6 pagesInternational audienceWe prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one

Thomann, Laurent

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A remark on the Schro¨dinger smoothing effectLaurent ThomannTo cite this version:Laurent Thomann. A remark on the Schro¨dinger smoothing effect. Asymptotic Analysis, IOSPress, 2010, 69 (1-2), pp.117–123. <hal-00422424v2>HAL Id: hal-00422424https://hal.archives-ouvertes.fr/hal-00422424v2Submitted on 25 Dec 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A REMARK ON THE SCHRO¨DINGER SMOOTHINGEFFECTbyLaurent ThomannAbstract. — We prove the equivalence between the smoothing effect for aSchro¨dinger operator and the decay of the associate spectral projectors. Wegive two applications to the Schro¨dinger operator in dimension one.Re´sume´. — On donne une caracte´risation de l’effet re´gularisant pour unope´rateur de Schro¨dinger par la de´croissance de ses projecteurs spectraux. Onen de´duit deux applications a` l’ope´rateur de Schro¨dinger en dimension un.1. IntroductionLet d ≥ 1, and consider the linear Schro¨dinger equation(1.1){i∂tu = Hu, (t, x) ∈ R× Rd,u(0, x) = f(x) ∈ L2(Rd),where H is a self-adjoint operator on L2(Rd).By the Hille-Yoshida theorem, the equation (1.1) admits a unique solutionu(t) = e−itHf ∈ C(R;L2(Rd)). Under suitable conditions on H, this solutionenjoys a local gain of regularity (in the space variable) : For all T > 0 thereexists C > 0 so that( ∫ T0‖Ψ(x) 〈H〉γ2 e−itHf‖2L2(Rd)d t) 12 ≤ C‖f‖L2(Rd),for some weight Ψ and exponent γ > 0.2000 Mathematics Subject Classification. — 35-XX.Key words and phrases. — Schro¨dinger equation, potential, smoothing effect.The author was supported in part by the grant ANR-07-BLAN-0250.2 LAURENT THOMANNThis phenomenon has been discovered by T. Kato [7] in the context of KdVequations. For the Schro¨dinger equation in the case H = −∆, it has beenproved by P. Constantin- J.-C. Saut [2], P. Sjo¨lin [11], L. Vega [12] andK. Yajima [13]. The variable coefficients case has been obtained by S. Do¨ı[3, 4, 5, 6].The more general results are due to L. Robbiano-C. Zuily [9, 10] for equationswith obstacles and potentials.Let H be a self adjoint operator on L2(Rd). It can be represented thanks tothe spectral measure byH =∫λdEλ.In the sequel we moreover assume that H ≥ 0. For N ≥ 0, we can then definethe spectral projector PN associated to H by(1.2) PN = 1[N,N+1[(H) =∫1[N,N+1[(λ)dEλ.Our main result is a characterisation of the smoothing effect by the decay ofthe spectral projectors. Denote by 〈H〉 = (1 +H2) 12 .Theorem 1.1 (Smoothing effect vs. decay). —Let γ > 0 and Ψ ∈ C(Rd,R). Then the following conditions are equivalent(i) There exists C1 > 0 so that for all f ∈ L2(Rd)(1.3)(∫ 2pi0‖Ψ(x) 〈H〉γ2 e−itHf‖2L2(Rd)d t) 12 ≤ C1‖f‖L2(Rd).(ii) There exists C2 > 0 so that for all N ≥ 1 and f ∈ L2(Rd)(1.4) ‖ΨPNf‖L2(Rd) ≤ C2N−γ2 ‖PNf‖L2(Rd).The interesting point is that we can take the same function Ψ and exponentγ > 0 in both statements (1.3) and (1.4).By the works cited in the introduction, in the case H = −∆ on Rd, (1.3) isknown to hold with γ = 12 and Ψ(x) = 〈x〉−12−ν , for any ν > 0.There is also a class of operators H on L2(Rd) for which (1.3) is well under-stood. Let V ∈ C∞(R,R+), and assume that for |x| large enough V (x) ≥C〈x〉k and that for any j ∈ Nd, there exists Cj > 0 so that |∂jxV (x)| ≤Cj〈x〉k−|j|. Then L. Robbiano and C. Zuily [9] show that the smoothing effect(1.3) holds for the operator H = −∆+V (x), with γ = 1k and Ψ(x) = 〈x〉−12−ν ,for any ν > 0.A REMARK ON THE SCHRO¨DINGER SMOOTHING EFFECT 3We now turn to the case of dimension d = 1, and consider the operatorH = −∆+ V (x). We make the following assumption on VAssumption 1. — We suppose that V ∈ C∞(R,R+), and that there exist2 < m ≤ k so that for |x| large enough(i) There exists C > 1 so that1C〈x〉k ≤ V (x) ≤ C〈x〉k.(ii) V ′′(x) > 0 and xV ′(x) ≥ mV (x) > 0(iii) For any j ∈ N, there exists Cj > 0 so that |∂jxV (x)| ≤ Cj〈x〉k−|j|.For instance V (x) = 〈x〉k with k > 2 satisfies Assumption 1.It is well known that under Assumption 1, the operator H has a self-adjointextension on L2(R) (still denoted by H) and has eigenfunctions(en)n≥1 whichform an Hilbertian basis of L2(R) and satisfyHen = λ2nen, n ≥ 1,with λn −→ +∞, when n −→ +∞.For N ∈ N the spectral projector PN defined in (1.2) can be written in thefollowing way. Let f =∑n≥1αnen ∈ L2(R), thenPNf =∑N≤λ2n<N+1αnen.Observe that we then have f =∑N≥0PNf.For such a potential, we can remove the spectral projectors in (1.4) and deducefrom Theorem 1.1Corollary 1.2. —Let γ > 0 and Ψ ∈ C(R,R). Let H = ∆ + V (x) so that V (x) = x2 or V (x)satisfies Assumption 1. Then the following conditions are equivalent(i) There exists C1 > 0 so that for all f ∈ L2(R)(1.5)(∫ 2pi0‖Ψ(x) 〈H〉γ2 e−itHf‖2L2(R)d t) 12 ≤ C1‖f‖L2(R).(ii) There exists C2 > 0 so that for all n ≥ 1(1.6) ‖Ψ en‖L2(R) ≤ C2λ−γn , ∀n ≥ 1.The statements (1.5) and (1.6) were obtained by K. Yajima & G. Zhang in[16] when Ψ is the indicator of a compact K ⊂ R and with γ = 1k .4 LAURENT THOMANNThe statement (1.5) holds for Ψ(x) = 〈x〉− 12−ν , by [9], but as far as we know,the bound (1.6) with this Ψ was unknown.With Theorem 1.1 we are also able to prove the following smoothing effectfor the usual Laplacian ∆ on R.Proposition 1.3. — Let Ψ ∈ L2(R). Then there exists C > 0 so that for allf ∈ L2(R)( ∫ 2pi0‖Ψ(x) 〈∆〉 14 e−it∆f‖2L2(R)d t) 12 ≤ C‖Ψ‖L2(R)‖f‖L2(R).From the works cited in the introduction, we have(∫R‖Ψ(x) 〈∆〉 14 e−it∆f‖2L2(R)d t) 12 ≤ C‖f‖L2(R),for Ψ(x) = 〈x〉 12−ν , for any ν > 0. Hence Proposition 1.3 shows that we canextend the class of the weights, but we are only able to prove local integrabilityin time.Notation. — We use the notation a . b if there exists a universal constantC > 0 so that a ≤ Cb.2. Proof of the resultsWe define the self adjoint operator A = [H] (entire part of H) byA =∫[λ]dEλ.Notice that we immediately have that A−H is bounded on L2(Rd).The first step in the proof of Theorem 1.1 is to show that we can replace e−itHby e−itA in (1.3)Lemma 2.1. — Let γ > 0 and Ψ ∈ C(Rd,R). Then the following conditionsare equivalent(i) There exists C1 > 0 so that for all f ∈ L2(Rd)(2.1)(∫ 2pi0‖Ψ(x) 〈H〉γ2 e−itAf‖2L2(Rd)d t) 12 ≤ C1‖f‖L2(Rd).(ii) There exists C2 > 0 so that for all f ∈ L2(Rd)(2.2)(∫ 2pi0‖Ψ(x) 〈H〉γ2 e−itHf‖2L2(Rd)d t) 12 ≤ C2‖f‖L2(Rd).A REMARK ON THE SCHRO¨DINGER SMOOTHING EFFECT 5Proof. — We assume (2.1) and we prove (2.2). Let f ∈ L2(Rd) and Definev = e−itHf . This function solves the problem(i∂t −A)v = (H −A)v, v(0, x) = f(x).Then by the Duhamel formulae−itHf = v = e−itAf − i∫ t0e−i(t−s)A(H −A)v ds= e−itAf − i∫ 2pi01{s<t}e−i(t−s)A(H −A)v ds.Therefore by (2.1) and Minkowski‖Ψ 〈H〉γ2 e−itHv‖L22piL2 . ‖Ψ 〈H〉γ2 e−itAv‖L22piL2+∫ 2pi0‖Ψ 〈H〉γ2 1{s<t}e−i(t−s)A(H −A)v‖L2tL2x ds. ‖f‖L2 +∫ 2pi0‖(H −A)v‖L2 ds.(2.3)Now use that the operator (H − A) : L2(Rd) → L2(Rd) is bounded, and by(2.3) we obtain‖Ψ 〈H〉γ2 e−itHv‖L22piL2 . ‖f‖L2 ,which is (2.2).The proof of the converse implication is similar.Proof of Theorem 1.1. — The proof is based on Fourier analysis in time. Thisidea comes from [8] and has also been used in [16], but this proof was inspiredby [1].(i) =⇒ (ii) : To prove this implication, we use the characterisation (2.1).From (1.2) and the definition of A, e−itAPNf = e−itNPNf . Hence it sufficesto replace f with PNf in (1.3) and (1.4) follows.(ii) =⇒ (i) : Again we will use Lemma 2.1. We assume (2.2) and we firstprove that(2.4) ‖Ψ 〈A〉γ2 e−itAf‖L2(0,2pi;L2(Rd)) . ‖f‖L2(Rd).Write f =∑N≥0 PNf , thenΨ 〈A〉γ2 e−itAf =∑N≥0e−iNt〈N〉γ2Ψ PNf.6 LAURENT THOMANNNow by Parseval in time‖Ψ 〈A〉γ2 e−itAf‖2L2(0,2pi) .∑N≥0〈N〉γ |Ψ PNf |2,and by integration in the space variable and (1.4)‖Ψ 〈A〉γ2 e−itAf‖2L2(0,2pi;L2(Rd)) .∑N≥0〈N〉γ‖Ψ PNf‖2L2(Rd).∑N≥0‖PNf‖2L2(Rd) = ‖f‖2L2(Rd),which yields (2.4).Now since the operator 〈A〉−γ/2〈H〉γ/2 is bounded on L2 and commutes withe−itA, we have by (2.4)‖Ψ 〈H〉γ2 e−itAf‖L2(0,2pi;L2(Rd)) == ‖Ψ 〈A〉γ2 e−itA(〈A〉− γ2 〈H〉γ2 f)‖L2(0,2pi;L2(Rd)). ‖〈A〉− γ2 〈H〉γ2 f‖L2(Rd). ‖f‖L2(Rd),which is (2.1).Proof of Corollary 1.2. — Let V satisfy Assumption 1. Then by [14, Lemma3.3] there exists C > 0 such that|λ2n+1 − λ2n| ≥ Cλ1− 2mn ,for n large enough. This implies that [λ2n] < [λ2n+1] for n large enough, becausem > 2 and λn −→ +∞. As a consequencePNf = αnen, with n so that N ≤ λ2n < N + 1,and this yields the result.We now consider V (x) = x2. In this case, the eigenvalues are the integersλ2n = 2n+ 1, and the claim follows.Remark 2.2. — With this time Fourier analysis, we can prove the followingsmoothing estimate for H which satisfies Assumption 1‖〈H〉 θ(q,k)2 e−itHf‖Lp(R;L2(0,T )) . ‖f‖L2(R),A REMARK ON THE SCHRO¨DINGER SMOOTHING EFFECT 7where θ is defined byθ(q, k) =2k (12 − 1q ) if 2 ≤ q < 4,12k − η for any η > 0 if q = 4,12 − 23 (1− 1q )(1− 1k ) if 4 < q <∞,4−k6k if q =∞.This was done in [16] with a slightly different formulation.Proof of Proposition 1.3. — By Theorem 1.1, we have to prove that the op-erator T defined byTf(x) = N14Ψ(x)1[N,N+1[(−∆)f(x),is continuous from L2(R) to L2(R) with norm independent of N ≥ 1. By theusual TT ∗ argument, it is enough to show the result for TT ∗.The kernel of T is K(x, y) = N14Ψ(x)FN (x− y) where(2.5) FN (u) =12pi∫eiuξ1[√N,√N+1[(|ξ|)dξ = 4cos(DNu)sin(CNu)u,with CN = (√N + 1−√N)/2 and DN = (√N + 1 +√N)/2.The kernel of TT ∗ is given byΛ(x, z) =∫K(x, y)K(z, y)dy,and by Parseval and (2.5)Λ(x, z) = N12Ψ(x)Ψ(z)∫FN (x− y)FN (z − y)dy=14N12Ψ(x)Ψ(z)∫ei(x−z)ξ1[√N,√N+1[(|ξ|)dξ= piN12Ψ(x)Ψ(z) cos(DN (x− z))sin(CN (x− z))x− z .Now, since CN . 1/√N and | sin(x)| ≤ |x|, we deduce that |Λ(x, z)| ≤C|Ψ(x)||Ψ(z)| (independent of N ≥ 1), and TT ∗ is continuous for Ψ ∈ L2(R).Acknowledgements. — The author would like to thank D. Robert for manyenriching discussions and the anonymous referee for valuable suggestionswhich improved this paper.8 LAURENT THOMANNReferences[1] N. Burq, G. Lebeau and F. Planchon. Global existence for energy critical wavesin 3-D domains. Journal of the A.M.S., (21) 831–845, 2008.[2] P. Constantin and J.-C. Saut. Local smoothing properties of dispersive equations.Journal American Mathematical Society (1988) 413–439.[3] S. Do¨ı. Smoothing effects of Schro¨dinger evolution group on Riemannian mani-folds. Duke Math. J. 82 (1996) 679–706.[4] S. Do¨ı. Smoothing effects for Schro¨dinger evolution equation and global behaviorof geodesic flow. Math. Ann. 318 (2000) 355–389.[5] S. Do¨ı. Remarks on the Cauchy problem for Schro¨dinger type equations. Comm.in PDE, 21 (1996) 163–178.[6] S. Do¨ı. Smoothness of solutions for Schro¨dinger equations with unbounded poten-tial. Publ. Res. Inst. Math. Sci 41 (2005), 1, 175–221.[7] T. Kato. On nonlinear Schro¨dinger equations, II. Hs-solutions and unconditionalwell-posedness, J. Anal. Math., 87 (1995) 281–306.[8] G. Mockenhaupt, A. Seeger and C. Sogge. Local smoothing of Fourier integraloperators and Carleson-Sjo¨lin estimates. J. Amer. Math. Soc., 6(1):65–130, 1993.[9] L. Robbiano, and C. Zuily. Remark on the Kato smoothing effect for Schro¨dingerequation with superquadratic potentials. Comm. Partial Differential Equations, 33(2008), no. 4-6, 718–727.[10] L. Robbiano, and C. Zuily. The Kato Smoothing Effect for Schro¨dinger equationswith unbounded potentials in exterior domains. Int. Math. Res. Notices , 15 (2009),1636–1698.[11] P. Sjo¨lin. Regularity of solution to the Schro¨dinger equation. Duke Math. J. 55(1987) 699–715.[12] L. Vega. Schro¨dinger equations, pointwise convergence to the initial data. Proc.Amer. Math. Soc. 102 (1988) 874–878.[13] K. Yajima. On smoothing property of Schro¨dinger propagator. Lectures notes inMath. 1450 Springer Verlag (1990) 20–35.[14] K. Yajima. Smoothness and non-smoothness of the fundamental solution of timedependent Schro¨dinger equations. Comm. Math. Phys. 180 (1996) no. 3, 605–629.[15] K. Yajima, and G. Zhang. Local smoothing property and Strichartz inequalityfor Schro¨dinger equations with potentials superquadratic at infinity. J. DifferentialEquations (2004), no. 1, 81–110.[16] K. Yajima, and G. Zhang. Smoothing property for Schro¨dinger equations withpotential superquadratic at infinity. Comm. Math. Phys. 221 (2001), no. 3, 573–590.Laurent Thomann, Universite´ de Nantes, Laboratoire de Mathe´matiquesJ. Leray, UMR CNRS 6629, 2, rue de la Houssinie`re, F-44322 Nantes Cedex 03, France. • E-mail : laurent.thomann@univ-nantes.frUrl : http://www.math.sciences.univ-nantes.fr/∼thomann/

A remark on the Schrödinger smoothing effect

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A remark on the Schrödinger smoothing effect

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