Growth of Sobolev norms for linear Schr{ö}dinger operators
aa r X i v : . [ m a t h . A P ] J un GROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGEROPERATORS
LAURENT THOMANN
Abstract.
We give an example of a linear, time-dependent, Schr¨odinger operator with optimalgrowth of Sobolev norms. The construction is explicit, and relies on a comprehensive study of thelinear Lowest Landau Level equation with a time-dependent potential. Introduction and main result
The aim of this paper is to present an example of linear, time-dependent, Schr¨odinger operatorwhich exhibits optimal polynomial growth of Sobolev norms. Moreover, this operator takes the form e H + L ( t ), where e H is an elliptic operator with compact resolvent and where the perturbation L ( t )is a small, time-dependent, bounded self-adjoint operator. Our construction is actually entirelyexplicit and it is based on the study of linear Lowest Landau Level equations (LLL) with a time-dependent potential.In Maspero-Robert [29], the authors study linear Schr¨odinger operators, obtain global well-posedness results and prove very precise polynomial bounds on the possible growth of Sobolevnorms under general conditions (see Assumption 1.1 below). We show here that these bounds areoptimal.Our setting is the following: consider the 2-dimensional harmonic oscillator H = − ( ∂ x + ∂ y ) + ( x + y ) = − ∂ z ∂ z + | z | , where z = x + iy , ∂ z = ( ∂ x − i∂ y ). This operator acts on the space e E = (cid:8) u ( z ) = e − | z | f ( z ) , f entire holomorphic (cid:9) ∩ S ′ ( C ) , and if we define the Bargmann-Fock space E by E = (cid:8) u ( z ) = e − | z | f ( z ) , f entire holomorphic (cid:9) ∩ L ( C ) , then the so-called special Hermite functions ( ϕ n ) n ≥ given by ϕ n ( z ) = z n √ πn ! e − | z | , form a Hilbertian basis of E , and are eigenfunctions of H , namely Hϕ n = 2( n + 1) ϕ n , n ≥ . Let 0 ≤ τ < ρ ( τ ) = − τ ) ∈ [1 / , ∞ ). We define the operator e H = ( H + 1) ρ ( τ ) , whichin turn defines the scale of Hilbert spaces (cid:0) e H s (cid:1) s ≥ by e H s = (cid:8) u ∈ L ( C ) , e H s/ u ∈ L ( C ) (cid:9) ∩ E , e H = E , Mathematics Subject Classification.
Key words and phrases.
Linear Schr¨odinger equation, time-dependent potential, growth of Sobolev norms,reducibility.The author is supported by the grants ”BEKAM” ANR-15-CE40-0001 and ”ISDEEC” ANR-16-CE40-0013. and we denote by L the Lebesgue measure on C .For a family (cid:0) L ( t ) (cid:1) t ∈ R of continuous linear mappings L ( t ) : e H s −→ e H s , we consider the following assumptions : Assumption 1.1. (cid:0) L ( t ) (cid:1) t ∈ R is a family of linear operators which satisfies: ( i ) One has t L ( t ) ∈ C b (cid:0) R ; L ( e H s ) (cid:1) for all s ≥ . ( ii ) For every t ∈ R , L ( t ) is symmetric w.r.t. the scalar product of e H , Z C v L ( t ) u dL = Z C u L ( t ) v dL, ∀ u, v ∈ e H . ( iii ) There exists ≤ τ < , such that the family (cid:0) L ( t ) (cid:1) t ∈ R is e H τ -bounded in the sense that t [ L ( t ) , e H ] e H − τ ∈ C b (cid:0) R , L ( e H s ) (cid:1) for all s ≥ . ( iv ) For all ℓ ∈ N , one has t L ( t ) ∈ C ℓb (cid:0) R ; L ( e H s ; e H s − ℓτ ) (cid:1) for all s ≥ . Finally, for s ≥
0, we consider the initial value problem ( i∂ t u = (cid:0) e H + L ( t ) (cid:1) u, ( t, z ) ∈ R × C ,u ( t, · ) | t = t = u ∈ e H s , (1.1)and we are able to state our main result : Theorem 1.2.
For any ǫ > , there exists a family of linear operators (cid:0) L ( t ) (cid:1) t ∈ R which satisfiesAssumption 1.1, so that for all s ≥ : ( i ) There exists C s > such that sup t ∈ R k L ( t ) k L ( e H s ) ≤ C s ǫ. ( ii ) The problem (1.1) is globally well-posed in e H s : for any u ∈ e H s , there exists a unique solution u ( t ) := U ( t, t ) u such that u ∈ C (cid:0) R , e H s (cid:1) to (1.1) . Moreover, U has the group property U ( t , t ) U ( t , t ) = U ( t , t ) , U ( t, t ) = I d , ∀ t, t , t ∈ R , and U is unitary in e H (cid:13)(cid:13) U ( t, t ) u (cid:13)(cid:13) e H = k u k e H , ∀ t ∈ R . ( iii ) Any solution to (1.1) , with initial condition u ∈ e H s , satisfies for all t ∈ R (cid:13)(cid:13) U ( t, t ) u (cid:13)(cid:13) e H s ≤ C k u k e H s h ǫ ( t − t ) i s − τ ) . ( iv ) There exists a nontrivial initial condition u ∈ ∩ k ≥ e H k such that the corresponding solutionto (1.1) satisfies for all t ∈ R (cid:13)(cid:13) U ( t, t ) u (cid:13)(cid:13) e H s ≥ c k u k e H s h ǫ ( t − t ) i s − τ ) . Actually, items ( ii ) and ( iii ) directly follow from [29, Theorem 1.5]. The novelty in our work isitem ( iv ) which shows that the upper bounds obtained in [29, Theorem 1.5] are optimal withoutfurther assumptions, even for small perturbations L ( t ), see item ( i ).It seems that the example of Theorem 1.2 is the first one which covers all the possible values of0 ≤ τ <
1, and it is noticeable the result of Theorem 1.2 is obtained for any value of 0 ≤ τ < τ = 0 (see also [28] for an alternative proof), and in [5, Appendix A] inthe case τ = 1 /
2, but it seems that the other cases were left open.
ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 3
We stress that our example is an operator which takes the form e H + L ( t ), where e H is a (timeindependent) elliptic operator with compact resolvent and L ( t ) a bounded self-adjoint operator.Moreover, this perturbation is small and satisfies indeed for all ℓ ≥ t ∈ R k ∂ ℓt L ( t ) k L ( e H s ; e H s − ℓτ ) ≤ C s,ℓ ǫ. If one allows unbounded perturbations, it is simpler to obtain growth of Sobolev norms, as it isshown by an elementary example given in Appendix A. In this latter context, growth of Sobolevnorms can occur even with time-independent operators.Observe that the ϕ n are the eigenfunctions of e H , namely e Hϕ n = 2 ρ ( τ ) ( n + 1) ρ ( τ ) ϕ n , n ≥ . Hence in our example, we see an exact correspondance between the asymptotics of the eigenvaluesof e H and the rate of growth for (1.1). If ρ > τ > / e H satisfies a gap condition, but in our example, ∂ ℓt L ( t ) is not regular enough (see item ( iv ) inAssumption 1.1) to meet the hypotheses of [29, Theorems 1.8 and 1.9], in which better upperbounds are obtained.Let us recall the following characterization of the Sobolev spaces e H s . By [18, Lemma C.1], forany s ≥
0, there exist c, C > u ∈ e H s c kh z i ρ ( τ ) s u k L ( C ) ≤ k u k e H s ≤ C kh z i ρ ( τ ) s u k L ( C ) , h z i = (1 + | z | ) / . (1.2)As a consequence, in the Bargmann-Fock space, a growth of Sobolev norm corresponds to a transferof energy in the physical space. In our example, the growth will be induced by a traveling wave.This is in contrast to the previous known examples [13, 5, 28], where the growth was inherited bya time-periodic phenomenon.We end this section by reviewing some results on the growth of linear Schr¨odinger equations onmanifolds with time-dependent potentials i∂ t u + ∆ u + V ( t, x ) u = 0 . (1.3)In [9] Bourgain proves a polynomial bound of the Sobolev norm for (1.3), when V ( t, x ) is a bounded(real analytic) potential. Moreover, when the potential is quasi-periodic in time he obtains in [10] alogarithmic bound (see also [12, 35, 16, 22], for more results on norm inflation phenomena in varioussettings). Delort [13] constructs an example with polynomial growth for the harmonic oscillatorperturbed by a (time-periodic) pseudo-differential operator of order zero. In [5], the authors givethe example of a time-periodic order one perturbation of the harmonic oscillator which inducespolynomial growth. We refer to [28, 24] for more examples with growth of norms and to [6] forbounds on abstract linear Schr¨odinger equations. Finally, let us mention the recent article [27] inwhich the authors obtain very precise results on the dynamics of a family of perturbations of theharmonic oscillator.2. The linear LLL equation with time-dependent potential
We now present our example more in details. Let W ∈ L ∞ ( R × C , R ) be a real-valued time-dependent potential and consider the linear equation ( i∂ t u − δHu = Π (cid:0) W ( t, z ) u (cid:1) , ( t, z ) ∈ R × C , δ ∈ R ,u ( t, · ) | t = t = u ∈ E , (2.1) LAURENT THOMANN where Π is the orthogonal projector on the space E (the kernel of Π is very explicit, see (2.7) below).The equation (2.1) is the linearization of the Lowest Landau Level equation i∂ t u − δHu = Π( | u | u ) , (2.2)which is used in the modeling of fast rotating Bose-Einstein condensates. See e.g. the introductionof [18] for physical motivation, and we refer to [1, 30, 18, 7, 8, 33] for the study of (2.1). Equa-tion (2.1) is a natural mathematical toy model, for which we can try to exhibit some particulardynamics.The dispersion parameter δ ∈ R does not play a role in the dynamics of equation (2.2). Actually, u solves (2.2) if and only if v = e iδtH u solves (2.2) with δ = 0. This comes from the crucial property e − itH Π (cid:0) e itH a e itH b e itH c (cid:1) = Π (cid:0) a b c (cid:1) , ∀ a, b, c ∈ E , see [19, Lemma 2.4 and Corollary 2.5]. However, the transformation v = e iδtH u does not preservethe left hand side of (2.1), that is why we must keep the parameter δ ∈ R in our study (nonethelesswe will see that it does not affect the dynamics of equation (2.1), excepted in the reducibility resultstated in Appendix A where we need δ = 0).In the sequel, by a time translation, we restrict to the case t = 0.In this section, we state global well-posedness results with optimal bounds on the growth of theSobolev norms for (2.1). We are also able to obtain reducibility results for (2.1), when W is a smallquasi-periodic potential, but these results are direct applications of [23], thus we have postponedthe statements to the Appendix A.2.1. Statement of the results.
Our first result concerns the global well-posedness of such anequation under general conditions on W . For s ≥
0, we denote by L ,s = (cid:8) u ∈ L ( C ) , h z i s u ∈ L ( C ) (cid:9) , h z i = (1 + | z | ) / the weighted Lebesgue space and L ,s E = L ,s ∩ E . Then our well-posedness result reads: Theorem 2.1.
Let δ ∈ R and W ∈ L ∞ ( R × C , R ) . For all u ∈ E , there exists a unique solution u ∈ C ( R , E ) to equation (2.1) . Moreover, for every t ∈ R , Z C | u ( t, z ) | dL ( z ) = Z C | u ( z ) | dL ( z ) . Furthermore, if for some s > , u ∈ L ,s E , then u ( t ) ∈ L ,s E for every t ∈ R . A natural question is the control of higher order Sobolev norms of the solution for large times andthis will be achieved, under some additional conditions on W . The notation W ∈ C ∞ b t (cid:0) R × R , R (cid:1) means that W is continuous and bounded in t and smooth in the variables ( x, y ). We stress thatderivation in the time variable is not needed. Theorem 2.2.
Let δ ∈ R and s ≥ . Assume that W ∈ C ∞ b t (cid:0) R × R , R (cid:1) is such that sup ≤ k ≤ (cid:6) s (cid:7) k ∂ kz W ( t, · ) k L ∞ ( C ) ≤ C , t ∈ R , (2.3) then any solution to (2.1) , with initial condition u ∈ L ,s E , satisfies for all t ∈ R kh z i s u ( t ) k L ( C ) ≤ C kh z i s u k L ( C ) h C t i s , where the constant C > only depends on s ≥ . ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 5
Condition (2.3) is rather natural in the space E . For instance, it is satisfied by the followingclass of potentials: assume that V ( t, · ) ∈ E , uniformly in t ∈ R , then W = | V | satisfies (2.3) forall k ≥
0, by Lemma C.2 and (2.9).This bound is indeed optimal as shown by the next result:
Theorem 2.3.
Let δ ∈ R . For all ǫ > , there exists W ǫ ∈ S ( R × R , R ) such that for all ≤ p ≤ ∞ , k W ǫ ( t, · ) k L p ( C ) ≤ ǫ , and such that for all k, j ≥ and uniformly in time k ∂ jz ∂ kz W ǫ ( t, · ) k L ∞ ( C ) ≤ ǫC jk , t ∈ R , and there exists a nontrivial initial condition u ∈ T k ≥ L ,k E such that the corresponding solutionto (2.1) satisfies for all s ≥ and t ∈ R kh z i s u ( t ) k L ( C ) ≥ c s kh z i s u k L ( C ) h ǫt i s . Moreover, we have the following equivalent, when t −→ ±∞kh z i s u ( t ) k L ( C ) ∼ c s ǫ s | t | s k u k L ( C ) . This result is a direct consequence of [33, Theorem 1.5 and Corollary 1.6] and we can make theexplicit choices u = √ ǫ (cid:0) ϕ + i √ ϕ (cid:1) , α = √ π ǫ, and W ǫ ( t, z ) = ǫ π (cid:12)(cid:12) − i √ e − iδt z + αt ) (cid:12)(cid:12) e −| e − iδt z + αt | . (2.4)Actually, in [33, Theorem 1.5 and Corollary 1.6] (see also [33, equation (2.4)]), unbounded trajec-tories where constructed for the system i∂ t u − δHu = Π( | v | u ) , ( t, z ) ∈ R × C ,i∂ t v − δHv = − Π( | u | v ) ,u (0 , · ) = u , v (0 , · ) = v , (2.5)and the idea is here to consider the second equation in (2.5) as given, and to interpret the term | v | in the first line as a given time-dependent potential.Notice that the growth of Sobolev norms is not obtained by a periodic potential as in [12, 5].Here, as it is shown in (2.4), the growth is exhibited by a time translation (more precisely, by amagnetic translation in the Bargmann-Fock space).In general, growth of Sobolev norms is a phenomenon which happens due to resonances of theequation. Recall that the dynamics of the cubic LLL equation i∂ t u = Π( | u | u ) is included in theso-called cubic resonant (CR) equation, which was derived in [17] as a resonant approximation ofNLS (we also refer to [19] for a comprehensive study of the (CR) equation).In the last result of this section we show that if W has additional spacial decay, then the possiblegrowth of the solution of (2.1) enjoys better controls : Theorem 2.4.
Let δ ∈ R and s ≥ and let W ∈ C ∞ b t (cid:0) R × R , R (cid:1) . ( i ) Assume that sup ≤ j ≤ k ≤ (cid:6) s (cid:7) k z k − j ∂ jz W ( t, · ) k L ∞ ( C ) ≤ C , t ∈ R , then any solution to (2.1) , with initial condition u ∈ L ,s E , satisfies for all t ∈ R kh z i s u ( t ) k L ( C ) ≤ C kh z i s u k L ( C ) h C t i , where the constant C > only depends on s ≥ . LAURENT THOMANN ( ii ) Let ǫ > . Assume that for all k ≥ and uniformly in time kh z i k ∂ kz W ( t, · ) k L ∞ ( C ) ≤ C k , t ∈ R , (2.6) then any solution to (2.1) , with initial condition u ∈ L ,s E , satisfies for all t ∈ R kh z i s u ( t ) k L ( C ) ≤ C kh z i s u k L ( C ) h t i ǫ , where the constant C > depends on W , s ≥ and ǫ > . This result is the analogous to [9, 12] in which similar bounds are obtained for the linearSchr¨odinger equation with time-dependent potential, but in our case the proof is much simpler.The result of Theorem 2.4 shows that growth of Sobolev norms can occur only if W is concen-trated in the region | z | ≫ t −→ ±∞ . This is typically the case with the example of thetraveling wave exhibited in Theorem 2.3 (see (2.4)).Under additional conditions on W (analyticity in time and quasi-periodicity) one can show thesolutions are indeed bounded, see Theorem B.1.2.2. Plan of the paper.
The rest of the paper is organized as follows. We end this section bygiving some notations. In Section 3 we study the linear LLL equation (2.1). We are then able toapply these results to prove Theorem 1.2 in Section 4. In Appendix A we give an another exampleof Schr¨odinger operator with unbounded orbits and in Appendix B we state a reducibility resultfor (2.1).2.3.
Some recalls and notations.
The harmonic oscillator H is defined by H = − ∂ z ∂ z + | z | , with the classical notations z = x + iy and ∂ z = 12 ( ∂ x − i∂ y ) , ∂ z = 12 ( ∂ x + i∂ y ) . Denote by ( ϕ n ) n ≥ the family of the special Hermite functions given by ϕ n ( z ) = z n √ πn ! e − | z | . The family ( ϕ n ) n ≥ forms a Hilbertian basis of E (see [37, Proposition 2.1]), and the ϕ n are theeigenfunctions of H , namely Hϕ n = 2( n + 1) ϕ n , n ≥ . We can show (see [18]) that Π, the orthogonal projection on E , is given by the formula(Π u )( z ) = 1 π e − | z | Z C e wz − | w | u ( w ) dL ( w ) , (2.7)where L stands for Lebesgue measure on C .For s ≥
0, we denote by L ,s = (cid:8) u ∈ L ( C ) , h z i s u ∈ L ( C ) (cid:9) , h z i = (1 + | z | ) / the weighted Lebesgue space and L ,s E = L ,s ∩ E . For s ≥
0, we define the harmonic Sobolev spacesby H s = (cid:8) u ∈ L ( C ) , H s/ u ∈ L ( C ) (cid:9) ∩ E , equipped with the natural norm k u k H s = k H s/ u k L ( C ) . Then by [18, Lemma C.1], we have H s = L ,s E with the equivalence of norms c kh z i s u k L ( C ) ≤ k u k H s ≤ C kh z i s u k L ( C ) , ∀ u ∈ L ,s E . (2.8) ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 7
Recall the hypercontractivity estimates (see [11] or [33, Lemma A.2] for the bounds without theoptimal constants which will be enough for our purpose) : for all 1 ≤ p ≤ q ≤ + ∞ and u ∈ e E (cid:16) q π (cid:17) /q k u k L q ( C ) ≤ (cid:16) p π (cid:17) /p k u k L p ( C ) . (2.9)In this paper c, C > Study of the linear LLL equation
Global existence.
To solve equation (2.1) we find a fixed point in a ball of E to F : u e − iδtH u − i Z t e − iδ ( t − s ) H (cid:0) Π( W u )( s ) (cid:1) ds. Let us sketch the proof: since e iτH is unitary in L , we have k F ( u )( t ) k L ≤ k u k L + Z t k Π( W u )( s ) k L ds ≤ k u k L + Ct sup s ∈ [0 ,t ] k u ( s ) k L k W k L ∞ , where we used the continuity of Π in L in the last line (for continuity results for Π we referto [18, Proposition 3.1]). Contraction estimates are obtained similarly, and this gives a local intime solution. Globalization can be obtained by the Gr¨onwall inequality since the equation is linear.The L norm of a solution is a conserved quantity, since the potential W is real valued.If moreover u ∈ L ,s E , we can prove the wellposedness in L ,s E , thanks to the following lemma,which we quote for future reference: Lemma 3.1.
Let W ∈ L ∞ ( C ) and v ∈ L ,s E , then kh z i s Π (cid:0) W v (cid:1) k L ( C ) ≤ C k W k L ∞ ( C ) kh z i s v k L ( C ) , (3.1) and kh z i s e iτH v k L ≤ C kh z i s v k L . (3.2) Proof.
The bound (3.1) is a consequence of [18, Proposition 3.1]). For (3.2), we use (2.8) and thefact that e iτH is unitary in H s : kh z i s e iτH v k L ≤ c k e iτH v k H s = c k v k H s ≤ C kh z i s v k L . (cid:3) Bounds on Sobolev norms: proof of Theorem 2.2.
Now that equation (2.1) is well-posed, let us inspect the behaviour of the norms of the solutions. For this we need a result, whichis an consequence of [33, Lemma 2.1] :
Lemma 3.2.
Let k ∈ N and let W ∈ C k ( R × R , R ) be a real valued function. Assume that u ∈ L ,k E satisfies i∂ t u − δHu = Π (cid:0) W u (cid:1) . Then ddt Z C | z | k | u ( t, z ) | dL ( z ) = − k X j =1 ( − j (cid:18) kj (cid:19) Im Z C z k z k − j | u ( t, z ) | (cid:0) ∂ jz W ( t, z ) (cid:1) dL ( z ) . (3.3) LAURENT THOMANN
Proof.
We compute ddt Z C | z | k | u | dL = 2 Re Z C | z | k u∂ t udL = 2 Im Z C | z | k u Π( W u ) dL + 2 δ Im Z C | z | k uHudL. Let us first show that Im Z C | z | k uHudL = 0. Since H = − ∂ z ∂ z + | z | , it remains to show that Im Z C | z | k u∂ z ∂ z udL = 0. Write u ( z ) = f ( z ) e − | z | , then Im Z C | z | k u∂ z ∂ z udL = Im Z C | z | k f e − | z | ∂ z ∂ z (cid:0) f e − | z | ) dL = − Im Z C | z | k f (cid:0) f + z∂ z f − | z | f ) e −| z | dL = − Im Z C z k +1 z k f ( ∂ z f ) e −| z | dL = 0 , by integrating by parts, hence the result. To complete the proof, we apply [33, Lemma 2.1]. (cid:3) We are now able to prove Theorem 2.2. By linearity, it is enough to consider the case kh z i k u k L ( C ) = 1.We use the identity (3.3). Then, since k ∂ jz W k L ∞ ( C ) ≤ C for all 1 ≤ j ≤ k , we deduce by H¨older ddt Z C | z | k | u | dL ≤ CC Z C h z i k − | u | dL ≤ CC (cid:16) Z C h z i k | u | dL (cid:17) − k (cid:16) Z C | u | dL (cid:17) k , therefore, using the conservation of the mass, ddt (cid:13)(cid:13) h z i k u (cid:13)(cid:13) L ( C ) ≤ CC (cid:13)(cid:13) h z i k u (cid:13)(cid:13) − k L ( C ) , which in turn implies, by time integration, kh z i k u ( t ) k L ( C ) ≤ (cid:0) kh z i k u k /kL ( C ) + CC | t | (cid:1) k ≤ C (1 + C | t | ) k , hence the result when k is an integer. The general case follows by interpolation.3.3. Bounds on Sobolev norms: proof of Theorem 2.4.
The proof is similar, excepted thatnow we have better controls on W .( i ) By (3.3) we have ddt Z C | z | k | u | dL ≤ CC Z C | u | dL, which implies the result by time integration : (cid:13)(cid:13) h z i k u (cid:13)(cid:13) L ( C ) ≤ (cid:13)(cid:13) h z i k u (cid:13)(cid:13) L ( C ) + CC | t |k u k L ( C ) . ( ii ) Here we assume that the stronger condition (2.6) holds. For σ ≥ s , we have the interpolationinequality kh z i s u ( t ) k L ( C ) ≤ kh z i σ u ( t ) k s/σL ( C ) k u ( t ) k − s/σL ( C ) , and we apply it with σ = s/ǫ , together with the bound obtained in ( i ). ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 9
Growth of Sobolev norms: proof of Theorem 2.3.
Let us define the magnetic transla-tions by the formula R α : ( u, v )( z ) (cid:0) u ( z + α ) e ( zα − zα ) , v ( z + α ) e ( zα − zα ) (cid:1) , α ∈ C , as well as the space rotations L θ : ( u, v )( z ) (cid:0) u ( e iθ z ) , v ( e iθ z ) (cid:1) , θ ∈ T . As it can be checked on the ( ϕ n ) n ≥ , we have e itH = e it L t for all t ∈ R . Now we refer to [33,Section 1.7.2]. The system i∂ t u − δHu = Π( | v | u ) , ( t, z ) ∈ R × C ,i∂ t v − δHv = − Π( | u | v ) ,u (0 , z ) = u ( z ) , v (0 , z ) = v ( z ) , admits the following explicit solutions:( u, v ) = (cid:0) e − iλt e − iδtH R αt U, e − iµt e − iδtH R αt V (cid:1) = (cid:0) e − i ( λ +2 δ ) t L − δt R αt U, e − i ( µ +2 δ ) t L − δt R αt V (cid:1) , with U = √ ǫ (cid:0) ϕ + i √ ϕ (cid:1) , V = √ ǫ (cid:0) ϕ − i √ ϕ (cid:1) , and λ = 7 ǫ π , µ = − ǫ π , α = √ π ǫ. It remains to check that W := | v | and u satisfy the assumptions and the conclusions of Theorem 2.3.On the one hand, for all t ∈ R , k u ( t ) k L = √ ǫ , and for s > k u k H s ≤ c s √ ǫ and by (1.2), k u ( t ) k H s = k R αt U k H s ≥ c kh z i s R αt U k L = c kh z − αt i s U k L , which implies that k u ( t ) k H s ≥ c √ ǫ h ǫt i s .On the other hand, we have the explicit expression W ( t, z ) = ǫ π (cid:12)(cid:12) − i √ e − iδt z + αt ) (cid:12)(cid:12) e −| e − iδt z + αt | = (cid:12)(cid:12) L − δt R αt V ( z ) (cid:12)(cid:12) . Therefore we have k W k L = k L − δt R αt V k L = k V k L = ǫ and from (2.9) we have k W k L ∞ = k L − δt R αt V k L ∞ = k V k L ∞ ≤ ǫ. Moreover, from Lemma C.2, we deduce k ∂ jz ∂ kz W k L ∞ ( C ) = k ∂ jz ∂ kz (cid:0) | L − δt R αt V | (cid:1) k L ∞ ( C ) ≤ C jk k L − δt R αt V k L ∞ ≤ ǫC jk , which was the claim. 4. Proof of Theorem 1.2
Some notations.
For 0 ≤ τ < ρ ( τ ) = − τ ) > e H = ( H + 1) ρ ( τ ) , where H is the harmonic oscillator defined by H = − ∂ z ∂ z + | z | . We then define the family of Hilbert spaces (cid:0) e H s (cid:1) s ≥ by e H s = (cid:8) u ∈ L ( C ) , e H s/ u ∈ L ( C ) (cid:9) ∩ E , e H = E . Recall that c kh z i ρ ( τ ) s u k L ( C ) ≤ k u k e H s ≤ C kh z i ρ ( τ ) s u k L ( C ) , h z i = (1 + | z | ) / . Observe also that e H s = H sρ (4.1)where H σ stands for the harmonic Sobolev space based on the harmonic oscillator H , and we have c kh z i s u k L ( C ) ≤ k u k H s ≤ C kh z i s u k L ( C ) . (4.2)4.2. Definition of the operator L ( t ) . Define the potential W ( t, z ) as follows: V = √ ǫ (cid:0) ϕ − i √ ϕ (cid:1) , W ( t, z ) = | R αt V ( z ) | = ǫ π (cid:12)(cid:12) − i √ z + αt ) (cid:12)(cid:12) e −| z + αt | , α = √ π ǫ, and with Lemma C.2, we show that all the derivatives of W are bounded uniformly in t ∈ R : k ∂ jz ∂ kz W ( t ) k L ∞ ( C ) = k ∂ jz ∂ kz (cid:0) | R αt V | (cid:1) k L ∞ ( C ) ≤ C jk k R αt V k L ∞ = C jk k V k L ∞ ≤ ǫC jk . (4.3)Now we define the mapping L ( t ) : e H s −→ e H s u e − it e H Π (cid:0) W ( t ) e it e H u (cid:1) = e − it ( H +1) ρ Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1) , (4.4)and we consider the initial value problem ( i∂ t u = (cid:0) e H + L ( t ) (cid:1) u, ( t, z ) ∈ R × C ,u ( t ) | t = t = u ∈ e H s . (4.5)4.3. Verification of Assumption 1.1.
We now prove that L ( t ) satisfies the required properties.( i ) Let us check that L ∈ C b (cid:0) R , L ( e H s ) (cid:1) , with norm k L ( t ) k L ( e H s ) ≤ C s ǫ . First, by (4.1) it isequivalent to show that L ∈ C b (cid:0) R , L ( H s ) (cid:1) . Then, since e it ( H +1) ρ is unitary in H s , and by (4.2) (cid:13)(cid:13) e − it ( H +1) ρ Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1)(cid:13)(cid:13) H s = (cid:13)(cid:13) Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1)(cid:13)(cid:13) H s ≤ C kh z i s Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1) k L ( C ) . Next, by (3.1) and (3.2) kh z i s Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1) k L ( C ) ≤ C k W ( t ) k L ∞ ( C ) kh z i s e it ( H +1) ρ u k L ( C ) ≤ C k W ( t ) k L ∞ ( C ) k e it ( H +1) ρ u k H s = C k W ( t ) k L ∞ ( C ) k u k H s . Recall that W ( t ) = | R αt V | , where V ∈ E , then k W ( t ) k L ∞ ( C ) = k V k L ∞ ≤ Cǫ . Putting all theprevious estimates toghether, we obtain (cid:13)(cid:13) e − it ( H +1) ρ Π (cid:0) W ( t ) e it ( H +1) ρ u (cid:1)(cid:13)(cid:13) H s ≤ C s ǫ k u k H s , hence the announced bound. The time-continuity of L follows from the previous estimates to-gether with the continuity of the translations for the Lebesgue measure and the fact that e it e H ∈C b (cid:0) R , L ( e H s ) (cid:1) .( ii ) The symmetry of L , w.r.t. the scalar product of e H = E , is a consequence of the symmetryof Π, the conjugation by the unitary operator e it ( H +1) ρ and the fact that W is a real valuedfunction.( iii ) Let us check that (cid:2) e H, L ( t ) (cid:3) is e H τ -bounded. By Lemma C.1, H and Π commute, thus (cid:2) e H, L ( t ) (cid:3) e H − τ = (cid:2) ( H + 1) ρ , L ( t ) (cid:3) ( H + 1) − ρτ = e − it ( H +1) ρ Π (cid:2) ( H + 1) ρ , W ( t ) (cid:3) ( H + 1) − ρτ e it ( H +1) ρ . ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 11
Recall that Π is bounded in all the H s spaces, as well as the operators e − it ( H +1) ρ . • Case s = 0. Let us first prove that Π (cid:2) ( H + 1) ρ , W ( t ) (cid:3) ( H + 1) − ρτ : E −→ E is bounded,uniformly in t ∈ R . For that, we use the Weyl-H¨ormander pseudo-differential calculus (we referto [32, 25] or to [31, Chapter 3] for a review of this theory). Denote by z = x + ix and ξ = ξ + ξ .For m ∈ R , we define the symbol class S m by S m = n a ∈ C ∞ ( R ; C ) : (cid:12)(cid:12) ∂ α x ∂ α x ∂ β ξ ∂ β ξ a ( x , x , ξ , ξ ) (cid:12)(cid:12) ≤ C α,β h| z | + | ξ |i m − β − β , ∀ α, β ∈ N o , and for a ∈ S m , we define its Weyl-quantization by the formula a w ( x, D ) u ( x ) = 1(2 π ) Z R Z R e i ( x − y ) · ξ a ( x + y , ξ ) u ( y ) dydξ, u ∈ S ( R ) . First, using the functional calculus associated to the operator H , we obtain that ( H + 1) ρ is apseudo-differential operator with symbol in S ρ . By (4.3), W ( t ) ∈ S uniformly in t ∈ R , andtherefore the commutator (cid:2) ( H + 1) ρ , W ( t ) (cid:3) is a pseudo-differential operator with symbol in S ρ − ,which in turn implies that (cid:2) ( H +1) ρ , W ( t ) (cid:3) ( H +1) − ρτ is a pseudo-differential operator with symbolin S (because 2 ρ − − ρτ = 0), hence it is bounded. • The proof in the general case s ≥ iv ) From (4.4), a direct computation gives ∂ t L ( t ) = e − it e H Π( ∂ t W ( t )) e it e H − ie − it e H Π (cid:2) e H, W ( t ) (cid:3) e it e H = e − it e H Π( ∂ t W ( t )) e it e H − i (cid:2) e H, L ( t ) (cid:3) . (4.6)From the expression of W , we deduce that sup t ∈ R k ∂ ℓt W ( t ) k L ∞ ( C ) ≤ C ℓ for all ℓ ≥
0. In particular,the first term in the right hand side of (4.6) is bounded e H s −→ e H s . In item ( iii ) we have shownthat [ e H, L ( t ) (cid:3) : e H s −→ e H s − τ is bounded, uniformly in t ∈ R . As a consequence, for all s ≥ L ∈ C b (cid:0) R ; L ( e H s ; e H s − τ ) (cid:1) . The general case ℓ ≥ Proof of Theorem 1.2.
We are now ready to complete the proof of Theorem 1.2. Considerthe problem (4.5) and for convenience, assume that t = 0. By (4.4), the equation (4.5) is equivalentto ( i∂ t v = Π (cid:0) W ( t ) v (cid:1) , ( t, z ) ∈ R × C ,v (0 , · ) = u ∈ e H s = H ρs , with the change of unknown v = e it ( H +1) ρ u . As a consequence we can directly apply the results ofSection 2 (case δ = 0) to this model.( i ) The fact that k L ( t ) k L ( e H s ) ≤ C s ǫ has already been shown in the previous paragraph.( ii ) For all s ≥
0, the problem (4.5) is globally well-posed, in e H s by Theorem 2.1. The groupproperty of U is a consequence of uniqueness, and its unitarity follows from the conservation of the L norm.( iii ) The upper bound is given by Theorem 2.2, namely, for all t ∈ R k u ( t ) k e H s ≤ C kh z i ρs u ( t ) k L ( C ) ≤ C kh z i ρs u k L ( C ) h ǫt i ρs ≤ C k u k e H s h ǫt i ρs , where ρ = − τ ) .( iv ) Consider the fonction u ∈ ∩ k ≥ L ,k E = ∩ k ≥ e H k given by Theorem 2.3, (see paragraph 3.4)then k u k e H s ≤ C √ ǫ and k u ( t ) k e H s ≥ c kh z i ρs u ( t ) k L ( C ) ≥ c √ ǫ h ǫt i ρs ≥ c k u k e H s h ǫt i ρs , hence the result. Notice that the items ( ii ) and ( iii ) also directly follow from the general result [29, Theorem 1.5]. Appendix A. A non-perturbative & time-independent example
In this section, we give another example of linear Schr¨odinger operator which yields unboundeddynamics, and which meets the assumptions (H0)-(H3) of [29]. This example differs from the oneexhibited in Theorem 1.2 in two main aspects : • it is a non-perturbative example : it is not a lower order perturbation of a time-independentelliptic differential operator ; • it is time-independent.However, with a change of unknown, we can obtain a time-dependent perturbation of a constantcoefficient self-adjoint elliptic operator, see Remark A.3 below.This example is very simple, and that is why we decided to develop it here. Actually themechanism involved in the norm inflation is the same as in Theorem 1.2: it is a traveling wavemeasured in a weighted L space. Actually, our example is close to the one developed in [5,Appendix A], after change of variables.On L ( R ) we define the operator usual harmonic oscillator H = − ∂ x + x . For 0 ≤ τ <
1, we set ρ ( τ ) = − τ ) ∈ [1 / , ∞ ). We define the operator e H = ( H + 1) ρ ( τ ) and the scale of Hilbert spaces (cid:0) e H s (cid:1) s ≥ by e H s = (cid:8) u ∈ L ( C ) , e H s/ u ∈ L ( R ) (cid:9) , e H = L ( R ) , endowed with the natural norm k u k e H s := k e H s/ u k L ( R ) . By [36, Lemma 2.4], we have the followingequivalence of norms k u k e H s ≡ kh x i ρs u k L ( R ) + k ( − ∂ x ) ρs/ u k L ( R ) . (A.1)Now, for ǫ >
0, we consider the problem, ( i∂ t u = − iǫ∂ x u, ( t, x ) ∈ R × R ,u (0 , · ) = u ∈ e H s . (A.2)In this framework, we are able to prove the following result for the operator iǫ∂ x in the spaces e H s : Lemma A.1. ( i ) One has iǫ∂ x ∈ L ( e H s +1 /ρ , e H s ) (cid:1) for all s ≥ , and k iǫ∂ x k L ( e H s +1 /ρ , e H s ) (cid:1) ≤ C s ǫ. ( ii ) The operator iǫ∂ x is symmetric on e H /ρ w.r.t. the scalar product of e H , Z R v ( iǫ∂ x u ) dx = Z R u ( iǫ∂ x v ) dx, ∀ u, v ∈ e H /ρ . ( iii ) The operator iǫ∂ x is e H τ -bounded in the sense that [ iǫ∂ x , e H ] e H − τ ∈ L ( e H s ) for all s ≥ . Therefore the operator i∂ x satisfies the assumptions (H0)-(H3) of [29].On the other hand, we have the following elementary result: Proposition A.2.
Let s ≥ , then ( i ) The problem (A.2) is globally well-posed in e H s , and the solution is explicitly given by u ( t, x ) = u ( x − ǫt ) . ( ii ) The following bounds hold true : for all u ∈ e H s and for all t ∈ R c h ǫt i s − τ ) k u k e H s ≤ k u ( t ) k e H s ≤ C h ǫt i s − τ ) k u k e H s . (A.3)This result is directly obtained using (A.1) and the expression ρ ( τ ) = − τ ) . ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 13
Remark
A.3 . Notice that the function v ( t ) = e − it e H u ( t ) is solution to ( i∂ t v − e Hv = − iǫ (cid:0) e − it e H ∂ x e it e H (cid:1) v, ( t, x ) ∈ R × R ,v (0 , · ) = v = u ∈ e H s , and satisfies the conclusions of Lemma A.1 and the bounds (A.3). This yields an example inthe spirit of the one exhibited in Theorem 1.2, but in the present case, the perturbation is oforder 1 /ρ ∈ (0 ,
2] instead of being of order 0.
Proof of Lemma A.1.
Item ( i ) is a direct consequence of (A.1) and ( ii ) is elementary.( iii ) It is convenient to introduce the Sobolev space based on the harmonic oscillator ( s ≥ H s = (cid:8) u ∈ L ( R ) , H s/ u ∈ L ( R ) (cid:9) , H = L ( R ) . Thanks to the pseudo-differential calculus associated to H (see also paragraph 4.3), we first provethat for all r >
0, [ i∂ x , ( H + 1) r ]( H + 1) − r + ∈ L (cid:0) L ( R ) (cid:1) . (A.4)Here the symbol class S m reads S m = n a ∈ C ∞ ( R ; C ) : (cid:12)(cid:12) ∂ αx ∂ βξ a ( x, ξ ) (cid:12)(cid:12) ≤ C α,β h| x | + | ξ |i m − β , ∀ α, β ∈ N o . The symbol of [ i∂ x , ( H + 1) r ], modulo terms in S r − , is given by the formula − i (cid:8) ξ, ( x + ξ + 1) r (cid:9) = − i∂ ξ ( ξ ) ∂ x (cid:0) ( x + ξ + 1) r (cid:1) + i∂ x ( ξ ) ∂ ξ (cid:0) ( x + ξ + 1) r (cid:1) = − i rx ( x + ξ + 1) r − ∈ S r − . Since the symbol of ( H + 1) − r + belongs to S − r +1 , we deduce (A.4).Next, for ρ > s ≥ H + 1) s [ i∂ x , ( H + 1) ρ ]( H + 1) − ρ + ( H + 1) − s == − [ i∂ x , ( H + 1) s ]( H + 1) − s + + [ i∂ x , ( H + 1) s + ρ ]( H + 1) − ( s + ρ )+ , and by applying (A.4) twice, we deduce that for all s ≥ i∂ x , ( H + 1) ρ ]( H + 1) − ρ + ∈ L (cid:0) H s (cid:1) . (A.5)Finally recall that e H = ( H + 1) ρ , thus (A.5) is equivalent to[ i∂ x , e H ] e H − τ ∈ L (cid:0) e H s (cid:1) , since τ = 1 − / (2 ρ ), which was the claim. (cid:3) Appendix B. On the reducibility of the linear LLL equation
We state here a reducibility result for the linear LLL equation. It turns out that the abstractreducibility result obtained in [23] can be applied to this model, which is close in many aspects tothe usual 1D cubic quantum harmonic oscillator with time-dependent potential. We consider thelinear equation ( i∂ t u − δHu = ǫ Π( W ( tω, z ) u ) , ( t, z ) ∈ R × C ,u (0 , z ) = u ( z ) , (B.1)where δ = 0, where ǫ > ω ∈ [0 , π ) n is the frequency vector,for some given n ≥
1. Up to a rescaling, we can assume that δ = 1. We assume in the sequel thatthe potential W : T n × C −→ R T n := ( R / π Z ) n ( θ, z ) W ( θ, z ) , is analytic in θ on | Im θ | < τ for some τ >
0, and C in x, y (where z = x + iy ), and we supposemoreover that there exists γ > C > θ ∈ T n and z ∈ C | W ( θ, z ) | ≤ C h z i − γ , | ∂ jz ∂ ℓz W ( θ, z ) | ≤ C, (B.2)for any 0 ≤ j, ℓ ≤ ω = 0, all the solutions to (B.1) are almost periodic in time. This can be proved byconstructing a Hilbertian basis ( ψ k ) k ≥ of E composed of eigenfunctions of the operator u Hu + ǫ Π( W (0 , z ) u ), such that Hψ k + ǫ Π (cid:0) W (0 , z ) ψ k (cid:1) = λ k ψ k , k ≥ . Then (B.1) can be solved by u ( t, z ) = + ∞ X k =0 c k e − itλ k ψ k ( z ) , u ( z ) = + ∞ X k =0 c k ψ k ( z ) , which shows that any solution to (B.1) is an infinite superposition of periodic functions, hence itis an almost-periodic function in time.For ω = 0, the reducibility theory adresses the question if, by the means of a time quasi-periodictransformation, one can reduce to the previous case. It turns out that for (B.1), it is the case fora large set of values ω ∈ Λ ǫ : Theorem B.1.
Assume that W satisfies (B.2) . Then there exists ǫ such that for all ≤ ǫ < ǫ there exists a set Λ ǫ ⊂ [0 , π ) n of positive measure and asymptotically full measure: Meas (Λ ǫ ) → (2 π ) n as ǫ → , such that for all ω ∈ Λ ǫ , the linear equation (B.1) reduces, in E , to a linear equationwith constant coefficients. We refer to [23, Theorem 7.1] for a more precise statement, giving in particular more informationon the transformation.Assume that ( θ, z ) V ( θ, z ) is analytic in θ on | Im θ | < τ , that V ( θ, · ) ∈ E for all θ ∈ T n , andsatisfies, for some γ >
0, the bound | V ( θ, z ) | ≤ C h z i − γ uniformly in θ ∈ T n . Then, by Lemma C.2, W = | V | satisfies (B.2). Such a potential even satisfies | ∂ jz ∂ ℓz W ( θ, z ) | ≤ C for all k, ℓ ∈ N (withoutadditional assumptions on V ∈ E ).We also have the following result on the dynamics of the solutions of (B.1) : Corollary B.2.
Assume that W is C ∞ in x, y with all its derivatives bounded and satisfying (B.2) .Let s ≥ and u ∈ H s . Then there exists ǫ > so that for all < ǫ < ǫ and ω ∈ Λ ǫ , there existsa unique solution u ∈ C (cid:0) R ; H s (cid:1) of (B.1) so that u (0) = u . Moreover, u is almost-periodic in timeand we have the bounds (1 − ǫC ) k u k H s ≤ k u ( t ) k H s ≤ (1 + ǫC ) k u k H s , ∀ t ∈ R , for some C = C ( s, ω ) . The result of Theorem B.1 can also be formulated in term of the Floquet operator. Consider theFloquet Hamiltonian operator, defined on
E ⊗ L ( T n ) by K := i n X k =1 ω k ∂ θ k + H + ǫ Π (cid:0) W ( θ, z ) · (cid:1) , then we can state Such a Hilbertian basis exists, since u Hu + ǫ Π( W (0 , z ) u ) is a self-adjoint operator with compact resolventin E . ROWTH OF SOBOLEV NORMS FOR LINEAR SCHR ¨ODINGER OPERATORS 15
Corollary B.3.
Assume that W satisfies (B.2) . There exists ǫ > so that for all < ǫ < ǫ and ω ∈ Λ ǫ , the spectrum of the Floquet operator K is pure point. We refer to [23, Section 7], where similar results are proven for the 1D quantum harmonicoscillator.For the reducibility of the periodic Schr¨odinger equation, we refer to [15] and for the reducibilityof the quantum harmonic oscillator in any dimension to [21, 26, 5] and we refer to [2, 3, 4] forthe reducibility for 1- d operators with unbounded perturbations. For references on the theory ofFloquet operators, see [14, 34].Finally, let us mention that, concerning the nonlinear cubic LLL equation, the abstract KAMresult of [23] was applied in [18, Theorem 4.3] in order to show the existence of invariant torii,and that the result of [20] was applied to show an almost global existence result for the cubic LLLequation. We refer to [18, Section 4.2] for more details.The arguments of [23, Section 7] can be directly applied to the equation (B.1), and we addressthe reader to this latter paper for the proofs of the previous results. Let us just sketch the idea :we expand u and ¯ u on the basis given by the special Hermite functions u = X j ≥ c j ϕ j , u = X j ≥ c j ϕ j . Then equation (B.1) reads as an autonomous Hamiltonian system in an extended phase space ˙ c j = − i ( j + 1) c j − iǫ∂ ¯ c j Q ( θ, c, ¯ c ) j ≥ c j = 2 i ( j + 1)¯ c j + iǫ∂ c j Q ( θ, c, ¯ c ) j ≥ θ j = ω j j = 0 , . . . , n ˙ Y j = − ǫ∂ θ j Q ( θ, z, ¯ z ) j = 0 , . . . , n (B.3)where Q is a quadratic functional in ( c, ¯ c ) given by Q ( θ, c, ¯ c ) = Z C W ( θ, z ) (cid:16) X j ≥ c k ϕ k ( z ) (cid:17)(cid:16) X j ≥ c j ϕ j ( z ) (cid:17) dL ( z ) , (B.4)and the Hamiltonian of the system (B.3) is n X j =1 ω j Y j + 2 X j ≥ ( j + 1) c j c j + Q ( θ, c, ¯ c ) . Then, we can check that (B.4) satisfies the assumptions of [23, Theorem 7.1]. The dispersiveestimate k ϕ n k L ∞ ( C ) ≤ Cn − / satisfied by the ( ϕ n ) n ≥ , is the key ingredient which allows to followthe lines of [23, Section 7]. Appendix C. Some technical results
Lemma C.1.
The operators H and Π commute.Proof. Recall that [Π u ]( z ) = 1 π e − | z | Z C e wz − | w | u ( w ) dL ( w ) , and that H = − ∂ z ∂ z + | z | . On the one hand, by integration by parts (cid:0) Π ∂ z ∂ z u (cid:1) ( z ) = 1 π e − | z | Z C e wz − | w | ∂ w ∂ w u ( w ) dL ( w )= 1 π e − | z | Z C ∂ w ∂ w (cid:0) e wz − | w | (cid:1) u ( w ) dL ( w )= −
12 Π u ( z ) − z Π( wu )( z ) + 14 Π( | w | u )( z ) , thus Π Hu ( z ) = 2Π u ( z ) + 2 z Π( wu )( z ) . On the other hand ∂ z Π u ( z ) = − π z e − | z | Z C e wz − | w | u ( w ) dL ( w ) + 1 π e − | z | Z C e wz − | w | wu ( w ) dL ( w )= − z u ( z ) + Π (cid:0) zu (cid:1) ( z ) , then ∂ z ∂ z Π u ( z ) = −
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Lemma C.2.
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