Low frequency asymptotics and local energy decay for the Schr{\"o}dinger equation

We prove low frequency resolvent estimates and local energy decay for the Schr{\"o}dinger equation in an asymptotically Euclidean setting. More precisely, we go beyond the optimal estimates by comparing the resolvent of the perturbed Schr{\"o}dinger operator with the resolvent of the free Laplacian. This gives the leading term for the developpement of this resolvent when the spectral parameter is close to 0. For this, we show in particular how we can apply the usual commutators method for generalized resolvents and simultaneously for different operators. Finally, we deduce similar results for the large time asymptotics of the corresponding evolution problem.


Introduction and statement of the main results
Let d ě 2. We consider on R d the Schrödinger equation where f P L 2 and P is a general Laplace operator.More precisely we set P " ´1 wpxq div Gpxq∇, (1.2) where wpxq and the symmetric matrix Gpxq are smooth and uniformly positive functions: there exists C ě 1 such that for all x P R d and ξ P R d we have We assume that P is associated to a long range perturbation of the flat metric.This means that Gpxq and wpxq are long range perturbations of Id and 1, respectively, in the sense that for some ρ 0 Ps0, 1s there exist constants C α ą 0, α P N d , such that for all x P R d , ˇˇB α pGpxq ´Idq ˇˇ`ˇˇB α pwpxq ´1q ˇˇď C α x ´ρ0 ´|α| . (1.3) Here and everywhere below we use the standard notation x " p1 `|x| 2 q 1 2 .We also denote by ∆ G the Laplace operator in divergence form corresponding to G: This definition of P includes in particular the cases of the free Laplacian, a Laplacian in divergence form, or a Laplace-Beltrami operator.We recall that the Laplace-Belbrami operator associated to a metric g " pg j,k q 1ďj,kďd is given by where |gpxq| " |detpgpxqq| and pg j,k pxqq 1ďj,kďd " gpxq ´1.Then P g is of the form (1.2) with w " |g| 1 2 and G " |g| 1 2 g ´1.
2010 Mathematics Subject Classification.47N50, 47A10, 35B40, 47B44, 35J05. 1 After a Fourier transform with respect to time, (1.1) can be rewritten as a frequency dependent (stationary) problem.In this paper, we are mainly interested in the contribution of low frequencies.More precisely, we study the behavior of the corresponding resolvent and its powers when the spectral parameter approaches 0.Then, using the already known results for the contribution of high frequencies, we will discuss the large time behavior of the solution of (1.1).
The operator P is defined on L 2 with domain H 2 .Its spectrum is the set R `of nonnegative real numbers.We are interested in the properties of the resolvent pP ´ζq ´1 (and its powers) when ζ is close to R `.The limiting absorption principle (limit of the resolvent when ζ goes to some λ ą 0) is an important topic in mathematical physics and is now well understood.In particular, it is known that if K is a compact subset of C ˚, then for n P N ˚and δ ą n ´1 2 the operator x ´δ pP ´ζq ´n x ´δ is uniformly bounded in LpL 2 q for ζ P KzR `.From this result, we can deduce that the contribution of a compact interval of positive frequencies for the time dependant problem decays faster than any negative power of time in suitable weighted L 2 -spaces.
The contribution of high frequencies for (1.1) depends on the properties of pP ´ζq ´n for ζ large (Repζq " 1 and 0 ă Impζq ! 1).These properties depend themselves on the geometry of the problem, and more precisely on the classical trajectories of the corresponding Hamiltonian problem.We always have as much decay for the solution of (1.1) as we wish if we allow a loss of regularity for the initial data.This decay is in fact uniform in weighted L 2spaces under the usual non-trapping condition.We denote by φ t the geodesic flow corresponding to the metric G ´1 on R 2d » T ˚Rd .For px 0 , ξ 0 q P R 2d and t P R we set φ t px 0 , ξ 0 q " pxpt, x 0 , ξ 0 q, ξpt, x 0 , ξ 0 qq.Then we have non-trapping if all the classical trajectories escape to infinity: @px 0 , ξ 0 q P R d ˆpR d z t0uq, |xpt, x 0 , ξ 0 q| Ý ÝÝÝ Ñ tÑ˘8 `8. (1.4) We set C `" tζ P C : Impζq ą 0u , D " tζ P C : |ζ| ď 1u , D `" D X C `.
Under the assumption (1.4), it is known that for n P N ˚and δ ą n ´1 2 there exists c ą 0 such that for ζ P CzpR `Y Dq we have . (1.5) The proof is based on semiclassical analysis.We refer for instance to [RT87] for a Schrödinger operator with a potential, to [Bur02] for a general compactly supported perturbation of the Laplacian in an exterior domain and to [Bou11a] for a long range perturbation of the flat metric.
The analysis of low frequencies is more recent.We first recall that given R ą 0 the behavior of the localized resolvent for the free Laplacian at ζ P CzR `is given by (1.6) Estimates of the resolvent near 0 for a long range perturbation of the free Laplacien have first been proved in [Bou11b] (operator in divergence form), [BH10] (Laplace-Beltrami operator) and [Bou11a] (estimates for the powers of the resolvent).Earlier papers also considered the limiting absorption principle at zero energy in some particular settings (see for instance [Wan06,DS09] and references therein).For a similar result in a non-selfadjoint setting we also refer to [KR17], and in a more general geometrical setting we mention [GH08, GH09,GHS13] and [BR15].
The optimal estimates for these powers have finally been proved in the recent paper [BB21].More precisely, it is proved that the estimates for the resolvent of the Schrödinger operator P are the same as for the free Laplacian in (1.6).
In this paper we go beyond this optimal estimate and give the asymptotic profile of pP ´ζq ´1 at the limit ζ Ñ 0, in the sense that the difference between the resolvent and the profile is smaller than the resolvent or the profile themselves.
Such asymptotic expansions of the resolvent at the low frequency limit have already been studied for a Schrödinger operator with potential.We refer for instance to [JK79].We also mention the more recent papers [Wan20] and [Aaf21] for complex-valued potentials.The difficulty in these cases is that one might have an eigenvalue or a resonance at the bottom of the spectrum, which gives a singularity for the resolvent.This is why these results require much stronger decay assumption on the potential.
We already know that the size of the powers of the resolvent for the Schrödinger operator is the same as for the free Laplacian P 0 " ´∆.We prove that, at the first order, they are actually given by the powers of this model operator modified by the factor w.More precisely, our main result is the following.
Theorem 1.1.Let ρ 1 P r0, ρ 0 r, n P N ˚and δ ą n `1 2 .There exists C ą 0 such that for ζ P DzR `we have This proves that for ζ close to 0 the difference between pP ´ζq ´n and pP 0 ´ζq ´nw is smaller that pP 0 ´ζq ´nw (see (1.6)).We deduce in particular that pP ´ζq ´n behaves in weighted spaces exactly as pP 0 ´ζq ´nw at the low frequency limit.As a corollary, we recover the optimal estimate for the resolvent as given in [BB21].
Corollary 1.2.Let n P N ˚and δ ą n `1 2 .There exists C ą 0 such that for (1.7) As usual for this kind of resolvent estimates, the proof will rely in particular on the Mourre commutators method.To prove our result we show that this method can be applied with much more flexibility than usual.
We have to apply the result simultaneously for P and P 0 .One of the difficulty is that P is selfadjoint the weighted space L 2 w " L 2 pw dxq while P 0 is selfadjoint on L 2 .Thus, unless w " 1, the operators P and P 0 are not selfadjoint on the same Hilbert space.
For this reason, we do not estimate the resolvent of P in L 2 w but stay in the usual L 2 space.Then P is no longer selfadjoint, but we can rewrite its resolvent as pP ´ζq ´1 " p´∆ G ´ζwq ´1w. (1.8) Now the difficulty is that p´∆ G ´ζwq ´1 is not a resolvent in the usual sense, and in particular its derivatives are no longer given by its powers.We will see that it is not necessary to apply the Mourre method to a resolvent.We will just see p´∆ G ´ζwq

´1
as the inverse of a parameter-dependant dissipative operator.In particular, even if we discuss a selfadjoint operator, our proof never really uses this selfadjointness and our method is robust with respect to non-selfadjoint (dissipative) perturbations.This is important in the perspective to apply the same method to different models.Finally, we do not apply the Mourre method to a power of the resolvent of some operator, but to the product of some different parameter-dependant operators.Some of the factors will be of the form p´∆ G ´ζwq ´1 as discussed above, there will be resolvents of P 0 , but we will also have the factor w which appears in (1.8) and factors comming from the difference p´∆ G ´ζwq ´p´∆ ´ζq.
The smallness at infinity of the corresponding coefficients given by (1.3) will play a crucial role in the proof of Theorem 1.1.In particular, it is usual to use decaying weights on both sides of the resolvent, but here we will also have to use the weights which appear between the resolvents.
Note that replacing pP ´ζq ´1 by p´∆ G ´ζwq ´1w is not just a technical issue.It is really p´∆ G ´ζwq ´1 that we can compare with p´∆ ´ζq ´1, and (1.8) explains the additional factor w in the estimates of Theorem 1.1.Now we discuss one of the important applications of the resolvent estimates, namely the analysis of the large time behavior for the time dependent problem (1.1).
After Theorem 1.1, it is expected that for large times the solution of (1.1) should behave in weighted spaces like a solution of the free Schrödinger equation, with a different initial condition.
The model problem is where f 0 P L 2 .The L 2 -norm of the solution u 0 ptq is constant but, given R ą 0, there exists a constant C ą 0 such that if f 0 is compactly supported in the ball BpRq then the energy of the solution u 0 of the free Schrödinger equation satisfies Moreover this estimate is optimal (see [BB21]).The local energy decay has been proved for various perturbations of this model case, see for instance [Rau78,Tsu84].For a long range perturbation of the metric and under the non-trapping condition, local energy decay has been proved in [Bou11a,BH12] with a loss of size Opt ε q.The optimal decay at rate Opt ´d 2 q has then been proved in [BB21].
Again, our purpose is to go further and to give the large time asymptotic profile for the solution u of (1.1).Since the contribution of high frequencies decays very fast under the non-trapping condition, the large time behavior of u depends on the contribution of low frequencies.Then, with Theorem 1.1 we will see that for large times the solution u looks like a solution of the free Schrödinger equation (1.9): Theorem 1.3.Assume that the non-trapping condition (1.4) holds.Let ρ 1 P r0, ρ 0 r and δ ě d 2 `2.There exists C ě 0 such that for t ě 0 we have This statement says that for t large the solution u of (1.1) is close in weighted spaces to the solution of (1.9) with f 0 " wf .In particular, since we know that e ´itP 0 w decays like t ´d 2 in LpL 2,δ , L 2,´δ q, we recover the optimal local energy decay for u.
Corollary 1.4.Assume that the non-trapping condition (1.4) holds.Let δ ě d 2 `2.There exists C ě 0 such that for t ě 0 we have Organization of the paper.After this introduction, we give in Section 2 the main arguments for the proofs of Theorem 1.1.The proofs of the intermediate results are then given in the following three sections.In particular we improve and apply the commutators method in Section 5. Finally we prove Theorem 1.3 in Section 6.

Strategy for low frequency asymptotics
In this section we explain how Theorem 1.1 is proved.We only give the main steps, and the details will be postponed to the following three sections.
2.1.Difference of the resolvents.We recall that the operator P was defined on L 2 by (1.2), with domain H 2 .This is a non-negative and selfadjoint operator on L 2 w , and its resolvent pP ´ζq ´1 is well defined for any ζ P CzR `with norm distpζ, R `q´1 in LpL 2 w q.
In order to have consistent notation, we also set P 0 pzq " ´∆ ´z2 and R 0 pzq " p´∆ ´z2 q ´1.
For n P N ˚and z P D `we set R rns pzq " |z| 2n pP ´z2 q ´nw ´1 " |z| 2n `Rpzqw ˘n´1 Rpzq (2.1) and R rns 0 pzq " |z| 2n R 0 pzq n .Since w defines a bounded operator on the weighted space L 2,δ " L 2 p x 2δ dxq, the estimate of Theorem 1.1 is equivalent, for a possibly different constant C ą 0, to (2.4) These operators are now products of resolvents of the form Rpzq or R 0 pzq, with inserted factors w, θ 0 pzq or θ 1 pzq.The additional smallness in (2.2) compared to the estimates of R rms pzq or R rms 0 pzq alone will come from the smallness (in a suitable sense) of the factors θ 0 pzq and θ 1 pzq.
The estimate (2.2) and hence Theorem 1.1 are then consequences of the following result.
Proposition 2.2.Let ρ 1 P r0, ρ 0 r.Let n 1 , n 2 P N ˚, σ P t0, 1u and δ ą n 1 `n2 ´σ `1 2 .Then there exists C ą 0 such that for z P D `we have (2.5) 2.2.Estimates given by the commutators method.It will be the purpose of Section 5 to prove that we can apply the Mourre commutators method to operators of the form (2.4).
It is usual for a Schrödinger operator that this method gives uniform estimates for the resolvent near a positive frequency.Near 0, the size of the weighted resolvent is as required uniform with respect to the imaginary part of the spectral parameter, but the estimate blows up if its real part also goes to 0.
It is standard that an important role is played by the generator of dilations (2.6)Here we will not apply the commutators method directly with the operator A 0 as the conjugate operator.Since P pzq is a small perturbation of P 0 pzq only at infinity, we will use as in [BB21] a version of A 0 localized at infinity.More precisely, for some χ P C 8 0 pR d , r0, 1sq equal to 1 on a neighborhood of 0, we consider the operator (2.7) Its domain is the set of u P L 2 such that p1 ´χpxqqpx ¨∇qu P L 2 in the sense of distributions.This is also a selfadjoint operator on L 2 and for θ P R, u P L 2 and x P R d we have pe ´iθAχ uqpxq " detpd x φ θ χ pxqq 1 2 upφ θ χ pxqq. (2.8) where θ Þ Ñ φ θ χ is the flow corresponding to the vector field p1 ´χpxqqx.For r P D `and x P R d we set χ r pxq " χprxq.We will work with the operator A r " A χr .For z P D we set χ z " χ |z| and A z " A χz . (2.9) With the rescaled versions of the resolvents, the estimates given by the commutators method read as follows.

2.3.
Elliptic regularity in the low frequency Sobolev spaces.Theorem 2.3 is not enough to prove Proposition 2.2.As in [Bou11a,BR14,Roy18] we use the gain of regularity to get some smallness when z is close to 0.
For z P D `and r " |z| we have the resolvent identity Rpzq ´Rpirq " pz 2 `r2 qRpirqwRpzq " pz 2 `r2 qRpzqwRpirq. (2.12) These factors Rpirq will give the required regularity.Then we will use the weights x ´δ to recover, in the end, estimates in LpL 2 q.
The following two propositions will be proved in Section 4.
Proposition 2.4.Let ρ P r0, ρ 0 r.Let n 1 , n 2 P N ˚and σ P t0, 1u.Let s 1 , s 2 P " 0, d 2 " , δ 1 ą s 1 and δ 2 ą s 2 .There exists C ą 0 such that for z P D `and r " |z| we have We observe that in Proposition 2.2 we work in weighted spaces, and the weight is given by negative powers of x.But for the commutators method in Theorem 2.3 we need negative powers of the generator of dilations A z , which also contains derivatives.Thus we also have to use the regularity of Rpirq to turn estimates with weights A z ´δ into estimates with x ´δ .
There exist N 0 P N and C ą 0 such that if N ě N 0 then for z P D `and r " |z| we have To prove these two results, we will work in rescaled Sobolev spaces.We set D " ?´∆ and, for r Ps0, 1s, we define D r " D{r.
Assume that in (2.5) we replace R rn 1 s pzq and R rn 2 s 0 pzq by terms of the form (2.19) and (2.21), respectively.Then it is enough to prove that for some (2.23) Given ρ Psρ 1 , ρ 0 r, this is a consequence of Proposition 2.4 applied with δ 1 " δ 2 " δ and , n 1 `n2 ´σ˙. (2.24) Now assume that in (2.5) we replace R rn 1 s pzq and R rn 2 s 0 pzq by terms of the form (2.20) and (2.22), where N can be chosen as large as we wish.By (2.11), (2.13) and (2.15) applied with s as in (2.24) we have for Then we consider the case where R rm 1 s pzq is replaced by a term of the form (2.20) and R rm 2 s 0 pzq is replaced by a term of the form (2.21).In this case we have to estimate an operator of the form where ν 1 ď n 1 , m 2 ě n 2 , and N 1 can be chosen arbitrarily large.If m 2 is too small, we cannot apply (2.15) on the right of R rν 1 s pzq (to which we apply Theorem 2.3).Then we proceed with more resolvent identities.More precisely, we apply (2.19)-(2.20) to R rν 1 s pzq, replacing R rN s pirqwR rνs pzq by R rνs pzqwR rN s pirq in (2.20).Now we have to estimate terms of the form (2.23) or with N, N 1 large, ν ď n 1 and m 2 ě n 2 .For such a term, we apply Theorem 2.3 to the factor R rνs pzq, and then (2.13) and (2.16) on each side.
Finally, if R rn 1 s pzq is replaced by a term of the form (2.19) and R rn 2 s 0 pzq by a term of the form (2.22) we proceed as in the previous case.We omit the details.

Preliminary results
In this section we give some preliminary results which will be used in the next two sections.We fix ρ P r0, ρ 0 r and ρ Psρ, ρ 0 r.
We fix an integer d 0 greater than d 2 .For κ ě 0 we denote by S ´κ the set of smooth functions φ such that (3.2) Proposition 3.1 explains how the weights which appear in the resolvent estimates can be used to convert some regularity into a power of the small spectral parameter z.As a particular case of (3.2), we record the following estimates.
Lemma 3.4.Let s P " 0, d 2 " and δ ą s.There exists C ą 0 such that for r Ps0, 1s we have } x ´δ } LpH s r ,L 2 q ď C r s and } x ´δ } LpL 2 ,H ´s r q ď C r s .With Proposition 3.1 we also see that the decay of the coefficients in (1.3) gives smallness for the operators θ σ pzq defined in (2.3).Proposition 3.5.Let ρ 1 P r0, ρs and s P . There exists C ą 0 which only depends on s, ρ 1 and ρ such that for z P D `we have In particular, for any s P ‰ ´d 2 , d 2 " we have Proof.The first estimate directly follows from Proposition 3.1 applied with κ " ρ 1 and η " ρ ´ρ1 ą 0. Then for j, k P t1, . . ., du we have which gives the estimate on P pzq ´P0 pzq.With ρ 1 " 0 this gives the last property since }P 0 pzq} LpH s`1 z ,H s´1 z q " 1.In Proposition 4.2 below, we will apply Proposition 3.5 with ρ 1 " 0 because we can only pay two derivatives.Because of this, the difference between P pzq and P 0 pzq is not small even for z close to 0, unless }G ´Id} S ´ρ is.Since we have not assumed that this is the case, we will write the perturbation G ´Id as a sum of a small perturbation and a compactly supported contribution which will be handled differently.
Proof.Let φ P C 8 0 be equal to 1 on a neighborhood of 0. For ε ą 0 and x P R d we set φ ε pxq " φpεxq.Then pG ´Idqφ ε is always compactly supported, and on the other hand, }pG ´Idqp1 ´φε q} S ´ρ À ε ρ 0 ´ρ .We conclude by choosing ε small enough and by setting G 0 " pG ´Idqφ ε and G 8 " Id `pG ´Idqp1 ´φε q.
3.2.Commutators.All along the proofs of the following two sections we are going to use commutators of the different operators involved with the operators of multiplication by the variables x j and the generator of dilations localized at infinity A z .
Let T be a linear map on the Schwartz space S. For r Ps0, 1s and j P t1, . . ., du we set ad rx j pT q " T rx j ´rx j T : S Ñ S. For z P D `we set ad j,z " ad |z|x j .Then for µ " pµ 1 , . . ., µ d q P N d we set (notice that ad rx j and ad rx k commute for j, k P t1, . . ., du) ad µ rx " ad µ 1 rx 1 ˝¨¨¨˝ad µ d rx d .We fix χ P C 8 0 equal to 1 on a neighborhood of 0 and we define A χ by (2.7) and then A z by (2.9).
We set ad 0,z pT q " ad Az pT q " T A z ´Az T : S Ñ S. Finally, for N P N we set I N " Ť N k"0 t0, . . ., du k , and for J " pj 1 , . . ., j k q P I N (with k P t0, . . ., N u and j 1 , . . ., j k P t0, . . ., du) we set ad J z pT q " `ad j 1 ,z ˝¨¨¨˝ad j k ,z ˘pT q.And if for some s 1 , s 2 P R the operator ad J z pT q defines a bounded operator from H s 1 z to H s 2 z for all J P I N , then we set We write (3.3)Note that we can rewrite A χ as Then the commutators of A χ with derivatives and multiplication operators are given by rV, A χ s " ip1 ´χqx ¨∇V, (3.4) and rB j , A χ s " ´ip1 ´χqB j `ipB j χqpx ¨∇q `id 2 pB j χq `i 2 `Bj px ¨∇χq ˘. (3.5) By induction on k P N we get in particular Lemma 3.7.Let N P N and s P R. Let ρ 1 P r0, ρs.There exists C ą 0 such that the following assertions hold for all z P D `.
For G ´Id we observe that, by (3.4) and Proposition 3.1, This gives the estimate on pG ´Idq.The estimates on pw ´1q, G and w are similar.With (3.5) applied with χ z (and (3.4)) we can check by induction on m P N that for z P D `we have ad m iAz pB j q " p1 ´χz q m B j `bj , N P N and ρ 1 P r0, ρs.There exists C ą 0 such that for z P D `we have }P pzq} then for σ P t0, 1u we also have Finally, it is known that the commutators method that we will use to prove Theorem 2.3 is based on the positivity of the commutator between the real part of the operator under study and the conjugate operator (see (H5) in Definition 5.1 below).In Section 5 we will use the following result.For z P D `we set P R pzq " ´∆G ´wRepz 2 q (3.7) and Kpzq " rP R pzq, iA z s ´2p1 ´χz q `PR pzq `Repz 2 q ˘. (3.8) Proposition 3.9.(i) There exists C ą 0 such that the commutator rP R pzq, A z s extends to a bounded operator from Proof.The first statement follows from Lemma 3.7 as Proposition 3.8.We prove the second property.We have Kpzq " r´∆ G , iA z s `2p1 ´χz q∆ G ´Repz 2 qrw, iA z s `2p1 ´χz qRepz 2 qpw ´1q.
The contributions of the last two terms are estimated in LpL 2 q with (3.4) and the decay of w ´1 and x ¨∇w.For the terms involving ∆ G we write (3.9) For j, k P t1, . . ., du we have LpL 2 ,H ´1 z q À |z| 2 .This gives the estimate for the contribution of the first term in the right-hand side of (3.9).The third term is estimated similarly.For the second we write , and finally we observe that }B j χ z } 8 À |z| to prove that the last term in (3.9) is also of size Op|z| 2 q in LpH 1 z , H ´1 z q.The proof is complete.We finish this paragraph with general considerations about commutators in an abstract setting.Let H be a Hilbert space and let K be a reflexive Banach space densely and continuously embedded in H.We identify H with its dual.
We denote by LpK, K ˚q the space of semilinear maps from K to its dual K ˚.We similarly define LpK ˚, Kq.In particular, LpH, H ˚q is identified with LpHq.
We consider a selfadjoint operator A on H with domain D H Ă H (endowed with the graph norm).Then A can also be seen as an operator A H P LpD H , Hq.Moreover, for ϕ P H we have ϕ P D H if and only if A Hϕ P H and in this case Aϕ " A Hϕ.We set (3.10)By restriction, A defines an operator A K on K with domain D K .Then D K is endowed with the graph norm of A K .We can see A K as an operator in LpD K , Kq and A K maps K ˚to D K. We set and for ϕ P D K ˚we set A K ˚ϕ " A Kϕ.We have Moreover, for K 0 P tK, H, K ˚u we have and for ϕ P D K 0 we have A K0 ϕ " A K 0 ϕ.
Let K 1 , K 2 P tK, H, K ˚u.We set C 0 A pK 1 , K 2 q " LpK 1 , K 2 q and for S P LpK 1 , K 2 q we set ad 0 A pSq " S.Then, by induction on n P N ˚, we say that S P C n A pK 1 , K 2 q if S P C n´1 A pK 1 , K 2 q and the commutator ad n´1 A pSqA K 1 ´AK 2 ad n´1 A pSq P LpD K 1 , D K2 q extends to an operator ad n A pSq in LpK 1 , K 2 q.Then we set We write C n pK 1 q for C n pK 1 , K 1 q.We also write Cn A pK 1 , K 2 q instead of C n A pK 1 , K 2 q for semi-linear operators.
The general properties which will be used in the sequel are the following.
(i) For S P C 1 A pK 1 , K 2 q we have S ˚P C 1 A pK 2 , K 1 q and ad A pS ˚q " ´ad A pSq ˚. (ii) A pK 1 , K 3 q and ad A pS 2 S 1 q " S 2 ad A pS 1 q `ad A pS 2 qS 1 . (3.12) Proof.The first statement is clear.Let ϕ P D K 1 .We have Sϕ P K 2 and so Sϕ belongs to D K 2 and (3.11) follows.Then, applying S 2 to (3.11) gives Since S 1 ϕ P D K 2 we similarly have S 2 S 1 ϕ P D K 3 and This proves that S 2 S 1 P C 1 A pK 1 , K 3 q with ad A pS 2 S 1 q given by (3.12).We finally recall from [BR14] the following result.
(i) Let δ P r´N, N s.There exists C ą 0 such that for S P C N A pHq we have The first statement is [BR14, Proposition 5.12] and second easily follows from [BR14, Proposition 5.13].

Elliptic regularity
In this section we prove Propositions 2.4 and 2.5.The parameter ρ P r0, ρ 0 r is fixed by these statements.We also fix ρ Psρ, ρ 0 r.
Proposition 2.4 will be given by (4.4) while Proposition 2.5 will follow from Proposition 4.3.(ii)and Proposition 4.4.
Let s P R. For r Ps0, 1s the resolvent R 0 pirq " r ´2pD 2 r `1q ´1 defines a bounded operator from H s´1 r to H s`1 r with norm r ´2.More generally, if we set * , then there exists c 0 ą 0 such that for s P R and z P D I we have Then, for k P N ˚and s, s 1 P R such that s 1 ´s ď 2k we have Our first purpose is to prove a similar property for Rpzq.By the usual elliptic regularity this holds for any fixed z P D `, the difficulty is to get uniform estimates for z close to 0.
We cannot extend (4.1) to Rpzq in full generality.We begin with the case s " 0.
Proposition 4.1.There exists c ą 0 such that for all z P D I we have More generally, for N P N there exists c N ą 0 such that for z P D I we have Proof.Let z P D I and ϑ z P " ´π 3 , π 3 ‰ be such that argpzq " π 2 `ϑz .The operator e ´iϑz P pzq defines an operator in LpH 1 z , H ´1 z q uniformly in z P D I .Moreover for u P H 1 z we have Re e ´iϑz P pzqu, u z .The Lax-Milgram Theorem gives the first estimate.Now let N P N. For J P I N , we can write ad J z pRpzqq as a sum of terms of the form Rpzqad J 1 z pP pzqqRpzq . . .ad J k z pP pzqqRpzq where k P N and J 1 , . . ., J k P I N .The general statement follows from (3.3) and Proposition 3.8.
On the other hand, we have a result similar to (4.1) if G is a small perturbation of the flat metric and s is not too large: . There exist γ ą 0 and c ą 0 such that if }G ´Id} S ´ρ ď γ then for z P D I we have More generally, for N P N there exists c N ą 0 such that for z P D I we have Proof.Let c 0 ą 0 be given by (4.1).If }G ´Id} S ´ρ is small enough, then by Proposition 3.5 applied with ρ 1 " 0 there exists r 0 Ps0, 1s such that for z P D I with |z| ď r 0 we have Then For z P D I with |z| ě r 0 we use the standard elliptic estimates, and the first estimate is proved.The second estimate follows as in the proof of Proposition 4.1.
The first part of the following result with z 1 " i |z| gives Proposition 2.4.With z " z 1 and s 1 " s 2 " 0 it also gives Theorem 2.3 for z P D I (without any weight).The second part of the result gives Proposition 2.5 with zx δ instead of A δ .

˚.
(i) There exists C ą 0 such that for z P D `and z 1 P D I with |z| " |z 1 | we have
We can check that this gives (4.9) if one of the minima is equal to the first argument.
Finally, the proof of (4.8) similarly follows from (4.12), the fact that it is already proved for R 8 and, for k P t1, . . ., n 1 u, (4.16) and (4.11) applied with s 1 " 0 and s 2 " s.
To finish the proof of Proposition 2.5 we have to replace rx δ by A z δ in (4.5)-(4.8).For this we use again the elliptic regularity to compensate the derivatives with appear in A z δ .
Proposition 4.4.Let δ ě 0 and let n be an even positive integer at least equal to δ.
Then there exists C ą 0 such that all r Ps0, 1s we have Moreover, the same estimates hold with R rns pirq and w replaced by R rns 0 pirq and 1. Proof.We prove the first estimate, the second is similar.We start by proving by induction on k P N that for n ě k and µ P N d we have (4.17) The case k " 0 is given by Proposition 4.1 (we use the convention that R r0s pirqw " Id).
Let k P N ˚, n ě k and µ P N d .We can write ad µ rx `Rrns pirqw ˘as a sum of terms of the form ad µ 1 rx `Rrn´1s pirqw ˘ad µ 2 rx `Rpirqw ˘where µ 1 `µ2 " µ.For such a term we have For the contribution of j P t0, . . ., k ´1u we apply the induction assumption, Proposition 4.1 and (3.4) to get a uniform bound in LpL 2 q.Now we consider the term corresponding to j " k.We have The contribution of the first term is estimated as before (note that x ¨∇χ r is uniformly bounded).Now let ℓ P t1, . . ., du.By Proposition 4.1 again, the operator r ´1D ℓ ad µ 2 rx `Rpirqw ˘extends to a uniformly bounded operator in LpL 2 q.On the other hand, by (3.6) we have " rx ℓ rx ´k ad µ 1 rx `Rrn´1s pirqw ˘pA r ´ip1 ´χr qq k´1 ` rx ´k ad rx j `ad µ 1 rx `Rrn´1s pirq ˘˘pA r ´ip1 ´χr qq k´1 .
Both terms are estimated with the induction assumption, and (4.17) is proved.With µ " 0 this gives the first estimate of the proposition when δ is an even integer.The general case follows by interpolation.

The Commutators method
In this section we prove Theorem 2.3.The proof relies on the abstract positive commutators method.Compared to the already known versions, we show that we can apply the result to operators like Rpzq even though they are not exactly resolvents, and that the estimates for the powers of the resolvent can in fact be applied to a product of different operators.Notice that we will not use the selfadjointness of the original operator P .The method is naturally adapted to dissipative operators.5.1.Abstract uniform estimates.Let H and K be as in the beginning of Section 3.2.
For Q P LpK, K ˚q we have Q ˚P LpK, K ˚q.We set RepQq " pQ `Q˚q {2 and ImpQq " pQ ´Q˚q {2i.We similarly define the real and imaginaly parts of R P LpK ˚, Kq.We say that Q P LpK, K ˚q is non-negative if for all ϕ P K we have Qϕ, ϕ K ˚,K ě 0, and that R P LpK, K ˚q is non-negative if for all ψ P K ˚we have ψ, Rψ K ˚,K ě 0. Finally we say that Q is dissipative if ImpQq ď 0.
We consider Q P LpK, K ˚q with negative imaginary part: there exists c 0 ą 0 such that Q `:" ´ImpQq ě c 0 I, where I P LpK, K ˚q is the natural embedding.By the Lax-Milgram Theorem, Q has an inverse in LpK ˚, Kq.
Let A be a selfadjoint operator on H.We use the notation of Section 3.2.Definition 5.1.Let N P N ˚and Υ ě 1.We say that A is to Q up to order N if the following conditions are satisfied.(H1) For ϕ P K we have }ϕ} H ď Υ }ϕ} K .(H2) For all θ P r´1, 1s the propagator e ´iθA P LpHq defines by restriction a bounded operator on K.
A pH,Kq ď Υ, and for ϕ P H we have }Πϕ} K ď Υ }Πϕ} H , (c) Q K has an inverse R K P LpK ˚, Kq which satisfies }Π K R K } LpK ˚,Kq ď Υ and }R K Π K} LpK ˚,Kq ď Υ. (H5) There exists β P r0, Υs such that if we set M " iad A pQq `βQ `P LpK, K ˚q, then in the sense of quadratic forms on H we have The main assumption in this definition is (H5).The uniform estimates given by the commutators method are the following.We give a proof adapted to this setting in Section 5.4.
(i) Let δ ą 1 2 .There exists C ą 0 which only depends on Υ and δ such that (5.1) (ii) Assume that N ě 2 and let δ 1 , δ 2 ě 0 be such that δ 1 `δ2 ă N ´1.There exists C ą 0 which only depends on N , Υ, δ 1 and δ 2 such that (5.2) (iii) Assume that N ě 2 and let δ P ‰ 1 2 , N " .There exists C ą 0 which only depends on N , Υ and δ such that (5.4) We explain the notation of Definition 5.1 on the model case, namely the free Laplacian with the generator of dilation (2.6) as the commutator.To get estimates on H " L 2 for the resolvent p´∆ ´ζq ´1 with Impζq ą 0 and Repζq close to some E ą 0, we choose Q " p´∆ ´ζq (seen as a bounded operator from K " H 1 to H ´1 » K ˚, this last identification being semilinear) and in particular we have `" Impζq.Then we set Π " 2 s p´∆ ´Eq ě ´E∆, the commutators method give in particular a uniform bound in L 2 for A ´δ p´∆ ´ζq ´1 A ´δ , from which we can deduce an estimate for the resolvent in LpL 2,δ , L 2,´δ q.Our proof in the next paragraph is a perturbation of this model case with ζ " z 2 and E of order |z| 2 .5.2.Application to the Schrödinger operator.In this paragraph we apply the abstract commutators method to prove uniform estimates for Rpzq.For z P D I , Theorem 2.3 follows from Proposition 4.3 applied with z 1 " z and s 1 " s 2 " 0. Thus, it is enough to prove Theorem 2.3 for z in ) .
We prove all the intermediate estimates for z P D R and, in the end, we will deduce Theorem 2.3 for z P D Ŕ by a duality argument.We begin with estimates for a single resolvent.
Proposition 5.3.Let δ ą 1 2 and δ 1 , δ 2 P R.There exists C ą 0 such that for z P D R we have (5.5) (5.8) To prove Proposition 5.3, we apply Theorem 5.2 to |z| ´2 P pzq (seen as an operator in LpH 1 z , H ´1 z ) uniformly in z P D R and for any N P N ˚.Then Proposition 5.3 is a consequence of Theorem 5.2 and Proposition 5.4 below.
In the proof of Proposition 5.4 we will use the Helffer-Sjöstrand formula.Let A be a selfadjoint operator on a Hilbert space H, m ě 2 and let φ P C 8 pRq be such that φ pkq pτ q À C k τ ´k´κ for some κ ą 0 and for all k P t0, . . ., m `1u.Then we have where λ is the Lebesgue measure on C and for some ψ P C 8 0 pR, r0, 1sq supported on r´2, 2s and equal to 1 on r´1, 1s we have defined the almost analytic extension φ of φ by φpτ `iµq " ψ ˆµ τ ˙m ÿ k"0 φ pkq pτ q piµq k k! .
Proposition 5.4.Let N P N.There exist χ P C 8 0 and Υ ě 1 such that for all z P D R the operator A z defined by (2.9) is Υ-conjugate to |z| ´2 P pzq P LpH 1 z , H ´1 z q up to order N .
‚ Assumption (H1) is clear in our setting and (H2) follows from (2.8).For any χ P C 8 0 , the fact that |z| 2 P pzq is uniformly in C N `1 Az pH 1 z , H ´1 z q is given by Proposition 3.8.Finally, Q `" ´ImpP pzqq " Impz 2 qw, so Q `belongs to C 1 Az pH 1 z , H ´1 z q uniformly in z by Lemma 3.7.This gives (H3).‚ Now we construct the operator Π z which appears in (H4) and (H5).For z P D R we have already set P R pzq " ´∆G ´wRepz 2 q.We similarly define P 0 R pzq " ´∆ ´Repz 2 q.These two operators can be seen as selfadjoint operators on L 2 with domain H 2 or as bounded operators from H 1 z to H ´1 z .Let φ P C 8 0 pR, r0, 1sq be equal to 1 on r´1, 1s and supported in s ´2, 2r.For η Ps0, 1s we set By the Helffer-Sjöstrand formula (5.9) (applied with m ě 3) and the resolvent identity, the difference Π η,z ´Π0 η,z can be rewritten as We can check that for z P D `and ζ P 5DzR `we have (5.10) On the other hand, as in the proof of Proposition 3.5 we can check that This proves Since Bζ φ is supported in 5D and decays faster than |Impζq| 2 near the real axis, we deduce (5.11) There also exists C ą 0 such that for all z P D `and η Ps0, 1s we have (5.12) ‚ By a compactness argument (we can also use Proposition 3.1), there exists χ P C 8 0 equal to 1 on a neighborhood of 0 and such that where C ą 0 is given by (5.12).Then for all z P D `and η Ps0, 1s we have (5.13) ‚ We have defined Kpzq in (3.8).By (5.12) and Proposition 3.9 there exists C 1 ą 0 such that (5.14) x ´ρ 2 φp´∆ ´τ 2 0 q is compact as an operator from L 2 to H 1 and φp ´∆´τ 2 0 16η 2 0 q goes weakly to 0 as η 0 goes to 0, there exists We also have Π 2η 0 ,z " Π 2η 0 ,z Π 1,z , so (5.14) and (5.15) give ‰ is compact, we can choose η 0 so small that (5.16) holds for any z P D R .By (5.16), (5.11) and (5.14) there exists r 0 Ps0, 1s such that for z P D R with |z| ď r 0 we have (5.17) We set D R " z P D R : |z| ě r 0 ( .
Since 2Repz 2 q ě |z| 2 we get after composition by Π η 0 ,z on both sides This gives (H5) with Π z " Π η 0 ,z .‚ the Helffer-Sjöstrand formula as above and Proposition 3.9 we have We set where w min " min xPR wpxq ą 0. Then Q K pzq " i `P pzq ´QK pzq ˘" Impz 2 qpw ´wmin q is non-negative, Q K pzq is invertible and by the functional calculus we have As for (5.10) we obtain similar estimates in LpH ´1 z , H 1 z q.Finally, since Π z " Π 2η 0 ,z Π z we have }Π z u} H 1 z ď }Π 2η 0 ,z } LpL 2 ,H 1 z q }Π z u} L 2 for all u P L 2 .With (5.12) this gives (H4) and the proof is complete.

Multiple resolvent estimates.
In this paragraph we generalize the uniform estimates for the powers of a resolvent.Compared to the usual setting, we also consider a product of different resolvents.In fact, we can consider the product of any finite sequence of operators having a suitable behavior with respect to the conjugate operator.Everything is based on the following abstract lemma.
Lemma 5.5.Let H be a Hilbert space.Let n P N ˚, T 1 , . . ., T n P LpHq and T " T 1 . . .T n .Let N P N ˚.
For j P t0, . . ., nu we consider on H a (possibly unbounded) selfadjoint operator Θ j ě 1, and Π j , Π j P LpHq such that Π j `Πj " Id H .For j P t1, . . ., nu we assume that there exist ν j ě 0, σ j P r0, ν j s and a collection C j " tC j ; pC j,δ 1 ,δ 2 q; pC j,δ qu of constants such that for δ 1 , δ 2 ě 0 with δ 1 `δ2 ă N ´νj and δ P rσ j , N s we have Assume that N ą ν.We set Π ´" Π 0 and Π `" Π ǹ .There exists a collection of constants C " tC; pC δ ´,δ `q; pC δ q; pC δ qu which only depend on the constants C j , 1 ď j ď n and such that for δ ´, δ `ě 0 such that δ ´`δ `ă N ´ν we have for δ P rσ ´, N r we have and finally, for δ P rσ `, N r we have (5.26) Proof.The result is proved by induction on n P N ˚, the case n " 1 being the assumption.
It is important that the constants in the conclusion of the lemma only depend on the constants in the assumptions.Thus if for some operators T j pzq, 1 ď j ď n, the estimates (5.19)-(5.21)are independant of the parameter z, then so are the estimates (5.23)-(5.26).
We will usually apply Lemma 5.5 with Θ j " A , Π j " 1 R ˚pAq and Π j " 1 R `pAq, where A is the conjugate operator.
With Proposition 5.3 and Lemma 5.5 we can prove Theorem 2.3.Notice that we have used all the assumptions of Definition 5.1 to prove Proposition 5.3, but for the rest of the proof we no longer need a conjugate operator and only use the estimates of Proposition 5.3.
Proof of Theorem 2.3.For z P D R we apply Lemma 5.5 with factors T j of the form w or |z| 2 Rpzq and constants independant of z.For factors T j " w we take ν j " σ j " 0 by Lemma 3.7 and Proposition 3.11, while for factors T j " |z| 2 Rpzq we can choose ν j " 1 and any σ j P ‰ 2 , 1 ‰ by Proposition 5.3.Then the assumptions of Lemma 5.5 hold uniformly in z P D R .In particular, (5.23) gives (2.10) for z P D R .
We similarly prove, for z P D R , (5.27) Taking the adjoint in (2.10) and (5.27), we get (2.10) and (2.11) for z P D Ŕ , and the proof of Theorem 2.3 is complete.
5.4.Proof of the abstract resolvent estimates.In this paragraph we prove Theorem 5.2.The strategy is inpired by the original papers [Mou81, JMP84, Jen85] and the earlier dissipative versions [Roy10, BR14, Roy16], but we need a proof adapted to our setting.We use the notation introduced in Paragraph 5.1.
For ε P r0, 1s we set Q ε " Q ´iεΠ ˚M Π P LpK, K ˚q.By (H5), Q ε has a negative imaginary part.We set R ε " Q ´1 ε P LpK ˚, Kq.We prove estimates on R ε for ε Ps0, 1s.At the limit ε Ñ 0 this will give estimates for R " Q ´1.Note that by Assumptions (H3)-(H4) and Proposition 3.10 we have Q ε P C1 A pK, K ˚q.In the following proposition, we check that R ε also has a nice behavior with respect to A.
Proposition 5.6.(i) D K is dense in K.
(ii) For ε Ps0, 1s we have R ε P C1 A pK ˚, Kq with ad A pR ε q " ´Rε ad A pQ ε qR ε .(iii) R ε maps D H to D K and D K to D H for all ε Ps0, 1s.
Proof.‚ Assumption (H2) holds for any θ P R and the restriction of e ´iθA defines a one-parameter group pT K pθqq θPR on K. Taking the adjoint also gives a one-parameter group pT Kpθqq θPR on K ˚, and for all θ P R the restriction of T Kpθq to H is e itA .Since H is dense in K ˚, we can check that pT Kpθqq is strongly continuous on K ˚.Then pT K pθqq is weakly continuous, and hence strongly continuous (see [EN00, Th.I.5.8]).Finally we check that the generator of pT K pθqq is A K , defined on the domain D K .This gives in particular the first statement by [EN00, Th.II.1.4].‚ There exists C ě 1 and ω ě 0 such that }T K pθq} LpKq ď Ce ω|θ| for all θ P R (see [EN00, Prop.I.5.5]).Then ([EN00, Th.II.1.10])for |Impλq| ą ω we have λ P ρpA K q and In particular A K pA K ´iµq ´1 and ´iµpA K ´iµq ´1 go strongly to 0 and Id K , respectively, as µ goes to ˘8. ‚ For µ ą ω we set A K pµq " ´iµA K pA K ´iµq ´1 P LpKq.In LpK ˚, Kq we have and in LpK, K ˚q, This goes strongly to ´ad A pQ ε q as µ Ñ `8.Then, taking the strong limit in (5.28) gives in LpD K ˚, D K˚q This proves the second statement.By Proposition 3.10, R ε maps D K ˚(and in particular D H ) to D K .We similarly prove that R ε maps D H to D K , so R ε also maps D K to D H.
The Mourre method relies on the so-called quadratic estimates.Here we will use the following version: Proposition 5.7.Let Q P LpK, K ˚q be dissipative.We assume that Q has an inverse R P LpK ˚, Kq.Let Q`P LpK, K ˚q be such that 0 ď Q`ď ´Imp Qq.Then we have The second estimate is similar.
Remark 5.8.Given two Banach spaces K 1 and K 2 , T 1 P LpK 1 , Kq and T 2 P LpK 2 , Kq, we have by the Cauchy-Schwarz inequality With Assumption (H5) we can apply the quadratic estimates to R ε .This gives the following properties.
Proposition 5.9.Let K 0 P tK, H, K ˚u.Let Θ P LpK, K 0 q.There exists C ą 0 which only depends on Υ and such that for all ε Ps0, 1s we have LpK 0 ,K 0 q ˙(5.30) and Proof.‚ By (H5) we have εΠ ˚Π ď εΥRepΠ ˚M Πq ď ´ΥImpQ ε q, so we can apply Proposition 5.7 with Q " ΥQ ε and Q`" εΠ ˚Π.This gives With (H4) we obtain for ϕ P K This gives the first part of (5.29).Similarly, Taking the adjoint concludes the proof of (5.29).‚ We have Q ε " Q K ´iQ K ´iεΠ ˚M Π.With the resolvent identity we have in LpK 0 , Kq (5.32)By Remark 5.8, (H4) and Proposition 5.7 applied with Q`" Q K ď ´ImpQ ε q we have LpK 0 ,K 0 q .On the other hand, by (H4), (H3) and (5.29), LpK 0 ,K 0 q .The first term in (5.32) is estimated by (H4), and the first part of (5.30) follows.As above, we prove the same estimate for R ε and get the second part by taking the adjoint.Finally, (5.30) and (5.29) give (5.31).Now we can prove the first part of Theorem 5.2: Proof of Estimate (5.1).Without loss of generality, we can assume that δ P ‰ 1 2 , 1 ‰ .‚ For ε P r0, 1s we set Θ ε " A ´δ εA δ´1 .This defines a bounded selfadjoint operator on H and by the functional calculus we have where we denote by a prime the derivative with respect to ε.We set F ε " Θ ε R ε Θ ε .By (5.33) and Proposition 5.9 applied with Θ " Θ ε we get for ε Ps0, 1s and hence }F ε } LpHq À 1 ε . (5.34) The derivative of F is given by LpHq ˘. (5.35) For the last term we write in LpK, K ˚q By Proposition 5.9 and (H3)-(H4) for M we have With (5.33) and Proposition 5.9 we get LpHq `}F ε } LpHq .
Thus for ε small enough the operator Q N,ε " Qε `Pε is invertible and its inverse R N,ε is given by R N,ε " Rε ´R ε P ε pId K `R ε P ε q ´1 Rε .We deduce (5.37).‚ For the last statement we observe that in LpK, K ˚q we have As in Proposition 5.6 we can check that R N,ε P C1 A pK ˚, Kq with ad A pR N,ε q " ´RN,ε ad A pQ N,ε qR N,ε .We deduce in LpK, Now we can finish the proof of Theorem 5.2.
Proof of Estimate (5.2).Let ε N be given by Proposition 5.10.For ε Ps0, ε N s we set in LpHq F N,ε " A δ 1 e εA 1 R ´pAqR N,ε 1 R `pAqe ´εA A δ 2 .Then in the strong sense we have By Proposition 5.10 and the functional calculus we deduce Since N ´δ1 ´δ2 ´2 ą ´1, this proves that F N,ε is bounded in LpHq uniformly in ε Ps0, ε N s.
By Proposition 5.10 we have }F 1,ε } LpHq À ε ´1 2 .On the other hand we have (5.40) By interpolation we have For the second term in (5.40) we use (H3) and Proposition 5.10.Finally, (5.41) We similarly get a uniform bound for 1 R `pAqR ˚ A ´η. Taking the adjoint gives (5.42) ‚ For I Ă R we write A I for 1 I pAq.We prove that we have, uniformly in n, m P N, › › A rn,n`1r RA rm,m`1r › › LpHq À 1. (5.43) We observe that for any λ P R the operator A ´λ is also Υ-conjugated to Q up to order N , so the estimates (5.1) and (5.2) hold with A replaced by A ´λ uniformly in λ.In particular, with (5.1) applied to A ´n we get (5.43) when n " m.This also holds with R replaced by R ˚.For the general case we write A rn,n`1r RA rm,m`1r " A rn,n`1r A s´8,ms RA rm,m`1r `Arn,n`1r A sm,`8r R ˚Arm,m`1r `Arn,n`1r A sm,`8r pR ´R˚q A rm,m`1r .
The first two terms are estimated by (5.41) and (5.42) applied with A ´m instead of A.
For the third term we observe that R ´R˚" 2R ˚Q`R is non-negative, so by Remark 5.8 we have LpHq .We can apply (5.43) already proved when n " m to R and R ˚, which concludes the proof of (5.43) when n ‰ m.With (5.2) we finally get (5.3).The proof of (5.4) is similar.

Local energy decay
In this section we show how the local energy decay of Theorem 1.3 can be deduced from the resolvent estimates given by Theorem 1.1.
Proof of Theorem 1.3.‚ Let f P S and µ Ps0, 1s.All along the proof we use the notation ζ for τ `iµ, where τ is a variable in R. For t ą 0 we have e ´itP f " 1 2iπ ż R e ´itζ pP ´ζq ´1f dτ.
We similarly define u 0 ˚,µ with P replaced by P 0 and f replaced by f 0 " wf .‚ Let m P N ˚such that d `ρ1 2 ă m ă d `ρ1 2 `1.
We have δ ą m `1 2 .After integrations by parts and using the uniform estimates for the resolvent of P far from its spectrum, we see that where the constant hidden in the symbol À is independant of µ.Similarly, using (1.5) to estimate the derivatives of pP ´ζq ´1 near the positive real axis, we obtain We have similar estimates for u 0 ´,µ ptq and u 0 `,µ ptq.‚ By integrations by parts we have pitq m´1 `ulow ptq ´u0,low ptq ˘" 1 2iπ ż R e ´itζ θ pm´1q µ pτ q dτ, where we have set θ µ pτ q " χ low pτ q `pP ´ζq ´1 ´pP 0 ´ζq ´1w ˘f.
By Theorem 1.1 we have, uniformly in µ ą 0, ,m p|z| xq ¨∇ `|z| c j,m p|z| xq, where b j,m : R d Ñ C d and c j,m : R d Ñ C are smooth and compactly supported.Then multiplications by p1 ´χz q m , b j,m p|z| xq and c j,m p|z| xq define bounded operators on H s z uniformly in z P D `for any s P R.This is clear for s P N and the general case follows by interpolation and duality.This gives the last statement.With Lemma 3.7 and (3.3) we deduce the following result.Proposition 3.8.Let s P ‰ ´d 2 , d 2 "

› › A
rn,n`1r pR ´R˚q A rm,m`1r › › LpHq ď › › A rn,n`1r pR ´R˚q A rn,n`1r rm,m`1r pR ´R˚q A rm,m`1r

first step is to rewrite the difference R rms pzq ´Rrms 0 pzq as a sum of products of factors Rpzq and R 0 pzq. Lemma 2.1. For n P N ˚and z P D `we have
P N d and s P R the operator D α " p´iB x q α defines an operator from H s r to H Let z P D `.We set r " |z| and ẑ " z{r.Let n P N ˚.With (2.12) we can prove by induction on N P N that [BR14]ter conjugation by O r (see (2.18)), the following statement is Proposition 7.2 in[BR14].