SCATTERING FOR A NONLINEAR SCHR¨ODINGER EQUATION WITH A POTENTIAL

. We consider a 3d cubic focusing nonlinear Schr¨odinger equation with a potential i where V is a real-valued short-range potential having a small negative part. We ﬁnd criteria for global well-posedness analogous to the homogeneous case V “ 0 [10, 5]. Moreover, by the concentration-compactness approach, we prove that if V is repulsive, such global solutions scatter.


Introduction.
1.1. Setup of the problem. We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential iB t u`∆u´V u`|u| 2 u " 0, up0q " u 0 P H 1 , (NLS V ) where u " upt, xq is a complex valued function on RˆR 3 . We assume that V " V pxq is a time independent real-valued short range potential having a small negative part.
To be precise, we define the potential class K 0 as the norm closure of bounded and compactly supported functions with respect to the global Kato norm }V } K :" sup |V pyq| |x´y| dy, and denote the negative part of V by V´pxq :" minpV pxq, 0q.
Throughout this paper, we assume that and }V´} K ă 4π. (1.2) By the assumptions p1.1q and p1.2q, the Schrödinger operator H "´∆`V has no eigenvalues, and the solution to the linear Schrödinger equation iB t u`∆u´V u " 0, up0q " u 0 (1.3)

YOUNGHUN HONG
satisfies the dispersive estimate [2] and Strichartz estimates. As a consequence, a solution uptq to (1.3) scatters in L 2 (see Lemma 2.9), in the sense that there exists u˘P L 2 such that lim tÑ˘8 }uptq´e it∆ u˘} L 2 " 0.
On the other hand, Holmer-Roudenko [10] and Duyckaerts-Holmer-Roudenko [5] obtained the sharp criteria for global well-posedness and scattering for the homogeneous 3d cubic focusing nonlinear Schrödinger equation iB t u`∆u`|u| 2 u " 0, up0q " u 0 P H 1 (1.4) in terms of conservation laws of the equation. Here, by homogeneity, we mean that V " 0. Motivated by the linear and nonlinear scattering results, it is of interest to investigate the effect of a potential perturbation on the scattering behavior of solutions to the nonlinear equation pNLS V q.
By the assumptions p1.1q and p1.2q, the Cauchy problem for pNLS V q is locally well-posed in H 1 . Moreover, every H 1 solution obeys the mass conservation law, M ruptqs " ż R 3 |uptq| 2 dx " M ru 0 s and the energy conservation law, The goal of this paper is to find criteria for global well-posedness and scattering in terms of the above two conserved quantities. Here, we say that a solution uptq to pNLS V q scatters in H 1 (both forward and backward in time) if there exist ψ˘P H 1 such that lim tÑ˘8 }uptq´e´i tH ψ˘} H 1 " 0.
Note that by the linear scattering (Lemma 2.9), if the solution uptq to pNLS V q scatters in H 1 , then there exist ψ0 P L 2 such that lim tÑ˘8 }uptq´e it∆ ψ0 } L 2 " 0.

1.2.
Criteria for global well-posedness. In the first part of this paper, we find criteria for global well-posedness. As in the homogeneous case pV " 0q, such criteria can be obtained from the variational problem that gives the sharp constant for the Gagliardo-Nirenberg inequality, When V " 0, the sharp constant is attained at the ground state Q solving the nonlinear elliptic equation ∆Q´Q`Q 3 " 0.
(1.5) The following proposition is analogous to the variational problem in the inhomogeneous case. Proposition 1.1 (Variational problem). Suppose that V satisfies p1.1q and p1.2q. piq If V´" 0, then the sequence tQp¨´nqu nPN maximizes W V puq, where Q is the ground state for the elliptic equation (1.5). piiq If V´‰ 0, then there exists a maximizer Q P H 1 solving the elliptic equation

6)
Moreover, Q satisfies the Pohozhaev identities, where H 1{2 is defined by the spectral theorem and A related classical problem is to prove existence of ground states in the semiclassical setting [6,1], which is, by change of variables, equivalent to p´∆`V p ¨qqu `ω 2 u ´|u | 2 u " 0 (1. 8) for sufficiently small ą 0, where V is smooth and inf xPR 3 pω 2`V p xqq ą 0. In [1], considering the equation (1.8) as a perturbation of ∆u`pω 2`V p0qqu´|u| 2 u " 0, the authors found a ground state using a perturbation theorem in critical point theory. On the other hand, the ground state Q in Proposition 1.1 piiq is obtained via the concentration-compactness approach based on profile decomposition [8,9]. From this, we obtain a ground state even when V´is not point-wisely bounded, while V´is still small in the global Kato norm.
Remark 1.2. piq The ground state Q is special in that it satisfies the "exact" Pohozhaev identities. In general, solutions to p1.6q satisfy the Pohozhaev identities with extra terms (see Section 4.2). These exact identities will be crucially used to find criteria for global well-posedness. piiq It is of interest to investigate uniqueness of the ground state Q as in the free potential case [17,18,19]. However, we do not pursue this problem in this paper, since uniqueness is not necessary in proving our main theorem.
To state the main results, we need to introduce the following notation, where E 0 rus is the energy without a potential Our first main theorem provides criteria for global well-posedness in terms of the mass-energy ME and a critical number α.
Let uptq be the solution to pNLS V q with initial data u 0 P H 1 .
piq If then uptq exists globally in time, and piiq If during the maximal existence time.
1.3. Criteria for scattering. The second part of this paper is devoted to investigating the dynamical behavior of global solutions in Theorem 1.3 piq. In the homogeneous case, Duyckaerts, Holmer and Roudenko [5] proved that every global solution in Theorem 1.3 piq has finite Sp 9 H 1{2 q norm (see (2.1)) and, as a consequence, it scatters in H 1 . Motivated by this work, we formulate the following scattering conjecture for the perturbed equation pNLS V q. H 1{2 q-norm, and it scatters in H 1 .
To prove the scattering conjecture, we employ the robust concentration-compactness approach. This method has been developed by Colliander-Keel-Staffilani-Takaoka-Tao for the 3d quintic defocusing nonlinear Schrödinger equation and Kenig-Merle for the energy-critical focusing nonlinear Schrödinger and wave equations [13,14]. It has been successfully applied to solve scattering problems in various settings.
The method of concentration-compactness can be adapted to pNLS V q as follows. We assume that the scattering conjecture is not true, and the there is a threshold mass-energy ME c that is strictly less than ME. Then, we attempt to deduce a contradiction in three steps.
Step 1. Construct a special solution u c ptq, called a minimal blow-up solution, at the threshold between scattering and non-scattering regimes.
Step 2. Prove that the solution u c ptq is precompact in H 1 .
Step 3. Eliminate a minimal blow-up solution by the localized virial identities and the sharp Gagliardo-Nirenberg inequality.
First, assuming that the scattering conjecture is false, we construct a minimal blow-up solution (Step 1) and show that it satisfies the compactness properties (Step 2). Theorem 1.6 (Minimal blow-up solution). If Conjecture 1.5 fails, then there exists a global solution u c ptq such that M ru c,0 sEru c,0 s ă ME, }u c,0 } L 2 }H 1{2 u c,0 } L 2 ă α and }u c ptq} Sp 9 H 1{2 q " 8, where u c,0 " u c p0q. Moreover, u c ptq is precompact in H 1 .
The proof of Theorem 1.6 depends heavily on linear profile decomposition. However, since a potential perturbation breaks the symmetries of the both linear and the nonlinear Schrödinger equation, we need to modify the linear profile decomposition (Proposition 5.1) and its applications. We remark that similar modifications appear in [16], where the authors established scattering for the defocusing energy critical nonlinear Schrödinger equation in the exterior of a strictly convex obstacle.
For the scattering conjecture, we give a partial answer by eliminating a minimal blow-up solution (Step 3), provided that a potential is repulsive. Theorem 1.7 (Scattering, when V is repulsive). Suppose that V satisfies p1.1q and p1.2q. We also assume that V ě 0, x¨∇V pxq ď 0 and x¨∇V P L 3{2 . If To prove Theorem 1.7, we terminate a minimal blow-up solution employing the localized virial identity where χ P C 8 c is a radially symmetric function such that χpxq " |x| 2 for |x| ď 1 and χpxq " 0 for |x| ě 2, and χ R :" R 2 χpR q for R ą 0 (see Proposition 7.1). To this end, the right hand side of p1.9q has to be coercive. However, it may not be coercive due to the last term in p1.9q, (1.10) The repulsive condition guarantees p1.10q to be non-negative. The repulsiveness assumption on the potential V in Theorem 1.7 is analogous to the convexity of the obstacle Ω in [16]. In both cases, once wave packets are reflected by a potential or a convex obstacle, they never be refocused. However, unlike the obstacle case, if the confining part of a potential is not strong, then the dynamics of wave packets may not be changed much. Indeed, scattering for the linear equation An interesting open question is whether the repulsive condition in Theorem 1.7 is necessary for large data scattering in nonlinear settings. For this question, we address the following remarks.
Note that scattering for the linear Schrödinger equation (1.3) can be obtained without using the virial identities. Thus, the localized virial identity may not be the best tool to eliminate a minimal blow-up. piiiq In the presence of a repulsive potential, one cannot construct a finite-time blow-up via the virial identity as in [10,5]. Indeed, in this approach, negativity of p1.10q is required. pivq In the defocusing case, scattering is proved in the presence of more general magnetic potentials via the interaction Morawetz inequality [4].
1.4. Organization of the paper. In §2, we collect preliminary estimates to deal with a linear operator e itp∆´V q , and record relevant local theories. In §3, we solve the variational problem (Proposition 1.1). In §4, using the variational problem, we obtain the upper-bound versus lower-bound dichotomy (Theorem 1.3). In §5-7, we carry out the concentration-compactness argument with several modifications to overcome the broken symmetry. To this end, in §5, we establish the linear profile decomposition associated with the scaled linear propagator (Proposition 5.1). Then, we construct a minimal blow-up solution (Theorem 1.6) in §6. Finally, in §7, we prove scattering by excluding the minimal blow-up solution, provided that the potential is repulsive (Theorem 1.7).
1.5. Notations. We denote by NLS V ptqu 0 the solution to pNLS V q with the initial data u 0 . For r ą 0 and a P R 3 , we define V r,a :" 1 r 2 V p¨´a r q and H r,a :"´∆`V r,a .

Preliminaries.
2.1. Strichartz estimates and norm equivalence. We record preliminary tools to analyze the perturbed linear propagator e´i tH " e itp∆´V q .
First, we recall the dispersive estimate for the linear propagator e´i tH , but for simplicity, we assume that the negative part of a potential is small.
Proof. By Beceanu-Goldberg [2], it suffices to show that H doesn't have an eigenvalue or a nonnegative resonance. By Lemma A.1, H is positive, and thus it has no negative eigenvalue. Moreover, by Ionescu-Jerison [11], there is no positive eigenvalue or resonance.
By the arguments of Keel-Tao [12] and Foschi [7] in the abstract setting, one can derive Strichartz estimates from the dispersive estimate and unitarity of the linear propagator e´i tH . For notational convenience, we introduce the following definitions. We say that an exponent pair pq, rq is called 9 H s -admissible (in 3d) if 2 ď q, r ď 8 and 2 q`3 r " 3 2´s .
We define the Strichartz norm by Here, 2´is an arbitrarily preselected and fixed number ă 2; similarly for 3`. If the time interval I is not specified, we take I " R.
Remark 2.2. The ranges of exponent pairs in the Sp 9 H 1{2 q-norm and the S 1 p 9 H´1 {2 qnorm are chosen to satisfy the conditions in Theorem 1.4 of Foschi [7]. Note that p2, 3q is not included in S 1 p 9 H´1 {2 q, since it is not H´1 2 -admissible. If pq, rq " p4, 6q and pq,rq " p 4 3 , 6q, the sharp condition holds. Otherwise, pq, rq and pq,rq satisfy the non-sharp condition.
Remark 2.5. Keel-Tao and Foschi assumed the natural scaling symmetry (see (12) of [12] and Remark 1.5 of [7]). However, the same proof works without the scaling symmetry.
The following lemma says that the standard Sobolev norms and the Sobolev norms associated with H are equivalent for some exponent r. This norm equivalence lemma is crucial to establish the local theory for the perturbed nonlinear Schrödinger equation pNLS V q in Section 2.2.

YOUNGHUN HONG
For the proof, we need the Sobolev inequality associated with H.
This implies that }pa`Hq´s 2 f } L q À }f } L p with p, q, s in Lemma 2.7.
Indeed, since the heat kernel operator e´t H obeys the gaussian heat kernel estimate (see the proof of Lemma 2.7), these bounds follow from Sikora-Wright [22].
Combining the above two claims, we obtain that for 1 ă r ă 8 when Re z " 0 and for 1 ă r ă 3 2 when Re z " 1. Finally, applying the Stein-Weiss complex interpolation, we prove the norm equivalence lemma.
Remark 2.8. The range of exponent r in (2.2) is known to be sharp when s " 1 [21].
As an application of Strichartz estimates and the norm equivalence, we obtain the linear scattering. Lemma 2.9 (Linear scattering). piq Suppose that V P K 0 X L 3{2 and }V´} K ă 4π. Then, for any ψ P L 2 , there existψ˘P L 2 such that piiq If we further assume that V P W 1,3{2 , then for any ψ P H 1 , there existψ˘P H 1 such that Proof. piq Observe that if uptq solves with initial data ψ, then it solves the integral equation Applying Strichartz estimates, we obtain where in the last step, we used the fact that }uptq} L 2 tPR L 6 x " }e it∆ ψ} L 2 tPR L 6 x À }ψ} L 2 ă 8 (by Strichartz estimates). Hence, the limits ψ˘" lim tÑ˘8 e itH e it∆ ψ exist in L 2 . Now, repeating the above estimates, we prove that piiq For scattering in H 1 , we need to use the norm equivalence lemma, since the linear propagator e itH and the derivative don't commute. First, by the norm equivalence, we get Applying the Strichartz estimates and the norm equivalence again, we obtain that x À }φ} H 1 . Therefore, the limits ψ˘" lim tÑ˘8 e itH e it∆ ψ exist in H 1 . Moreover, repeating the above estimates, we show that pe it∆ ψé´i tHψ˘q Ñ 0 in H 1 as t Ñ˘8.
Remark 2.10 (Scaling and spatial translation). Note that the implicit constants for the above estimates are independent of the scaling and translation V pxq Þ Ñ V r0,x0 " For example, let c " cpV q ą 0 be the sharp constant for Strichartz estimate. Then, by Strichartz estimate for e itp∆´V q , we have Since r 0 , x 0 and f are arbitrarily chosen, this proves that cpV r0,x0 q " cpV q for all r 0 ą 0 and x 0 P R 3 .

2.2.
Local theory. Now we present the local theory for the perturbed equation pNLS V q. We note that the statements and the proofs of the following lemmas are similar to those for the homogeneous equation pNLS 0 q (see [10,Section 2]). The only difference in the proofs is that the norm equivalence (Lemma 2.6) is used in several steps.
We claim that SpL 2 ;Iq . Indeed, by Strichartz estimates and the norm equivalence, we obtain Iq . Therefore, taking sufficiently small T ą 0, we conclude that Φ u0 is a contraction on Lemma 2.12 (Small data). For A ą 0, there exists δ sd " δ sd pAq ą 0 such that if }u 0 } 9 H 1{2 ď A and }e´i tH u 0 } Sp 9 H 1{2 q ď δ sd , then the solution uptq is global in 9 H 1{2 . Moreover, Proof. Let Φ u0 be in Lemma 2.11. By Strichartz estimates and the norm equivalence, . By the Kato Strichartz estimate and the Sobolev inequality (Lemma 2.7), x , and by Strichartz estimates, the norm equivalence and the fractional Leibniz rule, Therefore, we obtain that It follows from the local well-posedness (Lemma 2.11) that if a solution is uniformly bounded in H 1 during its existence time, then it exists globally in time. However, uniform boundedness is not sufficient for scattering. For instance, in the homogeneous case pV " 0q, there are infinitely many non-scattering periodic solutions [3]. The following lemma provides a simple condition for scattering. If uptq has finite Sp 9 H 1{2 q norm, then uptq scatters in H 1 as t Ñ˘8.
Proof. We define Indeed, such limits exist in H 1 , since by the norm equivalence and Strichartz estimates, x Ñ 0 (by Hölder inequality) as t 1 , t 2 Ñ˘8. Hence, ψ˘is well-defined. Then, repeating the estimates in p2.3q, we conclude that Lemma 2.14 (Long time perturbation lemma). For A ą 0, there exist 0 " 0 pAq ą 0 and C " CpAq ą 0 such that the following holds: Let uptq P C t pR; H 1 x q be a solution to pNLS V q. Suppose thatũptq P C t pR; H 1 x q is a solution to the perturbed pNLS V q iũ t´Hũ`|ũ | 2ũ " e satisfying }ũ} Sp 9 H 1{2 q ď A, }e´i pt´t0qH pupt 0 q´ũpt 0 qq} Sp 9 H 1{2 q ď 0 and }e} S 1 p 9 H´1 {2 q ď 0 . Then, }u} Sp 9 H 1{2 q ď C " CpAq. Proof. We omit the proof, since it is similar to that for [10, Proposition 2.3]. Indeed, as we observed in the proofs of the previous lemmas, one can easily modify the proof of [10, Proposition 2.3] using the norm equivalence (Lemma 2.6).
3. Variational problem. In this section, we prove Proposition 1.1. Precisely, we will find a maximizer or a maximizing sequence for the nonlinear functional Nonnegative potential. We will show Proposition 1.1 piq. If V ě 0, then one can find a maximizing sequence simply by translating the ground state Q for the nonlinear elliptic equation Indeed, the sharp constant for the standard Gagliardo-Nirenberg inequality is given by the ground state Q, precisely, Collecting all, we conclude that lim nÑ8 W V pQp¨´nqq ą W V puq, @u P H 1 .
Therefore, we conclude that tQp¨´nqu 8 n"1 is a maximizing sequence for W V puq.

3.2.
Potential having a negative part. We prove Proposition 1.1 piiq by two steps. First, we find a maximizer. Then, we show the properties of the maximizer.
Lemma 3.1 (Profile decomposition [9, Proposition 3.1]). If tu n u 8 n"1 is a bounded sequence in H 1 , then there exist a subsequence of tu n u 8 n"1 (still denoted by tu n u 8 n"1 ), functions ψ j P H 1 and spatial sequences tx j n u 8 n"1 such that for J ě 1, The profiles are asymptotically orthogonal: For j ‰ k, |x j n´x k n | Ñ 8 as n Ñ 8 and for 1 ď j ď J, R J n p¨`x j n q á 0 weakly in H 1 . (3.1) The remainder sequence is asymptotically small: Moreover, the decomposition obeys the asymptotic Pythagorean expansion We also use the following elementary lemma.

YOUNGHUN HONG
Let tu n u 8 n"1 be a maximizing sequence. Note that Lemma 3.1 cannot be directly applied to the sequence tu n u 8 n"1 , because tu n u 8 n"1 may not be bounded in H 1 . Hence, instead of tu n u 8 n"1 , we consider the following sequence. For each n, we pick α n , r n ą 0 such that }α n u n pr n q} 2 L 2 " α 2 n r 3 n }u n } 2 L 2 " 1, }H 1{2 rn α n u n pr n q} 2 L 2 " α 2 n r n }H 1{2 u n } L 2 " 1, where H r "´∆`1 r 2 V pr q. Since W V pαuq " W V puq, replacing tu n u 8 n"1 by tα n u n u 8 n"1 , we may assume that }u n pr n q} L 2 " 1 and }H 1{2 rn u n pr n q} L 2 " 1. Set u n " u n pr n q. Then, tũ n u 8 n"1 is a bounded sequence in H 1 , because by the norm equivalence, Step 1. ψ j " 0 for all j ě 2) We will show that ψ j " 0 for all j ě 2. For contradiction, we assume that ψ j ‰ 0 for some j ě 2. Extracting a subsequence, we may assume that r n Ñ r 0 P r0,`8s and x 1 n Ñ x 1 0 P R 3 Y t8u. By Lemma 3.1, we have 1 2 }ψ j } 2 9 H 1 ď }R 1 n } 2

9
H 1 ď C, 3) for all sufficiently large n. Let First, p3.4q follows from the asymptotic Pythagorean expansion in Lemma 3.1. For p3.5q, we write By p3.1q, the third term is o n p1q. It suffices to show that the last term is o n p1q. If r n Ñ r 0 P p0,`8q and x 1 n Ñ x 1 0 P R 3 , then ż where the last step follows from p3.2q and
By Lemma 3.2, it follows from p3.3q, p3.4q, p3.5q and p3.6q that This contradicts to the maximality of tu n u 8 n"1 . (Step 2. R 1 n Ñ 0 in H 1 ) Passing to a subsequence, we may assume that lim nÑ8 }R 1 n } H 1 exists. For contradiction, we assume that As in the proof of p3.5q, one can show that ż Moreover, by the asymptotic smallness of the remainder in Lemma 3.1, passing to a subsequence, we have }R 1 n } L 4 Ñ 0. Therefore, we get which contradicts maximality of tu n u 8 n"1 . Therefore, we should have R 1 n Ñ 0 in H 1 . (Step 3. Convergence of tx n u 8 n"1 and tr n u 8 n"1 ) So far, we proved that, passing to a subsequence, u n pxq " ψpr n x´x n q, where r n Ñ r 0 P r0,`8s and x n Ñ x 0 P R 3 Y t8u. Suppose that r n Ñ 0, r n Ñ`8 or x n Ñ 8. Then, by the "free" Gagliardo-Nirenberg inequality and the assumption, we have On the other hand, since V´‰ 0, there exist x˚P R 3 and a small ą 0 such that ş R 3 V Q 2 p¨´x˚ qdx ă 0. Thus, it follows that Combining two inequalities, we deduce a contradiction.

Pohozhaev identities.
For ω ą 0, let Q ω be a strong solution to Multiplying p3.9q by Q ω (and px¨∇Q ω q), integrating and applying integration by parts, we get Solving it as a system of equations for }H 1{2 Q ω } 2 , we see that the extra term should be zero.
but if V has nontrivial negative part, Then, it follows from the Gagliardo-Nirenberg inequality and the energy conservation law that where f pxq " x 2 2´x 3 3α and gptq " }u 0 } L 2 }H 1{2 uptq} L 2 . Observe that f pxq is concave for x ě 0 and it has a unique maximum at x " α, f pαq " α 2 6 " ME. Moreover, by H 1 -continuity of solutions to pNLS V q, gptq is continuous. Therefore, we conclude that either gptq ă α or gptq ą α.   Comparability of gradient and energy). In the situation of Theorem 1.3 piq, we have 2Eru 0 s ď }H 1{2 uptq} 2 L 2 ď 6Eru 0 s, @t P R. Proof. The first inequality is trivial. For the second inequality, by the energy conservation law, we obtain By the Gagliardo-Nirenberg inequality (with c GN " 4 3α ) and Theorem 1.3 piq, we obtain Therefore, by the energy conservation law, we conclude that Corollary 4.2 (Small data scattering). If }u 0 } H 1 is sufficiently small, then uptq " NLS V ptqu 0 scatters in H 1 as t Ñ˘8.

Proposition 4.3 (Existence of wave operators). If
then there exists unique u 0 P H 1 , obeying the assumptions in Theorem 1.3 piq, such that lim Proof. For sufficiently small ą 0, choose T " 1 such that }e´i tH ψ`} Sp 9 H 1{2 ;rT,`8qq ď . Then, as we proved in Lemma 2.12, one can show that the integral equation has a unique solution such that }x∇yu} SpL 2 ;rT,`8qq ď 2}ψ`} H 1 and }u} Sp 9 H 1{2 ;rT,`8qq ď 2 . Observe that by Strichartz estimates and the norm equivalence, Since ą 0 is arbitrarily small, this proves that }uptq´e´i tH ψ`} H 1 Ñ 0 as t Ñ`8. By the energy conservation law and Lemma 4.1, we obtain that M rupT qsErupT qs " lim Moreover, we have lim tÑ`8 Hence, for sufficiently large T , upT q satisfies the assumptions in Theorem 1.3 piq, which implies that uptq is a global solution in H 1 . Let u 0 " up0q. Then, uptq " NLS V ptqu 0 satisfies p4.1q for positive time. By the same way, one can show p4.1q for negative time.
5. Linear profile decomposition associate with a perturbed linear propagator. We establish the linear profile decomposition associated with a perturbed linear propagator. This profile decomposition will play a crucial role in construction of a minimal blow-up solution.
Proposition 5.1 (Linear profile decomposition). Suppose that r n " 1, r n Ñ 0 or r n Ñ 8. If tu n u 8 n"1 is a bounded sequence in H 1 , then there exist a subsequence of tu n u 8 n"1 (still denoted by tu n u 8 n"1 ), functions ψ j P H 1 , time sequences tt j n u 8 n"1 and spatial sequences tx j n u 8 n"1 such that for every J ě 1, u n " J ÿ j"1 e it j n Hr n pψ j p¨´x j n qq`R J n , (5.1) where H rn "´∆`V rn "´∆`1 rn V pr n q. The time sequences and the spatial sequences have the following properties. For every j, t j n " 0 or t j n Ñ 8, and x j n " 0 or x j n Ñ 8. and for 1 ď j ď J, pe´i t j n Hr n R J n qp¨`x j n q á 0 weakly in H 1 .

(5.5)
The remainder sequence is asymptotically small: Moreover, we have the asymptotic Pythagorean expansion: First, we prove the profile decomposition in the case that the potential V effectively disappears by scaling.
Proof of Proposition 5.1 when r n Ñ 0 or r n Ñ`8. By the profile decomposition associated with the free linear propagator [5,Proposition], tu n u 8 n"1 has a subsequence (but still denoted by tu n u 8 n"1 ) such that satisfying the properties in Proposition 5.1 with V " 0. Note that in p5.9q, we may assume that time sequences tt j n u 8 n"1 and spatial sequences tx j n u 8 n"1 satisfy p5.2q and p5.3q. Indeed, passing to a subsequence, we may assume that t j n Ñ t j P R Y t8u and x j n Ñ x j P R 3 Y t8u. If t j ‰ 8 (x j ‰ 8, resp), we replace e´i t j n ∆ ψ j (ψ j p¨´x j n q, resp) in p5.9q by e´i t j ∆ ψ j (ψ j p¨´x j q, resp). Then, this modified profile decomposition satisfies p5.2q as well as other properties in Proposition 5.1. Similarly, one can also modify p5.9q so that p5.3q holds. Now, replacing e´i t∆ by e itHr n , we write the profile decomposition u n " J ÿ j"1 e it j n Hr n pψ j p¨´x j n qq`R J n , (5.10) whereR J n " R J n`J ÿ j"1 e it j n Hr n pψ j p¨´x j n qq´e´i t j n ∆ pψ j p¨´x j n qq.
We claim that p5.10q has the desired properties. We will show p5.6q only. Indeed, the other properties can be checked easily by the properties obtained from p5.9q.
To this end, we observe that uptq " e it∆ u 0 solves the integral equation }e it j n Hr n pψ j p¨´x j n qq} 4 (5.15) For arbitrary small ą 0, let ψ j P C 8 c such that }ψ j´ψj } H 1 ď {J. Replacing ψ j by ψ j in p5.15q with Op q-error, one may assume that ψ j P C 8 c . First, we observe that }e it j n Hr n pψ j p¨´x j n qq} 4 Indeed, each cross term of its left left hand side is of the form ż R 3 e it j 1 n Hr n pψ j1 p¨´x j1 n qqe it j 2 n Hr n pψ j2 p¨´x j2 n qq e it j 3 n Hr n pψ j3 p¨´x j3 n qqe it j 4 n Hr n pψ j4 p¨´x j4 n qqdx. (5.16) If there is one j k such that t j k n Ñ 8, for example, say t j1 n Ñ 8, by the dispersive estimate, the Sobolev inequality and the norm equivalence, we have |p5.16q| ď }e it j 1 n Hr n pψ j1 p¨´x j1 n qq} L 4 ź k"2,3,4 }e it j k n Hr n pψ j k p¨´x j k n qq} L 4 Thus, for ą 0, there exists J 1 " 1 such that }R J1 n } L 4 ď for all sufficiently large n. Hence, we obtain }e it j n Hr n pψ j p¨´x j n qq} 4 }e it j n Hr n pψ j p¨´x j n qq} 4 6. Construction of a minimal blow-up solution. We define the critical massenergy ME c by the supremum over all such that Here, ME c is a strictly positive number. Indeed, by the Sobolev inequality, Strichartz estimates, the norm equivalence and comparability of gradient and energy (Proposition 4.1), we have M ru 0 sEru 0 s. Hence, it follows from the small data scattering (Corollary 4.2) that (6.1) holds for all sufficiently small ą 0. Note that the scattering conjecture (Conjecture 1.5) is false if and only if ME c ă ME.
In this section, assuming that the scattering conjecture fails, we construct a global solution having infinite Strichart norm }¨} Sp 9 H 1{2 q at the critical mass-energy ME c . Theorem 6.1 (Minimal blow-up). If Conjecture 1.5 is false, there exists u c,0 P H 1 such that M ru c,0 sEru c,0 s " ME c , }u c,0 } L 2 }H 1{2 u c,0 } L 2 ă α and }u c ptq} Sp 9 H 1{2 q " 8, where u c ptq is the solution to pNLS V q with initial data u c,0 .
For each j, if t j n Ñ 8, by Proposition 4.3 (with V " 0), we getψ j P H 1 such that If t j n " 0, we setψ j " ψ j . Replacing each linear profile by the nonlinear profile, we define the approximation ofũ n ptq by where v j pt, xq " NLS 0 ptqψ j . Letw J n ptq " NLS 0 ptqw J n p0q. We will show that there exists A 0 ą 0, independent of J, such that }w J n ptq} Sp 9 H 1{2 q ď A 0 (6.4) for all n ě n 0 " n 0 pJq. Indeed, we have E 0 rw J n p0qs " J ÿ j"1 E 0 rv j p´t j n ,¨´x j n qs`o n p1q (by orthogonality of pt j n , x j n q) " J ÿ j"1 E 0 re´i t j n ∆ ψ j p¨´x j n qs`o n p1q (by p6.3q when t j n Ñ 8) " J ÿ j"1 E Vr n re it j n Hr n pψ j p¨´x j n qqs`o n p1q (by the argument in p5.12q) ď E Vr n rũ n,0 s`o n p1q " r´1 n E V ru n,0 s`o n p1q (by Corollary 5.2).
Next, we claim that there exists A 1 ą 0, independent of J, such that }w J n ptq} Sp 9 H 1{2 q ď A 1 (6.7) for all n ě n 1 " n 1 pJq. To see this, we observe that w J n ptq solves iB t w J n`∆ w J n`| w J n | 2 w J n " e, where e " |w J n | 2 w J n´J ÿ j"1 |v j pt´t j n , x´x j n q| 2 v j pt´t j n , x´x j n q.
set u c,0 "ψ 1 . Then, by the argument in [10], one can show that u c,0 satisfies the desired properties in Theorem 6.1.
Proposition 6.2 (Precompactness of a minimal blow-up solution). Let u c ptq be in Theorem 6.1. Then K :" tu c ptq : t P Ru is precompact in H 1 .
Proof. Let tt n u 8 m"1 be a sequence in R. Passing to a subsequence, we may assume that t n Ñ t˚P r´8,`8s. If t˚‰ 8, then u c pt n q Ñ u c pt˚q in H 1 . Suppose that t˚" 8. Applying Proposition 5.1 to tu c pt n qu 8 n"1 , we write u c pt n q " J ÿ j"1 e it j n H pψ j p¨´x j n qq`R J n .
If ψ j ‰ 0 for some j ě 2 by the argument in the proof of [10, Proposition 5.5], one can deduce a contradiction. Therefore, we have u c pt n q " e it 1 n H pψ 1 p¨´x 1 n qq`R 1 n . If x 1 n Ñ 8, approximating e it 1 n H pψ 1 p¨´x 1 n qq " pe it 1 n p´∆`V p¨`x 1 n qq ψ 1 qp¨´x 1 n q by the nonlinear profile pNLS 0 p´t 1 n qqψ 1 p¨´x 1 n q as in the proof of Theorem 6.1, one can deduce a contradiction from the homogeneous case (Theorem 1.7). Hence, x 1 n " 0. It remains to show R 1 n Ñ 0 in H 1 and t 1 n " 0. The proof is very close to that of [10, Proposition 5.5], so we omit the proof. Lemma 6.3 (Precompactness implies uniform localization). Suppose that K :" tuptq : t P Ru is precompact in H 1 . Then, for any ą 0, there exists R " Rp q ą 1 such that sup tPR ż |x|ěR |∇upt, xq| 2`| upt, xq| 2`| upt, xq| 4 dx ď .
Proof. The proof follows from exactly the same argument in [10], so we omit it.
7. Extinction of a minimal blow-up solution. Finally, we prove Theorem 1.7 eliminating a minimal blow-up solution via the localized vial identities. Proposition 7.1 (Localized virial identities). Let χ P C 8 c pR 3 q. Suppose that uptq is a solution to pNLS V q. Then, p∇χ¨∇V q|u| 2 dx, .
Proof. By the equation and by integration by parts, we get p∇χ¨∇uqūdx.
Appendix A. Positivity of the Schrödinger operator. The Schrödinger operator H is positive definite when the negative part of a potential is small.