STRICHARTZ ESTIMATES FOR N -BODY SCHR¨ODINGER OPERATORS WITH SMALL POTENTIAL INTERACTIONS

. In this paper, we prove Strichartz estimates for N -body Schr¨odinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the V pS -norm of Koch and Tataru [19]. As an application, we prove scattering for N -body Schr¨odinger operators.


Introduction. Let d ¥ 3. We consider the N -body Schrödinger equation in
where Under suitable assumptions, H N is a self-adjoint operator, and thus Stone's theorem implies that the equation is globally well-posed in L 2 x , and that the N -body Schrödinger propagator is unitary, i.e., The purpose of this article is to investigate time decay properties of solutions to the N -body Schrödinger equation (1).
When there is no potential interaction, using the explicit integral kernel formula for the free Schrödinger propagator e it∆x , one can show the dispersive estimate As a consequence, the space-time norm bounds, namely Strichartz estimates, hold for all dN -dimensional admissible pairs pq, rq [16]. Here, an exponent pair pq, rq is called D-dimensional admissible if 2 ¤ q, r ¤ V, pq, r, Dq $ p2, V, 2q and 2 q D r D 2 .
In a physical perspective, it is natural to ask whether similar estimates hold in the presence of interaction potentials. In the one-body case, the research on the decay properties of the propagator e itp∆x¡V pxqq has a long history (see [15,3,4,21,11,9,10,1], the survey [22] and the references therein). In the N -body case, the dispersive and Strichartz estimates follow from the kernel estimate in [8] for a large class of potential interactions. However, they are valid only on a finite time interval. To the best of the author's knowledge, there is no such result for the N -body equation on the whole time interval.
In this paper, as a first step, we establish global-in-time Strichartz estimates for an N -body Schrödinger operator, assuming small particle interactions.
wherex α is the pN ¡ 1q spatial variables except the α-th variable x α , i.e., x . Indeed, we will prove an even stronger space-time norm bound in the V p ∆x -norm introduced by Koch and Tataru [19] (see Section 4 for the definition and the properties). Theorem 1.2 (Strichartz estimates for an N -body Schrödinger operator in the V p ∆x -norm). Let d ¥ 3 and p p1, 2q. There exists a small number ¡ 0 such that if }V } x . As an application, we prove scattering for an N -body Schrödinger operator with rough small interactions.
Corollary 1 (Scattering). Let d ¥ 3 and, let ¡ 0 be a small constant given in We prove Strichartz estimates for an N -body Schrödinger operator via the endpoint Strichartz estimates for the free linear propagator. Motivated by the simple proof of Strichartz estimates for e itp∆¡V pxqq in Section 2, we introduce the Strichartz estimates for the N -particle free propagator e it∆x freezing pN ¡ 1q spatial variables (Proposition 1). However, as described in Section 3, such estimates are not sufficient due to lack of symmetries of the space-time norms with respect to interchanges of variables. In Section 4 and 5, we solve this problem using the V p ∆x -norm, particularly taking an advantage of the symmetry of the norm. The novelty of this article is to use of the U p -and V p -spaces for many-body systems.
Our main result is the first step to study dispersive and Strichartz estimates for N -body Schrödinger operators with general short range interactions, as well as the related scattering problem [23,12]. The spectral properties of an N -body Schrödinger operator are much more complicated without smallness assumption. Nevertheless, our approach might be useful to include large interactions in the future.
We also note that other Strichartz estimates, introduced in this paper, can be employed for general dispersive equations having several spatial variables. For instance, Strichartz estimates with frozen variables (Proposition 1) are used in the study of dispersive equations in the Heisenberg picture [5]. In [7], X. Chen and Holmer proved the Klainerman-Machedon conjecture on the BBGKY hierarchy, which is a key step for rigorous derivation of the nonlinear Schrödinger equation [18,17,6]. One of their key estimates is the many-particle Strichartz estimates in the so-called Bourgain space X s,b [7, Lemma 4.1]. The Strichartz estimates in the V p ∆x (in Section 4) sharpen the bounds in X s,b by 0 in that Strichartz estimates in the X s,b space do not cover the endpoint Strichartz estimates, while those in the V p ∆x -space do.

2.
Single-particle case. The easiest way of proving Strichartz estimates for a linear propagator in the presence of small potentials is to use the endpoint Strichartz estimates for the free linear propagator. In this section, in order to illustrate this strategy, we provide a proof of Strichartz estimates in the single-particle case.
For the proof, recall the Strichartz estimates for the free linear propagator e it∆ in [16].

Theorem 2.2 (Strichartz estimates).
There exists c 0 ¡ 0 such that for d-admissible pairs pq, rq and pq,rq, where L r,s is the Lorentz norm (see [2] for the definition).
Proof of Theorem 2.1. It suffices to prove the theorem in the endpoint case pq, rq p2, 2d d¡2 q, since the full set of Strichartz estimates can be obtained by interpolating with the trivial bound }e itp∆¡V q u 0 } L V tR L 2

YOUNGHUN HONG
Let uptq e itp∆¡V q u 0 . Then, it solves the integral equation Applying the endpoint Strichartz estimates (Theorem 2.2) to the Duhamel formula and the Hölder inequality in the Lorentz spaces (see [2]), we get ( Thus, the endpoint Strichartz estimate for e itp∆¡V q follows from the smallness assumption and the embedding L r,2 ã Ñ L r for r ¥ 2. 3. Strichartz estimates for e it∆x with frozen spatial variables. We observed that the endpoint Strichartz estimates for the free linear propagator imply Strichartz estimates for a linear propagator with a small perturbation. However, it is easy to see from the Duhamel formula (see (4) below) that the dN -dimensional endpoint Strichartz estimates (2) do not suit well in the N -body case. Instead, the following Strichartz estimates seem more natural for our purpose.
Proposition 1 (Strichartz estimates for e it∆x with frozen variables). Let d ¥ 3, and let c 0 ¡ 0 be the constant for Theorem 2.2. Then, for 1 ¤ α ¤ N and ddimensional admissible pairs pq, rq and pq,rq, we have Proof. The proof is identical to that of [5, Theorem 3.1], but we give a proof for completeness of the paper.
We identify a complex-valued function f pxq : xα of one-spatial variable in L r,2 xα . Let r ¥ 2. Using unitarity of the linear propagator e it∆x α in L 2 xα , Then, by the dispersive estimate we obtain }e it∆x u 0 } L r xα pL 2 xα q À 1 |t| dp 1 The proposition follows from Theorem 10.1 in [16] with B 0 L 2 xα pL 2 xα q, B 1 L 1 xα pL 2 xα q, H L 2 xα pL 2 xα q and σ d 2 .

5359
Proposition 1 is, however, still not sufficient to complete the proof of Theorem 1.2. To see this, we consider the Duhamel formula the N -body Schrödinger equation (1), We define the rotation operator R αβ by (5) Then, by Proposition 1, we can estimate one term in the integral term in (4), with the exponents in (3) as follows: Rα 0β0 where in the first identity, we used that the linear propagator e ipt¡sq∆x commutes with a rotation. However, one cannot estimate other integral terms, with pα, βq $ pα 0 , β 0 q, in (6) by the same norm }R α0β0 ¤ } 4. Strichartz estimates for e it∆x in the V p ∆x -norm. To overcome the problem mentioned in the previous section, we look for a space-time norm that plays the role of the rotated space-time norm }R αβ ¤ } L q tR L r,2 xα L 2 xα in (7) as well as dominates such norms for all 1 ¤ α β ¤ N . It turns out that the U p ∆x -and the V p ∆x -spaces of Koch and Tataru (see (8)) have the desired properties.

A brief review of the
We denote by up¨Vq the limit of uptq as t Ñ¨V if it exists. Let 1 ¤ p V. We call aptq : where tt k u K k0 Z, tφ k u K¡1 k0 H, φ 0 0 and°K ¡1 k0 }φ k } p H 1. We define the atomic space by We define V p  piiq The embedding U p ã Ñ V p ¡,rc is continuous.
pivq The embedding V p ¡,rc ã Ñ U q is continuous. Lemma 4.3 (Duality; Proposition 2.7 and Theorem 2.8 in [13]). Let 1 p V. piq For u U p and v V p I and a partition t : tt k u K k0 Z, we define B t pu, vq : There exists a unique complex number Bpu, vq such that for any ¡ 0, there exists t Z such that for every t I t, |B t Ipu, vq ¡ Bpu, vq| The bilinear form B : U p ¢ V p I Ñ C satisfies |Bpu, vq| ¤ }u} U p }v} V p . piiq pU p q ¦ V p I in the sense that T : V p I Ñ pU p q ¦ , defined by T pvq Bp¤, vq, is an isometric isomorphism.
Let 1 ¤ p V. For a self-adjoint operator S, we define U p S (and V p ¡,rc,S respectively) by e i¤S U p (and e i¤S V p ¡,rc , respectively) with the norm }u} U p S : }e ¡itS u} U p }u} V p S : }e ¡itS u} V p , respectively¨.
In this paper, we choose H L 2 x and S ∆ x . For notational convenience, we Strichartz estimates for e it∆x in the V p ∆x -norm. We now establish Strichartz estimates in the V p ∆x -norm, which are the key estimates to prove the main theorem. In the literature, the U p S -space is typically chosen to be the solution space of the given equation. However, we employ the V p ∆x -space for the space of solutions to the N -body Schrödinger equation (1), because it is easier to prove the integral estimate (Proposition 4). Indeed, by Lemma 4.1 and 4.2, one can replace the V p ∆x -norms in the following propositions by U p ∆x -norms with arbitrarily small loss of the exponent p, and such losses are acceptable for our purpose.
First, we prove homogeneous Strichartz estimate in V p ∆x .
Proof. The total variation of e ¡it∆x p1 r0, Vq e it∆x u 0 q 1 r0, Vq u 0 is simply u 0 . Hence, by definition, we have }1 r0, Vq e it∆x u 0 } V p ∆x }u 0 } L 2 x . Next, we prove the transference principle.
Proposition 3 (Transference principle). Let d ¥ 1, p p1, 2q, q ¥ 2 and X be a Banach space. If a function u : R Ñ X satisfies the bound }e it∆x u 0 } L q tR X À }u 0 } L 2 x , then }u} L q tR X À }u} V p ∆x .
Proof. By Lemma 4.2 piiq and density, it suffices to show that }u} L q tR X À }u} U 2 ∆x for an U 2 ∆ -atom aptq °K k1 1 rt k¡1 ,t k q e it∆x φ k¡1 . Using that the interval rt k¡1 , t k q's are mutually disjoint, we write Then, by the assumption and the embedding 2 ã Ñ q for q ¥ 2, it follows that As a consequence, we show that the V p ∆x -norm dominates the space-time norms in Proposition 1 and (2). Corollary 2. Let d ¥ 1, p p1, 2q, and R be a rotation in R 3N x . Then, we have for a d-dimensional admissible pair pq, rq and 1 ¤ α ¤ N . Moreover, }Ru} L q tR L r,2 x ¤ c 1 }u} V p ∆x for a dN -dimensional admissible pair pq, rq.
Finally, we prove the integral estimate analogous to (7).
Proposition 4 (Integral estimate in V p ∆x ). Let d ¥ 3 and p p1, 2q. Then, we have Proof. For notational convenience, we denote We will estimate w by duality. Since we only expect w V p ¡ , not w V p , we considerwptq wp¡tq. Note that w V p ¡ if and only ifwptq V p , and that by Lemma 4.3 piiq, Hence, by Lemma 4.3 piq and density, it suffices to show that for any fine partition of unity t tt j u J j0 Z and any U p I -atom aptq °K k1 1 rs k¡1 ,s k q φ k¡1 . Here, including s 0 , ¤ ¤ ¤ , s K if necessary, we may assume that t tt j u J j0 contains s 0 , ¤ ¤ ¤ , s K . Moreover, due to the cut-off 1 r0, Vq in (9), we also assume that t 0 ¡V, t J V and t 1 , ¤ ¤ ¤ , t J¡1 p¡V, 0q.
Inserting this sum into B t pa,wq in (10), we get a simpler sum B t pa,wq Ķ k1 xφ k¡1 ,wps k q ¡wps k¡1 qy L 2 x .
By (9) withwptq wp¡tq, unitarity of the linear propagator e is∆x and change of the variables by the rotation R αβ , we write where in the last step, we used that u V p ∆x is right-continuous. Then, applying the Hölder inequality, Proposition 1 and Corollary 2, we obtain .
It remains to show that Indeed, by the definition of the norm, given ¡ 0 and k, there exists a partition t k tt j k u J k j k 1 in the interval r¡s k , ¡s k¡1 q such that }1 rt k¡1 ,t k q u} p V p ∆x J ķ j k 1 }e itj k ∆x upt j k q ¡ e itj k ¡1∆x upt j k ¡1 q} p L 2 K .
Since K k1 t k is also a partition of unity of R, by the definition of the norm, we have Ķ k1 }1 rt k¡1 ,t k q u} p V p ∆x Ķ k1 J ķ j k 1 }e itj k ∆x upt j k q ¡ e itj k ¡1∆x upt j k ¡1 q} p Since is arbitrary, this completes the proof.

5.
Proof of Theorem 1.2 and Corollary 1. Now, we are ready to prove our main results.
Proof of Corollary 1. We consider only for the positive time. It suffices to show that u : lim tÑ V e ¡it∆x e ¡itH N u 0 exists in L 2 x as t Ñ V. Indeed, by the Duhamel formula (4), }e ¡it2∆x e ¡it2H N u 0 ¡ e ¡it1∆x e ¡it1H N u 0 } L 2 x ¤1 ¤α β¤N » t2 t1 e ¡is∆x pV px α ¡ x β qe ¡isH N u 0 qds By by the rotation R αβ , Proposition 1 and Corollary 1.2, we prove that » t2 t1 e ¡is∆x pV px α ¡ x β qe ¡isH N u 0 qds L 2 x Rαβ » t2 t1 e ¡is∆x pV px α ¡ x β qe ¡isH N u 0 qds L 2 x » t2 t1 e ¡is∆x pV p c 2x α qpR αβ e ¡isH N u 0 qqds as t 1 , t 2 Ñ V. Thus, we conclude that the limit exists.