Thierry Jecko

Jecko, Thierry

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use ofngueolvante auxundarylanck’sboundsing alassicalsetting,r smoothcultiesork. WeC. R. Acad. Sci. Paris, Ser. I 338 (2004) 545–548Partial Differential Equations/Mathematical PhysicsFrom classical to semiclassical non-trapping behaviourThierry JeckoIRMAR, Université de Rennes I, campus Beaulieu, 35042 Rennes cedex, FranceReceived 27 October 2003; accepted after revision 27 January 2004Presented by Gilles LebeauAbstractFor the semiclassical Schrödinger operator with smooth long-range potential, we prove in a new way, makingsemiclassical measures, that the boundary values of its resolvent at non-trapping energies are bounded by O(1/h), h beingthe semiclassical parameter.To cite this article: T. Jecko, C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméNon-capture : du classique au semi-classique. Pour l’opérateur de Schrödinger semi-classique avec potentiel lisse à loportée, on montre d’une manière nouvelle, au moyen de mesures semi-classiques, que les valeurs au bord de sa résénergies non-captives sont de taille O(1/h), oùh est le paramètre semi-classique.Pour citer cet article : T. Jecko, C. R. Acad.Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. IntroductionConcerning the Schrödinger operator with smooth long-range potential, it is well known that the bovalues of the resolvent at positive energies exist (cf. [2]). At the semiclassical level with respect to Pconstanth, it is known that these boundary values at energyλ > 0 are O(h−1) if and only if λ is non-trappingfor the associated classical flow. While the necessity of the non-trapping condition was proved in [9], theO(h−1) for the boundary values of the resolvent was derived from the non-trapping condition in [7,5] usemiclassical version of Mourre’s commutator method and in [8] (in greater generality) using a semicMourre estimate. For the latter result, an alternative approach was introduced by Burq in [1], in a generalbut for compactly supported pertubation of the Laplacian. Our purpose here is to adapt Burq’s approach folong-range potentials.Our main motivation concerns the corresponding result for matricial operators, for which there are diffito apply Mourre’s theory. It is thus interesting to investigate another strategy in a simple and close framewE-mail address: thierry.jecko@univ-rennes1.fr (T. Jecko).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2004.01.022546 T. Jecko / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 545–548togetherroth,rs,ing, forieldingfollow.owns ofalso think it is useful to present a rather short and elementary proof, based on pseudodifferential calculuswith the use of semiclassical measures.Let us now introduce some notation and the result. We denote by‖ · ‖ and〈· , ·〉 the natural norm and scalaproduct, respectively, in L2(Rd ;C) with d  1 and by	x the Laplacian inRd . Let V ∈ C∞(Rd;R), satisfying,for someρ > 0,∀α ∈Nd , ∀x ∈Rd, ∣∣∂αx V (x)∣∣=Oα(〈x〉−ρ−|α|), (1)where〈x〉 = (1+ |x|2)1/2. Let h ∈]0;h0], for someh0 > 0. The semiclassical Schrödinger operator with smolong-range potential is given byPh := −h2	x + V (x), acting in L2(Rd;C). It is well known thatPh is self-adjoint on the domainD of the Laplacian (see [2]). Denoting its resolvent byR(z) := (Ph − z)−1, with z in theresolvent set ofPh, we know from [2] that it has boundary valuesR(λ± i0), for λ ∈]0;+∞[, as bounded operatofrom L2s (Rd;C) to L2−s(Rd ;C), for s > 1/2. Here L2s (Rd;C) denotes the weighted L2 space of mesurableC-valued functionsf onRd such thatx → 〈x〉sf (x) belongs to L2(Rd;C). We denote byp(x, ξ) := |ξ |2+V (x),(x, ξ) ∈ T ∗Rd , the symbol ofPh and byφt the associated Hamilton flow onT ∗Rd . An energyλ is non-trappingfor p if∀(x, ξ) ∈ p−1(λ), limt→−∞∣∣φt(x, ξ)∣∣=+∞ and limt→+∞∣∣φt(x, ξ)∣∣=+∞. (2)Our goal is to give a new proof of the following result, already obtained in [7,5,8].Theorem 1.1. Under the previous assumptions, let I ⊂]0;+∞[ be a compact interval of non-trapping energiesfor p. Then, for small enough h0 and any s > 1/2, there exists Cs > 0 such that, uniformly for λ ∈ I and h ∈]0;h0],∥∥〈x〉−sR(λ± i0)〈x〉−s∥∥  Csh−1. (3)2. Proof of Theorem 1.1: semiclassical trappingWithout assuming the non-trapping condition, we first study the situation, called semiclassical trappwhich (3) is false. Then we show that this semiclassical trapping contradicts the non-trapping condition (2), ya proof of Theorem 1.1 by contradiction. The situation here is similar to that in [1], the strategy of which weHowever, new ingredients and new results appear in the present Note.Possibly after extraction of subsequences, the negation of (3) for some> 1/2 and forR(λ + i0) impliesthe existence of sequences(hn)n ∈]0;h0]N tending to zero,(fn)n of nonzero functions of the domainD, and(zn)n ∈CN with (zn)→ λ > 0 and 0 (zn)/hn→ r  0, such that‖〈x〉−sfn‖ = 1 and‖〈x〉s (Phn − zn)fn‖ =o(hn). We can also assume that the previous bounded sequence(〈x〉−sfn)n in L2 is pure and we denote byµs itssemiclassical measure. It is a finite, non-negative Radon measure onT ∗Rd , satisfying, for anya ∈C∞0 (T ∗Rd),limn→∞〈awhn〈x〉−sfn, 〈x〉−sfn〉= ∫T ∗Rda(x, ξ)µs(dx dξ)=: µs(a). (4)Hereawh the Weylh-quantization of the symbola. Let us mention that we shall use at many places well knproperties of semiclassical measures, of the Weylh-pseudodifferential calculus, and of the functional calculuHelffer–Sjöstrand. For details, we refer to [3,4,6].Take a ∈ C∞0 (T ∗Rd) with support disjoint fromp−1(λ), the sequencea〈x〉−2s(p − zn)−1 ∈ C∞0 (T ∗Rd ) isbounded, since(zn)→ λ. Therefore〈awhn〈x〉−sfn, 〈x〉−sfn〉 tends to 0 asn→∞, since‖〈x〉s (Phn − zn)fn‖ =o(hn). This means thatµs is supported inp−1(λ).According to [1], we expect that the Poisson bracket (in the distributional sense){p, 〈x〉2sµs} equalsr〈x〉2sµs .But it turns out thatr = 0 in our case. Indeed,T. Jecko / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 545–548 547rearo(hn)=〈〈x〉s (Phn − zn)fn, 〈x〉−sfn〉= 〈(Phn − zn)fn, fn〉= 〈fn, (Phn − zn)fn〉= 〈fn, (Phn − zn)fn〉+ 〈fn,2i(zn)fn〉= o(hn)+ 〈fn,2i(zn)fn〉,yielding‖fn‖2(zn)/hn = o(1). If lim (zn)/hn = r = 0 then‖fn‖2→ 0. Sinces  0, 1= ‖〈x〉−sfn‖2  ‖fn‖2.We arrive at a contradiction. Thereforer = 0 and we get the following result.Proposition 2.1. The measure 〈x〉2sµs is invariant under φt , that is {p, 〈x〉2sµs} = 0.Proof. We follow [1]. For anya ∈C∞0 (T ∗Rd),〈ih−1n[Phn, awhn]fn,fn〉= 〈ih−1n awhnfn, (Phn − zn)fn〉− 〈ih−1n awhn(Phn − zn)fn, fn〉= (2(zn)/hn)〈awhnfn, fn〉+ o(1)= o(1), (5)〈ih−1n[Phn, awhn]fn,fn〉= 〈{p,a}whnfn, fn〉+O(hn)= µs(〈x〉2s{p,a})+ o(1). ✷Looking for other properties ofµs , we learn from [4] that, iflimR→+∞ lim supn→∞∫|x|>R〈x〉−2sf 2n dx = 0, (6)then the total mass ofµs equals limn→∞ ‖〈x〉−sfn‖2. In particular,µs is nonzero. From (1), we see that theexistsc > 0 such that{p,x · ξ} c onp−1(λ), for |x| large enough. ‘Quantizing’ this fact carefully, in a similway as in [8], we shall show a stronger version of (6), namely (10).Let R > 0 and let1{|x|>R} be the characteristic function of the set{(x, ξ) ∈ T ∗Rd; |x| > R} =: T ∗Rd \ B∗R .Let χ0 ∈ C∞0 (R;R) such that 0 χ0  1, χ0 = 0 on ]−∞;1/3[, and χ0 = 1 on ]2/3;+∞[. For R largeenough and anyδ ∈]0;min(1;ρ)[, whereρ appeared in (1), we can define, nearp−1(λ) \ B∗R , the symbola∞(x, ξ) := x̂ · ξ̂ − |x|−δ(χ0(x̂ · ξ̂ )−χ0(−x̂ · ξ̂ )), wherex̂ := x/|x|. It is easy to show that, nearp−1(λ) \B∗R , a∞is a smooth, bounded function such that, form= 0,∀(α,β) ∈N2d, ∃Cαβ > 0; ∀(x, ξ) ∈ p−1(λ) \B∗R,∣∣∂αx ∂βξ a∞(x, ξ)∣∣  Cαβ〈x〉−m−|α|. (7)Let τ ∈ C∞0 (R;R) with 0  τ  1, τ = 1 on [−R;R], suppτ ⊂ [−R − 1;R + 1], and letχ(x) := τ (|x|). ForRlarge enough, there is, thanks to (1), somec > 0 such that{p,a∞}  2c〈x〉−1−δ and{p, (1− χ)2}a∞  0 nearp−1(λ) \B∗R . Let θ ∈C∞0 (R;R+) with support sufficiently close toλ. By the Gårding inequality (cf. [3]),θ(Phn)({p, (1− χ)2}a∞)whnθ(Phn)  〈x〉−m θ(Phn)Õ(hn)θ(Phn)〈x〉−m, (8)for any m ∈ N, whereÕm(hn) denotes a bounded operator, the norm of which isOm(hn). By the h-pseudo-differential calculus, for symbols satisfying (7) withm= 1,θ(Phn)(1− χ){p,a∞}whn(1− χ)θ(Phn)  c〈x〉−(1+δ)/2θ(Phn)(1− χ)2 θ(Phn) 〈x〉−(1+δ)/2+ 〈x〉−1 θ(Phn)Õ(hn)θ(Phn)〈x〉−1. (9)Since ih−1n [Phn, ((1−χ)2a∞)whn ] = {p, (1−χ)2a∞}whn+〈x〉−1Õ(hn)〈x〉−1 and‖χ̃θ(Phn)〈x〉s−1 〈x〉−sfn‖ =O(1)for χ̃ ∈ C∞0 (Rd), we obtain, usingδ  1, (8), and (9),〈ih−1n[Phn,((1− χ)2a∞)whn]θ(Phn)fn, θ(Phn)fn〉 (c/2)∥∥(1− χ) θ(Phn)〈x〉−(1+δ)/2fn∥∥2+O(hn).Sincea∞ is a bounded symbol, the l.h.s. of the last inequality tends to zero as in (5), yieldinglim∥∥1{|x|>R}θ(Phn)〈x〉−(1+δ)/2fn∥∥2= 0. (10)n→∞548 T. Jecko / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 545–548hatsionsometry,06 (1988)46 (1987).Notice that this and the energy localization of thefn show that(〈x〉−(1+δ)/2fn)n is bounded in L2. It is also pureand its semiclassical measureµ(1+δ)/2 satisfies, fora ∈ C∞0 (T ∗Rd), µ(1+δ)/2(a)= µs(〈x〉2s−1−δa). The limit (10)implies thatµ(1+δ)/2 is supported inB∗R . Thusµs have compact support included inp−1(λ)∩B∗R and (6) holds truefor the sequence(〈x〉−(1+δ)/2fn)n. Therefore, this sequence converges to the total mass ofµ(1+δ)/2. Sinces > 1/2,we can chooseδ  s, yielding‖〈x〉−(1+δ)/2fn‖2  ‖〈x〉−sfn‖2= 1, for all n. This shows thatµs is nonzero.Now, assume thatλ is a non-trapping energy. Sinceµs has compact support inp−1(λ) ∩ B∗R , we can findg ∈ C∞0 (T ∗Rd ) with g = 1 onp−1(λ) ∩B∗R . By the non-trapping condition (2),a(x, ξ) := −∫ +∞0 g ◦ φt (x, ξ)dtis a well-defined, smooth function nearp−1(λ) such that{p,a} = g. Sinceµs has compact support,µs(〈x〉2s )=µs(〈x〉2s{p,a})= 0, by Proposition 2.1, leading to a contradiction. This ends the proof of Theorem 1.1.What would happen, ifs  1/2? If λ is non-trapping,µs = 0 and the previous arguments show tlimn→∞ ‖〈x〉−(1+δ)/2fn‖2 = 0 which does not contradict a priori‖〈x〉−sfn‖ = 1. This appears forV = 0, forwhich eachλ > 0 is non-trapping, and fors ∈ [0;1/2[ : givenk ∈Rd \ {0}, χ ∈C∞0 (]1;2[;R) with∫Rχ2= 1, anddenoting bymd the Lebesgue measure of the(d − 1)-dimensional unit sphere, takefn(x) := 1√mdeih−1n k·x〈x〉s |x|(1−d)/2 1√nχ(|x|/n).AcknowledgementsThe author thanks M. Klein for hospitality at the university of Potsdam (Germany) and for fruitful discusand advice.References[1] N. Burq, Semiclassical estimates for the resolvent in non trapping geometries, Int. Math. Res. Notices 5 (2002) 221–241.[2] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global GeSpringer-Verlag, 1987.[3] J. Derezínski, C. Gérard, Scattering Theory of Classical and QuantumN -Particle Systems, Springer-Verlag, 1997.[4] P. Gérard, E. Leichtnam, Ergodic properties of eigenfunctions for the Dirichlet problem, Duke Math. J. 71 (2) (1993).[5] C. Gérard, A. Martinez, Principe d’absorption limite pour des opérateurs de Schrödinger à longue portée, C. R. Acad. Sci. Paris 3121–123.[6] D. Robert, Autour de l’approximation semi-classique, Birkhäuser, 1987.[7] D. Robert, H. Tamura, Semiclassical estimates for resolvents and asymptotics for total cross-section, Ann. Inst. H. Poincaré415–442.[8] A. Vasy, M. Zworski, Semiclassical estimates in asymptotically Euclidean scattering, Comm. Math. Phys. 212 (1) (2000) 205–217[9] X.P. Wang, Semiclassical resolvent estimates forN -body Schrödinger operators, J. Funct. Anal. 97 (1991) 466–483.

From classical to semiclassical non-trapping behaviour

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From classical to semiclassical non-trapping behaviour

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