Energy-critical NLS with potentials of quadratic growth

Consider the global wellposedness problem for nonlinear Schr\"odinger equation \[ i\partial_t u = [-\tfrac{1}{2} \Delta + V(x)] u \pm |u|^{4/(d-2)} u, \ u(0) \in \Sigma(\mathbf{R}^d), \] where $\Sigma$ is the weighted Sobolev space $\dot{H}^1 \cap |x|^{-1} L^2$. The case $V(x) = \tfrac{1}{2}|x|^2$ was recently treated by the author. This note generalizes the results to a class of"approximately quadratic"potentials. We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we apply more general tools from microlocal analysis, including a Fourier integral parametrix of Fujiwara.

Consider the Hamiltonian PDE whose flow preserves the energy The equation is defocusing if µ = 1 and focusing if µ = −1. The potential is assumed to satisfying th following three conditions: (V1) V = V (x) is smooth and nonnegative.
(V2) V is subquadratic in the sense that (V3) V (x) ≥ δ|x| 2 for some δ > 0. These hypotheses on V ensure that for some constant δ > 0. Therefore Σ is the energy space for the linear and defocusing problems, and is also the form domain Q(H) = D(H 1/2 ) for the positive operator H = − 1 2 ∆ + V . It will sometimes be more convenient to work with the norm f Q(H) := H 1/2 f L 2 = ( ∇f 2 L 2 + V 1/2 f 2 L 2 ) 1/2 , because it is preserved by the propagator e −itH . This norm is clearly equivalent to the original norm on Σ.
The adjective "energy-critical" refers to the fact that if we ignore the potential V in the equation, the Cauchy problem is invariant under the scaling u → u λ (t, x) = λ − d−2 2 u(λ −2 t, λ −1 x). One makes this comparison because solutions to the equation (1.2) can be approximated by those to (1.3) in certain limiting regimes. Roughly speaking, if a solution u to (1.2) is highly concentrated at some point x 0 , it sees the potential V as approximately a constant V (x 0 ), and for short times the behavior of u is basically controlled by equation (1.3). We made this heuristic rigorous for the quadratic potential in [15]; see also Section 5. Such concentrated solutions must be considered when studying the wellposedness of (1.2). Therefore, it is helpful to first understand the scale-invariant problem. That equation is hypothesized to have the following behavior: dx dt ≤ C(E ∆ (u 0 )) < ∞.
Moreover, there exist functions u ± ∈Ḣ 1 (R d ) such that lim t→±∞ u(t) − e ± it∆ 2 u ± Ḣ1 = 0, and the correspondences u 0 → u ± (u 0 ) are homeomorphisms ofḢ 1 . When µ = −1, one also has global wellposedness and scattering provided that where the ground state solves the elliptic equation 1 2 ∆ + |W | This conjecture has been studied intensively over the last fifteen years, and we summarize the state of the art in the following Theorem 1.1. Conjecture 1.1 holds for the defocusing equation. For the focusing equation, the conjecture holds for radial initial data when d ≥ 3, and for all initial data when d ≥ 5.
As the potentials under consideration cause the Hamiltonian to have purely discrete spectrum, one does not have global spacetime bounds of the form (1.4) even for the linear equation i∂ t u = (− 1 2 ∆ + V )u. Moreover, the semilinear harmonic oscillator i∂ t u = (− 1 2 ∆ + 1 2 |x| 2 )u + |u| p u is known [4,Prop. 3.1] to have periodicin-time solutions for p < 4/(d − 2). These considerations motivate the following for any compact interval I ⊂ R.
The restriction on the kinetic energy focusing case is necessary, for as in the case of the harmonic oscillator, we have: , and ∇u 0 L 2 ≤ ∇W L 2 , then the solution to (1.2) blows up in finite time.
To prove this one need only make notational changes to the discussion in [15,Section 7], and we refer the reader to there for details.
The main result of this paper extends our previous study [15], which considered the special potential V (x) = 1 2 |x| 2 for general data in the energy space, and in turn builded on the treatment of spherically symmetric data by Killip-Visan-Zhang [23]. As stated previously, our study of global wellposedness for equation (1.2) leans on the results for the scale-invariant equation. In the interest of succinctness we elect to state our main result as a conditional one; by Theorem 1.1, however, it is actually unconditional except in the focusing case for nonradial data in dimensions d = 3 and 4. This paper also has an ancestor in the work of Carles [5], who considered a large class of subquadratic potentials for the energy-subcritical problem Taking initial data in Σ, he established global wellposedness in the defocusing case when 4/d ≤ p < 4/(d − 2) and in the focusing case when 0 < p < 4/d. Carles did not require that V be bounded from below, and also allowed V = V (t, x) to depend on time. Prior to that work, Oh [25] had shown large data global existence in the focusing case when p < 4/d and the potential is time-independent and subquadratic.
We consider a more restricted class of potentials but focus on the subtleties at the exponent p = 4/(d − 2). In the absence of a potential, scale-invariance of the energy forces the guaranteed lifespan of local solutions to depend on the shape of the initial data instead of just on its energy. Control of the energy alone therefore does not rule out blowup.
Although our equation does not actually have scaling symmetry, it nonetheless contains the same essential difficulties as in the scale-invariant problem. For if we take initial data of the form u λ 0 = λ −(d−2)/2 φ(λ −1 ·) for a fixed Schwartz function φ, and take λ very small, the energy E(u λ 0 ) barely depends on λ. In Section 5, we shall see that if u λ is the solution to (1.2) with u λ (0) = u λ 0 and one restricts to a time window |t| λ 2 , then u λ can be approximated in critical spacetime norms by with v(0) = φ. Therefore, just as in the scale-invariant case, solutions to (1.2) with bounded energies can accumulate nontrivial spacetime norm over arbitrarily small time frames.
As in the case of the harmonic oscillator, to prove Theorem (1.2) we apply the modern incarnation of the induction on energy paradigm. This general strategy had been adapted previously to different critical equations [19,20,14,13,12,24]. See also the list of references following Theorem 1.1 for the pioneering instances of induction on energy. The main ingredients of the argument are: • Stability theory. One needs a statement to the following effect: ifũ approximately solves equation (1.2) with error sufficiently small in Strichartz norms, then there is an exact solution u to (1.2) with the same initial data asũ, and which is close toũ in critical spacetime norms. • Linear profile decomposition. This provides a way to quantify the failure of compactness of the embedding and provides a way to decompose a bounded sequence {f n } n ⊂ Q(H) as f n = j φ j n , where the profiles φ j n are asymptotically pairwise independent and reflect the "symmetries" of the problem.
• Analysis of scaling limits. A typical profile in the profile decomposition will look something like φ n = N (d−2)/2 n φ(N n ·) where either N n ≡ 1 or lim n N n = ∞ (this is an oversimplification but captures the general idea). We will show that in the latter case, for n large enough the solution u n to (1.2) with u n (0) = φ n behaves so similarly to a solution to the globally wellposed equation (1.3) that, by stability theory, u n itself must have finite spacetime norm on a length-1 time interval. This essentially rules out blowup for equation (1.2) when the initial data is highly concentrated at a point.
• Induction on energy. Originally introduced by Bourgain [2] and refined by the "I-team" [6], Keraani [18], Kenig-Merle [17], and others, the idea is to assume that global wellposedness of (1.2) fails for some initial data, and consider the smallest energy E c such that solutions u with E(u) ≥ E c fail to exist globally. This energy threshold is positive by the small data theory. Using the profile decomposition, the induction hypothesis that solutions with energy smaller than E c do exist globally, and the scaling limit analysis, one proves the existence of a blowup solution u c with energy E(u c ) = E c , and which must simultaneously obey an impossibly strong compactness property.
We import the arguments given for the harmonic oscillator in [15], making mostly notational changes save for one key difference. There, we exploited the classical formula for the linear fundamental solution: cos t−xy) (Mehler's Formula [7]).
No such explicit formula is available for more general potentials. Nonetheless, Fujiwara showed [9,10] that for a rather general class of subquadratic potentials, the propagator e −itH can be represented for small t as an oscillatory integral operator where for small t the integral kernel is close to that of the free propagator e it∆ 2 in a quantitative sense. In fact, Fujiwara considered time-dependent potentials.
Using this representation and some basic facts about oscillatory integrals due to Asada-Fujiwara [1], we are able to obtain suitable replacements for the steps in [15] that relied on the precise form of Mehler's formula.
Outline of paper. In Section 2 we set our notation and review some basic estimates regarding equation (1.2). We also recall Fujiwara's construction of the propagator as an oscillatory integral and record some oscillatory integral estimates of Asada-Fujiwara. Section 3 states some standard (but vital) local theory. Section 4 discusses the linear profile decomposition mentioned above, focusing on how to modify the arguments that previously invoked Mehler's formula.
The scaling limit analysis of Section 5 and the compactness arguments of Section 6 parallel the ones given in [15]. As will be the case throughout the paper, we describe mainly the required adjustments and refer to [15] for the rest of the details.
Acknowledgements. The author is indebted to his advisors Rowan Killip and Monica Visan for their helpful discussions as well as their feedback on the paper. This work was supported in part by NSF grants DMS-0838680 (RTG), DMS-1265868 (PI R. Killip), DMS-0901166, and DMS-1161396 (both PI M. Visan).

Notation and basic estimates. We write
We will sometimes use the more compact notation f p . If I ⊂ R d is an interval, the mixed Lebesgue norms on I × R d are defined by where one regards f (t) = f (t, ·) as a function from I to L r (R d ).
Throughout the paper we shall use the capital letters D and X to denote the operators f → −i∂f and f → xf , respectively. Introduce the following function spaces The notation B(R d × R d ) goes back to Schwartz [27] and is equivalent to the more modern notation S 0 |α|ρ+|β|δ for all multiindices α and β. Note, however, that in general B k does not coincide with any S k ρ,δ . We recall Fujiwara's construction of the fundamental solution for H. Recall that the symbol H(ξ, x) = 1 2 |ξ| 2 + V (x) defines the Hamiltonian flow Suppose that V is subquadratic in the sense that |D k V (x)| ≤ C k for all k ≥ 2. Then the vector field (−∇ x H, ∇ ξ H) is Lipschitz, hence complete, and we may regard x and ξ as well-defined functions x(t, y, η), ξ(t, y, η) of t and the initial data.
Remark. To get the second statement from the first, one invokes the Hadamard global inverse function theorem to see that (y, η) → (x, y) is a diffeomorphism for 0 = t sufficiently small. According to this result, when |t| ≤ δ 0 and t = 0 we can define the action where (x(τ ), ξ(τ )) is the unique trajectory with x(0) = y and x(t) = x.
Theorem 2.2 (Fundamental solution [9,10]). Let V be subquadratic as in the previous proposition. Then there exists δ 0 > 0 such that: • The action S(t, x, y) is well-defined by (2.2) for all 0 < |t| < δ 0 and satisfies where the term ω(t, ·, ·) belongs to B 2 uniformly for |t| ≤ δ 0 . That is, there exist constants C k such that for all k ≥ 0. The above integral representation immediately yields a dispersive estimate: We call a pair (q, r) admissible if q ≥ 2 and 2 By interpolation, this norm controls u L q t L r x for all admissible pairs (q, r). Define denotes the Hölder dual of (q k , r k ). Lemma 2.4 (Strichartz [16]). Let I be a compact time interval containing t 0 , and let u : I × R d → C be a solution to the inhomogeneous Schrödinger equation Then there is a constant C, depending only on the length of the interval I, such that u S(I) ≤ C( u 0 L 2 + F N (I) ).
Proof. This follows from [16] as a consequence of two ingredients: the dispersive estimate of the previous corollary, and the unitarity of e −itH on L 2 (R d ).
As V is nonnegative, we have access to the following spectral multipler theorem due to Hebisch [11]: The following norm equivalence was first proven for the quadratic potential by Killip-Visan-Zhang [23, Lemma 2.7]. Using the coercivity hypothesis (V3), we adapt their result to the potentials considered here. Proposition 2.6 (Equivalence of norms). For any 1 < p < ∞ and s ∈ [0, 1], we have To prove this we shall need the following fact, which is classical when V is quadratic; we verify it at the end for the sake of completeness.
. Then the space of smooth vectors for H is precisely Schwartz class: Proof of Proposition 2.6. We show first that By hypothesis, there is some On the other hand, the parabolic maximium principle implies 0 ≤ e −tH (x, y) ≤ e −tH δ (x, y) Combining this with the identity This yields half of (2.4). Specializing to the case s = 1 and writing −∆ = 2(H − V ), we obtain The rest of the argument is imported directly from [23], and is included to make the discussion self-contained. To show that (−∆) s H −s f p p f p for all s ∈ [0, 1], we use analytic interpolation. It suffices to verify that for all Schwartz f and g. By homogeneity, we may assume f p = g p ′ = 1. Put By the spectral theorm, F (z) is bounded and continuous on the closed strip {0 ≤ Re(z) ≤ 1} and analytic on its interior. By the special case (2.5) and Theorem 2.5, Hadamard's three-lines lemma implies that |F (z)| ≤ C on the whole strip. Thus Writing This completes the proof of the proposition modulo the lemma.
To prove the opposite inclusion, we show by an induction argument the equivalent assertion that We have the following identities: Define for each n ≥ 1 the following statements: Assume that they hold for some n. For u ∈ D(H n ) and m ∈ B, use (2.7) and the statements P 1 (n), P 2 (n) to see that H(mu) ∈ D(H n−1 ), so mu ∈ D(H n ) and P 1 (n + 1) holds since m was chosen arbitrarily in B. Similar reasoning shows that P 2 (n) and P 3 (n) imply P 2 (n + 1), and that P 1 (n), P 2 (n), P 3 (n) yield P 3 (n + 1). Hence, by induction these statements hold for all n ≥ 1.
Next, apply (2.4) in the special case s = 1, p = 2 to see that Suppose u ∈ D(H n ) and n ≥ 2. We have By induction, V Hu ∈ D(H n−2 ), while P 1 (n), P 2 (n), and P 3 (n − 1) imply that the second and third terms also belong to D(H n−2 ). Thus V u ∈ D(H n−1 ) Summing up, we find that These mapping properties, together with the coercivity hypothesis (V3), immediately yield the claim (2.6).
Thanks to this norm equivalence, H γ inherits many properties of the fractional derivative (−∆) γ , including Sobolev embedding: Similarly, the fractional chain and product rules carry over to the current setting: Using Proposition 2.6 and the Christ-Weinstein fractional product rule for (−∆) γ (e.g. [28]), we obtain

FIO technology.
We recall some properties of Fourier integral operators tailored to the Schrödinger equation. The operators we shall use were developed by Fujiwara [8] and Asada-Fujiwara [1].
and satisfies the nondegeneracy condition Given a phase φ(x, y) and an amplitude a(x, y) ∈ B(R d x × R d y ), we define for each λ = 0 the integral operator Remark. Asada and Fujiwara studied more general oscillatory integral operators of the form where a(x, θ, y) ∈ B(R n x × R m θ × R n y ) and the phase φ satisfies the nondegeneracy condition The integral operator A(λ) considered above corresponds to the case m = 0.
The operator A(λ) is bounded on L 2 . More precisely, Fujiwara proved: Let φ be a phase function. By the global inverse function theorem, the maps which is easily checked to be canonical in the sense that dξ ∧dx = dη ∧dy. The map χ(y, η) = (x(y, η), ξ(y, η)) is the canonical transformation generated by the phase function φ(x, y).
For a smooth symbol p ∈ B k (R d x × R d θ × R d y ) and λ = 0, let Op(p, λ) denote the the (semiclassical) pseudodifferential operator These operators obey the following Egorov-type relation: , be an amplitude, and A(λ) the corresponding Fourier integral operator. Let χ : R 2d → R 2d be the canonical transformation generated by φ. Let for some Fourier integral operator R(λ) with phase function φ. The operator norm of R(λ) satisfies

Local Theory
We record some standard local-wellposedness results for (1.2). These are direct translations of the theory for the scale-invariant equation (1.3). By Lemma 2.8 and Corollaries 2.9 and 2.10, essentially the same proofs as in that case will work here. We refer the reader to [22] for those proofs.
Proposition 3.1 (Local wellposedness). Let u 0 ∈ Σ(R d ) and fix a compact time interval 0 ∈ I ⊂ R. Then there exists a constant η 0 = η 0 (d, |I|) such that whenever η < η 0 and there exists a unique solution u : I × R d → C to (1.1) which satisfies the bounds Corollary 3.2 (Blowup criterion). Suppose u : (T min , T max ) × R d → C is a maximal lifespan solution to (1.1), and fix T min < t 0 < T max . If T max < ∞, then and that for some 0 < ε < ε 0 (E, L) one has Then there exists a unique solution u : I × R d → C to (1.1) with u(t 0 ) = u 0 and which further satisfies the estimates where 0 < c = c(d) < 1 and C(E, L) is a function which is nondecreasing in each variable.

Concentration compactness
Let 0 ∈ I be a compact interval so that |I| ≤ δ 0 , where δ 0 is the constant in Theorem 2.2. As is now standard in the analysis of energy-critical equations, the induction on energy argument relies on a linear profile decomposition for the Strichartz inequality In [15] we obtained such a decomposition when H = − 1 2 ∆ + 1 2 |x| 2 . Let us highlight the main modification required to adapt that proof to the present setting. One of the key steps in both proofs is to compare the linear evolutions of a spatially localized initial state under the propagators e −itH and e it∆ 2 with and without a potential, respectively (see Proposition 4.4 below). For the harmonic oscillator we relied on the Mehler formula to decompose where m t (x) = exp(i( cos t−1 2 sin t )x 2 ). While this factorization clearly manifests the relation between the two propagators, a more robust method is needed to deal with the more general potentials considered here. Instead we work directly with the integral representation from Theorem (2.2) and apply oscillatory integral estimates.
The rest of the strategy for the harmonic oscillator can be imported after essentially notational changes. We shall state the main definitions and lemmas to indicate the general flow but refer to [15] for their proofs.
Definition 4.1. A frame is a sequence (t n , x n , N n ) ∈ I × R d × 2 N conforming to one of the following scenarios: (1) N n ≡ 1, t n ≡ 0, and x n ≡ 0.
(2) N n → ∞ and N −1 n V (x n ) 1/2 → r ∞ ∈ [0, ∞). Remark. The quantity N −1 n V (x n ) 1/2 is the analog of the ratio N −1 n |x n | that was considered in [15]. These parameters will specify the temporal center, spatial center, and (inverse) length scale of a function. The hypothesis that V grows essentially quadratically ensures that |x n | N n , which reflects the fact that we only consider functions obeying some uniform bound in Q(H), and such functions cannot be centered arbitrarily far from the origin. We need to augment the frame {(t n , x n , N n )} with an auxiliary parameter N ′ n , which corresponds to a sequence of spatial cutoffs adapted to the frame.

Definition 4.2.
An augmented frame is a sequence (t n , x n , N n , N ′ n ) ∈ I × R d × 2 N × R belonging to one of the following types: ( The frame {(t n , x n , N n )} is the underlying frame.
Given an augmented frame (t n , x n , N n , N ′ n ), we define scaling and translation operators on functions of space and of spacetime by We also define spatial cutoff operators S n by   (2) Suppose F is of type (2b) and f n ∈ L q tḢ 1,r Here H 1,r (R d ) andḢ 1,r (R d ) denote the inhomogeneous and homogeneous L r Sobolev spaces, respectively, equipped with the norms Proposition 4.2 (Inverse Strichartz). Let I be a compact interval containing 0 of length at most δ 0 , and suppose f n is a sequence of functions in Q(H) satisfying Then, after passing to a subsequence, there exists an augmented frame F = {(t n , x n , N n , N ′ n )} and a sequence of functions φ n ∈ Q(H) such that one of the following holds: (1) F is of type 1 (i.e. N n ≡ 1) and φ n = φ where φ ∈ Q(H) is a weak limit of f n in Q(H). (2) F is of type 2, either t n ≡ 0 or N 2 n t n → ±∞, and φ n = e itnH G n S n φ where φ ∈Ḣ 1 (R d ) is a weak limit of G −1 n e −itnH f n inḢ 1 . Moreover, if F is of type (2a), then φ also belongs to L 2 (R d ).
The functions φ n have the following properties: Proof. We recall that the proof of the analogous result in [15, Section 4.1] used the following ingredients: • Littlewood-Paley theory adapted to the operator H = − 1 2 ∆ + 1 2 |x| 2 , which relied heavily on Theorem 2.5.
• A refined Strichartz inequality, proved using the Littlewood-Paley theory.
• Convergence properties of equivalent and orthogonal frames, in particular, the comparison of the linear flows generated by the Hamiltonians for the free particle and the harmonic oscillator, when acting on concentrated initial data. This was the only place in the proof that relied on the exact form of Mehler's formula (1.6). The reader will easily verify that adapting the first two to our situation requires little more than replacing all instances of |x| 2 /2 in the proofs with V . In the following section, we supply the details for the third. Once suitable analogues for these components are obtained, the rest of the proof carries over without difficulty, and we refer the reader to [15] for the details.

Proposition 4.3 (Linear profile decomposition).
Let 0 ∈ I be an interval with |I| ≤ δ 0 , and let f n be a bounded sequence in Q(H). After passing to a subsequence, there exists J * ∈ {0, 1, . . . }∪{∞} such that for each finite 1 ≤ j ≤ J * , there exist an augmented frame F j = {(t j n , x j n , N j n , (N j n ) ′ )} and a function φ j with the following properties.
• Either t j n ≡ 0 or (N j n ) 2 (t j n ) → ±∞ as n → ∞. • φ j belongs to Q(H), H 1 , orḢ 1 depending on whether F j is of type 1, (2a), or (2b), respectively. For each finite J ≤ J * , we have a decomposition where G j n , S j n are theḢ 1 -isometry and spatial cutoff operators associated to F j . Writing φ j n for e it j n H G j n S j n φ j , this decomposition has the following properties: Whenever j = k, the frames {(t j n , x j n , N j n )} and {(t k n , x k n , N k n )} are orthogonal: Proof. The argument is similar to the one for as in [15,Proposition 4.14]. One inductively applies inverse Strichartz to extract the frames F j and profiles φ j . To prove the decoupling assertion (4.11), one uses the convergence lemmas discussed in the next section, which completely parallel the ones used in [15].
The proof of this relies on the fact that e −itH conjugates D and X according to Heisenberg's equations. We used this stronger assertion to treat the focusing equation in an earlier draft of [15], but were unable to extend it to the present setting. It would be interesting to know whether one has separate decoupling of kinetic and potential energies for more general potentials.
If any of the above statements fail, we say that F 1 and F 2 are orthogonal. Note that replacing the N 1 n in the second and third expressions above by N 2 n yields an equivalent definition of orthogonality.
Two augmented frames are said to be equivalent if their underlying frames are equivalent.
(The last limit exists by the definition of a frame.) Let G M n , G N n be the scaling and translation operators associated with the frames F M and F N respectively. Then the sequence (e −it N n H G N n ) −1 e −it M n H G M n converges in the strong operator topology on B(Σ, Σ) to the operator U ∞ defined by . This is what we shall do. We proceed in two steps. Recall from Theorem 2.2 that the phase in the Fourier integral formula for e −itH is the classical action and has the form S(t, x, y) = |x−y| 2 2t + tω(t, x, y), ω(t, ·, ·) ∈ B 2 .
First we extract the lowest order term from the remainder. This additional information will reveal the limit of the sequence once everything has been expressed in terms of oscillatory integrals. Convergence will then follow from the oscillatory integral theory of Section 2.2.
The leading terms of the action will be obtained by replacing the classical trajectories with straight lines in the integral (2.2). Proceeding in the spirit of Fujiwara [9], we have the following lemma.
Lemma 4.5. Let H(ξ, x) = 1 2 |ξ| 2 + V (x) with V subquadratic, and let S(t, x, y) be the action (which is well-defined for all x and y so long as |t| ≤ δ 0 where δ 0 is the constant in Theorem 2.2). Then Proof. Start by rewriting the ODE system (2.1) in integral form: (4.14) As ∂ x V grows at most linearly, Gronwall's inequality implies that for all initial data y, η we have |x(t)| ≤ C(1 + |y| + |tη|).
Referring to the definition (2.2) of the action, we estimate the error incurred by replacing the true trajectory by the straight line path from y to x. Rearranging the above expression for x(t), we have For τ between 0 and t, hence |x(τ )| ≤ C(1 + |x| + |y|). The preceding computations reveal that By the fundamental theorem of calculus, (4.16) Next, by combining the first line of (4.14) with (4.15), we find that It is easy to see that second and third terms are bounded by O(t(1 + |x| + |y|)). Therefore, Combining this with (4.16) establishes the lemma.
Theorem 2.2 and Lemma 4.5 imply that these quantities obey the following estimates: We need the following adaptation of [9,Proposition 4.15].
We now verify the limit (4.13). As e iM 2 n tn ∆ 2 → e it∞ ∆ 2 strongly, it suffices to show that converges to 0 for all f ∈ Σ. By Lemma 4.6 we may assume f ∈ C ∞ c . The above difference may be written as In the remainder of this section we collect other lemmata regarding equivalent and orthogonal frames. They can be proved in much the same manner as their counterparts in [15,Section 4.2].
n , x n , M n , M ′ n )} and {t N n , y n , N n , N ′ n )} be equivalent augmented frames. Let S M n , S N n be the associated spatial cutoff operators. Then whenever φ ∈ H 1 if the frames conform to case (2a) and φ ∈Ḣ 1 if they conform to case (2b) in Definition 4.2.
Proof. Run an approximation argument using Lemma 4.1 in the manner of [15,Corollary 4.9].
The following "approximate adjoint" identity is the analogue of [15,Lemma 4.10].
Lemma 4.8. Suppose the frames {(t M n , x n , M n )} and {(t N n , y n , N n )} are equivalent. Put t n = t M n − t N n . Then for f, g ∈ Σ we have The proof of Lemma 4.6 yields the following commutator estimate: We have The claim then follows from Cauchy-Schwarz and the above estimate.
The next lemma is a converse to Proposition 4.4.
c by a change of variables and the dispersive estimate, thus for general f ∈ Σ by a density argument. Therefore (G N n ) −1 e −itnH G M n f converges weakly inḢ 1 to 0. We consider next the case where M 2 n t n → t ∞ ∈ R. The orthogonality of F M and F N implies that either N −1 n M n converges to 0 or ∞, or M n |x n −y n | diverges as n → ∞. In either case, one verifies easily that the operators (G N n ) −1 G M n converge to zero in the weak operator topology on B(Ḣ 1 ,Ḣ 1 ). Applying Proposition 4.4, we see that Proof. If φ ∈ C ∞ c , then S M n φ = φ for all large n, and the claim follows from Lemma 4.9. The case of general φ in H 1 orḢ 1 then follows from an approximation argument similar to the one used in the proof of Corollary 4.7.

The case of concentrated initial data
With the main complications of this paper behind us, we sketch the rest of the wellposedness argument in the remaining two sections. The next step is to rule out blowup for equation (1.2) when the initial data is highly concentrated in space. Assume that Conjecture 1.1 holds. Let F = {(t n , x n , N n , N ′ n )} be an augmented frame with t n ∈ I and N n → ∞, such that either t n ≡ 0 or N 2 n t n → ±∞; that is, F is type (2a) or (2b) in Definition 4.2. Let G n ,G n , and S n be the associated operators as defined in (4.1) and (4.2). Suppose φ belongs to H 1 orḢ 1 depending on whether F is type (2a) or (2b) respectively. Then, for n sufficiently large, there is a unique solution u n : I × R d → C to the defocusing equation (1.2), µ = 1, with initial data u n (0) = e itnH G n S n φ.
This solution satisfies a spacetime bound lim sup n→∞ S I (u n ) ≤ C(E(u n )).
Suppose in addition that {(q k , r k )} is any finite collection of admissible pairs with 2 < r k < d. Then for each ε > 0 there exists Assuming also that ∇φ L 2 < ∇W L 2 and E ∆ (φ) < E ∆ (W ), we have the same conclusion as above for the focusing equation (1.2), µ = −1.
Proof sketch. We only give a rough idea as one can proceed just as in Proposition 5 of [15] and replace every instance of |x n | 2 /2 wih V (x n ). The idea is to show that for n large enough, one can fashion a sufficiently accurate approximate solutionũ n on the interval I in the sense of Proposition 3.3, such that S I (ũ n ) are bounded. This bound will then be transferred to the exact solution u n by the stability theory.
While u n remains highly concentrated (over time scales on the order of N −2 n ), it will be approximated by a modified solution to the scale-invariant equation (1.3) (whose solutions admit global spacetime bounds). By the time this approximation breaks down, the solution u n will have dispersed to such an extent that the evolution of u n is essentially linear.
We make this heuristic precise. If t n ≡ 0, let v be the global solution to (1.3) furnished by Conjecture 1.1 with v(0) = φ. If N 2 n t n → ±∞, let v be the (unique) solution to (1.3) which scatters inḢ 1 to e it∆ 2 φ as t → ∓∞. Note the reversal of signs.
The approximate solution is obtained as follows. LetG n , S n be the frame operators as defined in (4.1) and (4.2), and define for each n a Littlewood-Paley cutoff where ϕ : R → R denotes a smooth function equal to 1 on the ball B(0, 1) and supported in B(0, 1.1). Fix a large T > 0, and define Inside the "window of concentration",ṽ T n is essentially a modulated solution to (1.3) with cutoffs applied in both space, to place the solution in C t Σ x , and frequency, to enable taking an extra derivative in the error analysis for the stability theory. The time translation by t n is needed to undo the time translation built into the operatorG n ; see (4.1).
Essentially the same computations as in [15]   where δ 0 is the constant in Theorem 2.2. In the focusing case, assume also that E c < E ∆ (W ) and ∇u n (t n ) L 2 < ∇W L 2 . Then there exists a subsequence such that u n (t n ) converges in Q(H).
Proof. We refer to the presentation following Proposition 6.1 in [15]. The proof uses a local smoothing estimate for the propagator e −itH , which can be obtained just as in Corollary 2.10 of [15]. In the focusing case, one also uses energy trapping arguments (see Section 7 of [15]) to see that the hypotheses are in fact equivalent to H 1/2 u n (t n ) L 2 < ∇W L 2 .
Let v : (−T min , T max ) be the maximum-lifespan solution to (1.2) with v(0) = φ. By comparing v(t, x) with the solutions u n (t + t n , x) and applying Proposition 3.3, we see that S (0,δ0/2) (v) = S (−δ0/2,0) (v) = ∞. Thus −δ 0 /2 ≤ −T min < T max ≤ δ 0 /2. But the orbit {v(t)} t∈(−Tmin,Tmax) is a precompact subset of Σ, by Proposition 6.1, so there is some sequence of times t n increasing to T max such that v(t n ) converges in Σ to some ψ. By considering a local solution with initial data ψ and invoking stability theory, we see that v can actually be extended to some larger interval (−T min , T max + η), in contradiction to the maximality of v.