |u| 2 ∈ L 2 t Ḣ1 y L 1

x . In view of the one-dimensional inequality

f L ∞ y f 1/3 L 1 y ∂ y f 2/3 L 2 y ,
we infer the analogue of the property established in [START_REF] Vega | Bilinear virial identities and applications[END_REF], u ∈ L 6 t L ∞ y L 2 x . On the other hand, the conservation of mass and energy yields

u ∈ L ∞ t H 1/2-α y Σ 1/2+α x , 0 α 1 2 .
Sobolev embedding yields (d -n = 1)

u ∈ L ∞ t L 1/α y Σ 1/2+α x , 0 α 1 2 .
By interpolation, we infer

u ∈ L 6/α t L 1/(α(1-α)) y Σ 1/2+δ x , 0 < α < 1 2 , δ = (1 -α) 1 2 + α - 1 2 > 0.
The lemma is straightforward, since u(t, x, y) = e itH1 f (x) ⊗ e itH2 g(y).

Following the formal argument given in [START_REF] Ginibre | An introduction to nonlinear Schrödinger equations[END_REF], the above lemma suggests that for σ 1/(d -n), long range effects are present in (1.3). Assume λ = 1 for simplicity. Let u ∈ C([T, ∞); L 2 (R d )) be a solution of (1.3) such that there exists u + ∈ L 2 (R d ) with u + (x, y) = f (x)g(y) and u(t) -e -itH u + L 2 -→ t→+∞ 0.

Formal computations indicate that necessarily, u + ≡ 0 and u ≡ 0: the linear and nonlinear dynamics are no longer comparable, due to long range effects. To see this, let for F = e -itH1 ψ 1 |e -itH1 f | 2σ e -itH1 f and G = F 2 ψ 2 |F 2 g| 2σ F 2 g. With the change of variable y → ty, the above integral is equal to

ψ = ψ 1 ⊗ ψ 2 with ψ 1 ∈ C ∞ 0 (R n ), ψ 2 ∈ C ∞ 0 (R d-n ),
1 t (d-n)σ R n F (t, x)dx R d-n
G(y)dy . Now F is periodic in time, with arbitrary mean value. But t → 1/t (d-n)σ is not integrable, so the above quantity is not integrable in time, unless u + = 0. The conservation of mass then implies u ≡ 0.

1 .

 1 Motivation. It is well known that the solution to(1.1)

Theorem 1 . 1 .

 11 Let d 1, λ ∈ C, σ > 0 with σ < 2/(d -2) if d 3.

t 2 ∆ 6 .

 26 u(t) -Existence of long range effects. Suppose σ 1/d. If u ∈ C([T, ∞); L 2 (R d ) solution to (1.4) and u + ∈ L 2 (R d ) satisfy

6. 2 .

 2 The case d = 4. In addition to the global existence result stated in Proposition 1.4, Proposition 1.9 yields the information u

hence

  Theorem 1.10 in the case d = 3.

4 .

 4 Let d 2, 1 n d -1, and 0 s 1. Then for all u 0 ∈ L 2 y (R d-n ; Σ sx ), all (d -n)-admissible pairs (p 1 , r 1 ) and (p 2 , r 2 ), there exists C r1,r2 such that for all interval I ∋ 0, the solution to i∂ t u = Hu + F, u |t=0 = u 0 ,

  The case d = 3. We now have the a priori information|∇ y | 1/2 |u| 2 ∈ L 2We then proceed as in the cased = 4. Notice that (4, 4) is 2-admissible. For B ∈ {Id, A 1 , A 2 , A 3 },and we note that 0 < θ < 1/2 since 1 < σ < 2. We interpolate again to introduce the quantity controlled by Morawetz estimate, Notice that y ∈ R 2 , so by Sobolev embedding,

						u	L ∞ t L	8(3σ-2) 2-σ y	x L 2	u L ∞ t H 1 y L 2 x	u 0 Σ ,
	where the last inequality stems from Proposition 1.4. We can then conclude as in the case
	d = 4, by considering first
								h(t) =
														3
	y L 2 x ). Theorem 1.10 follows for d = 4, thanks to the classical yields A 4 u ∈ L p (R; L r arguments recalled in Section 4.
	6.3. t L 2 y L 1 x . In
	view of Sobolev embedding, we infer
			u 2 L 4 t L 8 y L 2 x	= |u| 2	L 2 t L 4 y L 1 x	|∇ y | 1/2 |u| 2	L 2 t L 2 y L 1 x	< ∞.
					Bu L 4 t L 4 y L 2 x			t u 0 Σ + |u| 2σ Bu L 4/3 u 0 Σ + u 2σ L 4σ t L 4σ y L ∞ x	L Bu L 4 4/3 y L 2 x t L 4 y L 2 x .
	We interpolate the nonlinear potential by
			u L 4σ t L 4σ y L ∞ x	u θ L	8σ 2-σ t	L	8σ 2-σ y	L 2 x	u 1-θ L 4 t L 4 y H 1 x	, θ =	2(σ -1) 3σ -2	,
	u	L	8σ 2-σ t	L	8σ 2-σ y	L 2 x	u 1-α L ∞ t L	8(3σ-2) 2-σ y	L 2 x	u α L 4 t L 8 y L 2 x	, α =	2 -σ 2σ	∈ (0, 1).

  and t 2 t 1 T . By assumption,ψ, e it2H u(t 2 ) -e it1H u(t 1 ) = -i t2 t1 e -itH ψ, |u| 2σ u (t) dtgoes to zero as t 1 , t 2 → +∞. The above lemma implies that for t → +∞, we havee -itH ψ, |u| 2σ u (t) ≈ 1 t (d-n)(σ+1)

	R n	F (t, x)dx	R d-n	G	y t	dy,

Acknowledgments. P. Antonelli and J. Drumond Silva would like to thank the kind hospitality of the Institute of Mathematics at the Université Montpellier 2, where part of this work was developed. The authors are grateful to Nicola Visciglia for fruitful discussions.

This work was supported by the French ANR project SchEq (ANR-12-JS01-0005-01). J. Drumond Silva was partially supported by the Center for Mathematical Analysis, Geometry and Dynamical Systems-LARSys through the Fundac ¸ão para a Ciência e Tecnologia (FCT/Portugal) Program POCTI/FEDER. 1 H 1 = e -i t 2 ∆ u(t) -u - H 1 -→

For α ∈ (0, 1/2) to be fixed, and (p, r) a 1-admissible pair to fix too, write

Hx

,

where

, and η ∈ (0, 1). We have used Hölder inequality, with

Recalling that (p, r) is 1-admissible, we infer that necessarily,

Now let 0 < ε ≪ 1, and define η by the identity

Since σ > 2, for ε sufficiently small, this defines indeed η ∈ (0, 1). Then, up to decreasing ε, fix 0 < α < 1/2 so that

To define all the parameters, we now recall that r is given by ( 6.3): we check that 2 < r < 6, with r ≈ 6. Once all these parameters are fixed, Strichartz inequality yields

y Hx . Following the same argument as in other dimensions, we infer u ∈ L p (R; L r y H x ), and thus u ∈ L p1 (R; L r1 y H x ) for all 1-admissible pairs (p 1 , r 1 ). For B ∈ {A 0 , . . . , A 4 }, we also have, on any interval I = [t 0 , t 1 ],

where we have used the embedding

x ) for all 1-admissible pairs (p 1 , r 1 ). Theorem 1.10 then follows in the case d = 2. where F 2 stands for the partial Fourier transform with respect to y.

APPENDIX A. EXISTENCE OF LONG RANGE EFFECTS: FORMAL