aa r X i v : . [ m a t h . A P ] J un THE ENERGY-CRITICAL QUANTUM HARMONICOSCILLATOR
CASEY JAO
Abstract.
We consider the energy critical nonlinear Schr¨odinger equation indimensions d ≥ V ( x ) = | x | . Whenthe nonlinearity is defocusing, we prove global wellposedness for all initialdata in the energy space Σ, consisting of all functions u such that both ∇ u and xu belong to L . This result extends a theorem of Killip-Visan-Zhang[23], which treats the radial case. For the focusing problem, we obtain globalwellposedness for all data satisfying an analogue of the usual size restriction interms of the ground state W . The proof uses the concentration compactnessvariant of the induction on energy paradigm. In particular, we develop alinear profile decomposition adapted to the propagator exp[ it ( ∆ − | x | )] forbounded sequences in Σ. Contents
1. Introduction 1Acknowledgements 62. Preliminaries 62.1. Notation and basic estimates 62.2. Littlewood-Paley theory 92.3. Local smoothing 103. Local theory 124. Concentration compactness 124.1. An Inverse Strichartz Inequality 134.2. Convergence of linear propagators 214.3. End of Proof of Inverse Strichartz 254.4. Linear profile decomposition 265. The case of concentrated initial data 286. Palais-Smale and the proof of Theorem 1.2 367. Proof of Theorem 1.3 468. Bounded linear potentials 47References 501.
Introduction
We study the initial value problem for the energy-critical nonlinear Schr¨odingerequation on R d , d ≥
3, with a harmonic oscillator potential:(1.1) ( i∂ t u = ( − ∆ + | x | ) u + µ | u | d − u, µ = ± ,u (0) = u ∈ Σ( R d ) . The equation is defocusing if µ = 1 and focusing if µ = −
1. Solutions to this PDEconserve energy, which is defined as(1.2) E ( u ( t )) = Z R d h |∇ u ( t ) | + | x | | u ( t ) | + d − d µ | u ( t ) | dd − i dx = E ( u (0)) . Indeed (1.1) can be viewed as defining the (formal) Hamiltonian flow of E . Theterm “energy-critical” refers to the fact that if we ignore the | x | / u ( t, x ) u λ ( t, x ) := λ − d − u ( λ − t, λ − x )preserves both the equation and the energy. We take our initial data in the weightedSobolev space Σ, which is the natural space of functions associated with the energyfunctional. This space is equipped with the norm(1.4) k f k = k∇ f k L + k xf k L = k f k H + k f k L ( | x | dx ) We will frequently employ the notation H = − ∆ + | x | , F ( z ) = µ | z | d − z. Let us first clarify what we mean by a solution.
Definition.
A (strong) solution to (1.1) is a function u : I × R d → C that belongsto C t ( K ; Σ) for every compact interval K ⊂ I , and that satisfies the Duhamelformula(1.5) u ( t ) = e − itH u (0) − i Z t e − i ( t − s ) H F ( u ( s )) ds for all t ∈ I. The hypothesis on u implies that F ( u ) ∈ C t,loc L dd +2 x ( I × R d ). Consequently,the rightside above is well-defined, at least as a weak integral of tempered distributions.Equation (1.1) and its variants i∂ t u = ( − ∆ + V ) u + F ( u ) , V = ± | x | , F ( u ) = ±| u | p u, p > p < / ( d − V = | x | / F ( u ) = | u | p u, p < / ( d −
2) when the potential V ( x ) = | x | / F ( u ) = −| u | p u . In [6], he also studied [6] the case of an anisotropic harmonicoscillator with V ( x ) = P j δ j x j / , δ j ∈ { , , − } .There has also been interest in more general potentials. The paper [25] proveslong-time existence in the presence of a focusing, mass-subcritical nonlinearity F ( u ) = −| u | p u, p < /d when V ( x ) is merely assumed to grow subquadratically(by which we mean ∂ α V ∈ L ∞ for all | α | ≥ time-dependent subquadratic potentials V ( t, x ). Taking initial data in Σ, he estab-lished global existence and uniqueness when 4 /d ≤ p < / ( d −
2) for the defocusingnonlinearity and 0 < p < /d in the focusing case.This paper studies the energy-critical problem p = 4 / ( d − HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 3 the initial data u . Therefore, one cannot pass directly from local wellposedness toglobal wellposedness using conservation laws as in the subcritical case. This issueis most evident if we temporarily discard the potential and consider the equation(1.6) i∂ t u = − ∆ u + µ | u | d − u, u (0) = u ∈ ˙ H ( R d ) , d ≥ , which has the Hamiltonian E ∆ ( u ) = Z |∇ u | + µ d − d | u | dd − dx. We will refer to this equation in the sequel as the “potential-free”’, “translation-invariant”, or “scale-invariant” problem. Since the spacetime scaling (1.3) preservesboth the equation and the ˙ H norm of the initial data, the time of existence guar-anteed by the local wellposedness theory cannot depend merely on k u k ˙ H . Onecannot iterate the local existence argument to obtain global existence because witheach iteration the solution could conceivably become more concentrated in spacewhile remaining bounded in ˙ H , so that the duration of local existence could de-crease with each iteration. The scale invariance makes the analysis of (1.6) highlynontrivial.We mention equation (1.6) because the original equation increasingly resembles(1.6) as the initial data concentrates at a point; see sections 4.2 and 5 for more pre-cise statements concerning this sort of limit. Hence, one would expect the essentialdifficulties in the energy-critical NLS to also manifest themselves in the energy-critical harmonic oscillator. Understanding the scale-invariant problem is thereforean important step toward understanding the harmonic oscillator. The last fifteenyears have witnessed intensive study of the former, and the following conjecturehas been verified in all but a few cases: Conjecture 1.1.
When µ = 1 , solutions to (1.6) exist globally and scatter. Thatis, for any u ∈ ˙ H ( R d ) , there exists a unique global solution u : R × R d → C to (1.6) with u (0) = u , and this solution satisfies a spacetime bound (1.7) S R ( u ) := Z R Z R d | u ( t, x ) | d +2) d − dx dt ≤ C ( E ∆ ( u )) < ∞ . Moreover, there exist functions u ± ∈ ˙ H ( R d ) such that lim t →±∞ k u ( t ) − e ± it ∆2 u ± k ˙ H = 0 , and the correspondences u u ± ( u ) are homeomorphisms of ˙ H .When µ = − , one also has global wellposedness and scattering provided that E ∆ ( u ) < E ∆ ( W ) , k∇ u k L < k∇ W k L , where the ground state W ( x ) = | x | d ( d − ) d − ∈ ˙ H ( R d ) solves the elliptic equation ∆ + | W | d − W = 0 . Theorem 1.1.
Conjecture 1.1 holds for the defocusing equation. For the focusingequation, the conjecture holds for radial initial data when d ≥ , and for all initialdata when d ≥ .Proof. See [2, 9, 26, 31] for the defocusing case and [16, 21] for the focusing case. (cid:3)
CASEY JAO
One can formulate a similar conjecture for (1.1); however, as the linear propaga-tor is periodic in time, one only expects uniform local-in-time spacetime bounds.
Conjecture 1.2.
When µ = 1 , equation (1.1) is globally wellposed. That is, foreach u ∈ Σ there is a unique global solution u : R × R d → C with u (0) = u . Thissolution obeys the spacetime bound (1.8) S I ( u ) := Z I Z R d | u ( t, x ) | d +2) d − dx dt ≤ C ( | I | , k u k Σ ) for any compact interval I ⊂ R .If µ = − , then the same is true provided also that E ( u ) < E ∆ ( W ) and k∇ u k L ≤ k∇ W k L . In [23], Killip-Visan-Zhang verifed this conjecture with µ = 1 and sphericallysymmetric initial data. By adapting an argument of Bourgain-Tao for the equationwithout potential (1.6), they proved that the defocusing problem (1.1) is globallywellposed, and also proved scattering for the repulsive potential. We consider onlythe confining potential. In this paper, we remove the assumption of spherical sym-metry for the defocusing harmonic oscillator, and also establish global wellposednessfor the focusing problem under the assumption that Conjecture (1.1) holds for alldimensions. Specifically, we prove Theorem 1.2.
Assume that Conjecture 1.1 holds. Then Conjecture 1.2 holds.
By Theorem 1.1, this result is conditional only in the focusing situation fornonradial data in dimensions 3 and 4. Moreover, in the focusing case we haveessentially the same blowup result as for the potential-free NLS with the sameproof as in that case; see [21]. We recall the argument in Section 7.
Theorem 1.3 (Blowup) . Suppose µ = − and d ≥ . If u ∈ Σ satisfies E ( u ) HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 5 (1) Prove the existence of a minimal counterexample (where “minimal” will bedefined shortly).(2) Show that this counterexample violates properties obeyed by all solutions.Let us elaborate a little on these steps. The energy E ( u ) of a solution (whichequals the energy of the initial data) will serve as our induction parameter. Bythe local wellposedness theory (which we review in Section 3), uniform spacetimebounds hold for all solutions with sufficiently small energy E ( u ). Assuming thatTheorem 1.2 fails, we obtain a positive threshold 0 < E c < ∞ such that (1.8) holdswhenever E ( u ) < E c and fails when E ( u ) > E c . The first step described abovewould have us construct a solution u c : I × R d → C on a bounded interval I with S I ( u c ) = ∞ , and whose energy equals precisely the critical threshold E c . Thus“minimal” in our setting refers to minimal energy.We will carry out a variant of this strategy that is better adapted to what we aretrying to prove. Since the spacetime estimates of interest are local-in-time, it sufficesto prevent the blowup of spacetime norm on arbitrarily small time intervals. Thatis, we need only show that for each energy E > 0, there exists some L = L ( E ) > S I ( u ) ≤ C ( E ) whenever E ( u ) ≤ E and | I | ≤ L . To prove this statement, wework not with minimal-energy blowup solutions directly but rather with the Palais-Smale compactness theorem (Proposition 6.1) that would beget such solutions. Ourargument will ultimately reduce the question of global wellposedness for (1.1) tothat of global wellposedness and scattering for the potential-free equation (1.6). Ineffect, we shall discover that the only scenario where blowup could possibly occuris when the solution is highly concentrated at a point and behaves like a solutionto (1.6).This paradigm of recovering the potential-free NLS in certain limiting regimes isby now well-known and has been applied to the study of other equations. See [19, 20,14, 13, 12, 24] for adaptations to gKdV, Klein-Gordon, and NLS in various domainsand manifolds. While the particulars are unique to each case, a common key stepis to prove an appropriate compactness theorem in the style of Proposition 6.1. Asin the previous work, our proof of that proposition uses three main ingredients.The first prerequisite is a local wellposedness theory that gives local existenceand uniqueness as well as stability of solutions with respect to perturbations ofthe initial data or the equation itself. In our case, local wellposedness will followfrom familiar arguments employing the dispersive estimate satisfied by the linearpropagator e − itH , as well the fractional product and chain rules for the operators H γ , γ ≥ 0. We review the relevant results in Section 3.We also need a linear profile decomposition for the Strichartz inequality(1.9) k e − itH f k L d +2) d − t,x . k H f k L x . Such a decomposition in the context of energy-critical Schr¨odinger equations wasfirst proved by Keraani [17] in the translation-invariant setting for the free particleHamiltonian H = − ∆, and quantifies the manner in which a sequence of func-tions f n with k H / f n k L bounded may fail to produce a subsequence of e − itH f n converging in the spacetime norm. The defect of compactness arises in Keraani’scase from a noncompact group of symmetries of the inequality (1.9), which includesspatial translations and scaling. In our setting, there are no obvious symmetriesof (1.9); nonetheless, compactness can fail and in Section 4 we formulate a profiledecomposition for (1.9) when H is the Hamiltonian of the harmonic oscillator. CASEY JAO The final ingredient is an analysis of (1.1) when the initial data is highly concen-trated in space, corresponding to a single profile in the linear profile decompositionjust discussed. In Section 5, we show that blowup cannot occur in this regime. Thebasic idea is that while the solution to (1.1) remains highly localized in space, itcan be well-approximated up to a phase factor by the corresponding solution to thescale-invariant energy-critical NLS(1.10) ( i∂ t + ∆) u = ±| u | d − u. By the time this approximation breaks down, the solution to the original equationwill have dispersed and can instead be approximated by a solution to the linearequation ( i∂ t − H ) u = 0. We use as a black box the nontrivial fact (which is stilla conjecture in several cases) that solutions to (1.6) obey global spacetime bounds.By stability theory, the spacetime bounds for the approximations will be transferredto the solution for the original equation and will therefore preclude blowup.We have chosen to focus on the concrete potential V ( x ) = | x | mainly forconcreteness. In a forthcoming paper we will indicate how to extend the mainresult to a more general class of subquadratic potentials. Acknowledgements. The author is indebted to his advisors Rowan Killip andMonica Visan for their helpful discussions as well as their feedback on the pa-per. 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).2. Preliminaries Notation and basic estimates. We write X . Y to mean X ≤ CY forsome constant C . Similarly X ∼ Y means X . Y and Y . X . Denote by L p ( R d )the Banach space of functions f : R d → C with finite norm k f k L p ( R d ) = (cid:18)Z R d | f | p dx (cid:19) p . We will sometimes use the more compact notation k f k p . If I ⊂ R d is an interval,the mixed Lebesgue norms on I × R d are defined by k f k L qt L rx ( I × R d ) = Z I (cid:18)Z R d | f ( t, x ) | r dx (cid:19) qr dt ! q = k f ( t ) k L qt ( I ; L rx ( R d )) , where one regards f ( t ) = f ( t, · ) as a function from I to L r ( R d ).The operator H = − ∆ + | x | is positive on L ( R d ). Its associated heat kernelis given by Mehler’s formula [10]:(2.1) e − tH ( x, y ) = e ˜ γ ( t )( x + y ) e sinh( t )∆2 ( x, y )where ˜ γ ( t ) = 1 − cosh t t = − t O ( t ) as t → . By analytic continuation, the associated one-parameter unitary group has the in-tegral kernel(2.2) e − itH f ( x ) = 1(2 πi sin t ) d Z e i sin t (cid:16) x y cos t − xy (cid:17) f ( y ) dy. HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 7 Comparing this to the well-known free propagator(2.3) e it ∆2 f ( x ) = πit ) d Z e i | x − y | t f ( y ) dy, we obtain the relation(2.4) e − itH f = e iγ ( t ) | x | e i sin( t )∆2 ( e iγ ( t ) | x | f )where γ ( t ) = cos t − 12 sin t = − t O ( t ) as t → . Mehler’s formula immediately implies the local-in-time dispersive estimate(2.5) k e − itH f k L ∞ x . | sin t | − d k f k L . For d ≥ 3, call a pair of exponents ( q, r ) admissible if q ≥ q + dr = d . Write k f k S ( I ) = k f k L ∞ t L x + k f k L t L dd − x with all norms taken over the spacetime slab I × R d . By interpolation, we see thatthis norm controls the L qt L rx norm for all other admissible pairs. We let k F k N ( I ) = inf {k F k L q ′ t L r ′ x + k F k L q ′ t L r ′ x : ( q k , r k ) admissible , F = F + F } , where ( q ′ k , r ′ k ) is the H¨older dual to ( q k , r k ). Lemma 2.1 (Strichartz estimates) . Let I be a compact time interval containing t ,and let u : I × R d → C be a solution to the inhomogeneous Schr¨odinger equation ( i∂ t − H ) u = F. Then there is a constant C = C ( | I | ) , depending only on the length of the interval,such that k u k S ( I ) ≤ C ( k u ( t ) k L + k F k N ( I ) ) . Proof. This follows from the dispersive estimate (2.5), the unitarity of e − itH on L , and general considerations; see [15]. By partitioning time into unit intervals,we see that the constant C grows at worst like | I | (which corresponds to the timeexponent q = 2). (cid:3) It will be convenient to introduce the operators which represent the time evolu-tion of the momentum and position operators under the linear propagator. Theseare well-known in the literature and were used in [4] or [23], for example. We define P ( t ) = e itH i ∇ e − itH = i ∇ cos t − x sin tX ( t ) = e itH xe − itH = i ∇ sin t + x cos t. (2.6)One easily verifies the identity k P ( t ) f k L + k X ( t ) f k L = k P ( t ) f k L + k P ( t + π ) f k L = k f k . We use the fractional powers H γ of the operator H , defined via the Borel func-tional calculus, as a substitute for the usual derivative ( − ∆) γ , which does notcommute with the linear propagator e − itH . We have trivially that k H f k L ∼ k ( − ∆) f k L + k| x | f k L ∼ k f k Σ . CASEY JAO Perhaps less obvious is the fact that this equivalence generalizes to other L p normsand other powers of H . Using complex interpolation, Killip, Visan, and Zhangshowed that this is the case: Lemma 2.2 ([23, Lemma 2.7]) . For ≤ γ ≤ and < p < ∞ , one has k H γ f k L p ( R d ) ∼ k ( − ∆) γ f k L p ( R d ) + k| x | γ f k L p ( R d ) . As a consequence, H γ inherits many properties of ( − ∆) γ , including Sobolevembedding: Lemma 2.3 ([23, Lemma 2.8]) . Suppose γ ∈ [0 , and < p < d γ , and define p ∗ by p ∗ = p − γd . Then k f k L p ∗ ( R d ) . k H γ f k L p ( R d ) . Similarly, the fractional chain and product rules carry over to the current setting: Corollary 2.4 ([23, Proposition 2.10]) . Let F ( z ) = | z | d − z . For any ≤ γ ≤ and < p < ∞ , k H γ F ( u ) k L p ( R d ) . k F ′ ( u ) k L p ( R d ) k H γ f k L p ( R d ) for all p , p ∈ (1 , ∞ ) with p − = p − + p − . Using Lemma 2.2 and the Christ-Weinstein fractional product rule for ( − ∆) γ (e.g. [30]), we obtain Corollary 2.5. For γ ∈ (0 , , r, p i , q i ∈ (1 , ∞ ) with r − = p − i + q − i , i = 1 , , wehave k H γ ( f g ) k r . k H γ f k p k g k q + k f k p k H γ g k q . The exponent γ = is particularly relevant to us, and it will be convenient touse the notation L qt Σ rx ( I × R d ) for the space of functions f with norm k f k L qt Σ rx = k H f k L qt L rx . The superscript on Σ is assumed to be 2 if omitted. We shall need the followingrefinement of Fatou’s Lemma due to Br´ezis and Lieb: Lemma 2.6 (Refined Fatou [3]) . Fix ≤ p < ∞ , and suppose f n is a sequence offunctions in L p ( R d ) such that sup n k f n k p < ∞ and f n → f pointwise. Then lim n →∞ Z R d || f n | p − | f n − f | p − | f | p | dx = 0 . Finally, we record an analogue of the H¨ormander-Mikhlin Fourier multiplier the-orem proved by Hebisch [11]. It enables a Littlewood-Paley theory adapted to H ,as discussed in the next section. Theorem 2.1. If F : R → C is a bounded function which obeys the derivativeestimates | ∂ k F ( λ ) | . k | λ | − k for all ≤ k ≤ d + 1 , then the operator F ( H ) , defined initially on L via the Borel functional calculus, isbounded from L p to L p for all < p < ∞ . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 9 Littlewood-Paley theory. Owing largely to Theorem 2.1, we can importthe basic results of Littlewood-Paley theory with little effort, the only change beingthat one replaces Fourier multipliers with spectral multipliers. We fashion two kindsof Littlewood-Paley projections, one using compactly supported bump functions,and the other based on the heat kernel of H . The parabolic maximum principleimplies that(2.7) 0 ≤ e − tH ( x, y ) ≤ e t ∆2 ( x, y ) = πt ) d/ e − | x − y | t . Fix a smooth function ϕ supported in | λ | ≤ ϕ ( λ ) = 1 for | λ | ≤ 1, and let ψ ( λ ) = ϕ ( λ ) − ϕ (2 λ ). For each dyadic number N ∈ Z , which we will often referto as “frequency,” define P H ≤ N = ϕ ( p H/N ) , P HN = ψ ( p H/N ) , ˜ P H ≤ N = e − H/N , ˜ P HN = e − H/N − e − H/N . The associated operators P H H>N , etc. are defined in the usual manner. Remark. As the spectrum of H is bounded away from 0, by choosing ϕ appropri-ately we can arrange for P < = 0; thus we will only consider frequencies N ≥ P ∆ ≤ N = ϕ ( p − ∆ /N ) P ∆ N = ψ ( p − ∆ /N ) , (2.8) ˜ P ∆ ≤ N = e ∆ / N ˜ P ∆ N = e ∆ / N − e /N . (2.9)denote the classical Littlewood-Paley projections. From the maximum principle weobtain the pointwise bound(2.10) | ˜ P HN f ( x ) | + | ˜ P H ≤ N f ( x ) | . ˜ P ∆ ≤ N | f | ( x ) + ˜ P ∆ ≤ N/ | f | ( x ) . To reduce clutter we usually suppress the superscripts H and ∆ when it is clearfrom the context which type of projection we are using. For the rest of this section, P ≤ N and P N denote P H ≤ N and P HN , respectively. Lemma 2.7 (Bernstein estimates) . For f ∈ C ∞ c ( R d ) , < p ≤ q < ∞ , s ≥ , onehas the Bernstein inequalities k P ≤ N f k p . k ˜ P ≤ N f k p , k P N f k p . k ˜ P N f k p (2.11) k P ≤ N f k p + k P N f k p + k ˜ P ≤ N f k p + k ˜ P N f k p . k f k p (2.12) k P ≤ N f k q + k P N f k q + k ˜ P ≤ N f k q + k ˜ P N f k q . N dp − dq k f k p (2.13) N s k P N f k p ∼ k H s P N f k p (2.14) k P >N f k p . N − s k H s P >N f k p . (2.15) In (2.13) , the estimates for ˜ P ≤ N f and ˜ P N f also hold when p = 1 , q = ∞ . Further, f = X N P N f = X N ˜ P N f (2.16) where the series converge in L p , < p < ∞ . Finally, we have the square functionestimate k f k p ∼ k ( X N | P N f | ) / k p . (2.17) Proof. The estimates (2.11) follow immediately from Theorem 2.1. To see (2.12),observe that the functions ϕ ( p · /N ) , e −· /N satisfy the hypotheses of Theorem 2.1uniformly in N . Next use (2.7) together with Young’s convolution inequality to get(2.18) k ˜ P ≤ N f k q + k ˜ P N f k q . N dq − dp k f k p for 1 ≤ p ≤ q ≤ ∞ . From (2.11) we obtain the rest of (2.13). Now consider (2.14). Let ˜ ψ be a fattenedversion of ψ so that ˜ ψ = 1 on the support of ψ . Put F ( λ ) = λ s ˜ ψ ( √ λ ). ByTheorem 2.1, the relation ψ = ˜ ψψ , and the functional calculus, k N − s H s P N f k p = k F ( H/N ) P N f k p . k P N f k p . The reverse inequality follows by considering F ( x ) = λ − s ˜ ψ ( λ ).We turn to (2.16). The equality holds in L by the functional calculus and thefact that the spectrum of H is bounded away from 0. For p = 2, choose q and0 < θ < p − = 2 − (1 − θ ) + q − θ . By (2.12), the partial sum operators S N ,N = X N The following local smoothing lemma and its corollarywill be needed when proving properties of the nonlinear profile decomposition inSection 6. Lemma 2.8. If u = e − itH φ, φ ∈ Σ( R d ) , then Z I Z R d |∇ u ( x ) | h R − ( x − z ) i − dx dt . R (1 + | I | ) k u k L ∞ t L x k H / u k L ∞ t L x . with the constant independent of z ∈ R d and R > .Proof. We recall the Morawetz identity. Let a be a sufficiently smooth function of x ; then for any u satisfying the linear equation i∂ t u = ( − ∆ + V ) u , one has ∂ t Z ∇ a · Im( u ∇ u ) dx = Z a jk Re( u j u k ) dx − Z | u | a jjkk dx − Z | u | ∇ a · ∇ V dx (2.19)We use this identity with a ( x ) = h R − ( x − z ) i and V = | x | , and compute a j ( x ) = R − ( x j − z j ) h R − ( x − z ) i , a jk ( x ) = R − (cid:20) δ jk h R − ( x − z ) i − R − ( x j − z j )( x k − z k ) h R − ( x − z ) i (cid:21) ∆ a ( x ) ≤ − R − h R − ( x − z ) i . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 11 As ∆ a ≤ 0, the right side of (2.19) is bounded below by R − Z h R − ( x − z ) i − h |∇ u | − | R − ( x − z ) h R − ( x − z ) i · ∇ u | i dx − R Z | u | R − ( x − z ) h R − ( x − z ) i · x dx ≥ R − Z |∇ u ( x ) | h R − ( x − z ) i − dx − R − Z | u | | x | dx. Integrating in time and applying Cauchy-Schwarz, we get R − Z I Z R d h R − ( x − z ) i − |∇ u ( t, x ) | dxdt . sup t ∈ I R − Z R − ( x − z ) h R − ( x − z ) i | u ( t, x ) ||∇ u ( t, x ) | dx + R Z I Z R d | x || u | dxdt . R − (1 + | I | ) k u k L ∞ t L x k H / u k L ∞ t L x . This completes the proof of the lemma. (cid:3) Corollary 2.9. Fix φ ∈ Σ( R d ) . Then for all T, R ≤ , we have k∇ e − itH φ k L t,x ( | t − t |≤ T, | x − x |≤ R ) . T d +2) R d +23( d +2) k φ k Σ k e − itH φ k L d +2) d − t,x . When d = 3 , we also have k∇ e − itH φ k L t L x ( | t − t |≤ T, | x − x |≤ R ) . T R k e − itH φ k L t,x k φ k Σ Proof. The proofs are fairly standard (see [32] or [24]), and we present just the proofof the second claim, which is slightly more involved. Let E the region {| t − t | ≤ T, | x − x | ≤ R } . Norms which do not specify the region of integration are takenover the spacetime slab {| t − t | ≤ T } × R . By H¨older, k∇ e − itH φ k L t L x ( E ) ≤ k∇ e − itH φ k L t,x ( E ) k∇ e − itH φ k L t L x ( E ) . By H¨older and Strichartz, k∇ e − itH φ k L t L x ( E ) . T k∇ e − itH φ k L t L x . T k φ k Σ . (2.20)We now estimate k∇ e − itH φ k L t,x . Let N ∈ N be a dyadic number to be chosenlater, and decompose k∇ e − itH φ k L t,x ( E ) ≤ k∇ e − itH P H ≤ N φ k L t,x ( E ) + k∇ e − itH P H>N φ k L t,x ( E ) . For the low frequency piece, apply H¨older and the Bernstein inequalities to obtain k∇ e − itH P H ≤ N φ k L t,x . T R k∇ e − itH P H ≤ N φ k L t,x . T R N k e − itH φ k L t,x . For the high-frequency piece, apply local smoothing and Bernstein: k∇ e − itH P H>N φ k L t,x . R k P H>N φ k L k H φ k Σ . R N − k φ k Σ . Optimizing in N , we obtain k∇ e − itH φ k L t,x . T R k e − itH φ k L t,x k φ k Σ . Combining this estimate with (2.20) yields the conclusion of the corollary. (cid:3) Local theory We record some standard results concerning local-wellposedness for (1.1). Theseare direct analogues of the theory for the scale-invariant equation (1.6). By Lemma 2.3and Corollaries 2.4 and 2.5, we can use essentially the same proofs as in that case.We refer the reader to [22] for those proofs. Proposition 3.1 (Local wellposedness) . Let u ∈ Σ( R d ) and fix a compact timeinterval ∈ I ⊂ R . Then there exists a constant η = η ( d, | I | ) such that whenever η < η and k H e − itH u k L d +2) d − t L d ( d +2) d x ( I × R d ) ≤ η, there exists a unique solution u : I × R d → C to (1.1) which satisfies the bounds k H u k L d +2) d − t L d ( d +2) d x ( I × R d ) ≤ η and k H u k S ( I ) . k u k Σ + η d +2 d − . Corollary 3.2 (Blowup criterion) . Suppose u : ( T min , T max ) × R d → C is a maxi-mal lifespan solution to (1.1) , and fix T min < t < T max . If T max < ∞ , then k u k L d +2) d − t,x ([ t ,T max )) = ∞ . If T min > −∞ , then k u k L d +2) d − t,x (( T min ,t ]) = ∞ . Proposition 3.3 (Stability) . Fix t ∈ I ⊂ R an interval of unit length and let ˜ u : I × R d → C be an approximate solution to (1.1) in the sense that i∂ t ˜ u = Hu ± | ˜ u | d − ˜ u + e for some function e . Assume that (3.1) k ˜ u k L d +2) d − t,x ≤ L, k H u k L ∞ t L x ≤ E, and that for some < ε < ε ( E, L ) one has (3.2) k ˜ u ( t ) − u k Σ + k H e k N ( I ) ≤ ε, Then there exists a unique solution u : I × R d → C to (1.1) with u ( t ) = u andwhich further satisfies the estimates (3.3) k ˜ u − u k L d +2) d − t,x + k H (˜ u − u ) k S ( I ) . C ( E, L ) ε c where < c = c ( d ) < and C ( E, L ) is a function which is nondecreasing in eachvariable. Concentration compactness In this section we discuss some concentration compactness results for the Strichartzinequality k e − itH f k L d +2) d − t,x ( I × R d ) ≤ C ( | I | , d ) k f k Σ , culminating in the linear profile decomposition of Proposition 4.14. Our profiledecomposition resembles that of Keraani [17] in the sense that each profile livesat a well-defined location in spacetime and has a characteristic length scale. But HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 13 since the function space Σ lacks both translation and scaling symmetry, the precisedefinitions of our profiles will be more complicated.Keraani considered the analogous Strichartz estimate k e it ∆ f k L d +2) d − t,x ( R × R d ) . k f k ˙ H ( R d ) . Recall that in that situation, if f n is a bounded sequence in ˙ H with nontrivial linearevolution, then one has a decomposition f n = φ n + r n where φ n = e it n ∆ G n φ , G n arecertain unitary scaling and translation operators on ˙ H (defined as in (4.1)), and φ is a weak limit of G − n e − it n ∆ f n in ˙ H . The “bubble” φ n is nontrivial and decouplesfrom the remainder r n in various norms. By applying this decomposition inductivelyto the remainder r n , one obtains the full collection of profiles constituting f n .We follow the general presentation in [22, 32]. Let f n ∈ Σ be a bounded sequence.Using a variant of Keraani’s argument, we seek to obtain an ˙ H -weak limit φ interms of f n and write f n = φ n + r n where φ n is defined analogously as before by“moving the operators onto f n .” However, we need to modify this procedure inlight of two issues.The first problem is that while f n belong to Σ, an ˙ H weak limit of a sequencelike G − n e it n H f n need only belong to ˙ H . Indeed, the ˙ H isometries G − n will ingeneral have unbounded norm as operators on Σ because of the | x | spatial weight,which penalizes very wide functions. To define φ n , we need to introduce suitablychosen spatial cutoffs to obtain functions in Σ.Secondly, to establish the various orthogonality assertions we must understandhow the linear propagator e − itH interacts with the ˙ H symmetries of translationand scaling in certain limits. We study this interaction in Section 4.2. In particular,the convergence lemmas proved there serve as a substitute for the scaling relation e it ∆ G n = G n e iN n t ∆ where G n φ = N d − n φ ( N n ( · − x n )) . They can also be regarded as a precise form of the heuristic stated in the intro-duction that as we scale the initial data to concentrate at a point x , the potential V ( x ) = | x | / V ( x ); hence for short times the linear propagator e − itH can beapproximated up to a phase factor by the free particle propagator. In Section 5 weshall see a nonlinear version of this statement.4.1. An Inverse Strichartz Inequality. Unless indicated otherwise, 0 ∈ I in thissection will denote a fixed interval of length at most 1, and all spacetime normswill be taken over I × R d .Suppose f n is a sequence of functions in Σ with nontrivial linear evolution e − itH f n . The following refined Strichartz estimate shows that there must be a“frequency” N n which makes a nontrivial contribution to the evolution. Proposition 4.1 (Refined Strichartz) . k e − itH f k L d +2) d − t,x . k f k d +2 Σ sup N k e − itH P N f k d − d +2 L d +2) d − t,x Proof. We quote essentially verbatim the proof of Refined Strichartz for the freeparticle propagator ([32] Lemma 3.1). Write f N for P N f , where P N = P HN unless indicated otherwise. When d ≥ 6, we apply the square function estimate (2.17),H¨older, Bernstein, and Strichartz to get k e − itH f k d +2) d − L d +2) d − t,x ∼ (cid:13)(cid:13)(cid:13) ( X N | e − itH f N | ) / (cid:13)(cid:13)(cid:13) d +2) d − d +2) d − = Z Z ( X N | e − itH f N | ) d +2 d − dx dt . X M ≤ N Z Z | e − itH f M | d +2 d − | e − itH f N | d +2 d − dx dt . X M ≤ N k e − itH f M k d − L d +2) d − t,x k e − itH f M k L d +2) d − t,x k e − itH f N k d − L d +2) d − t,x k e − itH f N k L d +2) dt,x . sup N k e − itH f N k d − L d +2) d − t,x X M ≤ N M k e − itH f M k L d +2) d − t L d ( d +2) d x k f N k L . sup N k e − itH f N k d − L d +2) d − t,x X M ≤ N M k f M k L x k f N k L x . sup N k e − itH f N k d − L d +2) d − t,x X M ≤ N MN k H / f M k L k H / f N k L x . sup N k e − itH f N k d − L d +2) d − t,x k f k . The cases d = 3 , , (cid:3) The next proposition goes one step further and asserts that the sequence e − itH f n with nontrivial spacetime norm must in fact contain a bubble centered at some( t n , x n ) with spatial scale N − n . We first introduce some vocabulary and notationwhich will help make the presentation more systematic. Adapting terminology fromIonescu-Pausader-Staffilani [14], we define Definition 4.1. A frame is a sequence ( t n , x n , N n ) ∈ I × R d × N conforming toone of the following scenarios:(1) N n ≡ , t n ≡ 0, and x n ≡ N n → ∞ and N − n | x n | → r ∞ ∈ [0 , ∞ ).Informally, the parameters t n , x n , N n will specify the temporal center, spatialcenter, and (spatial) frequency of a function. The condition that | x n | . N n reflectsthe fact that we only consider functions obeying some uniform bound in Σ, and suchfunctions cannot be centered arbitrarily far from the origin. We need to augmentthe frame { ( t n , x n , N n ) } with an auxiliary parameter N ′ n , which corresponds to asequence 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 × N × R belonging to one of the following types:(1) N n ≡ , t n ≡ , x n ≡ , N ′ n ≡ N n → ∞ , N − n | x n | → r ∞ ∈ [0 , ∞ ), and either(2a) N ′ n ≡ r ∞ > 0, or(2b) N / n ≤ N ′ n ≤ N n , N − n | x n | ( N n N ′ n ) → 0, and N n N ′ n → ∞ if r ∞ = 0. HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 15 Given an augmented frame ( t n , x n , N n , N ′ n ), we define scaling and translationoperators on functions of space and of spacetime by( G n φ )( x ) = N d − n φ ( N n ( x − x n ))( ˜ G n f )( t, x ) = N d − n f ( N n ( t − t n ) , N n ( x − x n )) . (4.1)We also define spatial cutoff operators S n by(4.2) S n φ = (cid:26) φ, for frames of type 1 (i.e. N n ≡ ,χ ( N n N ′ n · ) φ, for frames of type 2 (i.e. N n → ∞ ) , where χ is a smooth compactly supported function equal to 1 on the ball {| x | ≤ } .An easy computation yields the following mapping properties of these operators:lim n →∞ S n = I strongly in ˙ H and in Σ , lim sup n →∞ k G n k Σ → Σ < ∞ . (4.3)For future reference, we record a technical lemma that, as a special case, as-serts that the Σ norm is controlled almost entirely by the ˙ H norm for functionsconcentrating near the origin. Lemma 4.2 (Approximation) . Let ( q, r ) be an admissible pair of exponents with ≤ r < d , and let F = { ( t n , x n , N n , N ′ n ) } be an augmented frame of type 2. (1) Suppose F is of type 2a in Definition 4.2. Then for { f n } ⊆ L qt H ,rx ( R × R d ) ,we have lim sup n k ˜ G n S n f n k L qt Σ rx . lim sup n k f n k L qt H ,rx . (2) Suppose F is of type 2b and f n ∈ L qt ˙ H ,rx ( R × R d ) . Then lim sup n k ˜ G n S n f n k L qt Σ rx . lim sup n k f n k L qt ˙ H ,rx . Here H ,r ( R d ) and ˙ H ,r ( R d ) denote the inhomogeneous and homogeneous L r Sobolevspaces, respectively, equipped with the norms k f k H ,r = kh∇ik L r ( R d ) , k f k ˙ H ,r = k|∇| f k L r ( R d ) . Proof. By time translation invariance we may assume t n ≡ 0. Using Lemma 2.2,we see that it suffices to bound k∇ ˜ G n S n f n k L qt L rx and k| x | ˜ G n S n f n k L qt L rx separately.By a change of variables, the admissibility condition on ( q, r ), H¨older, and Sobolevembedding (which necessitates the restriction r < d ), we have k∇ ˜ G n S n f n k L qt L rx = k∇ [ N d − n f n ( N n t, N n ( x − x n )) χ ( N ′ n ( x − x n ))] k L qt L rx . k ( ∇ f n )( t, x ) k L qt L rx + N ′ n N n k f n ( t, x ) k L qt L rx ( R ×{| x |∼ NnN ′ n } ) . k∇ f n k L qt ˙ H ,rx . To estimate k| x | ˜ G n S n f n k L qt L rx we distinguish the two cases. Consider first the casein which f n ∈ L qt H ,rx . Using the bound | x n | . N n and a change of variables, weobtain k| x | ˜ G n S n f n k L qt L r . N d n k f n ( N n t, N n ( x − x n )) k L r . k f n k L qt L r . k f n k L qt H ,rx . Now consider the second case where f n are merely assumed to lie in L qt ˙ H ,rx . Foreach t , we use H¨older and Sobolev embedding to get k| x | ˜ G n S n f n k rL rx = N dr − d − rn Z | x | . NnN ′ n | x n + N − n x | r | f n ( N n t, x ) | r dx . N dr − dn h N − rn | x n | r + N − rn ( N n N ′ n ) r i Z | x | . NnN ′ n | f n ( N n t, x ) | r dx . N dr − dn h N − rn | x n | r ( N n N ′ n ) r + ( N ′ n ) − r i k∇ f n ( N n t ) k rL rx . By the hypotheses on the parameter N ′ n in Definition 4.2, the expression inside thebrackets goes to 0 as n → ∞ . After integrating in t and performing a change ofvariables, we conclude k| x | ˜ G n S n f n k L qt L rx . c n k f n k L qt ˙ H ,rx where c n = o (1) as n → ∞ . This completes the proof of the lemma. (cid:3) Proposition 4.3 (Inverse Strichartz) . Let I be a compact interval containing oflength at most , and suppose f n is a sequence of functions in Σ( R d ) satisfying < ε ≤ k e − itH f n k L d +2) d − t,x ( I × R d ) . k f n k Σ ≤ A < ∞ . 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 ∈ Σ such that one of the following holds: (1) F is of type 1 (i.e. N n ≡ ) and φ n = φ where φ ∈ Σ is a weak limit of f n in Σ . (2) F is of type 2, either t n ≡ or N n t n → ±∞ , and φ n = e it n H G n S n φ where φ ∈ ˙ H ( R d ) is a weak limit of G − n e − it n H f n in ˙ H . Moreover, if F is oftype 2a, then φ also belongs to L ( R d ) .The functions φ n have the following properties: (4.4) lim inf n k φ n k Σ & A (cid:0) εA (cid:1) d ( d +2)8 (4.5) lim n →∞ k f n k dd − dd − − k f n − φ n k dd − dd − − k φ n k dd − dd − = 0 . (4.6) lim n →∞ k f n k − k f n − φ n k − k φ n k = 0 Proof. The proof will occur in several stages. First we identify the parameters t n , x n , N n , which define the location of the bubble φ n and its characteristic size,and quickly dispose of the case where N n ≡ 1. The treatment of the case where N n → ∞ will be more involved, and we proceed in two steps. We define theprofile φ n and verify the assertions (4.4) and (4.6). Passing to a subsequence, wemay assume that the sequence N n t n converges in [ −∞ , ∞ ]. If the limit is infinite,decoupling (4.5) in the L dd − norm will also follow.After a brief interlude in which we study certain operator limits, we finish thecase where the sequence N n t n tends to a finite limit. We show that the time pa-rameter t n can actually be redefined to be identically zero after making a negligiblecorrection to the profile φ n , and verify in Lemma 4.13 that the modified profile HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 17 satisfies property (4.5) in addition to (4.4) and (4.6). One can show a posteriori that the original profile φ n also obeys this last decoupling condition.By Proposition 4.1, there exist frequencies N n such that k P N n e − itH f n k L d +2) d − t,x & ε d +24 A − d − . The comparison of Littlewood-Paley projectors (2.11) implies k ˜ P N n e − itH f n k L d +2) d − t,x & ε d +24 A − d − where ˜ P N = e − H/N − e − H/N denote the projections based on the heat kernel.By H¨older, Strichartz, and Bernstein, ε d +24 A − d − . k ˜ P N n e − itH f n k L d +2) d − t,x . k ˜ P N n e − itH f n k d − d L d +2) dt,x k ˜ P N n e − itH f n k d L ∞ t,x . ( N − n A ) d − d k ˜ P N n e − itH f n k d L ∞ t,x . Therefore, there exist ( t n , x n ) ∈ I × R d such that(4.7) | e − it n H ˜ P N n f n ( x n ) | & N d − n A ( εA ) d ( d +2)8 . The parameters t n , x n , N n will determine the center and width of a bubble. Weobserve first that the boundedness of f n in Σ limits how far the bubble can livefrom the spatial origin. Lemma 4.4. We have | x n | ≤ C A,ε N n . Proof. Put g n = | e − it n H f n | . By the kernel bound (2.10), N d − n A ( εA ) d ( d +2)8 . | ˜ P N n e − it n H f n ( x n ) | . ˜ P ∆ ≤ N n g n ( x n ) + ˜ P ∆ ≤ N n / g n ( x n ) . Thus one of the terms on the right side is at least half as large as the left side, andwe only consider the case when˜ P ∆ ≤ N n g n ( x n ) & N d − n A ( εA ) d ( d +2)8 since the argument with N n replaced by N n / P ∆ ≤ N n g n is essentially constant over length scales of order N − n , so if it is largeat a point x n then it is large on the ball | x − x n | ≤ N − n . More precisely, when | x − x n | ≤ N − n we have˜ P ∆ ≤ N n / g n ( x ) = N dn d (2 π ) d Z g n ( x − y ) e − N n | y | dy = N dn d (4 π ) d Z g n ( x n − y ) e − N n | y + x − xn | dy ≥ e − N dn d (4 π ) d Z g n ( x n − y ) e − N n | y | dy = e − − d ˜ P ∆ ≤ N n g n ( x n ) & N d − n A ( εA ) d ( d +2)8 . On the other hand, the mapping properties of the heat kernel imply that k ˜ P ∆ ≤ N n / g n k Σ . (1 + N − n ) A. Thus, A & k ˜ P ∆ ≤ N n / g n k Σ & k x ˜ P ∆ ≤ N n / g n k L ( | x − x n |≤ N − n ) & | x n | N − d n N d − n A ( εA ) d ( d +2)8 , which yields the claim. (cid:3) Case 1 . Suppose the N n have a bounded subsequence, so that (passing to asubsequence) N n ≡ N ∞ . The x n ’s stay bounded by 4.4, so after passing to asubsequence we may assume x n → x ∞ . We may also assume t n → t ∞ since theinterval I is compact. The functions f n are bounded in Σ, hence (after passing toa subsequence) converge weakly in Σ to a function φ .We show that φ is nontrivial in Σ. Indeed, h φ, e it ∞ H ˜ P N ∞ δ x ∞ i = lim n h f n , e it ∞ H ˜ P N ∞ δ x ∞ i = lim n →∞ [ e − it n H ˜ P N ∞ f n ( x n ) + h f n , ( e it ∞ H − e it n H ) ˜ P N ∞ δ x n i + h f n , e it ∞ H ˜ P N n ( δ x ∞ − δ x n ) i ] . Using the heat kernel bounds (2.10) and the fact that, by the compactness of theembedding Σ ⊂ L , the sequence f n converges to φ in L , one verifies easily thatthe second and third terms on the right side vanish. So |h φ, e it ∞ H ˜ P N ∞ δ x ∞ i| = lim n →∞ | e − it n H ˜ P N ∞ f n ( x n ) | & N d − ∞ ε d ( d +2)8 A − ( d − d +4)8 . On the other hand, by H¨older and (2.10), |h φ, e it ∞ H ˜ P N ∞ δ x ∞ i| ≤ k e − it ∞ H φ k L dd − k ˜ P N ∞ δ x ∞ k L dd +2 . k φ k Σ N d − ∞ . Therefore k φ k Σ & ε d ( d +2)8 A − ( d − d +4)8 . Set φ n ≡ φ, and define the augmented frame ( t n , x n , N n , N ′ n ) ≡ (0 , , , L dd − . As the embeddingΣ ⊂ L is compact, the sequence f n , which converges weakly to φ ∈ Σ, convergesto φ strongly in L . After passing to a subsequence we obtain convergence pointwisea.e. The decoupling (4.5) now follows from Lemma 2.6. This completes the casewhere N n have a bounded subsequence. Case 2 . We now address the case where N n → ∞ . The main nuisance is thatthe weak limits φ will usually be merely in ˙ H ( R d ), not in Σ, so defining the profiles φ n will require spatial cutoffs.As the functions N − ( d − / n ( e − it n H f n )( N − n · + x n ) are bounded in ˙ H ( R d ), thesequence has a weak subsequential limit(4.8) N − d − n ( e − it n H f n )( N − n · + x n ) ⇀ φ in ˙ H ( R d ) . By Lemma 4.4, after passing to a further subsequence we may assume(4.9) lim n →∞ N − n | x n | = r ∞ < ∞ and lim n →∞ N n t n = t ∞ ∈ [ −∞ , ∞ ] . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 19 It will be necessary to distinguish the cases r ∞ > r ∞ = 0, corresponding towhether the frame { ( t n , x n , N n ) } is type 2a or 2b, respectively. Lemma 4.5. If r ∞ > , the function φ defined in (4.8) also belongs to L .Proof. By (4.8) and the Rellich-Kondrashov compactness theorem, for each R ≥ N − d − n ( e − it n H f n )( N − n · + x n ) → φ in L ( {| x | ≤ R } ) . By a change of variables, N − d − n ( e − it n H f n )( N − n · + x n ) k L ( | x |≤ R ) = N n k e − it n H f n k L ( | x − x n |≤ RN − n ) . k xe − it n H f n k L whenever | x n | ≥ N n r ∞ and RN − n ≤ r ∞ , so we have uniformly in R ≥ n k N − d − n ( e − it n H f n )( N − n · + x n ) k L ( | x |≤ R ) . sup n k e − it n H f n k Σ . . Therefore k φ k L = lim R →∞ k φ k L ( | x |≤ R ) . (cid:3) Remark. The claim fails if r ∞ = 0. Indeed, if φ ∈ ˙ H ( R d ) \ L ( R d ), then f n = N ( d − / n φ ( N n · ) χ ( · ) are bounded in Σ, and N − ( d − / n f n ( N − n · ) = φ ( · ) χ ( N − n · )converges strongly in ˙ H to φ .Next we prove that φ is nontrivial in ˙ H . Lemma 4.6. k φ k ˙ H & A (cid:0) εA (cid:1) d ( d +2)8 .Proof. From (2.10) and (4.7), N d − n A (cid:0) εA (cid:1) d ( d +2)8 . ˜ P ∆ ≤ N n | e − it n H f n | ( x n ) + ˜ P ∆ ≤ N n / | e − it n H f n | ( x n ) , so one of the terms on the right is at least half the left side. Suppose first that˜ P ∆ ≤ N n | e − it n H f n | ( x n ) & N d − n A (cid:0) εA (cid:1) d ( d +2)8 . Put ˇ ψ = ˜ P ∆ ≤ δ = e ∆ δ . Since ˇ ψ is Schwartz, |h| φ | , ˇ ψ i L | ≤ k φ k ˙ H k ˇ ψ k ˙ H − . k φ k ˙ H . On the other hand, as the absolute values N − d − n | e − it n H f n | ( N − n · + x n ) convergeweakly in ˙ H to | φ | , we have h| φ | , ˇ ψ i L = lim n h N − d − n | e − it n H f n | ( N − n · + x n ) , ˇ ψ i L = lim n ˜ P ∆ ≤ N n | e − it n H f n | ( x n ) & A (cid:0) εA (cid:1) d ( d +2)8 . from which the claim follows. Similarly if˜ P ∆ ≤ N n / | e − it n H f n | ( x n ) & N d − n A (cid:0) εA (cid:1) d ( d +2)8 , then we obtain k φ k ˙ H ∼ k φ (2 · ) k ˙ H & N d − n A (cid:0) εA (cid:1) d ( d +2)8 . (cid:3) Having extracted a nontrivial bubble φ , we are ready to define the φ n . The basicidea is to undo the operations applied to f n in the definition (4.8) of φ . However,we need to first apply a spatial cutoff to embed φ in Σ.With the frame { ( t n , x n , N n ) } defined according to (4.7), we form the augmentedframe { ( t n , x n , N n , N ′ n ) } with the cutoff parameter N ′ n chosen according to thesecond case of Definition 4.2. Let G n , S n be the ˙ H isometries and spatial cutoffoperators associated to { ( t n , x n , N n , N ′ n ) } . Set(4.10) φ n = e it n H G n S n φ = e it n H [ N d − n φ ( N n ( · − x n )) χ ( N ′ n ( · − x n ))] . We now verify that the φ n satisfy the various properties claimed in the proposition. Lemma 4.7. A (cid:0) εA (cid:1) d ( d +2)8 . lim inf n →∞ k φ n k Σ ≤ lim sup n →∞ k φ n k Σ . .Proof. By the definition of the Σ norm and a change of variables, k φ n k Σ = k G n S n k Σ ≥ k S n φ k ˙ H . Hence Lemma 4.6 and the remarks following Definition 4.2 together imply the lowerbound lim inf n k φ n k Σ & A (cid:0) εA (cid:1) d ( d +2)8 . The upper bound follows immediately from the case ( q, r ) = ( ∞ , 2) in Lemma 4.2. (cid:3) We verify the decoupling property (4.6). By the Pythagorean theorem, k f n k − k f n − φ n k − k φ n k = 2 Re( h f n − φ n , φ n i Σ )= 2 Re( h e − it n H f n − G n S n φ, G n S n = 2 Re( h w n , G n S n φ i Σ ) . where w n = e − it n H f n − G n S n φ . By definition, h w n , G n S n φ i Σ = h w n , G n S n φ i ˙ H + h xw n , xG n S n φ i L . From (4.3) and the definition (4.8) of φ , it follows that G − n w n → H as n → ∞ . Hence lim n →∞ h w n , G n S n φ i ˙ H = lim n →∞ h G − n w n , S n φ i ˙ H = lim n →∞ h G − n w n , φ i ˙ H = 0 . We turn to the second component of the inner product. Fix R > 0, and estimate |h xw n , xG n S n φ i L |≤ Z {| x − x n |≤ RN − n } | xw n || xG n S n φ | dx + Z {| x − x − n | >RN − n } | xw n || xG n S n φ | dx = ( I ) + ( II )Perform a change of variable and drop the spatial cutoff S n , keeping in mind thebound | x n | . N n , to obtain( I ) . Z | x |≤ R | G − n w n || φ | dx → n → ∞ . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 21 Next, apply Cauchy-Schwartz and the upper bound of Lemma 4.7 to see that( II ) . Z {| x − x n | >RN − n } | xG n S n φ | dx . N − n Z R ≤| x | . NnN ′ n | x n + N − n x | | φ ( x ) | dx . ( N − n | x n | + N − n ( N ′ n ) − ) Z R ≤| x | . NnN ′ n | φ ( x ) | dx. Suppose that the frame { ( t n , x n , N n ) } is of type 2a, so that lim n N − n | x n | > 0. ByLemma 4.5 and dominated convergence, the right side above is bounded by Z R ≤| x | | φ ( x ) | dx → R → ∞ , uniformly in n . If instead { ( t n , x n , N n ) } is of type 2b, use H¨older to see that theright side is bounded by( N − n | x n | ( N n N ′ n ) + ( N ′ n ) − ) k φ k L dd − . By Sobolev embedding and the construction of the parameter N ′ n in Definition 4.2,the above vanishes as n → ∞ . In either case, we obtainlim R →∞ lim sup n →∞ ( II ) = 0 . Combining the two estimates and choosing R arbitrarily large, we conclude asrequired that lim n →∞ |h xw n , xG n S n φ i L | = 0 . To close this subsection, we verify the L dd − decoupling property (4.5) when N n t n → ±∞ . Assume first that the φ appearing in the definition (4.10) of φ n hascompact support. By the dispersive estimate (2.5) and a change of variables, wehave lim n →∞ k φ n k L dd − . | t n | − k G n φ k L dd +2 . ( N n | t n | ) − k φ k L dd +2 = 0 . The claimed decoupling follows immediately.For general φ in H or ˙ H (depending on whether lim n N − n | x n | is positive orzero), select ψ ε ∈ C ∞ c converging to φ in the appropriate norm as ε → 0. Then forall n large enough, we have k φ n k L dd − ≤ k e it n H G n S n [ φ − ψ ε ] k L dd − + k e it n H G n S n ψ ε k L dd − , and we once again have decoupling by Lemmas 2.3 and 4.2, and the special casejust proved. (cid:3) Convergence of linear propagators. To complete the proof of Proposi-tion 4.3, we need a more detailed understanding of how the linear propagator e − itH interacts with the ˙ H -symmetries G n associated to a frame in certain limiting sit-uations. The lemmas proved in this section are heavily inspired by the discussionsurrounding [20, Lemma 5.2], in which the authors prove analogous results relatingthe linear propagators of the 2D Schr¨odinger equation and the complexified Klein-Gordon equation − iv t + h∇i v = 0. We begin by introducing some terminology dueto Ionescu-Pausader-Staffilani [14]. Definition 4.3. We say two frames F = { ( t n , x n , N n ) } and F = { ( t n , x n , N n ) } (where the superscripts are indices, not exponents) are equivalent if N n N n → R ∞ ∈ (0 , ∞ ) , N n ( x n − x n ) → x ∞ ∈ R d , ( N n ) ( t n − t n ) → t ∞ ∈ R . If any of the above statements fails, we say that F and F are orthogonal . Notethat replacing the N n in the second and third expressions above by N n yields anequivalent definition of orthogonality. Remark. If F and F are equivalent, it follows from the above definition thatthey must be of the same type in Definition 4.1, and that lim n ( N n ) − | x n | andlim n ( N n ) − | x n | are either both zero or both positive.One interpretation of the following lemma and its corollary is that when actingon functions concentrated at a point, e − itH can be approximated for small t byregarding the | x | / e − it | x | e it ∆2 where x is the spatial center of the initial data. Lemma 4.8 (Strong convergence) . Suppose F M = ( t Mn , x n , M n ) , F N = ( t Nn , y n , N n ) are equivalent frames. Define R ∞ = lim n →∞ M n N n , t ∞ = lim n →∞ M n ( t Mn − t Nn ) , x ∞ = lim n →∞ M n ( y n − x n ) r ∞ = lim n M − n | x n | = lim n M − n | y n | . Let G Mn , G Nn be the scaling and translation operators attached to the frames F M and F N respectively. Then ( e − it Nn H G Nn ) − e − it Mn H G Mn converges in the strong operatortopology on B (Σ , Σ) to the operator U ∞ defined by U ∞ φ = e − it ∞ ( r ∞ )22 R d − ∞ [ e it ∞ ∆2 φ ]( R ∞ · + x ∞ ) . Proof. If M n ≡ 1, then by the definition of a frame we must have F M = F N = { (1 , , } , so the claim is trivial. Thus we may assume that M n → ∞ . Put t n = t Mn − t Nn . Using Mehler’s formula (2.4), we write( e − it Nn H G Nn ) − e − it Mn H G Mn = ( G Nn ) − e − it n H G Mn φ ( x )= ( M n N n ) d − e iγ ( t n ) | y n + N − n x | e iM n sin( tn )∆2 [ e iγ ( t n ) | x n + M − n ·| φ ]( M n N n x + M n ( y n − x n )) . where γ ( t ) = cos t − 12 sin t = − t + O ( t ) . We see that e iγ ( t n ) | x n + M − n ·| φ → e − it ∞ ( r ∞ )24 φ in Σ . Indeed, k∇ [ e iγ ( t n ) | x n + M − n ·| φ − e iγ ( t n ) | x n | φ ] k L = k∇ x [( e iγ ( t n )[ M − n | x | + M − n x n · x ] − φ ] k L . k t n ( M − n x + M − n x n ) φ k L + k ( e iγ ( t n )[ M − n | x | +2 M − n x n · x ] − ∇ φ k L . | t n | M − n k xφ k L + | t n || x n | M − n k φ k L + k ( e iγ ( t n )[ M − n | x | M − n x n · x ] − ∇ φ k L . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 23 As n → ∞ , the first two terms vanish because k xφ k + k φ k . k φ k Σ , while the thirdterm vanishes by dominated convergence. Dominated convergence also implies that k x [ e iγ ( t n ) | x n + M − n x | φ − e iγ ( t n ) | x n | φ ] k L → n → ∞ . On the other hand, since γ ( t n ) | x n | = − M n t n M − n | x n | + O ( M − n ) → − t ∞ ( r ∞ ) , it follows that k e iγ ( t n ) | x n + M − n ·| φ − e − it ∞ ( r ∞ )24 φ k Σ → e iM n sin( tn )∆2 → e it ∞ ∆2 in the strong operator topologyon B (Σ , Σ), we obtain e iM n sin( tn )∆2 [ e iγ ( t n ) | x n + M − n ·| φ ] → e − it ∞ ( r ∞ )24 e it ∞ ∆2 φ in Σ , and the full conclusion quickly follows. (cid:3) Corollary 4.9. Let { ( t Mn , x n , M n , M ′ n ) } and { ( t Nn , y n , N n , N ′ n ) } be augmented framessuch that { ( t Mn , x n , M n ) } and { ( t Nn , y n , N n ) } are equivalent. Let S Mn , S Nn be the as-sociated spatial cutoff operators as defined in (4.2) . Then (4.11) lim n →∞ k e − it Mn H G Mn S Mn φ − e − it Nn H G Nn S Nn U ∞ φ k Σ = 0 and (4.12) lim n →∞ k e − it Mn H G Mn S Mn φ − e − it Nn H G Nn U ∞ S Nn φ k Σ = 0 whenever φ ∈ H if the frames conform to case 2a and φ ∈ ˙ H if they conform tocase 2b in Definition 4.2.Proof. As before, the result is immediate if M n ≡ M n → ∞ . Suppose first that φ ∈ C ∞ c . Using theunitarity of e − itH on Σ, the operator bounds (4.3), and the fact that S Mn φ = φ forall n sufficiently large, we write the left side of (4.11) as k G Nn [( G Nn ) − e − i ( t Mn − t Nn ) H G Mn φ − S Nn U ∞ φ ] k Σ . k ( G Nn ) − e − i ( t Mn − t Nn ) H G Mn φ − S Nn U ∞ φ k Σ . k ( G Nn ) − e − i ( t Mn − t Nn ) H G Mn φ − U ∞ φ k Σ + k (1 − S Nn ) U ∞ φ k Σ which goes to zero by Lemma 4.8 and dominated convergence. This proves (4.11)under the additional hypothesis that φ ∈ C ∞ c .We now remove this crutch and take φ ∈ H or ˙ H depending on whether theframes are of type 2a or 2b in Definition 4.2, respectively. For each ε > 0, choose φ ε ∈ C ∞ c such that k φ − φ ε k H < ε or k φ − φ ε k ˙ H < ε , respectively. Then k e − it Mn H G Mn S Mn φ − e − it Nn H G Nn S Nn U ∞ φ k Σ ≤ k e − it Mn H G Mn S Mn ( φ − φ ε ) k Σ + k e − it n H G Mn S Mn φ ε − e − it Nn H G Nn S Nn U ∞ φ ε k Σ + k e − it Nn H G Nn S Nn U ∞ ( φ − φ ε ) k Σ In the limit as n → ∞ , the middle term vanishes and we are left with a quantityat most a constant timeslim sup n →∞ k G Mn S Mn ( φ − φ ε ) k Σ + lim sup n →∞ k G Nn S Nn U ∞ ( φ − φ ε ) k Σ . Applying Lemma 4.2 and using the mapping properties of U ∞ on ˙ H and H , wesee that lim sup n →∞ k e − it n H G Mn S Mn φ − e it Nn H G Nn S Nn U ∞ φ k Σ . ε for every ε > 0. This proves the claim (4.11). Similar considerations handle thesecond claim (4.12). (cid:3) Lemma 4.10. Suppose the frames { ( t Mn , x n , M n ) } and { ( t Nn , y n , N n ) } are equiva-lent. Put t n = t Mn − t Nn . Then for f, g ∈ Σ we have h ( G Nn ) − e − it n H G Mn f, g i ˙ H = h f, ( G Mn ) − e it n H G Nn g i ˙ H + R n ( f, g ) , where | R n ( f, g ) | ≤ C | t n |k G Mn f k Σ k G Nn g k Σ . Remark. We regard this as an “approximate adjoint” formula; note that e − itH isnot actually defined on all of ˙ H . It follows from Lemma 4.8 thatlim n →∞ h ( G Nn ) − e − it n H G Mn f, g i ˙ H = lim n →∞ h f, ( G Mn ) − e it n H G Nn g i ˙ H for fixed f, g ∈ Σ. The content of this lemma lies in the quantitative error bound. Proof. From the identities (2.6), we obtain the commutator estimate k [ ∇ , e − itH ] k Σ → L = O ( t ) . By straightforward manipulations, we obtain h ( G Nn ) − e − it n H G MN f, g i ˙ H = h f, ( G Mn ) − e it n H G Nn g i ˙ H + R n ( f, g )where R n ( f, g ) = h [ ∇ , e − it n H ] G Mn f, ∇ G Nn g i L − h∇ G Mn f, [ ∇ , e it n H ] G Nn g i L . Theclaim follows from Cauchy-Schwartz and the above commutator estimate. (cid:3) The next lemma is a converse to Lemma 4.8. Lemma 4.11 (Weak convergence) . Assume the frames F M = { ( t Mn , x n , M n ) } and F N = { ( t Nn , y n , N n ) } are orthogonal. Then, for any f ∈ Σ , ( e − it Nn H G Nn ) − e − it Mn H G Mn f → weakly in ˙ H . Proof. Put t n = t Mn − t Nn , and suppose that | M n t n | → ∞ . Then k ( G Nn ) − e − it n H G Mn f k L dd − → f ∈ C ∞ c by a change of variables and the dispersive estimate, thus for general f ∈ Σ by a density argument. Therefore ( G Nn ) − e − it n H G Mn f converges weakly in ˙ H to 0. We consider next the case where M n t n → t ∞ ∈ R . The orthogonality of F M and F N implies that either N − 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 Nn ) − G Mn convergeto zero in the weak operator topology on B ( ˙ H , ˙ H ). Applying Lemma 4.8, wesee that ( G Nn ) − e − it n H G Mn f = ( G Nn ) − G Mn ( G Mn ) − e − it n H G Mn f converges to zeroweakly in ˙ H . (cid:3) Corollary 4.12. Let { ( t Mn , x n , M n , M ′ n ) } and { ( t Nn , y n , N n , N ′ n ) } be augmentedframes such that { ( t Mn , x n , M n ) } and { ( t Nn , y n , N n ) } are orthogonal. Let G Mn , S Mn and G Nn , S Nn be the associated operators. Then ( e − it Nn H G Nn ) − e − it Mn H G Mn S Mn φ ⇀ in ˙ H whenever φ ∈ H if F M is of type 2a and φ ∈ ˙ H if F M is of type 2b. HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 25 Proof. If φ ∈ C ∞ c , then S Mn φ = φ for all large n , and the claim follows fromLemma 4.11. The case of general φ in H or ˙ H then follows from an approximationargument similar to the one used in the proof of Corollary 4.9. (cid:3) End of Proof of Inverse Strichartz. We return to the proof of Proposi-tion 4.3. Let us pause briefly to assess our progress. We have thus far identified aframe { ( t n , x n , N n , N ′ n ) } and an associated profile φ n such that the sequence N n t n has a limit in [ −∞ , ∞ ] as n → ∞ . The φ n were shown to satisfy properties (4.4)through (4.6) if either ( t n , x n , N n ) = (0 , , 1) or N n → ∞ and N n t n → ±∞ . Thus,it remains to prove that if N n → ∞ and N n t n remains bounded, then we maymodify the frame so that t n is identically zero and find a profile φ n correspondingto this new frame which satisfies all the properties asserted in the proposition. Thefollowing lemma will therefore complete the proof of Proposition 4.3. Lemma 4.13. Let f n ∈ Σ satisfy the hypotheses of Proposition 4.3. Suppose { ( t n , x n , N n , N ′ n ) } is an augmented frame with N n → ∞ and N n t n → t ∞ ∈ R as n → ∞ . Then there is a profile φ ′ n = G n S n φ ′ associated to the frame { (0 , x n , N n , N ′ n ) } such that properties (4.4) , (4.5) , and (4.6) hold with φ ′ n in placeof φ n .Proof. Let φ n = e it n H G n S n φ be the profile defined by (4.10). We have alreadyseen that φ n satisfies properties (4.4) and (4.6), and that φ = ˙H -w-lim n →∞ G − n e − it n H f n . As the sequence G − n f n is bounded in ˙ H , it has a weak subsequential limit φ ′ = ˙H -w-lim n →∞ G − n f n . For any ψ ∈ C ∞ c , we apply Lemma 4.10 with f = G − n e − it n H f n to see that h φ ′ , ψ i ˙ H = lim n →∞ h G − n f n , ψ i ˙ H = lim n →∞ h G − n e it n H G n G − n e − it n H f n , ψ i ˙ H = lim n →∞ h G − n e − it n H f n , G − n e − it n H G n ψ i ˙ H = h φ, U ∞ ψ i ˙ H , where U ∞ = s-lim n →∞ G − n e − it n H G n is the strong operator limit guaranteed byLemma 4.8. As U ∞ is unitary on ˙ H , we have the relation φ = U ∞ φ ′ .Put φ ′ n = G n S n φ ′ . By Corollary 4.9, k φ n − φ ′ n k Σ = k e it n H G n S n φ − G n S n U − ∞ φ k Σ → n → ∞ . Hence φ ′ n inherits property (4.4) from φ n . The same proof as for φ n shows thatΣ decoupling (4.6) holds as well. It remains to verify the last decoupling property(4.5). As G − n f n converges weakly in ˙ H to φ ′ , by Rellich-Kondrashov and a diag-onalization argument we may assume after passing to a subsequence that G − n f n converges to φ ′ almost everywhere on R d . By the Lemma 2.6, the observation thatlim n →∞ k G n S n φ ′ − G n φ ′ k dd − = 0, and a change of variables, we havelim n →∞ (cid:20) k f n k dd − dd − − k f n − φ ′ n k dd − dd − − k φ ′ n k dd − dd − (cid:21) = lim n →∞ (cid:20) k G − n f n k dd − dd − − k G − n f n − φ ′ k dd − dd − − k φ ′ k dd − dd − (cid:21) = 0 . (cid:3) Remark. As lim n →∞ k φ n − φ ′ n k Σ = 0, we see by Sobolev embedding that thedecoupling (4.5) also holds for the original profile φ n = e it n H G n S n φ with nonzerotime parameter t n .4.4. Linear profile decomposition. We are ready to write down the linear profiledecomposition. As before, I will denote a fixed interval containing 0 of length atmost 1, and all spacetime norms are taken over I × R d unless indicated otherwise. Proposition 4.14. Let f n be a bounded sequence in Σ . After passing to a subse-quence, there exists J ∗ ∈ { , , . . . } ∪ {∞} such that for each finite ≤ j ≤ J ∗ ,there exist an augmented frame F j = { ( t jn , x jn , N jn , ( N jn ) ′ ) } and a function φ j withthe following properties. • Either t jn ≡ or ( N jn ) ( t jn ) → ±∞ as n → ∞ . • φ j belongs to Σ , H , or ˙ H depending on whether F j is of type 1, 2a, or2b, respectively.For each finite J ≤ J ∗ , we have a decomposition (4.13) f n = J X j =1 e it jn H G jn S jn φ j + r Jn , where G jn , S jn are the ˙ H -isometry and spatial cutoff operators associated to F j .Writing φ jn for e it jn H G jn S jn φ j , this decomposition has the following properties: ( G Jn ) − e − it Jn H r Jn ˙ H ⇀ for all J ≤ J ∗ , (4.14) sup J lim n →∞ (cid:12)(cid:12)(cid:12) k f n k − J X j =1 k φ jn k − k r Jn k (cid:12)(cid:12)(cid:12) = 0 , (4.15) sup J lim n →∞ (cid:12)(cid:12)(cid:12) k f n k dd − L dd − x − J X j =1 k φ jn k dd − L dd − x − k r Jn k dd − L dd − x (cid:12)(cid:12)(cid:12) = 0 . (4.16) Whenever j = k , the frames { ( t jn , x jn , N jn ) } and { ( t kn , x kn , N kn ) } are orthogonal: (4.17) lim n →∞ N jn N kn + N kn N jn + N jn N kn | t jn − t kn | + q N jn N kn | x jn − x kn | = ∞ . Finally, we have (4.18) lim J → J ∗ lim sup n →∞ k e − it n H r Jn k L d +2) d − t,x = 0 , Remark. One can also show a posteriori using (4.17) and (4.18) the fact, whichwe will neither prove nor use, thatsup J lim n →∞ (cid:12)(cid:12)(cid:12) k e − itH f n k d +2) d − L d +2) d − t,x − J X j =1 k e − itH φ jn k d +2) d − L d +2) d − t,x − k e − itH w Jn k d +2) d − L d +2) d − t,x (cid:12)(cid:12)(cid:12) = 0 . The argument uses similar ideas as in the proofs of [17][Lemma 2.7] or Lemma 6.3;we omit the details. Proof. We proceed inductively using Proposition 4.3. Let r n = f n . Assume that wehave a decomposition up to level J ≥ A J = lim n k r Jn k Σ and ε J = lim n k e − it n H r Jn k L d +2) d − t,x . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 27 If ε J = 0, stop and set J ∗ = J . Otherwise we apply Proposition 4.3 to the sequence r Jn to obtain a frame ( t J +1 n , x J +1 n , N J +1 n , ( N J +1 n ) ′ ) and functions φ J +1 ∈ ˙ H , φ J +1 n = e it J +1 n H G J +1 n S J +1 n φ J +1 ∈ Σwhich satisfy the conclusions of Proposition 4.3. In particular φ J +1 is the ˙ H weaklimit of ( G J +1 n ) − e − it J +1 n H r Jn . Let r J +1 n = r Jn − φ J +1 n . By the induction hypothesis,(4.15) and (4.16) are satisfied with J replaced by J + 1. We also have( G J +1 n ) − e − it J +1 n H r J +1 n = [( G J +1 n ) − e − it J +1 n H r Jn − φ J +1 ] + (1 − S J +1 n ) φ J +1 . As n → ∞ , the first term goes to zero weakly in ˙ H while the second term goesto zero strongly. Thus (4.14) holds at level J + 1 as well. After passing to asubsequence, we may define A J +1 = lim n k r J +1 n k Σ and ε J +1 = lim n k e − itH r J +1 n k L d +2) d − t,x . If ε J +1 = 0, stop and set J ∗ = J + 1. Otherwise continue the induction. If thealgorithm never terminates, set J ∗ = ∞ . From (4.15) and (4.16), the parameters A J and ε J satisfy the inequality A J +1 ≤ A J [1 − C ( ε J A J ) d ( d +2)4 ] . If lim sup J → J ∗ ε J = ε ∞ > 0, then as A J are decreasing there would exist infinitelymany J ’s so that A J +1 ≤ A J [1 − C ( ε ∞ A ) d ( d +2)4 ] , which implies that lim J → J ∗ A J = 0. But this contradicts the Strichartz inequalitywhich dictates that lim sup J → J ∗ A J & lim sup J → J ∗ ε J = ε . We conclude thatlim J → J ∗ ε J = 0 . Thus (4.18) holds.It remains to prove the assertion (4.17). Suppose otherwise, and let j < k bethe first two indices for which F j and F k are equivalent. Thus F ℓ and F k areorthogonal for all j < ℓ < k . By the construction of the profiles, we have r j − n = e it jn H G jn S jn φ j + e it kn H G kn S kn φ k + X j<ℓ The next step in the proof of Theorem 1.2 is to establish wellposedness whenthe initial data consists of a highly concentrated “bubble”. The picture to keepin mind is that of a single profile φ jn in Proposition 4.14 as n → ∞ . In the nextsection we combine this special case with the profile decomposition to treat generalinitial data. Although we state the following result as a conditional one to permit aunified exposition, by Theorem 1.1 the result is unconditionally true in most cases. Proposition 5.1. Let I = [ − , . Assume that Conjecture 1.1 holds. Suppose F = { ( t n , x n , N n , N ′ n ) } is an augmented frame with t ∈ I and N n → ∞ , such that either t n ≡ or N n t n → ±∞ ; that is, F is type 2a or 2b in Definition 4.2. Let G n , ˜ G n , and S n bethe associated operators as defined in (4.1) and (4.2) . Suppose φ belongs to H or ˙ H depending on whether F is type 2a or 2b respectively. Then, for n sufficientlylarge, there is a unique solution u n : I × R d → C to the defocusing equation (1.1) , µ = 1 , with initial data u n (0) = e it n H 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 < r k < d . Then for each ε > there exists ψ ε ∈ C ∞ c ( R × R d ) such that (5.1) lim sup n →∞ X k k u n − ˜ G n [ e − itN − n | xn | ψ ε ] k L qkt Σ rkx ( I × R d ) < ε. Assuming also that k∇ φ k L < k∇ W k L and E ∆ ( φ ) < E ∆ ( W ) , we have the sameconclusion as above for the focusing equation (1.1) , µ = − . The proof proceeds in several steps. First we construct an approximate solutionon I in the sense of Proposition 3.3. Roughly speaking, when N n is large and t = O ( N − n ), solutions to (1.1) are well-approximated up to a phase factor bysolutions to the energy-critical NLS with no potential, which by Conjecture 1.1exist globally and scatter. In the long-time regime | t | >> N − n , the solution to(1.1) has dispersed and resembles a linear evolution e − itH φ (note that we are not HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 29 claiming scattering since we consider only a fixed finite time window). By patchingthese approximations together, we obtain an approximate solution over the entiretime interval I with arbitrarily small error as N n becomes large. We then invokeProposition 3.3 to conclude that for n large enough (1.1) admits a solution on I with controlled spacetime bound. The last claim about approximating the solutionby functions in C ∞ c ( R × R d ) will follow essentially from our construction of theapproximate solutions.We first record a basic commutator estimate. Throughout the rest of this section, P ≤ N , P N will denote the standard Littlewood-Paley projectors based on − ∆. Lemma 5.2 (Commutator estimate) . Let v be a global solution to ( i∂ t + ∆) v = F ( v ) , v (0) ∈ ˙ H ( R d ) where F ( z ) = ±| z | d − z . Then on any compact time interval I , lim N →∞ k P ≤ N F ( v ) − F ( P ≤ N v ) k L t H , dd +2 x ( I × R d ) = 0 Proof. We recall [29, Lemma 3.11] that as a consequence of the spacetime bound(1.7), ∇ v is finite in all Strichartz norms:(5.2) k∇ v k S ( R ) < C ( k v (0) k ˙ H ) < ∞ . Clearly it will suffice to show separately thatlim n →∞ k P ≤ N F ( v ) − F ( P ≤ N v ) k L t L dd +2 x = 0 , (5.3) lim n →∞ k∇ [ P ≤ N F ( v ) − F ( P ≤ N v )] k L t L dd +2 x = 0 . (5.4)Write k∇ [ P ≤ N F ( v ) − F ( P ≤ N v )] k L t L dd +2 x ≤ k∇ P >N F ( v ) k L t L dd +2 x + k∇ [ F ( v ) − F ( P ≤ N v )] k L t L dd +2 x . (5.5)As P >N = 1 − P ≤ N and k∇ F ( v ) k L t L dd +2 x . k v k d − L d +2) d − t,x k∇ v k L d +2) d − t L d ( d +2) d x ≤ C ( k v (0) k ˙ H ) , dominated convergence implies thatlim N →∞ k∇ P >N F ( v ) k L t L dd +2 x = 0 . To treat the second term on the right side of (5.5), observe first that with F ( z ) = | z | d − z , | F z ( z ) − F z ( w ) | + | F z ( z ) − F z ( w ) | . ( | z − w | ( | z | − dd − + | w | − dd − ) , ≤ d ≤ | z − w | d − , d ≥ . Combining this with the pointwise bound |∇ [ F ( v ) − F ( P ≤ N v )] | ≤ ( | F z ( v ) − F z ( P ≤ N v ) | + | F z ( v ) − F z ( P ≤ N v ) | ) |∇ v | + ( | F z ( P ≤ N v ) | + | F z ( P ≤ N v ) | ) |∇ P >N v | , H¨older, and dominated convergence, when d ≥ k∇ [ F ( v ) − F ( P ≤ N v )] k L t L dd +2 x . k| P >N v | d − |∇ v |k L t L dd +2 x + k| P ≤ N v | d − |∇ P >N v |k L t L dd +2 x . k P >N v k d − L d +2) d − t,x k∇ v k L d +2) d − t L d ( d +2) d x + k v k d − L d +2) d − t,x k P >N ∇ v k L d +2) d − t L d ( d +2) d x → N → ∞ . (5.6)If 3 ≤ d ≤ 5, the first term in the second line of (5.6) is replaced by k| P >N v | ( | v | − dd − + | P ≤ N v | − dd − ) |∇ v |k L t L dd +2 x ≤ k P >N v k L d +2) d − t,x k v k − dd − L d +2) d − t,x k∇ v k L d +2) d − t L d ( d +2) d x which goes to 0 by dominated convergence. This establishes (5.4). The proof of(5.3)is similar. Write k P ≤ N F ( v ) − F ( P ≤ N v ) k L t L dd +2 x ≤ k P >N F ( v ) k L t L dd +2 x + k F ( v ) − F ( P ≤ N v ) k L t L dd +2 x . By H¨older, Bernstein, and the chain rule, k P >N F ( v ) k L t L dd +2 x . N − k v | d − L d +2) d − t,x k∇ v k L d +2) d − t L d ( d +2) d x = O ( N − ) . Using Bernstein, H¨older, and Sobolev embedding, and the pointwise bound | F ( v ) − F ( P ≤ N v ) | . | P >N v | ( | v | d − + | P ≤ N v | d − ) , we obtain k F ( v ) − F ( P ≤ N v ) k L t L dd +2 x ≤ k ( | v | d − + | P ≤ N v | d − ) P >N v k L t L dd +2 x . | I | ( k∇ v k d − L ∞ t L x + + k∇ v k d − L ∞ t L x ) k∇ P >N v k L ∞ t L x . As v ∈ C t ˙ H x ( I × R d ), the orbit { v ( t ) } t ∈ I is compact in ˙ H ( R d ). The Rieszcharacterization of L compactness therefore implies that the right side goes to 0as N → ∞ . (cid:3) Now suppose that φ n = e it n H G n S n φ as in the statement of Proposition 5.1. If µ = − 1, assume also that k φ k ˙ H < k W k ˙ H , E ( φ ) < E ∆ ( W ). We first construct“quasi-approximate” solutions ˜ v n which obey all of the conditions of the Propo-sition 3.3 except possibly the hypothesis in (3.2) about matching initial data. Aslight modification of the ˜ v n will then yield genuine approximate solutions.If t n ≡ 0, let v be the solution to the potential-free problem (1.6) provided byConjecture 1.1 with v (0) = φ . If N n t n → ±∞ , let v be the solution to (1.6) whichscatters in ˙ H to e it ∆2 φ as t → ∓∞ . Note the reversal of signs.Put(5.7) ˜ N ′ n = ( N n N ′ n ) , HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 31 Let T > v Tn ( t ) = e − it | xn | ˜ G n [ S n P ≤ ˜ N ′ n v ]( t + t n ) | t | ≤ T N − n e − i ( t − T N − n ) H ˜ v Tn ( T N − n ) , T N − n ≤ t ≤ e − i ( t + T N − n ) H ˜ v Tn ( − T N − n ) , − ≤ t ≤ − T N − n The time translation by t n is needed to undo the time translation built into theoperator ˜ G n ; see (4.1). We will suppress the superscript T unless we need toemphasize the role of that parameter. Introducing the notation v n ( t, x ) = [ ˜ G n v ]( t + t n , x ) = N d − n v ( N n t, N n ( x − x n )) ,χ n ( x ) = χ ( N ′ n ( x − x n )) , where χ is the function used to define the spatial cutoff operator S n in (4.2), andusing the identity ˜ G n χ = χ n ˜ G n , we can also write the top expression in (5.8) as˜ v n ( t ) = e − it | xn | χ n P ≤ ˜ N ′ n N n v n , | t | ≤ T N − n . As discussed previously, inside the “potential-free” window ˜ v n is essentially amodulated solution to (1.6) with cutoffs applied in both space, to place the solutionin C t Σ x , and frequency, to enable taking an extra derivative in the error analysisbelow.On the time interval | t | ≤ T N − n , we use Lemma 4.2 and the fact that k v k L ∞ t ˙ H x . k φ k ˙ H to deduce lim sup n k ˜ v n k L ∞ t Σ x ( | t |≤ T N − n ) . k φ k ˙ H , therefore lim sup n k ˜ v n k L ∞ t Σ x ([ − , . k φ k ˙ H . (5.9)From (1.7), (5.9), and Strichartz, we obtain k ˜ v n k L d +2) d − t,x ([ − , × R d ) ≤ C ( k φ k ˙ H ) for n large . (5.10)Let e n = ( i∂ t − H )˜ v n − F (˜ v n ) . We show that lim T →∞ lim sup n →∞ k H e n k N ([ − , = 0 , (5.11)so that by taking T large enough the ˜ v n will satisfy the second error condition in(3.2) for all n sufficiently large. Our first task is to deal with the time interval | t | ≤ T N − n . Lemma 5.3. lim T →∞ lim sup n →∞ k H e n k N ( | t |≤ T N − n ) = 0 . Proof. When − T N − n ≤ t ≤ T N − n , we compute e n = e − it | xn | [ χ n P ≤ ˜ N ′ n N n F ( v n ) − χ d +2 d − n F ( P ≤ ˜ N ′ n N n v n )+ | x n | − | x | P ≤ ˜ N ′ n N n v n ) χ n + 12 ( P ≤ ˜ N ′ n N n v n )∆ χ n + ( ∇ P ≤ ˜ N ′ n N n v n ) · ∇ χ n ]= e − it | xn | [( a ) + ( b ) + ( c ) + ( d )] , and estimate each term separately in the dual Strichartz space N ( {| t | ≤ T N − n } ).Write( a ) = χ n P ≤ ˜ N ′ n N n F ( v n ) − χ d +2 d − n F ( P ≤ ˜ N ′ n N n v n )= χ n [ P ≤ ˜ N ′ n N n F ( v n ) − F ( P ≤ ˜ N ′ nNn v n )] + χ n (1 − χ d − n ) F ( P ≤ ˜ N ′ n N n v n )= ( a ′ ) + ( a ′′ ) . By the Leibniz rule and a change of variables, k∇ ( a ′ ) k L t L dd +2 x ( | t |≤ T N − n ) ≤ k∇ [ P ≤ ˜ N ′ n F ( v ) − F ( P ≤ ˜ N ′ n v )] k L t L dd +2 x ( | t |≤ T ) + k [ P ≤ ˜ N ′ n N n F ( v n ) − F ( P ≤ ˜ N ′ n N n v n )] ∇ χ n k L t L dd +2 x ( | t |≤ T N − n ) . (5.12)By Lemma 5.2, the first term disappears in the limit as n → ∞ . That lemma alsoapplies to the second term after a change of variables to give k [ P ≤ ˜ N ′ n N n F ( v n ) − F ( P ≤ ˜ N ′ n N n v n )] ∇ χ n k L t L dd +2 x ( | t |≤ T N − n ) . N ′ n k P ≤ ˜ N ′ n N n F ( v n ) − F ( P ≤ ˜ N ′ n N n v n ) k L t L dd +2 x ( | t |≤ T N − n ) . N ′ n N n k P ≤ ˜ N ′ n F ( v ) − F ( P ≤ ˜ N ′ n v ) k L t L dd +2 x ( | t |≤ T ) → n → ∞ . Therefore lim n →∞ k∇ ( a ′ ) k L t L dd +2 x ( | t |≤ T N − n ) = 0 . By changing variables, using the bound | x n | . N n , and referring to Lemma 5.2once more, k| x | ( a ′ ) k L t L dd +2 x . N n k P ≤ ˜ N ′ n N n F ( v n ) − F ( P ≤ ˜ N ′ n N n v n ) k L t L dd +2 x ( | t |≤ T N − n ) . k P ≤ ˜ N ′ n F ( v ) − F ( P ≤ ˜ N ′ n v ) k L t L dd +2 x ( | t |≤ T ) → n → ∞ . It follows from Lemma 2.2 thatlim n →∞ k H ( a ′ ) k L t L dd +2 x ( | t |≤ T N − n ) = 0 . To estimate ( a ′′ ), we use the Leibniz rule, a change of variables, H¨older, Sobolevembedding, the bound (5.2), and dominated convergence to obtain k∇ ( a ′′ ) k L t L dd +2 x . k| P ≤ ˜ N n N n v n | d − ∇ P ≤ ˜ N ′ n N n v n k L t L dd +2 x ( | t |≤ T N − n , | x − x n |∼ ( N ′ n ) − ) + N ′ n N n k P ≤ ˜ N ′ n N n v n k d +2 d − L ∞ t L dd − x . k∇ v k L d +2) d − t L d ( d +2) d x k P ≤ ˜ N ′ n v k d − L d +2) d − t,x ( | t |≤ T, | x |∼ NnN ′ n ) + O ( N ′ n N n ) . C ( E ( v ))( k P > ˜ N ′ n v k L d +2) d − t,x + k v k L d +2) d − t,x ( | t |≤ T, | x | & N n N ′ n ) ) d − + O ( N ′ n N n )= o (1) + O ( N ′ n N n ) . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 33 Similarly, k| x | ( a ′′ ) k L t L dd +2 x ∼ k F ( P ≤ ˜ N ′ n v ) k L d +2) d − t L dd − x ( | t |≤ T, | x |∼ N n N ′ n ) . ( k P > ˜ N ′ n v k L d +2) d − t L dd − x ( | t |≤ T ) + k v k L d +2) d − t L dd − x ( | t |≤ T, | x |∼ N n N ′ n ) ) d +2 d − = o (1) . Therefore lim N →∞ k H ( a ′′ ) k L t L dd +2 t,x ( | t |≤ T N − n ) = 0as well. This completes the analysis for ( a ).The estimates for ( b ) , ( c ) , ( d ) are less involved. For ( b ), note that on thesupport of the function we have (cid:12)(cid:12) | x n | − | x | (cid:12)(cid:12) = | x n − x || x n + x | ∼ N n ( N ′ n ) − . Thusby H¨older and Sobolev embedding, k∇ ( b ) k L t L x ( | t |≤ T N − n ) . N n N ′ n k∇ P ≤ ˜ N ′ n N n v n k L t L x ( | t |≤ T N − n ) + N n k P ≤ ˜ N ′ n N n v n k L t L x ( | t |≤ T N − n , | x − x n |∼ ( N ′ n ) − ) . ( N ′ n N n ) − k∇ v n k L ∞ t L x → n → ∞ . Using H¨older and Sobolev embedding, we have k| x | ( b ) k L t L x ( | t |≤ T N − n ) ∼ N n N ′ n k P ≤ ˜ N ′ n N n v n k L t L x ( | t |≤ T N − n , | x − x n | . ( N ′ n ) − ) . (cid:26) ( N ′ n ) − k∇ v n k L ∞ t L x , lim n →∞ N − n | x n | = 0 k v n k L ∞ t L x = O ( N − n ) , lim n →∞ N − n | x n | > , which vanishes as n → ∞ in either case. Thus k H / ( b ) k L t L x → 0. The term ( c )is dealt with similarly. For (d), use H¨older, Bernstein, and the definition (5.7) ofthe frequency cutoffs ˜ N ′ n to obtain k∇ ( d ) k L t L x ( | t |≤ T N − n ) . N ′ n k|∇| P ≤ ˜ N ′ n N n v n k L t L x + k|∇ P ≤ ˜ N ′ n N n v n | ( |∇| χ n ) k L t L x . (cid:20)(cid:16) N ′ n N n (cid:17) + (cid:16) N ′ n N n (cid:17) (cid:21) k∇ v n k L ∞ t L x → . Applying H¨older in the time variable, we get k| x | ( d ) k L t L x ( | t |≤ T N − n ) . N ′ n N n k∇ v n k L ∞ t L x → . This completes the proof of the lemma. (cid:3) Next, we estimate the error over the time intervals [ − , T N − n ] and [ T N − n , Lemma 5.4. lim T →∞ lim sup n →∞ k H e n k N ([ − ,T N − n ] ∪ [ T N − n , = 0 .Proof. We consider just the forward time interval as the other interval is treatedsimilarly. Since ˜ v Tn solves the linear equation, the error e n is just the nonlinearterm: e n = ( i∂ t − H )˜ v Tn − F (˜ v Tn ) = − F (˜ v Tn ) . By the chain rule (Corollary 2.4) and Strichartz, k H e n k N ([ T N − n , . k ˜ v Tn k d − L d +2) d − t,x ([ T N − n , k ˜ v Tn ( T N − n ) k Σ . By definition ˜ v Tn ( T N − n ) = e − iTN − n | xn | ˜ G n S n P ≤ ˜ N ′ n v ( T N − n − t n ), so Lemma 4.2implies thatlim sup n →∞ k ˜ v Tn ( T N − n ) k Σ . (cid:26) k v | L ∞ t ˙ H x , lim n →∞ N − n | x n | = 0 , k v k L ∞ t H x , lim n →∞ N − n | x n | > T →∞ lim sup n →∞ k ˜ v Tn k L ∞ T L dd − x ([ T N − n , = 0 . As we are assuming Conjecture 1.1, there exists v ∞ ∈ ˙ H so thatlim t →∞ k v ( t ) − e it ∆2 v ∞ k ˙ H x = 0 . Then one also has lim t →∞ lim sup n →∞ k P ≤ ˜ N ′ n v ( t ) − e it ∆2 v ∞ k ˙ H x = 0 , and Lemma 4.2 implies thatlim T →∞ lim sup n →∞ k ˜ v n ( T N − n ) − e − iTN − n | xn | G n S n ( e iT ∆2 v ∞ ) k Σ = 0 . An application of Strichartz and Corollary 4.9 yields˜ v n ( t ) = e − i ( t − T N − n ) H [˜ v n ( T N − n )]= e − i ( t − T N − n ) H [ e − iTN − n | xn | G n S n e iT ∆2 v ∞ ] + error= e − itH [ G n S n v ∞ ] + errorwhere lim T →∞ lim sup n →∞ k error k Σ = 0 uniformly in t . By Sobolev embedding,lim T →∞ lim sup n →∞ k ˜ v n k L ∞ t L dd − x ([ T N − n , = lim T →∞ lim sup n →∞ k e − itH [ G n S n v ∞ ] k L ∞ t L dd − x ([ T N − n , . A standard density argument using the dispersive estimate for e − itH shows thatthe last limit is zero. (cid:3) Lemmas 5.3 and 5.4 together establish (5.11). Lemma 5.5 (Matching initial data) . Let u n (0) = e it n H G n S n φ as in Proposi-tion 5.1. Then lim T →∞ lim sup n → ∞ k ˜ v Tn ( − t n ) − u n (0) k Σ = 0 . Proof. If t n ≡ 0, then by definition ˜ v Tn (0) = G n S n P ≤ N ′ n φ , so Lemma 4.2 and thedefinition (5.7) of the frequency parameter N ′ n implylim n →∞ k ˜ v Tn (0) − u n (0) k Σ . lim n →∞ (cid:26) k P >N ′ n φ k H , lim n →∞ N − n | x n | > k P >N ′ n φ k ˙ H , lim n →∞ N − n | x n | = 0 (cid:27) = 0Next we consider the case N n t n → ∞ ; the case N n t n → −∞ works similarly.Arguing as in the previous lemma and recalling that in this case, the solution v waschosen to scatter backward in time to e it ∆2 φ , for n large we have˜ v Tn ( − t n ) = e it n H [ G n S n φ ] + errorwhere lim T →∞ lim sup n →∞ k error k Σ → 0. The claim follows. (cid:3) HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 35 For each fixed T > 0, set ˜ u Tn ( t ) = ˜ v Tn ( t − t n ) , (5.13)which is defined for t ∈ [ − , T , this is anapproximate solution for all n sufficiently large in the sense of Proposition 3.3.Indeed, by (5.9) and (5.10), ˜ u Tn satisfy the hypotheses (3.1) with E . k φ k ˙ H and L = C ( k φ k ˙ H ). Lemmas 5.3, 5.4, 5.5, Sobolev embedding, and Strichartz showthat for any ε > 0, there exists T > u Tn satisfies the hypotheses (3.2) forall large n . Invoking Proposition 3.3, we obtain the first claim of Proposition 5.1concerning the existence of solutionsThe remaining assertion of Proposition 5.1 regarding approximation by smoothfunctions will follow from the next lemma. Recall that we use the notation k f k L qt Σ rx = k H f k L qt L rx . Lemma 5.6. Fix finitely many admissible ( q k , r k ) with ≤ r k < d . For every ε > , there exists a smooth function ψ ε ∈ C ∞ c ( R × R d ) such that for all k lim sup T →∞ lim sup n →∞ k ˜ v Tn − ˜ G n [ e − itN − n | xn | ψ ε ]( t + t n ) k L qkT Σ rkx ([ − , < ε. Proof. We continue using the notation defined at the beginning. Let˜ w Tn = e − it | xn | ˜ G n [ S n v ]( t + t n ) , | t | ≤ T N − n e − i ( t − T N − n ) H [ ˜ w Tn ( T N − n )] , t ≥ T N − n e − i ( t + T N − n ) H [ ˜ w Tn ( − T N − n )] , t ≤ − T N − n This is essentially ˜ v Tn in (5.8) without the frequency cutoffs. We observe first that˜ v Tn can be well-approximated by ˜ w Tn in spacetime:lim sup n →∞ k ˜ v Tn − ˜ w Tn k L qkt Σ rkx ([ − , = 0 , sup T > lim sup n →∞ k ˜ w Tn k L qkt Σ rkx ([ − , < ∞ . (5.14)Indeed by dominated convergence, k∇ ( v − P ≤ ˜ N ′ n v ) k L qkt L rkx ( R × R d ) → n → ∞ , thus (5.14) follows from Lemma 4.2 and the Strichartz inequality for e − itH .The next observation is that most of the spacetime norm of ˜ w Tn is concentratedin the time interval | t | ≤ T N − n :(5.15) lim T →∞ lim sup n →∞ k ˜ w Tn k L qkt Σ rkx ([ − , − T N − n ] ∪ [ T N − n , = 0 . To see this, it suffices by symmetry to consider the forward interval. Recall that v scatters forward in ˙ H to some e it ∆2 v ∞ . By Lemma 4.2,lim T →∞ lim sup n →∞ k ( ˜ G n S n v ( T N − n − t n ) − G n S n ( e iT ∆2 v ∞ ) k Σ = 0 . By Strichartz,lim T →∞ lim sup n →∞ k e iTN − n | xn | ˜ w Tn − e − i ( t − T N − n ) H [ G n S n ( e iT ∆2 v ∞ )] k L qkt Σ rkx ([ T N − n , = 0By Corollary 4.9 and Strichartz, for each T > n →∞ k e − i ( t − T N − n ) H [ G n S n ( e iT ∆2 v ∞ )] − e iT ( r ∞ )22 e − itH [ G n S n v ∞ ] k L qkt Σ rkx ([ T N − n , = 0 . For each ε > 0, choose v ε ∞ ∈ C ∞ c such that k v ∞ − v ε ∞ k ˙ H < ε . By the dispersiveestimate, k e − itH [ G n v ε ∞ ] k L qkt L rkx ([ T N − n , . T − qk k v ε ∞ k L r ′ kx Combining the above with Strichartz and Lemma 4.2, we getlim sup n →∞ k ˜ w Tn k L qkt Σ rkx ([ T N − n , . o (1) + ε + O ε,q k ( T − qk ) as T → ∞ . Taking T → ∞ , we findlim sup T →∞ lim sup n →∞ k ˜ w Tn k L qkt Σ rkx ([ T N − n , . ε for any ε > 0, thereby establishing (5.15).Choose ψ ε ∈ C ∞ c ( R × R d ) such that P Nk =1 k v − ψ ε k L qkt ˙ H ,rkx < ε . By combiningLemma 4.2 with (5.14) and (5.15), we getlim T →∞ lim sup n →∞ k ˜ v n ( t, x ) − e − it | xn | ˜ G n ψ ε ( t + t n ) k L qkt Σ rkx ([ − , . ε. This completes the proof of the lemma, hence Proposition 5.1. (cid:3) Remark. From the proof it is clear that that the proposition also holds if theinterval I = [ − , 1] is replaced by any smaller interval.6. Palais-Smale and the proof of Theorem 1.2 In this section we prove a Palais-Smale condition on sequences of blowing upsolutions to (1.1). This will quickly lead to the proof of Theorem 1.2.For a maximal solution u to (1.1), define S ∗ ( u, L ) = sup { S I ( u ) : I is an open interval with ≤ L } , where we set S I ( u ) = ∞ if u is not defined on I . All solutions in this section areassumed to be maximal. By the triangle inequality, finiteness of S ∗ ( u, L ) for some L implies finiteness for all L . SetΛ d ( E, L ) = sup { S ∗ ( u, L ) : u solves (1.1), µ = +1 , E ( u ) = E } Λ f ( E, L ) = sup { S ∗ ( u, L ) : u solves (1.1), µ = − , E ( u ) = E, k∇ u (0) k L < k∇ W k L } . Note as before that finiteness for some L is equivalent to finiteness for all L . Finally,define Λ d ( E ) = lim L → Λ d ( E, L ) , Λ f ( E ) = lim L → Λ f ( E, L ) , E d = { E : Λ d ( E ) < ∞} , E f = { E : Λ f ( E ) < ∞} . By the local theory, Theorem 1.2 is equivalent to the assertions E d = [0 , ∞ ) , E f = [0 , E ∆ ( W )) . Suppose Theorem 1.2 failed. By the small data theory, E d , E f are nonemptyand open, and the failure of Theorem 1.2 implies the existence of a critical energy E c > 0, with E c < E ∆ ( W ) in the focusing case such that Λ d ( E ) , Λ f ( E ) = ∞ for E > E c and Λ d ( E ) , Λ f ( E ) < ∞ for all E < E c . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 37 Define the spaces˙ X = L t,x ∩ L t Σ x ([ − , ] × R d ) , d = 3 L d +2) d − t,x ∩ L d +2) d t Σ d +2) d x ([ − , ] × R d ) , d ≥ . When d = 3, we also define˙ Y = ˙ X ∩ L t Σ x ([ − , ] × R ) . Proposition 6.1 (Palais-Smale) . Assume Conjecture 1.1 holds. Suppose that u n :( t n − , t n + ) × R d → C is a sequence of solutions with lim n →∞ E ( u n ) = E c , lim n →∞ S ( t n − ,t n ] ( u n ) = lim n →∞ S [ t n ,t n + ) ( u n ) = ∞ . In the focusing case, assume also that E c < E ∆ ( W ) and k∇ u n ( t n ) k L < k∇ W k L .Then there exists a subsequence such that u n ( t n ) converges in Σ .Proof of Proposition 6.1. By replacing u n ( t ) with u n ( t + t n ), we may assume t n ≡ u n (0) is bounded in Σ. Applying Proposition 4.14, after passing to asubsequence we have a decomposition u n (0) = J X j =1 e it jn H G n S n φ j + w Jn = J X j =1 φ jn + w Jn with the properties stated in that proposition. In particular, the remainder hasasymptotically trivial linear evolution:(6.1) lim J → J ∗ lim sup n →∞ k e − itH w Jn k L d +2) d − t,x , and we have asymptotic decoupling of energy:(6.2) sup J lim n →∞ | E ( u n ) − J X j =1 E ( φ jn ) − E ( w Jn ) | = 0 . Observe that lim inf n E ( φ jn ) ≥ 0. This is obvious in the defocusing case. In thefocusing case, (4.15) and the discussion in Section 7 imply thatsup j lim sup n k φ jn k Σ ≤ k u n k Σ < k∇ W k L , so the claim follows from Lemma 7.1. Therefore, there are two possibilities. Case 1 : sup j lim sup n →∞ E ( φ jn ) = E c .By combining (6.2) with the fact that the profiles φ jn are nontrivial in Σ, wededuce that J ∗ = 1 and u n (0) = e it n H G n S n φ + w n , lim n →∞ k w n k Σ = 0 . We will show that N n ≡ x n = 0 and t n = 0). Suppose N n → ∞ .Proposition 5.1 implies that for all large n , there exists a unique solution u n on [ − , ] with u n (0) = e it n H G n S n φ and lim sup n →∞ S ( − , ) ( u n ) ≤ C ( E c ). Byperturbation theory (Proposition 3.3),lim sup n →∞ S [ − , ] ( u n ) ≤ C ( E c ) , which is a contradiction. Therefore, N n ≡ , t jn ≡ , x jn ≡ 0, and u n (0) = φ + w n for some φ ∈ Σ. This is the desired conclusion. Case 2 : sup j lim sup n →∞ E ( φ jn ) ≤ E c − δ for some δ > E c , there exist solutions v jn : ( − , ) × R d → C with k v jn k L d +2) d − t,x ([ − , ]) . E c ,δ E ( φ jn ) . By standard arguments (c.f. [29, Lemma 3.11]), this implies the seemingly strongerbound(6.3) k v jn k ˙ X . E c ,δ E ( φ jn ) . In the case d = 3, we also have k v jn k ˙ Y . E ( φ jn ) . Put(6.4) u Jn = J X j =1 v jn + e − itH w Jn . We claim that for sufficiently large J and n , u Jn is an approximate solution in thesense of Proposition 3.3. To prove this claim, we check that u Jn has the followingthree properties:(i) lim J → J ∗ lim sup n →∞ k u Jn (0) − u n (0) k Σ = 0.(ii) lim sup n →∞ k u Jn k L d +2) d − t,x ([ − T,T ]) . E c ,δ J .(iii) lim J → J ∗ lim sup n →∞ k H e Jn k N ([ − , ]) = 0, where e n = ( i∂ t − H ) u Jn − F ( u Jn ) . There is nothing to check for part (i) as u Jn (0) = u n (0) by construction. Theverification of (ii) relies on the asymptotic decoupling of the nonlinear profiles v jn ,which we record in the following two lemmas. Lemma 6.2 (Orthogonality) . Suppose that two frames F j = ( t jn , x jn , N jn ) , F k =( t k , x kn , N kn ) are orthogonal, and let ˜ G jn , ˜ G kn be the associated spacetime scaling andtranslation operators as defined in (4.1) . Then for all ψ j , ψ k in C ∞ c ( R × R d ) , k ( ˜ G jn ψ j )( ˜ G kn ψ k ) k L d +2 d − t,x + k ( ˜ G jn ψ j ) ∇ ( ˜ G kn ψ k ) k L d +2 d − t,x + k| x | ( ˜ G jn ψ j )( ˜ G kn ψ k ) k L d +2 d − t,x + k| x | ( ˜ G jn ψ j )( ˜ G kn ψ k ) k L d +2 dt,x + k ( ∇ ˜ G jn ψ j )( ∇ ˜ G kn ψ k ) k L d +2 dt,x → as n → ∞ . When d = 3 , we also have k| x | ( ˜ G jn ψ j )( ˜ G kn ψ k ) k L t L x + k ( ∇ ˜ G jn ψ j )( ∇ ˜ G kn ψ k ) k L t L x → . Proof. The arguments for each term are similar, and we only supply the detailsfor the second term. Suppose N kn ( N jn ) − → ∞ . By the chain rule, a change ofvariables, and H¨older, k ( ˜ G jn ψ j ) ∇ ( ˜ G kn ψ k ) k L d +2 d − t,x = k ψ j ∇ ( ˜ G jn ) − ˜ G kn ψ k k L d +2 d − t,x ≤ k ψ j χ n k L d +2 d − t,x k∇ ψ k k L d +2) dt,x , HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 39 where χ n is the characteristic function of the support of ∇ ( ˜ G jn ) − ˜ G kn ψ k . As thesupport of χ n has measure shrinking to zero, we havelim n →∞ k ψ j χ n k L d +2) d − t,x = 0 . A similar argument deals with the case where N jn ( N kn ) − → ∞ . Therefore, we maysuppose that N kn N jn → N ∞ ∈ (0 , ∞ ) . Make the same change of variables as before, and compute ∇ ( ˜ G jn ) − ˜ G kn ψ k ( t, x ) = ( N kn N jn ) d ( ∇ ψ k )[ N kn N jn t + ( N kn ) ( t jn − t kn ) , N kn N jn x + N kn ( x jn − x kn )] . The decoupling statement (4.17) implies that( N kn ) ( t jn − t kn ) + N kn | x jn − x kn | → ∞ . Therefore, the supports of ψ j and ∇ ( ˜ G jn ) − ˜ G kn ψ k are disjoint for large n . (cid:3) Lemma 6.3 (Decoupling of nonlinear profiles) . Let v jn be the nonlinear solutionsdefined above. Then when d ≥ , k v jn v kn k L d +2) d − t,x + k v jn ∇ v kn k L d +2 d − t,x + k| x | v jn v kn k L d +2 d − t,x + k ( ∇ v jn )( ∇ v kn ) k L d +2) dt,x + k| x | v jn v kn k L d +2) dt,x → as n → ∞ . When d = 3 , the same statement holds with the last two expressionsreplaced by k ( ∇ v jn )( ∇ v kn ) k L t L x + k| x | v jn v kn k L t L x → . Proof. We spell out the details for the k v jn | x | v kn k L d +2 d − t,x term. Consider first the case d ≥ 4. As 2 < d +2) d < d , by Proposition 5.1 we can approximate v jn in ˙ X by testfunctions c jn ˜ G n ψ j , ψ j ∈ C ∞ c ( R × R d ) , c jn ( t ) = e − i ( t − tjn ) | xjn | . By H¨older and a change of variables, k v jn | x | v kn k L d +2 d − t,x ≤ k ( v jn − c jn ˜ G jn ψ j ) | x | v kn k L d +2 d − t,x + k| x | ˜ G jn ψ j ( v kn − c kn ˜ G kn ψ k ) k L d +2 d − t,x + k| x | ˜ G jn ψ j ˜ G kn ψ k k L d +2 d − t,x ≤ k ( v jn − c jn ˜ G jn ψ j ) k L d +2) d − t,x k v kn k ˙ X + k ψ j k L d +2) d − t,x k ( v kn − c kn ˜ G kn ψ k ) k ˙ X + k ( ˜ G jn ψ j ) | x | ( ˜ G kn ψ k ) k L d +2 d − t,x By first choosing ψ j , then ψ k , then invoking the previous lemma, one obtains forany ε > n →∞ k v jn | x | v kn k L d +2 d − t,x ≤ ε. When d = 3, we also approximate v jn in ˙ X (which is possible because the exponent in the definition of ˙ X is less than 3), and estimate k v jn | x | v kn k L t,x ≤ k ( v jn − c jn ˜ G jn ψ j ) | x | v kn k L t,x + k| x | ˜ G jn ψ j ( v kn − c kn ˜ G kn ψ k ) k L t,x + k| x | ˜ G jn ψ j ˜ G kn ψ k k L t,x ≤ k ( v jn − c jn ˜ G jn ψ j ) k L t,x k v kn k ˙ Y + k ψ j k L t L x k v kn − c kn ˜ G kn ψ k k ˙ X + k ( ˜ G jn ψ j ) | x | ( ˜ G kn ψ k ) k L t,x which, just as above, can be made arbitrarily small as n → ∞ . Similar approxima-tion arguments deal with the other terms. (cid:3) Let us verify Claim (ii) above. In fact we will show that(6.5) lim sup n →∞ k u Jn k ˙ X ([ − , ]) . E c ,δ J. First, we have S ( u Jn ) = Z Z | J X j =1 v jn + e − itH w Jn | d +2) d − dxdt . S ( J X j =1 v jn ) + S ( e − itH w Jn ) . By the properties of the LPD, lim J → J ∗ lim sup n →∞ S ( e − itH w Jn ) = 0. Recalling(6.3), we have S ( J X j =1 v jn ) = (cid:13)(cid:13)(cid:13) ( J X j =1 v jn ) (cid:13)(cid:13)(cid:13) d +2 d − L d +2 d − t,x ≤ ( J X j =1 k v jn k L d +2) d − t,x + X j = k k v jn v kn k L d +2 d − t,x ) d +2 d − . ( J X j =1 E ( φ jn ) + o J (1)) d +2 d − where for the last line we invoked Lemma 6.3. Since energy decoupling implieslim sup n →∞ P Jj =1 E ( φ jn ) ≤ E c , we obtain lim J → J ∗ lim sup n →∞ S ( u Jn ) . E c ,δ n →∞ ( k∇ u Jn k L d +2) dt,x + k| x | u Jn k L d +2) dt,x ) . E c ,δ J. This completes the verification of property (ii) in the case d ≥ 4. The case d = 3 isdealt with in a similar fashion. Remark. The above argument shows that for each J and each η > 0, there exists J ′ ≤ J such that lim sup n →∞ k J X j = J ′ v jn k ˙ X ([ − , ]) ≤ η. It remains to check property (iii) above, namely, that(6.6) lim J → J ∗ lim sup n →∞ k H / e Jn k N ([ − , ]) = 0 . Writing F ( z ) = | z | d − z , we decompose e Jn = [ J X j =1 F ( v jn ) − F ( J X j =1 v jn )] + [ F ( u Jn − e − itH w Jn ) − F ( u Jn )] = ( a ) + ( b ) . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 41 Consider (a) first. Suppose d ≥ 6. Using the chain rule ∇ F ( u ) = F z ( u ) ∇ u + F z ( u ) ∇ u and the estimates | F z ( z ) | + | F z ( z ) | = O ( | z | d − ) , | F z ( z ) − F z ( w ) | + | F z ( z ) − F z ( w ) | = O ( | z − w | d − ) , we compute |∇ ( a ) | . J X j =1 X k = j | v kn | d − |∇ v jn | . By H¨older, Lemma 6.3, and the induction hypothesis (6.3), k∇ ( a ) k L d +2) d +4 t,x . J X j =1 X k = j k| v kn ||∇ v jn |k d − L d +2 d − t,x k∇ v kn k d − d − L d +2) dt,x = o J (1)as n → ∞ . When 3 ≤ d ≤ 5, we have instead |∇ ( a ) | . J X j =1 X k = j | v kn ||∇ v jn | O ( (cid:12)(cid:12)(cid:12) J X k =1 v kn (cid:12)(cid:12)(cid:12) − dd − + | v jn | − dd − ) , thus k∇ ( a ) k L d +2) d +4 t,x . J J X j =1 k v jn k − dd − L d +2) d − t,x J X j =1 X k = j k| v kn ||∇ v jn |k L d +2 d − t,x = o J (1) . Similarly, writing | ( a ) | ≤ J X j =1 (cid:12)(cid:12)(cid:12) | v jn | d − − | J X k =1 v kn | d − (cid:12)(cid:12)(cid:12) | v jn | . J X j =1 X k = j | v jn || v kn | d − , we have k x ( a ) k L d +2) d +4 t,x J X j =1 X k = j k| x | v jn k d − d − L d +2) dt,x k| x | v jn v kn k d − L d +2 d − t,x = o J (1) . When 3 ≤ d ≤ | ( a ) | . J X j =1 X k = j | v jn | v kn | O ( (cid:12)(cid:12)(cid:12) J X k =1 v kn (cid:12)(cid:12)(cid:12) − dd − + | v jn | − dd − ) , hence also k| x | ( a ) k L d +2) d +4 t,x = o J (1) . Summing up, k H / ( a ) k L d +2) d +4 t,x . k∇ ( a ) k L d +2) d +4 t,x + k x ( a ) k L d +2) d +4 t,x = o J (1) . We now estimate (b), restricting temporarily to dimensions d ≥ 4. When d ≥ b ) = F ( u Jn − e − itH w Jn ) − F ( u Jn )= ( | u Jn − e − itH w Jn | d − − | u Jn | d − ) J X j =1 v jn − ( e − itH w Jn ) | u Jn | d − = O ( | e − itH w Jn | d − ) J X j =1 v jn − ( e − itH w Jn ) | u Jn | d − , and apply H¨older’s inequality: k| x | ( b ) k L d +2) d +4 t,x . k e − itH w Jn k d − L d +2) d − t,x k J X j =1 | x | v jn k L d +2) dt,x + k| x | u Jn k d − L d +2) dt,x k| x | e − itH w Jn k d − d − L d +2) dt,x k e − itH w Jn k d − L d +2) d − t,x (6.7)When d = 4 , b ) = ( e − itH w Jn ) O ( | u Jn | − dd − + | u Jn − e − itH w Jn | − dd − ) J X j =1 v jn − ( e − itH w Jn ) | u Jn | d − , thus k| x | ( b ) k L d +2) d +4 t,x . k e − itH w Jn k L d +2) d − t,x k| x | J X j =1 v jn k L d +2) dt,x ( k u Jn k − dd − L d +2) d − t,x + k e − itH w Jn k − dd − L d +2) d − t,x )+ k e − itH w Jn k L d +2) d − t,x k xu Jn k L d +2) dt,x k u Jn k − dd − L d +2) d − t,x . Using (6.5), Strichartz, and the decay property (6.1), we getlim J → J ∗ lim sup n →∞ k| x | ( b ) k L d +2) d +4 t,x = 0 . It remains to bound ∇ ( b ). By the chain rule, ∇ ( b ) . | e − itH w Jn | d − | (cid:12)(cid:12)(cid:12) J X j =1 ∇ v jn (cid:12)(cid:12)(cid:12) + | u Jn | d − |∇ e − itH w Jn | = ( b ′ ) + ( b ′′ ) . The first term ( b ′ ) can be handled in the manner of (6.7) above. We now concernourselves with (b”). Fix a small parameter η > 0, and use the above remark toobtain J ′ = J ′ ( η ) ≤ J such that k J X j = J ′ v jn k ˙ X ≤ η. HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 43 By the subadditivity of z 7→ | z | d − (which is true up to a constant when d = 4 , k ( b ′′ ) k L d +2) d +4 t,x = k| J X j =1 v jn + e − itH w Jn | d − |∇ e − itH w Jn |k L d +2) d +4 t,x . k e − itH w Jn k d − L d +2) d − t,x k H / e − itH w Jn k L d +2) dt,x + k J X j = J ′ v jn k d − L d +2) d − t,x k H / e − itH w Jn k L d +2) dt,x + C J ′ J ′ − X j =1 k∇ e − itH w Jn k d − d − L d +2) dt,x k| v jn ||∇ e − itH w Jn k d − L d +2 d − t,x . By Strichartz and the decay of e − itH w Jn in L d +2) d − t,x , the first term goes to 0 as J → ∞ , n → ∞ . By Strichartz and the definition of J ′ , the second term isbounded by η d − k w Jn k Σ which can be made arbitrarily small since lim sup n →∞ k w Jn k Σ is bounded uniformlyin J . To finish, we show that for each fixed j lim J → J ∗ lim sup n →∞ k| v jn |∇ e − itH w Jn k L d +2 d − t,x = 0 . (6.8)For any ε > 0, there exist ψ j ∈ C ∞ c ( R × R d ) such that if c jn = e − i ( t − tjn ) | xjn | then lim sup n →∞ k v jn − c jn ˜ G jn ψ j k L d +2) d − t,x ([ − , ]) < ε, Note that ˜ G jn ψ j is supported on the set {| t − t jn | . ( N jn ) − , | x − x jn | . ( N jn ) − } . Thus for all n sufficiently large, k v jn ∇ e − itH w Jn k L d +2 d − t,x ≤ k v jn − c jn ˜ G jn ψ j k L d +2) d − t,x k∇ e − itH w Jn k L d +2) dt,x + k ˜ G jn ψ j ∇ e − itH w Jn k L d +2 d − t,x . E c ε + k ( ˜ G jn ψ j ) ∇ e − itH w Jn k L d +2 d − t,x . By H¨older, noting that d +2 d − ≤ d ≥ k ( ˜ G jn ψ j ) ∇ e − itH w Jn k L d +2 d − t,x . ε ( N jn ) d − k∇ e − itH w Jn k L d +2 d − t,x ( | t − t jn | . ( N jn ) − , | x − x jn | . ( N jn ) − ) . N jn k∇ e − itH w Jn k L t,x ( | t − t jn | . ( N jn ) − , | x − x jn | . ( N jn ) − ) Since ( N jn ) − | x jn | = O (( N jn ) − ), Corollary 2.9 implies k v jn ∇ e − itH w Jn k L d +2 d − t,x . ε + C ε,E c k e − itH w Jn k L d +2) d − t,x . Sending n → ∞ , then J → J ∗ , then ε → d = 3, we estimate ( b ) instead with the L t L x dual Strichartz norm.Write( b ) = ( e − itH w Jn ) v jn O ( | u Jn | + | u Jn − e − itH w Jn | ) J X j =1 v jn − ( e − itH w Jn ) | u Jn | , and apply H¨older’s inequality: k| x | ( b ) k L t L x . k e − itH w Jn k L t,x k u Jn k L t,x k H / u Jn k L t L x + k e − itH w Jn k L t,x ( k u Jn k L t,x + k e − itH w Jn k L t,x ) k H J X j =1 v jn k L t L x . (6.9)Using (6.1) and (6.5), we havelim J → J ∗ lim sup n →∞ k| x | ( b ) k L t L x = 0 . It remains to bound ∇ ( b ). By the chain rule, ∇ ( b ) = O ( | u Jn − e − itH w Jn | − | u Jn | ) ∇ J X j =1 v jn + | u Jn | |∇ e − itH w Jn | = ( b ′ ) + ( b ′′ ) . The first term ( b ′ ) can be treated in the manner of k| x | ( b ) k L t L x above. We nowconcern ourselves with ( b ′′ ). Fix a small parameter η > 0, and use the above remarkto obtain J ′ = J ′ ( η ) ≤ J such that k J X j = J ′ v jn k ˙ X ≤ η. Thus by the triangle inequality and H¨older, k ( b ′′ ) k L t L x = k| J X j =1 v jn + e − itH w Jn | ( e − itH w Jn ) k L t L x . k e − itH w Jn k L t,x k H e − itH w Jn k L t L x + k| J X j = J ′ v jn | |∇ e − itH w Jn |k L t L x + C J ′ J ′ X j =1 k| v jn | ∇ e − itH w Jn k L t L x . k e − itH w Jn k L t,x k H e − itH w Jn k L t L x + k J X j = J ′ v jn k X k|∇ e − itH w Jn |k L t L x + C J ′ J ′ X j =1 k| v jn | ∇ e − itH w Jn k L t L x By Strichartz and the decay of e − itH w Jn in L t,x , the first term goes to 0 as J →∞ , n → ∞ . By Strichartz and the definition of J ′ , the second term is bounded by η k w Jn k ΣHE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 45 which can be made arbitrarily small since lim sup n →∞ k w Jn k Σ is bounded uniformlyin J . To finish, we show that for each fixed j lim J → J ∗ lim sup n →∞ k| v jn | ∇ e − itH w Jn k L t L x = 0 . By H¨older, k| v jn | ∇ e − itH w Jn k L t L x ≤ k v jn k L t,x k v jn ∇ e − itH w Jn k L t L x , so by (6.3) it suffices to showlim J → J ∗ lim sup n →∞ k v jn ∇ e − itH w Jn k L t L x = 0 . (6.10)For any ε > 0, there exists ψ j ∈ C ∞ c ( R × R ) and functions c jn ( t ) , | c jn | ≡ n →∞ k v jn − c jn ˜ G jn ψ j k L t,x ([ − , ]) < ε, Note that ˜ G jn ψ j is supported on the set {| t − t jn | . ( N jn ) − , | x − x jn | . ( N jn ) − } . Thus for all n sufficiently large, k v jn ∇ e − itH w Jn k L t L x ≤ k v jn − c jn ˜ G jn ψ j k L t,x k∇ e − itH w Jn k L t L x + k ˜ G jn ψ j ∇ e − itH w Jn k L t L x . E c ε + k ( ˜ G jn ψ j ) ∇ e − itH w Jn k L t L x . From the definition of the operators ˜ G jn , we have k ( ˜ G jn ψ j ) ∇ e − itH w Jn k L t L x . ε N n k∇ e − itH w Jn k L t L x ( | t − t jn | . ( N jn ) − , | x − x jn | . ( N jn ) − ) . Since ( N jn ) − | x jn | = O (( N jn ) − ), Corollary 2.9 implies k v jn ∇ e − itH w Jn k L t L x . ε + C ε k e − itH w Jn k L t,x k w Jn k Σ . Sending n → ∞ , then J → J ∗ , then ε → d = 3.By perturbation theory, lim sup n →∞ S ( − T,T ) ≤ C ( E c ) < ∞ , contrary to thePalais-Smale hypothesis. This rules out Case 2 and completes the proof of Propo-sition 6.1. (cid:3) Armed with Proposition 6.1, we can finish the proof of Theorem 1.2. Proof of Theorem 1.2. Suppose the theorem failed. In the defocusing case, there ex-ist E c ∈ (0 , ∞ ) and a sequence of solutions u n with E ( u n ) → E c and S ( − n , ( u n ) →∞ and S [0 , n ) ( u n ) → ∞ . The same is true in the focusing case except E c is restrictedto the interval (0 , E ∆ ( W )) and lim sup n k u n (0) k ˙ H < k W k ˙ H . By Proposition 6.1,after passing to a subsequence u n (0) converges in Σ to some φ . Let u ∞ be themaximal solution to (1.1) with u ∞ (0) = φ . By the stability theory and domi-nated convergence, lim n →∞ S [ − n , n ] ( u n ) = lim n →∞ S [ − n , n ] ( u ∞ ) = 0, which is acontradiction. (cid:3) Proof of Theorem 1.3 We begin by recalling some facts about the ground state W ( x ) = (1 + | x | d ( d − ) − d − ∈ ˙ H ( R d )This function satisfies the elliptic PDE ∆ W + W d − W = 0 . It is well-known (c.f. Aubin [1] and Talenti [27]) that the functions witnessing thesharp constant in the Sobolev inequality k f k L dd − ( R d ) ≤ C d k∇ f k L ( R d ) , are precisely those of the form f ( x ) = αW ( β ( x − x )) , α ∈ C , β > , x ∈ R d .For the reader’s convenience, we reiterate the definitions of the energy associatedto the focusing energy-critical NLS with and without potential: E ∆ ( u ) = Z R d |∇ u | − (1 − d ) | u | dd − dx,E ( u ) = E ∆ ( u ) + k xu k L . Lemma 7.1 (Energy trapping [16]) . Suppose E ∆ ( u ) ≤ (1 − δ ) E ∆ ( W ) . • If k∇ u k L ≤ k∇ W k L , then there exists δ > depending on δ such that k∇ u k L ≤ (1 − δ ) k W k L , and E ∆ ( u ) ≥ . • If k∇ u k L ≥ k∇ W k L then there exists δ > depending on δ such that k∇ u k L ≥ (1 + δ ) k∇ W k L , and k∇ u k L − k u k dd − L dd − ≤ − δ E ∆ ( W ) . Now suppose E ( u ) < E ∆ ( W ) and k∇ u k L ≤ k∇ W k L . The energy inequalitycan be written as k u k + (1 − d )( k W k dd − L dd − − k u k dd − L dd − ) ≤ k∇ W k L . By the variational characterization of W , the difference of norms on the left side isnonnegative; therefore k u k Σ ≤ k∇ W k L . Combining the above with conservation of energy and a continuity argument, weobtain Corollary 7.2. Suppose u : I × R d → C is a solution to the focusing equation (1.1) with E ( u ) ≤ (1 − δ ) E ∆ ( W ) . Then there exist δ , δ > , depending on δ , suchthat • If k u (0) k ˙ H ≤ k W k ˙ H , then sup t ∈ I k u ( t ) k Σ ≤ (1 − δ ) k W k ˙ H and E ( u ) ≥ . • If k u (0) k ˙ H ≥ k W k ˙ H , then inf t ∈ I k u ( t ) k Σ ≥ (1 + δ ) k W k ˙ H and k∇ u k − k u k dd − L dd − ≤ − δ E ∆ ( W ) . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 47 Proof of Theorem 1.3. Let u be the maximal solution to (1.1) with u (0) = u , E ( u ) < E ∆ ( W ) , k∇ u k ≥ k∇ W k . Let f ( t ) = R R d | x | | u ( t, x ) | dx . It can be shown [8] that f is C on the interval ofexistence and f ′′ ( t ) = Z |∇ u ( t, x ) | − | u ( t, x ) | dd − − | x | | u ( t, x ) | dx. By the corollary, f ′′ is bounded above by some fixed C < 0. Therefore f ( t ) ≤ A + Bt + C t for some constants A and B . It follows that u has a finite lifespan in both timedirections. (cid:3) Bounded linear potentials In this section we show, using a perturbative argument, that(8.1) i∂ t u = ( − ∆ + V ) u + | u | d − u, u (0) = u ∈ H ( R d )is globally wellposed whenever V is a real-valued function with V max := k V k L ∞ + k∇ V k L ∞ < ∞ . This equation defines the Hamiltonian flow of the energy functional(8.2) E ( u ( t )) = Z R d |∇ u ( t, x ) | + V | u ( t, x ) | + d − d | u | dd − dx = E ( u (0)) . Solutions to (8.1) also conserve mass : M ( u ( t )) = Z R d | u ( t, x ) | dx = M ( u (0)) . It will be convenient to assume V is positive and bounded away from 0. Thishypothesis allows us to bound the H norm of u purely in terms of E instead ofboth E and M , and causes no loss of generality because for sign-indefinite V wecould simply consider the conserved quantity E + CM in place of E , where C issome positive constant. Theorem 8.1. For any u ∈ H ( R d ) , (8.1) has a unique global solution u ∈ C t,loc H x ( R × R d ) . Further, u obeys the spacetime bounds S I ( u ) ≤ C ( k u k H , | I | ) for any compact interval I ⊂ R . As alluded to at the beginning of this section, the proof uses the strategy pio-neered by [29] and treats the term V u as a perturbation to (1.6), which is globallywellposed. Thus Duhamel’s formula reads(8.3) u ( t ) = e it ∆2 u ( t ) − i Z t e i ( t − s )∆2 [ | u ( s ) | d − u ( s ) + V u ( s )] ds. We record mostly without proof some standard results in the local theory of(8.1). Introduce the notation k u k X ( I ) = k∇ u k L d +2) d − t L d ( d +2) d x ( I × R d ) . Lemma 8.1 (Local wellposedness) . Fix u ∈ H ( R d ) , and suppose T > is suchthat k e it ∆2 u k X ([ − T ,T ]) ≤ η ≤ η where η = η ( d ) is a fixed parameter. Then there exists a positive T = T ( k u k H , η, V max ) such that (8.1) has a unique (strong) solution u ∈ C t H x ([ − T , T ] × R d ) . Further,if ( − T min , T max ) is the maximal lifespan of u , then k∇ u k S ( I ) < ∞ for every com-pact interval I ⊂ ( − T min , T max ) , where k · k S ( I ) is the Strichartz norm defined inSection 2.1.Proof sketch. Run the usual contraction mapping argument using the Strichartzestimates to show that I ( u )( t ) = e it ∆2 u − i Z t e i ( t − s )∆2 [ | u ( s ) | d − u ( s ) + V u ( s )] dx has a fixed point in a suitable function space. Estimate the terms involving V inthe L t L x dual Strichartz norm and choose the parameter T to make those termssufficiently small after using H¨older in time. (cid:3) Lemma 8.2 (Blowup criterion) . Let u : ( T , T ) × R d → C be a solution to (8.1) with k u k X (( T ,T )) < ∞ . If T > −∞ or T < ∞ , then u can be extended to a solution on a larger timeinterval. The key result we will rely on is the stability theory for the energy-critical NLS(1.6). Lemma 8.3 (Stability [28]) . Let ˜ u : I × R d → C be an approximate solution toequation (1.6) in the sense that i∂ t ˜ u = − ∆ u ± | ˜ u | d − ˜ u + e for some function e . Assume that (8.4) k ˜ u k L d +2) d − t,x ≤ L, k∇ u k L ∞ t L x ≤ E, and that for some < ε < ε ( E, L ) one has (8.5) k ˜ u ( t ) − u k ˙ H + k∇ e k N ( I ) ≤ ε, where k · k N ( I ) was defined in Section 2.1. Then there exists a unique solution u : I × R d → C to (1.6) with u ( t ) = u which further satisfies the estimates (8.6) k ˜ u − u k L d +2) d − t,x + k∇ (˜ u − u ) k S ( I ) . C ( E, L ) ε c where < c = c ( d ) < and C ( E, L ) is a function which is nondecreasing in eachvariable.Proof of Theorem 8.1. It suffices to show that for T sufficiently small dependingonly on E = E ( u ), the solution u to (8.1) on [0 , T ] satisfies an a priori estimate(8.7) k u k X ([0 ,T ]) ≤ C ( E ) . HE ENERGY-CRITICAL QUANTUM HARMONIC OSCILLATOR 49 From Lemma 8.2 and energy conservation, it will follow that u is a global solutionwith the desired spacetime bound.By Theorem 1.1, the equation( i∂ t + ∆) w = | w | d − w, w (0) = u (0) . has a unique global solution w ∈ C t,loc ˙ H x ( R × R d ) with the spacetime bound (1.7).Fix a small parameter η > , ∞ ) into J ( E, η ) intervals I j = [ t j , t j +1 ) so that(8.8) k w k X ( I j ) ≤ η. For some J ′ < J , we then have[0 , T ] = J ′ − [ j =0 ([0 , T ] ∩ I j ) . We make two preliminary estimates. By H¨older in time,(8.9) k V u k N ( I j ) + k∇ ( V u ) k N ( I j ) . C V T k u k L ∞ t H x ( I j ) ≤ ε for any ε provided that T = T ( E, V, ε ) is sufficiently small. Further, observe that(8.10) k e i ( t − tj )∆2 w ( t j ) k X ( I j ) ≤ η for η sufficiently small depending only on d . Indeed, from the Duhamel formula(8.11) w ( t ) = e i ( t − tj )∆2 w ( t j ) − i Z tt j e i ( t − s )∆2 ( | w | d − w )( s ) ds, Strichartz, and the chain rule, we find that k e i ( t − tj )∆2 w ( t j ) k X ( I j ) ≤ k w k X ( I j ) + c d k∇ ( | w | d − w ) k L t L dd +2 x ( I j ) ≤ η + c d k w k d +2 d − X ( I j ) ≤ η + c d η d +2 d − . Taking η sufficiently small relative to c d , we obtain (8.10).Now, choosing ε < η in (8.9) (and adjusting T accordingly), we use the Duhamelformula (8.3), Strichartz, H¨older, and (8.10) to obtain k u k X ( I ) ≤ k e it ∆2 u (0) k X ( I ) + c d k u k d +2 d − X ( I ) + C k V u k L t H x ( I ) ≤ η + c d k u k d +2 d − X ( I ) + C V T k u k L ∞ t H x ( I ) ≤ η + c d k u k d +2 d − X ( I ) By a continuity argument, k u k X ( I ) ≤ η. (8.12)Choosing ε sufficiently small in (8.9) so that the smallness condition (8.5) is satisfied,we apply Lemma 8.3 with k u (0) − w (0) k ˙ H = 0 to find that(8.13) k∇ ( u − w ) k S ( I ) ≤ C ( E ) ε c On the interval I , use (8.10), (8.13), and the usual estimates to obtain k u k X ( I ) ≤ k e i ( t − t u ( t ) k X ( I ) + c d k u k d +2 d − X ( I ) + C V T k u k L ∞ t H x ( I ) ≤ C ( E ) ε c + 2 η + c k u k d +2 d − X ( I ) + η, where the C ( E ) in the last line has absorbed the Strichartz constant c ; this re-definition of C ( E ) will cause no trouble because the number of times it will occurdepends only on E , d , and V . By by choosing ε sufficiently small relative to η andusing continuity, we find that k u k X ( I ) ≤ η. As before, by choosing T sufficiently small we obtain k∇ ( V u ) k ˙ N ( I ) ≤ ε k e i ( t − t [ u ( t ) − w ( t )] k X ( I ) ≤ C ( E ) ε c for any ε ≤ ε ( E, L ). 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