Scattering of solutions to nonlinear Schr\"odinger equations with regular potentials

In this paper, we prove the scattering for radial solutions to energy-critical nonlinear Schr\"odinger equations with regular potentials in defocusing case.

There are many important areas of application which motivate the study of nonlinear Schrödinger equations with potentials (Gross-Pitaevskii equation). In the most fundamental level, it arises as a mean field limit model governing the interaction of a plenty large number of weakly interacting bosons [22,32,40]. In a macroscopic level, it arises as the equation governing the evolution of the envelope of the electric field of a light pulse propagating in a medium with defects, see for instance, [19,20].
First, we recall some history on the scattering of solutions to (1.1) for small initial data.
zero as time goes to infinity. However, the linear Klein-Gordon equation with a potential (LKG) admits a family of periodic solutions with H 1 norms tending to zero. Thus solutions to NLKG with small data can not scatter to those periodic solutions to LKG, i.e. the wave operator is not complete.
Remark 1.2. The hypothesis (iii) is assumed to provide a dispersive estimate of e it∆ V . The condition given here is due to Journe, Soffer, Sogge [25]. There are many related works in this direction such as Rodnianski, Schlag [36], Schlag [38]. When d = 3, weaker assumption on V is available for the dispersive estimates, see for instance [4,10].
In our case, for 1 + 4 d < p < 1 + 4 d−2 , global well-posedness and scattering can be proved by interacting Morawetz identity, see for instance [7]. Thus we only need to consider the energycritical case. In the following, we prove global well-posedness and scattering for radial data in high dimensions (d ≥ 7). Theorem 1.1. Assume that V is radial, nonnegative, ∂ r V ≤ 0, and V satisfies Regular Potential Hypothesis. For d ≥ 7, p = 1 + 4 d−2 , λ = −1, u 0 ∈Ḣ 1 rad (R d ), (1.1) is globally wellposed and moreover, there exists u + ∈Ḣ 1 such that lim t→∞ u(t) − e it∆ V u + Ḣ1 = 0. Remark 1.3. A similar theorem is possible if V has a small negative part. The radial assumption for V is to ensure that every radial initial data evolves into a radial solution. If one considers non-radial data, the assumption ∂ r V ≤ 0 can be replaced by x · ∇V ≤ 0. Remark 1.4. Since V ≥ 0, the spectrum of −∆ V is included in [0, ∞). Since V ∈ L 2 , by Weyl's criterion, the essential spectrum of −∆ V is (0, ∞). The decay of V guarantees that there are no positive eigenvalues by Kato's theory. Moreover, it is known that there is no resonance for d ≥ 5. Therefore, for V in Theorem 1.1, (ii) in Regular Potential Hypothesis is equivalent to that 0 is not an eigenvalue of −∆ V . But this is true if V is non-negative. Therefore, (ii) is not needed in the presentation of Theorem 1.1. The facts that the equation is not scaling invariant and the energy space is homogeneous bring some new difficulties. As is known, the scaling invariance makes the bounded set in a homogeneous space noncompact. If the energy is not in the homogenous space, we can rule out one of the direction of the possible scaling such as what has been done in the study of scattering to nonlinear Klein-Gordon equations. If the equation is scaling invariant, the scaling will disappear when one does some estimates in the homogeneous space, which makes the analysis of limits of scaling not so important. In our case, because the energy lies inḢ 1 level, we have to handle two directions of the scaling. Meanwhile, the lack of scaling invariance makes the estimates sensitive to the varying of scaling. For instance, in the linear profile decomposition for the linear Schrödinger equation, the remainder term governed by the linear Schrödinger equation is asymptotically zero in Strichartz norms. However, if the scaling goes to infinity or zero, the remainder term tends to be a solution of free Schrödinger equation, for which whether it is asymptotically zero or not is not obvious. In order to overcome the difficulty, we prove two convergence results concerning the scaled Schrödinger operator and the free Schrödinger operator, namely Proposition 3.3 and 3.4. Proposition 3.3 gives the convergence of scaled Schrödinger operator to free Schrödinger operator in the strong operator topology. Proposition 3.4 proves the convergence in operator norm in a finite time interval. Although the strong operator topology convergence is weak, it is useful in proving profile decomposition since it is uniform in time. The operator norm convergence is essential in proving that the remainder is still asymptotically zero in Strichartz norms after taking a limit of scaling.
We assume d ≥ 7 because the Strichartz norm inḢ 1 level agrees with (−∆ V ) 1 2 u S 0 , where S 0 is the L 2 level Strichartz norm. However, for d ≤ 4, the two norms are not equivalent in general. The equivalence relation can compensate the loss of Leibnitz rule for (−∆ V ) 1 2 and the non-commutativity between ∇ and e it∆ V . In principle, the scattering for (1.1) when d = 5, 6 can be proved similarly, we rule out the two cases for technical problems. The focusing case can be dealt with similarly, in the subcritical case, see for instance [23].
The article is organized as follows. In Section 2, we give some estimates on Schrödinger operators and prove local well-posedness and stability theorem. In Section 3, we prove some important convergence lemmas concerning scaled Schrödinger operators and free Schrödinger operators, as an application, we give the linear profile decomposition. In section 4, Theorem 1.1 is proved by the compactness-contradiction arguments.
Notation and Preliminaries. We denote F V as the distorted Fourier transformation defined in Section 3. For s ∈ R, the fractional differential operator |∇| s is defined by We define the homogeneous Sobolev norms by and inhomogeneous Sobolev norms by The Besov norms are defined as follows: For a linear operator A from Banach space X to Banach space Y , we denote its operator norm by A L(X;Y ) . All the constants are denoted by C and they can change from line to line.
We use ε to denote some sufficiently small constant and it may vary from line to line. We use the notation b + and b − to stand for a number slightly less than b and a number slightly bigger than b respectively.
is a bounded operator from H η to H η for some α > d + 4, η > 0, with FV ∈ L 1 . Assume also that 0 is neither an eigenvalue nor a resonance of −∆ V . Then By the abstract theorem in Keel, Tao [26], one can prove: Suppose that V is the potential in Theorem 1.1. And assume that (p, q) and ( p, q) are Strichartz admissible with 2 ≤ p, q, p, q ≤ ∞ except the endpoint then we have where I is any interval containing t = 0, C is some constant depending only on V, d, p, q, In addition, we say (p, q) is aḢ 1 level Strichartz pair if We define the Strichartz norms to be We also define ∀ s ≥ 0, 2 Preliminaries on Schrödigner operators, local theory and stability theorem We consider the defocusing energy-critical NLS, namely Before going to the well-posedness theory, we recall some preliminaries on Schrödinger operators. Remark 5.3 in Chen, Magniez and Ouhabaz [6] proved the following result which implies the equivalence of (−∆ V ) 1 2 u p and ∇u p for some p.
From Lemma 2.1 and the complex interpolation, see for instance [10], we immediately deduce the following result.
Proof. The Sobolev embedding f 2d ≤ C ∆f 2 and Hölder inequality yield Thus it suffices to prove the inverse direction We prove it by contradiction. Suppose that (2.4) is false, then there exists {f n } ⊂Ḣ 2 such that Without loss of generality, we assume ∆f n 2 = 1. Then lim Since f n Ḣ2 is bounded, after extracting a subsequence, we may assume f n ⇀ f * weakly iṅ Indeed, by integrating by parts, one has For any ε > 0, choosing R > 0 sufficiently large, Hölder's inequality and Sobolev embedding Similarly we have Since the Sobolev embedding is compact on bounded domains, by extracting a subsequence, together with (2.7), (2.8) and (2.9), we obtain Then (2.6) follows. Therefore, we have proved Combining (2.5) and (2.10), with lim inf Hence we have ∆ V f * 2 = 0. By Hölder inequality and f * ∈ L 2d d−4 , there exists σ > 0 sufficiently large such that f * ∈ L 2 ( x −σ ). If f * = 0, then we see f * is an eigenfunction of −∆ V at zero when f * ∈ L 2 or a resonance when f * / ∈ L 2 . Both of these two cases contradict with the assumption Now we give the local wellposedness theorem, the existence of wave operator and stability theorem without proofs, since they are standard.

Lemma 2.3 (Local wellposedness).
For any u 0 ∈Ḣ 1 , there exists a unique maximal lifespan solution u to (2.1), with (T min , T max ) be the maximal existence time interval such that u ∈ Suppose that (T min , T max ) is the lifespan of u(t), then the energy Lemma 2.4 (Existence of the wave operator ). For any ϕ ∈Ḣ 1 , there exist positive constants Lemma 2.6 (Stability theorem). Let I ⊆ R be an interval and let t 0 ∈ I. Suppose thatũ is for some function e. If then there exists ε 0 depending on M, A, A ′ and d such that there exists a solution u to (2.1) with

Convergence lemmas and Linear profile decomposition
In order to establish the linear profile decomposition, we need to give some estimates. First, we will recall the spectral multiplier theorem and the distorted Fourier transformation.
The following spectral multiplier theorem is proved in Proposition 5.2 in [11].
In [1], Alsholm and Schmidt proved the existence of distorted Fourier transformation. We briefly describe their results.
Proposition 3.2 (Distorted Fourier transformation ). Assume that V is the potential in Theorem 1.1, then there exists a function ϕ(x, k) and a unitary operator F V in L 2 defined by Proof. We claim that for f ∈ H 1 , To verify (3.2), recall the Morawetz identity. Let u be a solution to i∂ t u + ∆ V u = 0, for a(x) sufficiently smooth, one has Taking a(x) = x , it is easy to see Therefore, integrating in time, by Hardy's inequality and complex interpolation (see for instance Lemma A.10 of [43]), we obtain (3.2). Now we prove (3.1). Take a cutoff function g ∈ C ∞ c (R) such that g(x) vanishes when |x| > 2, and g(x) equals one for |x| < 1. For ρ > 0, Hölder inequality, Corollary 2.1 and Proposition 3.1 yield Meanwhile, Proposition 3.2 and (3.2) indicate Therefore (3.1) follows by choosing ρ appropriately.
Lemma 3.1 can be used to prove the following corollary, which is important in proving the existence of the critical element.
Proof. It suffices to prove Strichartz estimate, Hölder inequality and Lemma 3.1 give thus finishing our proof.
The following approximate results are essential in proving the existence of the critical element Proof. Since we have then Corollary 2.1 with s = 1, p = 2 gives Similarly, by Lemma 2.2 and (3.5), for f ∈Ḣ 2 , we have Then by (3.5) and Strichartz estimates, it is direct to verify Hence by Strichartz estimates, (3.7) and Hölder inequality, we deduce v Ṡ1 (3.9) (3.10) First, we consider λ → 0. (3.7) and Hölder inequality yield Hence it suffices to show lim λ→0 V λ ∇u 2 Splitting the time interval R into two parts, by Hölder inequality, we have . I is easy to handle: x ≤ λ f Ḣ1 .
We give a local but uniform version of Proposition 3.3. As a preparation, we introduce an inhomogeneous Strichartz pair. It is elementary to verify that ifr = ( 2d d+2 ) − , 2 < q < ∞, then for r ∈ (2, ∞) defined by Then by Theorem 2.4 in [46], To avoid confusions, for (q, r) introduced above, we denote ∇u L q t L r x by u IH .
We now come to the last preparation, after which we will give the linear profile decomposition.
Suppose that h n , h j n ∈ (0, ∞), define the transformation T n , T j n as with the inverse transform of T j n being Lemma 3.2. If h n → 0 or ∞, g n ⇀ 0 inḢ 1 , then for ψ ∈Ḣ 1 , Proof. It is easy to verify T n ψ, T n g n Ḣ1 V = ∇ψ, ∇g n L 2 + h 2 n V (h n x)ψ, g n L 2 .

16)
If h n → 0, (3.16) gives our proposition. If h n → ∞, instead of (3.16), we use The linear profile decomposition is given below and we follow arguments in [24]. rad . Then up to extracting a subsequence there exists K ∈ N such that for each j ≤ K, there exist ϕ j ∈Ḣ 1 (R d ), and If v = 0, take K = 0. Otherwise for n large enough, there exists (t n , x n ) ∈ R × R d and nonnegative integer k n such that (3.20) By radial Gagliardo-Nirenberg inequality and Bernstein inequality, Take R n = R 0 2 −kn and let R 0 be sufficiently large such that Then by (3.20), x n satisfies |x n | ≤ R n and n v n (t n , h n x). By (3.21), Because |x n | ≤ R 0 h n , up to extracting a subsequence, we can assume h −1 n x n → x * for some constant vector x * ∈ R d . Since ψ n is bounded inḢ 1 , we can postulate ψ n ⇀ ψ inḢ 1 , then by If h n → 0 or ∞, we take (t 0 n , h 0 n ) = (t n , h n ), ϕ 0 = ψ. If h n → h ∞ > 0, then let then one has due to (3.23). When h n → 0 or ∞, as a consequence of Lemma 3.2 and the fact ψ n − ψ ⇀ 0 iṅ Therefore we have proved (3.24). Since the inner product is preserved with respect to t, thus Until now, we have accomplished the first step. Next, we treat w 1 n as v n and do the same work. If lim sup Otherwise we can find v 1 n and w 2 n such that there exist (t 1 n , h 1 n ) ∈ R × (0, ∞) and ϕ 1 ∈Ḣ 1 (R d ) for which Iteration for times gives the desired decomposition, the remaining work is to verify (3.17), (3.18) and (3.19). Firstly, (3.17) is a direct corollary of (3.19) and the fact lim sup Secondly, we prove (3.19) under (3.18). We claim for l < j, v l n (0), v j n (0) Ḣ1 V → 0, as n → ∞. (3.25) It is easy to verify Careful calculations with the help of (3.26) and (3.27) imply When h l n h j n → 0, (3.25) follows from where we have used (3.6).
Combining this with S l,m n ϕ m ⇀ 0 and (3.31), (3.32) gives which is a contradiction.
The linear profile decomposition enjoys more properties than addressed in Proposition 3.5.
We collect them below.
Proposition 3.6. Suppose that v n , v j n , w k n , h j n are the components of the profile decomposition in Proposition 3.5. Then there are only three cases for h j n namely, lim n→∞ h j n = 0, or lim n→∞ h j n = ∞ or h j n = 1 for all n. For any fixed t, the following energy decoupling property holds: Proof. The proof of (3.33) is standard except some modifications, see for instance [27]. In fact, the linear part of E(v n ) has been proved in (3.17). The nonlinear part can be proved with the Using Hölder inequality, we conclude that where (η, p) isḢ 1 −admissible pair, (γ, r) is L 2 −admissible pair, and Direct calculation shows (3.38) coincides with the choice of θ, thus (3.35) follows from (3.36).
As a direct consequence of (3.35) and Corollary 3.1, we have Define the solution to (2.1) is globally wellposed and u(t, x) Denote E * = sup{E : E ∈ E}. We aim to prove E * = ∞ by contradiction. Suppose that E * < ∞, then there exists a sequence of solution(up to time translations) to (2.1), such that E(u n ) ր E * , as n → ∞, and where I n denotes the maximal interval of u n including 0.
Apply the linear profile decomposition to u n (0), we get ϕ j , {(h j n , t j n )} for which (3.17), (3.18), (3.19) hold and Now we construct the corresponding nonlinear profiles. Suppose that U j n is a solution to (2.1) with initial data U j n (0) = e −it j n ∆ V T j n ϕ j , then U j n (t) satisfies If h j n → 0 or h j n → ∞, let u j (t, x) be a solution to If τ j ∞ = ±∞, then u j is given by the wave operator. If τ j ∞ ∈ R, then u j is given by the global well-posedness and scattering theorem in [48], and we have u j Ṡ1 (R×R d ) < ∞.
If h j n = 1, let u j be a solution to Again for τ j ∞ = ±∞, Lemma 2.4 gives the existence of u j . For τ j ∞ ∈ R, local Cauchy theory namely Lemma 2.3 provides the existence of u j at least in a small interval. We call u j nonlinear profile. Suppose I j = (T j min , T j max ) is the lifespan of u j , then by the definition of u j , we have u j ∈ C 0 tḢ 1 , then u j n has the lifespan I j n = ((h j n ) 2 T j min +t j n , (h j n ) 2 T j max + t j n ). Define The following two lemmas are standard, which can be easily obtained by using the wellposedness and scattering theory in Lemma 2.3 and Lemma 2.5 as well as Proposition 3.5.
Lemma 4.1. There exists j 0 ∈ N such that T j min = −∞, T j max = ∞ for j > j 0 and Proof. We prove it by contradiction. Suppose that for 1 < ∞, then together with Lemma 4.2, we have u j n exists globally for j ≥ 1. Thus u <k n + w k n exists globally. If we have verified that for n, k sufficiently large, u <k n + w k n is a perturbation of u n , then by the stability theorem, we can derive a contradiction. From Proposition 3.5 and (4.7) Denote τ j n = − t j n (h j n ) 2 . When t = 0, by (3.27), we can easily see Combining We claim that where F (u) = |u| 4 d−2 u. Suppose that the claim holds, then from (4.6)-(4.9) and the stability theorem, we will obtain for n sufficiently large, which contradicts with (4.1). Thus Lemma 4.3 follows. Therefore we only need to prove (4.9).
Note that it suffices to verify and lim k→∞ lim n→∞ ∇(F (u <k n + w k n ) − F (u <k n )) First, we prove (4.12). When lim n→∞ h j n = 0 or lim n→∞ h j n = ∞, u j n satisfies From the scattering theorem in [48], we have

Direct calculations show
For any ε > 0, take a functionũ j ∈ C ∞ c (R × R d ) for which ũ j − u j similar arguments work. Hence for h j n → ∞ and h j n → 0, we have proved (4.14) When h j n = 1, (4.14) is obvious. By (4.14) and triangle inequality, (4.12) can be reduced to Following the same arguments in [27,29], (4.15) and (4.13) can be further reduced to  By density arguments, we can assume If h j n = 1, then u j n (t, x) = u j (t − t j n , x), Hölder's inequality and Lemma 3.1 give Thus Corollary 3.2 yields (4.16).

Case 2.
If h j n → ∞, by (3.5), Hölder's inequality and smoothing effect of the free Schrödinger equation, we get   Moreover, {u c (t) : t ∈ R} is pre-compact inḢ 1 rad (R d ). Consequently, we have for any ε > 0, there exits a constant R ε > 0, such that for all t ∈ R, |x|≥Rε |∇u c | 2 + |u c | 2 |x| 2 + |u c | 2d d−2 dx < ε. Define a nonnegative radial function φ ∈ C ∞ c (R) with where u(t, x) is a solution to (2.1). Then direct calculations give d dt V R (t) = 2ℑ R dū ∇u · ∇φ R dx, (4.18) By the virial identity above, we can prove the nonexistence of the critical element thus yielding a contradiction, from which Theorem 1.1 follows.