Limiting Absorption Principle for

Schrödinger Operators with Oscillating Potentials

Thierry Jecko, Aiman Mbarek

Received: October 19, 2016 Revised: February 28, 2017

Communicated by Heinz Siedentop

Abstract. Making use of the localised Putnam theory developed in [GJ1], we show the limiting absorption principle for Schrödinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [GJ2]) that was already studied in [BD, DMR, DR1, DR2, MU, ReT1, ReT2]. Allowing long-range and short-range components and local singularities in the perturbation, we improve known results. A subclass of the considered potentials actually cannot be treated by the Mourre commutator method with the generator of dilations as conjugate operator. Inspired by [FH], we also show, in some cases, the absence of positive eigenvalues for our Schrödinger operators.

2010 Mathematics Subject Classification: 35J10, 35P25, 35Q40, 35S05, 47B15, 47B25, 47F05.

Contents

1. Introduction. 728 2. Oscillations. 735 3. Regularity issues. 738 4. The Mourre estimate. 743 5. Polynomial bounds on possible eigenfunctions with positive energy. 745 6. Local finitness of the point spectrum. 748 7. Exponential bounds on possible eigenfunctions with positive energy.749 8. Eigenfunctions cannot satisfy unlimited exponential bounds. 752 9. LAP at suitable energies. 755 10. Symbol-like long range potentials. 761 Appendix A. Standard pseudodifferential calculus. 762 Appendix B. Regularity w.r.t. an operator. 764 Appendix C. Commutator expansions. 765 Appendix D. Strongly oscillating term. 770 References 773

1. Introduction.

In this paper, we are interested in the behaviour near the positive real axis of the resolvent of a class of continuous Schrödinger operators. We shall prove a so called “limiting absorption principle”, a very useful result to develop the scattering theory associated to those Schrödinger operators. It also gives information on the nature of their essential spectrum, as a byproduct. The main interest of our study relies on the fact that we include some oscillating contribution in the potential of our Schrödinger operators. To set up our framework and precisely formulate our results, we need to introduce some notation. Let d ∈ ℕ^{∗}. We denote by ⟨·, ·⟩ and ∥ · ∥ the right linear scalar product and the norm in L^{2}(ℝ^{d}), the space of squared integrable, complex functions on ℝ^{d}. We also denote by ∥ · ∥ the norm of bounded operators on L^{2}(ℝ^{d}). Writing x = (x_{1}; ⋯; x_{d}) the variable in ℝ^{d}, we set

⟨x⟩ := (1 + ∑_{j=1}^{d} x_{j}^{2})^{1/2} .

Let Q_{j} the multiplication operator in L^{2}(ℝ^{d}) by x_{j} and P_{j} the self-adjoint realization of −i∂_{x_{j}} in L^{2}(ℝ^{d}). We set Q = (Q_{1}; ⋯; Q_{d})^{T} and P = (P_{1}; ⋯; P_{d})^{T}, where T denotes the transposition. Let

H_{0} = |P |^{2} := ∑_{j=1}^{d} P_{j}^{2} = P^{T} · P

be the self-adjoint realization of the nonnegative Laplace operator −Δ in L^{2}(ℝ^{d}). We consider the Schrödinger operator H = H_{0} + V (Q), where V (Q) is the multiplication operator by a real valued function V on ℝ^{d} satisfying the following

Assumption 1.1. Let α, β ∈]0; +∞[. Let ρ_{sr}, ρ_{lr}, ρ′_{lr} ∈]0; 1]. Let v ∈ C^{1}(ℝ^{d}; ℝ^{d}) with bounded derivative. Let κ ∈ C_{c}^{∞}(ℝ; ℝ) with κ = 1 on [−1; 1] and 0 ≤ κ ≤ 1. We consider functions V_{sr}, Ṽ_{sr}, V_{lr}, V_{c}, W_{αβ} : ℝ^{d} ⟶ ℝ such that V_{c} is compactly supported and V_{c}(Q) is H_{0}-compact, such that the functions ⟨x⟩^{1+ρ_{sr}}V_{sr}(x), ⟨x⟩^{1+ρ_{sr}} Ṽ_{sr}(x), ⟨x⟩^{ρ_{lr}}V_{lr}(x) and the distributions ⟨x⟩^{ρ′_{lr}}x · ∇V_{lr}(x) and ⟨x⟩^{ρ_{sr}}(v · ∇Ṽ_{sr})(x) are bounded, and

(1.1)  W_{αβ}(x) = w(1 − κ(|x|))|x|^{−β} sin(k|x|^{α})

with real w. Let V = V_{sr} + v · ∇Ṽ_{sr} + V_{lr} + V_{c} + W_{αβ}.

Under Assumption 1.1, V (Q) is H_{0}-compact. Therefore H is self-adjoint on the domain D(H_{0}) of H_{0}, which is the Sobolev space H^{2}(ℝ^{d}) of L^{2}(ℝ^{d})-functions such that their distributional derivative up to second order belong to L^{2}(ℝ^{d}). By Weyl’s theorem, the essential spectrum of H is given by the spectrum of H_{0}, namely [0; +∞[. Let A be the self-adjoint realization of the operator (P · Q + Q · P)/2 in L^{2}(ℝ^{d}). By the Mourre commutator method with A as conjugate operator, one has the following Theorem, which is a consequence of the much more general Theorem 7.6.8 in [ABG]:

Theorem 1.2. [ABG]. Consider the above operator H with w = 0 (i.e. without the oscillating part of the potential). Then the point spectrum of H is locally finite in ]0; +∞[. Furthermore, for any s > 1/2 and any compact interval I ⊂]0; +∞[, that does not intersect the point spectrum of H,

(1.2)  sup_{{ℜz∈I, \atop ℑz≠0}} ∥⟨A⟩^{−s}(H − z)^{−1}⟨A⟩^{−s}∥ < +∞ .

Remark 1.3. In [Co, CG], a certain class of potentials that can be written as the divergence of a short range potential (i.e. a potential like V_{sr}) were studied. Theorem 1.2 covers this case. We point out that the short range conditions (on V_{sr} and Ṽ_{sr}) can be relaxed to reduce to a Agmon-Hörmander type condition (see Theorem 7.6.10 [ABG] and Theorem 2.14 in [GM]). “Strongly singular” terms (more singular than our V_{c}) are also considered in Section 3 in [GM].

Remark 1.4. When w = 0, H has a good enough regularity w.r.t. A (see Section 3 and Appendix B for details) thus the Mourre theory based on A can be applied to get Theorem 1.2. But it actually gives more, not only the existence of the boundary values of the resolvent of H (which is implied by (1.2)) but also some Hölder continuity of these boundary values. It is wellknown that all this implies that the same holds true when the weight ⟨A⟩^{−s} are replaced by ⟨Q⟩^{−s} (see Remark 1.12 below for a sketch). Still for w = 0, under some assumption on the form [V_{c}, A] (roughly (8.1) below), it follows from [FH, FHHH1] that H has no positive eigenvalue.

Now, we turn on the oscillating part W_{αβ} of the potential and ask ourselves, which result from the above ones is preserved. To formulate our first main result, we shall need the following

Assumption 1.5. Let α, β > 0 and set β_{lr} = min(β; ρ_{lr}). Unless |α−1|+β > 1, we take α ≥ 1 and we take β and ρ_{lr} such that β + β_{lr} > 1 or, equivalently, β > 1/2 and ρ_{lr} > 1 − β. We consider a compact interval I such that I ⊂ ]0; k^{2}/4[, if α = 1 and β ∈]1/2; 1], else such that I ⊂]0; +∞[.

Remark 1.6. If β > 1, W_{αβ} can be considered as short range potential like V_{sr}. If α < β ≤ 1, W_{αβ} satisfies the long range condition required on V_{lr}. If α + β > 2 and β ≤ 1 then, for ϵ = α + β − 2, for some short range potentials V̂_{sr} V̌_{sr} (i.e. satisfying the same requirement as V_{sr}), for some κ̃ ∈ C_{c}^{∞}(ℝ; ℝ) with κ̃ = 1 on [−1/2; 1/2] and with support in [−1; 1], and for x ∈ ℝ^{d},

(1.3)  w(1 − κ̃(|x|))|x|^{−1}x · ∇V̌_{sr}(x) = kαW_{αβ}(x) + V̂_{sr}(x) ,

where V̌_{sr}(x) = −(1 − κ(|x|))|x|^{−1−ϵ} cos(k|x|^{α}). In all cases, Theorem 1.2 applies.

Remark 1.7. Our assumptions allow V to contain the function x ↦ |x|^{−β} sin(k|x|^{α}) with β < 2 + α. This function was considered in [BD, DMR, DR1, DR2, ReT1, ReT2]. Assumption 1.5 excludes the situation where 0 < β ≤ α < 1. A reason for this is given just after Proposition 2.1 in Section 2. It turns out that our results do not change if one replaces the sinus function in W_{αβ} by a cosinus function.

Let Π be the orthogonal projection onto the pure point spectral subspace of H. We set Π^{⊥} = 1 − Π. For any complex number z ∈ ℂ, we denote by ℜz (resp. ℑz) its real (resp. imaginary) part. Our first main result is the following limiting absorption principle (LAP).

Theorem 1.8. Suppose Assumptions 1.1 and 1.5 are satisfied. For any s > 1/2,

(1.4)  sup_{{ℜz∈I, \atop ℑz≠0}} ∥⟨Q⟩^{−s}(H − z)^{−1}Π^{⊥}⟨Q⟩^{−s}∥ < +∞ .

Remark 1.9. In the litterature, the LAP is often proved away from the point spectrum, as in Theorem 1.2. If I in (1.4) does not intersect the latter, one can remove Π^{⊥} in (1.4) and therefore get the usual LAP. But the LAP (1.4) gives information on the absolutely continuous subspace of H near possible embedded eigenvalues. When |α − 1| + β > 1 and I does not intersect the point spectrum of H, the Mourre theory gives a stronger result than Theorem 1.8 (cf. Theorem 1.2 and Remark 1.4).

Historically, LAPs for Schrödinger operators were first obtained by pertubation, starting from the LAP for the Laplacian H_{0}. Lavine initiated nonnegative commutator methods in [La1, La2] by adapting Putnam’s idea (see [CFKS] p. 60). Mourre introduced 1980 in [Mo] a powerful, non pertubative, local commutator method, nowadays called “Mourre commutator theory” (see [ABG, GGé, GGM, JMP, Sa]). Nevertheless it cannot be applied to potentials that contain some kind of oscillaroty term (cf. [GJ2]). In [Co, CG], the LAP was proved pertubatively for a class of oscillatory potentials. This result now follows from Mourre theory (cf. Remark 1.3). In [BD, DMR, DR1, DR2, ReT1, ReT2], the present situation with V_{c} = 0 and a radial long range contribution V_{lr} was treated using tools of ordinary differential equations and again a pertubative argument. Theorem 1.8 improves the results of these papers in two ways. First, we allow a long range (non radial) part in the potential. Second, the set V of values of (α; β), for which the LAP 0 2 1 4/5 2/3 1/2

α

1 β

[figure] [figure]

red

red red

green

blue blue blue

blue

[figure]

A B

[figure]

Figure 1. LAP. V = blue ∪ green.

(on some interval) holds true, is here larger. However, in the case α = 1, these old results provide a LAP also beyond k^{2}/4 in all dimension d, whereas we are able to do so only in dimension d = 1. For α = β = 1, the LAP at high enough energy was proved in [MU]. Another proof of this result is sketched in Remark 1.11 below. We point out that the discrete version of the present situation is treated in [Man]. We also signal that the LAP for continuous Schrödinger operators is studied in [Mar] by Mourre commutator theory but with new conjugate operators, including the one used in [N]. We also emphasize an alternative approach to the LAP based on the density of states. It seems however that general long range pertubations are not treated yet. We refer to [Ben] for details on this approach. In Fig. 1, we drew the set V in a (α; β)-plane. It is the union of the blue and green regions. The papers [BD, DMR, DR1, DR2, ReT1, ReT2] etablished the LAP in the region above the red and black lines and, along the vertical green line, above the point A = (1; 2/3). According to Remark 1.6, Theorem 1.2 shows the LAP in the blue region (above the red lines and the blue one). Both results are obtained without energy restriction. Theorem 1.8 covers the blue and green regions (the set V), with a energy restriction on the vertical green line. In [GJ2], the LAP with energy restriction is proved at the point B = (1; 1). In the red region (below the red lines), the LAP is still an open question. Recall that A is the self-adjoint realization of the operator (P · Q + Q · P)/2 in L^{2}(ℝ^{d}). We are able to get the following improvement of a main result in [GJ2].

Theorem 1.10. Let α = β = 1. Under Assumption 1.1 with Ṽ_{sr} = V_{c} = 0, take a compact interval I ⊂]0; k^{2}/4[. Then, for any s > 1/2,

(1.5)  sup_{{ℜz∈I, \atop ℑz≠0}} ∥⟨A⟩^{−s}(H − z)^{−1}Π^{⊥}⟨A⟩^{−s}∥ < +∞ .

Proof. In [GJ2], it was further assumed that, for any μ ∈ I, Ker(H − μ) ⊂ D(A). Thanks to Corollary 5.2, this assumption is superfluous. □

Remark 1.11. Note that Assumption 1.5 is satisfied for α = β = 1. In dimension d = 1, the above result is still true if I ⊂]k^{2}/4; +∞[. A careful inspection of the proof in [GJ2] shows that Theorem 1.10 holds true in all dimensions if I ⊂]a; +∞[, for large enough positive a (depending on |w|). If |w| is small enough, the mentioned proof is even valid on any compact interval I ⊂]0; +∞[. For nonzero potentials V_{c} and Ṽ_{sr}, we believe that one can adapt the proof in [GJ2] of Theorem 1.10.

Remark 1.12. It is well known that (1.5) implies (1.4). Let us sketch this briefly. It suffices to restrict s to ]1/2; 1[. Take θ ∈ C_{c}^{∞}(ℝ; ℝ) such that θ = 1 near I. Then, the bound (1.4) is valid if (H − z)^{−1} is replaced by (1 − θ(H))(H − z)^{−1}. The boundedness of the contribution of θ(H)(H − z)^{−1} to the l.h.s of (1.4) follows from (1.5) and from the boundedness of ⟨Q⟩^{−s}θ(H)⟨A⟩^{s}. To see the last property, one can write

⟨Q⟩^{−s}θ(H)⟨A⟩^{s} = ⟨Q⟩^{−s}θ(H)⟨P ⟩^{s}⟨Q⟩^{s} · ⟨Q⟩^{−s}⟨P ⟩^{−s}⟨A⟩^{s} .

The last factor is bounded by Lemma C.1 in [GJ2]. The boundedness of the other one is granted by the regularity of H w.r.t. ⟨Q⟩ (see Section 3) and the fact that θ(H)⟨P ⟩ is bounded.

Remark 1.13. It is well known that (1.4) implies the absence of singular continuous spectrum in I (see [RS4]). On this subject, we refer to [K, Rem] for more general results.

In Section 3, we show that the Mourre commutator method, with the generator A of dilations as conjugate operator, cannot be applied to recover Theorem 1.8 in his full range of validity V, neither the classical theory with C^{1,1} regularity (cf. [ABG]), nor the improved one with “local” C^{1+0} regularity (cf. [Sa]). Indeed the required regularity w.r.t. A is not valid on V. As pointed out in [GJ2], Theorem 1.10 cannot be proved with these Mourre theories for the same reason. We expect that the use of known, alternative conjugate operators (cf. [ABG, N, Mar]) does not cure this regularity problem. However, according to a new version of the paper [Mar], one would be able to apply the Mourre theory in a larger region than the blue region mentioned above, this region still being smaller than V (cf. Section 3). The given proof of Theorem 1.10 relies on a kind of “energy localised” Putnam argument. This method, which is reminiscent of the works [La1, La2] by Lavine, was introduced in [GJ1] and improved in [Gé, GJ2]. It was originally called “weighted Mourre theory” but it is closer to Putnam idea (see [CFKS] p. 60) and does not make use of differential inequalities as the Mourre theory. Note that, up to now, the latter gives stronger results than the former. It is indeed still unknown whether this “localised Putnam theory” is able to prove continuity properties of the boundary values of the resolvent. We did not succeed in applying the “localised Putnam theory” formulated in [GJ2] to prove Theorem 1.8. We believe that, again, the bad regularity of H w.r.t. A is the source of our difficulties (cf. Section 3). Instead, we follow the more complicated version presented in [GJ1], which relies on a Putnam type argument that is localised in Q and H, and use the excellent regularity of H w.r.t. ⟨Q⟩ (cf. Section 3). A byproduct of the proof of Theorem 1.2 is the local finitness (counting multiplicity) of the pure point spectrum of H in ]0; +∞[. Thus this local finitness holds true if |α − 1| + β > 1. We extend this result to the case where |α − 1| + β ≤ 1 in the following way: the above local finitness is valid in ]0; +∞[, if α > 1, and in ]0; k^{2}/4[, if α = 1 (cf. Corollary 6.2). In the papers [FHHH2, FH], polynomial bounds and even exponential bounds were proven on possible eigenvectors with positive energy. In our framework, those results fully apply when |α−1|+β > 1. Here we get the same polynomial bounds under the less restrictive Assumptions 1.1 and 1.5 (cf. Proposition 5.1). Concerning the exponential bounds, we manage to get them under Assumptions 1.1 and 1.5, but for α > 1 (see Proposition 7.1). In the papers [FHHH2, FH] again, the absence of positive eigenvalue is proven. In our framework, this result applies when α < β and when β > 1, provided that the form [(V_{c} + v · ∇Ṽ_{sr})(Q), iA] is H_{0}-form-lower-bounded with relative bound < 2 (see (8.1) for details). When α + β > 2 and β ≤ 1, it applies under the same condition, provided that the oscillating part of the potential is small enough (i.e. if |w| is small enough). Indeed, in that case, the form [(V_{c} + v · ∇Ṽ_{sr} + W_{αβ})(Q), iA] is H_{0}-form-lower-bounded with relative bound < 2. Inspired by those papers, we shall derive our second main result, namely

Theorem 1.14. Under Assumptions 1.1 and 1.5 with α > 1 when |α−1|+β ≤ 1, we assume further that the form [(V_{c} + v · ∇Ṽ_{sr})(Q), iA] is H_{0}-form-lower-bounded with relative bound < 2 (see (8.1) for details). Furthermore, we require that |w| is small enough if α + β > 2 and β ≤ 1/2. Then H has no positive eigenvalue.

Proof. The result follows from Propositions 7.1 and 8.2. □

Remark 1.15. Our proof is strongly inspired by the ones in [FHHH2, FH]. Actually, these proofs cover the cases β > 1, α < β, and the case where α + β > 2, β ≤ 1, and |w| is small enough. In the last case, namely when α > 1, β > 1/2, ρ_{lr} > 1 − β, and α + β ≤ 2, the main new ingredient is an appropriate control on the oscillatory part of the potential. In particular, in the latter case, we do not need any smallness on |w|.

Remark 1.16. In the case α = β = 1, assuming (8.1), we can show the absence of eigenvalue at high energy. This follows from Remark 7.3 and Proposition 8.2.

0  1  2

1/2

α

1 β

[figure]

red

red red

[figure]

green

[figure]

blue blue blue

blue, small |w|

blue, small |w|

Figure 2. No positive eigenvalue in blue ∪ green.

However an embedded eigenvalue does exist for an appropriate choice of V (see [FH, CFKS, CHM]).

Remark 1.17. Under the assumptions of Theorem 1.14, for any compact interval I ⊂]0; +∞[, the result of Theorem 1.8, namely (1.4), is valid with Π^{⊥} replaced by the identity operator. Indeed, for any compact interval I′ ⊂]0; +∞[ containing I in its interior, 1I_{I′}(H)Π = 0 by Theorem 1.14. In view of Remark 1.11, the LAP (1.5) is valid at high energy, when α = β = 1. Thanks to Remark 1.16, one can also remove Π^{⊥} in (1.5).

One can find many papers on the absence of positive eigenvalue for Schrödinger operators: see for instance [Co, K, Si, A, FHHH2, FH, IJ, RS4, CFKS]. They do not cover the present situation due to the oscillations in the potential. In Fig. 2, we summarise results on the absence of positive eigenvalue. In the blue region (above the red and blue lines), the result is granted by [FHHH2, FH], with a smallness condition below the blue line. Theorem 1.14 covers the blue and green regions (above the red lines), with a smallness condition below the black line. In Assumption 1.5 with |α − 1| + β ≤ 1, the parameter ρ_{lr}, that controls the behaviour at infinity of the long range potential V_{lr}, stays in a β-dependent region. One can get rid of this constraint if one chooses a smooth, symbol-like function as V_{lr}, as seen in the next

Theorem 1.18. Assume that Assumption 1.1 is satisfied with |α − 1| + β ≤ 1 and β > 1/2. Assume further that V_{lr} : ℝ^{d} ⟶ ℝ is a smooth function such that, for some ρ_{lr} ∈]0; 1], for all γ ∈ ℕ^{d},

sup_{x∈ℝ^{d}}|⟨x⟩^{ρ_{lr}+|γ|}(∂_{x}^{γ}V_{lr})(x)| < +∞ .

Take α = 1. Then the LAP (1.4) holds true on any compact interval I such that I ⊂]0; k^{2}/4[, if d ≥ 2, and such that I ⊂]0; +∞[\{k^{2}/4}, if d = 1. Take α > 1. Then the LAP (1.4) holds true on any compact interval I ⊂ ]0; +∞[. If, in addition, [(V_{c} + v · ∇Ṽ_{sr})(Q), iA] is H_{0}-form-lower-bounded with relative bound < 2 (see (8.1) for details), then H has no positive eigenvalue. In particular, (1.4) holds true with Π^{⊥} removed.

Remark 1.19. We expect that our results hold true for a larger class of oscillatory potential provided that the “interference” phenomenon exhibited in Section 2 is preserved. In particular, we do not need that W_{αβ} is radial.

We point out that there still are interesting, open questions on the Schrödinger operators studied here. Concerning the LAP, for α = 1, it is expected that (1.4) is false near k^{2}/4. Note that the Mourre estimate is false there, when β = 1 (see [GJ2]). The validity of (1.4) beyong k^{2}/4 is still open, even at high energy when β < 1. Concerning the existence of positive eigenvalue, again for α = 1, it is known in dimension d = 1 that there is at most one at k^{2}/4 if β = 1 (see [FH]). It is natural to expect that this is still true for d ≥ 2 and β = 1. We do not know what happens for α = 1 > β. In Section 2, we analyse the interaction between the oscillations in the potential W_{αβ} and the kinetic energy operator H_{0}. In Section 3, we focus on regularity properties of H w.r.t. A and to ⟨Q⟩ and discuss the applicability of the Mourre theory and of the results from the papers [FHHH2, FH]. In Section 4, in some appropriate energy window, we show the Mourre estimate, which is still a crucial result. We deduce from it polynomial bounds on possible eigenvectors of H in Section 5. This furnishes the material for the proof of Theorem 1.10. In Section 6, we show the local finitness of the point spectrum in the mentioned energy window. In the case α > 1, we show exponential bounds on possible eigenvectors in Section 7 and prove the absence of positive eigenvalue in Section 8. Independently of Sections 7 and 8, we prove Theorem 1.8 in Section 9. Section 10 is devoted to the proof of Theorem 1.18. Finally, we gathered well-known results on pseudodifferential calculus in Appendix A, basic facts on regularity w.r.t. an operator in Appendix B, known results on commutator expansions and technical results in Appendix C, and an elementary, but lengthy argument, used in Section 2, in Appendix D. Aknowledgement: The first author thanks V. Georgescu, S. Golénia, T. Hargé, I. Herbst, and P. Rejto, for interesting discussions on the subject. Both authors express many thanks to A. Martin, who allowed them to access to some result in his work in progress. Both authors are particularly grateful to the anonymous referee for his constructive and fruitful report.

2. Oscillations.

In this section, we study the oscillations appearing in the considered potential V. It is convenient to make use of some standard pseudodifferential calculus, that we recall in Appendix A. As in [GJ2], our results strongly rely on the interaction of the oscillations in the potential with localisations in momentum (i.e. in H_{0}). This interaction is described in the following two propositions. The oscillating part of the potential V occurs in the potential W_{αβ} as described in Assumption 1.1. By (1.1), for some function κ ∈ C_{c}^{∞}(ℝ; ℝ) such that κ = 1 on [−1; 1] and 0 ≤ κ ≤ 1, W_{αβ} = w(2i)^{−1}(e_{+}^{α} − e_{−}^{α}), where

(2.1)  e_{±}^{α} : ℝ^{d} ⟶ ℂ , e_{±}^{α}(x) = (1 − κ(|x|))e^{±ik|x|^{α}} .

Let g_{0} be the metric defined in (A.2).

Proposition 2.1. [GJ2]. Let α = 1. For any function θ ∈ C_{c}^{∞}(ℝ; ℂ), there exist smooth symbols a_{±} ∈ S(1; g_{0}), b_{±}, c_{±} ∈ S(⟨x⟩^{−1}⟨ξ⟩^{−1}; g_{0}) such that

(2.2)  e_{±}^{α}θ(H_{0}) = a_{±}^{w}e_{±}^{α} + b_{±}^{w}e_{±}^{α} + e_{±}^{α}c_{±}^{w}

and, near the support of 1 − κ(| · |), a_{±} is given by

a_{±}(x; ξ) = θ(|ξ ∓ αk|x|^{α−2}x|^{2}) = θ(|ξ ∓ k|x|^{−1}x|^{2}) .

In particular, if θ has a small enough support in ]0; k^{2}/4[, then, for any ϵ ∈ [0; 1[, the operator θ(H_{0})⟨Q⟩^{ϵ} sin(k|Q|)θ(H_{0}) extends to a compact operator on L^{2}(ℝ^{d}), and it is bounded if ϵ = 1.

Remark 2.2. In dimension d = 1, the last result in Proposition 2.1 still holds true if θ has small enough support in ]0; +∞[\{k^{2}/4} (see [GJ2]).

Proof of Proposition 2.1. See Lemma 4.3 and Proposition A.1 in [GJ2]. □ In any dimension d ≥ 1, for 0 < α < 1, the above phenomenon is absent. A careful inspection of the proof of (2.2) shows that it actually works if 0 < α < 1. But, in constrast to the case α = 1, the principal symbol of θ(H_{0})⟨Q⟩^{ϵ} sin(k|Q|)θ(H_{0}), which is given by

ℝ^{2d} ∋ (x; ξ) ↦ (2i)^{−1}θ(|ξ|^{2})(a_{+} − a_{−})(x; ξ) ,

is not everywhere vanishing, for any choice of nonzero θ with support in ]0; +∞[. The conditions “|ξ|^{2} in the support of θ” and “|ξ ∓ αk|x|^{α−2}x|^{2} in the support of θ” are indeed compatible for large |x|. In this setting, namely for 0 < α < 1 and d ≥ 1, one can give the following, more precise picture with the help of an appropriate pseudodifferential calculus. Take a nonzero, smooth function θ with compact support in ]0; +∞[. For ϵ ∈]0; 1[, on L^{2}(ℝ^{d}), the operator

θ(H_{0})⟨Q⟩^{ϵ} sin(k|Q|^{α})θ(H_{0}) (resp. θ(H_{0}) sin(k|Q|^{α})θ(H_{0}))

is unbounded (resp. is not a compact operator). Indeed, for the function κ given in (2.1), the multiplication operator

(1 − κ(|Q|)) sin(k|Q|^{α})

is a pseudodifferential operator with symbol in S(1; g_{α}) for the metric g_{α} defined in (A.2). By pseudodifferential calculus for this admissible metric g_{α}, the symbol of

θ(H_{0})⟨Q⟩^{ϵ}(1 − κ(|Q|)) sin(k|Q|^{α})θ(H_{0}) , namely

θ(|ξ|^{2})#⟨x⟩^{ϵ}(1 − κ(|x|)) sin(k|x|^{α})#θ(|ξ|^{2}) ,

is not a bounded symbol. Thus, the operator is unbounded on L^{2}(ℝ^{d}), while

θ(H_{0})⟨Q⟩^{ϵ}κ(|Q|) sin(k|Q|^{α})θ(H_{0})

is compact since its symbol θ(|ξ|^{2})#⟨x⟩^{ϵ}κ(|x|) sin(k|x|^{α})#θ(|ξ|^{2}) tends to 0 at infinity. Still for the metric g_{α}, the symbol of

θ(H_{0})(1 − κ(|Q|)) sin(k|Q|^{α})θ(H_{0})

is θ(|ξ|^{2})#(1 − κ(|x|)) sin(k|x|^{α})#θ(|ξ|^{2}), that does not tend to zero at infinity. Therefore θ(H_{0})(1 − κ(|Q|)) sin(k|Q|^{α})θ(H_{0}) is not a compact operator, whereas so is θ(H_{0})κ(|Q|) sin(k|Q|^{α})θ(H_{0}).

Remark 2.3. The difference between the cases α = 1 and 0 < α < 1 sketched just above explains why we exclude the case β ≤ α < 1 in our results. Recall that the case 0 < α < β ≤ 1 is covered by Theorem 1.2 (cf. Remark 1.6).

In the case α > 1, one can relax the localisation to get compactness as seen in

Proposition 2.4. Let α > 1. For any real p ≥ 0, there exist ℓ_{1} ≥ 0 and ℓ_{2} ≥ 0 such that ⟨P ⟩^{−ℓ_{1}}⟨Q⟩^{p}(1 − κ(|Q|)) sin(k|Q|^{α})⟨P ⟩^{−ℓ_{2}} extends to a compact operator on L^{2}(ℝ^{d}). In particular, so does θ(H_{0})⟨Q⟩^{p}(1 − κ(|Q|)) sin(k|Q|^{α})θ(H_{0}), for any p and any θ ∈ C_{c}^{∞}(ℝ; ℂ).

Proof. The proof is rather elementary and postponed in Appendix D. Appropriate ℓ_{1} and ℓ_{2} depend on p, α, and on the dimension d. For instance, one can choose ℓ_{1} and ℓ_{2} greater than 1 plus the integer part of (α − 1)^{−1}(p + d). □

Remark 2.5. Take θ ∈ C_{c}^{∞}(ℝ; ℂ), τ ∈ C_{c}^{∞}(ℝ^{d}; ℂ) such that τ = 1 near zero, and α > 1. The smooth function

(x; ξ) ↦ (1 − τ(x))θ(|ξ ∓ αk|x|^{α−2}x|^{2}) ,

does not belong to S(m; g) for any weight m associated to the metric g_{0}. So we cannot use the proof of Proposition 2.1 in this case. The proof of Proposition 2.4 shows that the oscillations manage to transform a decay in ⟨P ⟩ in one in ⟨Q⟩. This is not suprising if one is aware of the following, one dimensional formula (see eq. (VII. 5; 2), p. 245, in [Sc]), pointed out by V. Georgescu. For any m ∈ ℕ, there exist λ_{0}, ⋯, λ_{2m} ∈ ℂ such that

∀x ∈ ℝ , (1 + x^{2})^{m} e^{iπx^{2}} = ∑_{j=0}^{2m} λ_{j} \frac{d^{j}}{dx^{j}} e^{iπx^{2}} .

Note that the result of Proposition 2.4 is false for α ≤ 1 by Proposition 2.1 and the discussion following it.

3. Regularity issues.

In this section, we focus on the regularity of H w.r.t. the generator of dilations A and also the multiplication operator ⟨Q⟩. We explain, in particular, why neither the Mourre theory with A as conjugate operator nor the results in [FHHH2, FH] on the absence of positive eigenvalue can be applied to H in the full framework of Assumption 1.5. Fig. 3 below provides, in the plane of the parameters (α, β), a region where those external results apply and another where they do not. We denote, for k ∈ ℕ, by H^{k}(ℝ^{d}) or simply H^{k}, the Sobolev space of L^{2}(ℝ^{d})- functions such that their distributional derivatives up to order k belong to L^{2}(ℝ^{d}). Using the Fourier transform, it can be seen as the domain of the operator ⟨P ⟩^{k}. The dual space of H^{k} can be identified with ⟨P ⟩^{−k}L^{2}(ℝ^{d}) and is denoted by H^{−k}. Recall that A is the self-adjoint realisation of (P ·Q+Q·P)/2 in L^{2}(ℝ^{d}). It is well known that the propagator ℝ ∋ t ↦ exp(itA), generated by A, acts on L^{2}(ℝ^{d}) as

(exp(itA)f)(x) = e^{td/2}f(e^{t}x) .

It preserves all the Sobolev spaces H^{k}, thus the domain D(H) = D(H_{0}) = H^{2} of H and H_{0}. The regularity spaces C^{k}(A), for k ∈ ℕ^{∗} ∪ {∞}, are defined in Appendix B. By Theorem B.3, H ∈ C^{1}(A) if and only if the form [H, A], defined on D(H)∩D(A), extends to a bounded form from H^{2} to H^{−2}, that is, if and only if there exists C > 0 such that, for all f, g ∈ H^{2},

(3.1)  |⟨f , [H, A]g⟩| ≤ C · ∥f∥_{H^{2}} · ∥g∥_{H^{2}} .

Before studying the regularity of H w.r.t. A, it is convenient to first show that H is very regular w.r.t. ⟨Q⟩. This latter property relies on the fact that V (Q) commutes with ⟨Q⟩.

Lemma 3.1. Assume that Assumptions 1.1 and 1.5 are satisfied.

(1) For i, j ∈ {1; ⋯; d}, the operators H_{0}, ⟨P ⟩, ⟨P ⟩^{2}, P_{i}, and P_{i}P_{j} all

belong to C^{∞}(⟨Q⟩) and D(⟨Q⟩⟨P ⟩) = D(⟨P ⟩⟨Q⟩).
(2) H ∈ C^{∞}(⟨Q⟩).

(3) For θ ∈ C_{c}^{∞}(ℝ; ℂ), for i, j ∈ {1; ⋯; d}, the bounded operators θ(H_{0}), P_{i}θ(H_{0}), P_{i}P_{j}θ(H_{0}), θ(H), P_{i}θ(H), and P_{i}P_{j}θ(H) belong to C^{∞}(⟨Q⟩),

and we have the inclusion θ(H)D(⟨Q⟩) ⊂ D(⟨P ⟩⟨Q⟩) ∩ D(H_{0}).

Proof. See Appendix C. □ The form [H, A] is defined on D(H) ∩ D(A) by ⟨f, [H, iA]g⟩ = ⟨Hf, Af⟩ − ⟨Af, Hf⟩. Let χ_{c} ∈ C_{c}^{∞}(ℝ^{d}; ℝ) such that χ_{c} = 1 on the compact support of V_{c}. By statement (1) in Lemma 3.1, the form [H, A] coincides, on D(⟨P ⟩⟨Q⟩) ∩ D(H_{0}), with the form [H, iA]′ given by

⟨f , [H, iA]′g⟩ = ⟨f , [H_{0}, iA]′g⟩ + ⟨f , [V_{sr}(Q), iA]′g⟩ + ⟨f , [V_{c}(Q), iA]′g⟩

+ ⟨f , [V_{lr}(Q), iA]′g⟩ + ⟨f , [W_{αβ}(Q), iA]′g⟩ (3.2)

+ ⟨f , [(v · ∇Ṽ_{sr})(Q), iA]′g⟩ , where ⟨f , [H_{0}, iA]′g⟩ = ⟨f , 2H_{0}g⟩, ⟨f , [V_{lr}, iA]′g⟩ = −⟨f , Q · (∇V_{lr})(Q)g⟩,

⟨f , [V_{sr}(Q), iA]′g⟩ = ⟨V_{sr}(Q)Qf , iP g⟩ + ⟨iP f , V_{sr}(Q)Qg⟩

+ d⟨f , V_{sr}(Q)g⟩ , (3.3)

⟨f , [V_{c}(Q), iA]′g⟩ = ⟨V_{c}(Q)f , χ_{c}(Q)Q · iP g⟩
+ ⟨χ_{c}(Q)Q · iP f , V_{c}(Q)g⟩

+ d⟨f , V_{c}(Q)g⟩ , (3.4)

⟨f , [(v · ∇Ṽ_{sr})(Q), iA]′g⟩ = ⟨Ṽ_{sr}(Q)f , (P · v(Q))(Q.P + 2^{−1}d)g⟩

+ ⟨(P · v(Q))(Q.P + 2^{−1}d)f , Ṽ_{sr}(Q)g⟩ (3.5)

⟨f , [W_{αβ}(Q), iA]′g⟩ = ⟨W_{αβ}(Q)Qf , iP g⟩ + ⟨iP f , W_{αβ}(Q)Qg⟩

+ d⟨f , W_{αβ}(Q)g⟩ . (3.6)

Here ⟨V_{sr}(Q)Qf , iP g⟩ means ∑_{j=1}^{d}⟨V_{sr}(Q)Q_{j}f , iP_{j}g⟩. Thanks to Assumption 1.1, we see that the forms [V_{sr}(Q), iA], [V_{c}(Q), iA], [(v · ∇Ṽ_{sr})(Q), iA]′, and [V_{lr}(Q), iA] are bounded on F and associated to a compact operator from F to its dual F′, for F given by H^{1}(ℝ^{d}), H^{2}(ℝ^{d}), H^{2}(ℝ^{d}) again, and L^{2}(ℝ^{d}), respectively. In particular, (3.1) holds true with H replaced by H − W_{αβ}(Q). This proves that H − W_{αβ}(Q) ∈ C^{1}(A).

Proposition 3.2. Assume Assumption 1.1 with w ≠ 0 and |α − 1| + β < 1. Then H ∉ C^{1}(A).

Remark 3.3. The Mourre theory with conjugate operator A requires a C^{1,1}(A) regularity for H, a regularity that is stronger than the C^{1}(A) regularity (cf. [ABG], Section 7). Thus this Mourre theory cannot be applied to prove our Theorem 1.8, by Proposition 3.2. As mentioned in Remark 1.6, Theorem 1.2 applies if |α − 1| + β > 1. In fact, the proof of this theorem relies on the fact that, in that case, H has actually the C^{1,1}(A) regularity. According to [Mar], H would have the C^{1,1}(A′) regularity for some other conjugate operator A′ if 2α + β > 3. Concerning the proof of the absence of positive eigenvalue in [FHHH2, FH], it is assumed in those papers that (3.1) holds true for H replaced by V. Proposition 3.2 shows that this assumption is not satisfied if |α − 1| + β < 1. In particular, our Theorem 1.14 is not covered by the results in [FHHH2, FH]. If |α − 1| + β < 1, the form [H, A] is not bounded from H^{2} to H^{−2}. However, we shall prove in Proposition 4.6 that, for appropriate function θ, the form θ(H)[H, A]θ(H) does extend to a bounded one on L^{2}(ℝ^{d}). This will give a meaning to the Mourre estimate and we shall prove its validity. Although H ∉ C^{1}(A), we shall be able to prove the “virial theorem” (see Proposition 6.1). Finally, we note that the proof of Theorem 4.15 in [GJ2] (and also the one of our Theorem 1.10) uses at the very begining that H ∈ C^{1}(A). We did not see how to modify this proof when H ∉ C^{1}(A). This explains why we chose to use the ideas of [GJ1] to prove Theorem 1.8 (see Section 9).

Proof of Proposition 3.2. Thanks to the considerations preceeding Proposition 3.2, we know that H − W_{αβ}(Q) ∈ C^{1}(A). Thus, for w ≠ 0, H ∈ C^{1}(A) if and only if the bound (3.1) holds true with H replaced by W_{αβ}(Q). Let w ≠ 0 and (α; β) such that 2|α−1|+β < 1. Let ϵ ∈]2|α−1|; 1−β +|α−1|]. We set, for all x ∈ ℝ^{d},

f(x) = (1 − κ(|x|)) · |x|^{|α−1|−2^{−1}(d+ϵ)}
and  g(x) = −(1 − κ(|x|)) · |x|^{1−α−2^{−1}(d+ϵ)} · cos(k|x|^{α}) .

Notice that f ∈ H^{2}, f ∈ D(Q · P) = D(A), and g ∈ H^{2}. Furthermore, there exists f_{1} ∈ L^{2}(ℝ^{d}) such that, for all x ∈ ℝ^{d},

x · ∇g(x) = f_{1}(x) + kα(1 − κ(|x|))|x|^{1−2^{−1}(d+ϵ)} · sin(k|x|^{α}) .

For n ∈ ℕ^{∗}, let g_{n} : ℝ^{d} ⟶ ℝ be defined by g_{n}(x) = κ(n^{−1}|x|)g(x). It belongs to H^{2}(ℝ^{d}). By the dominated convergence theorem, the sequence (g_{n})_{n} converges to g in H^{2}(ℝ^{d}). Moreover the following limits exist and we have

⟨iP f , W_{αβ}(Q)Qg⟩ = lim_{n→∞}⟨iP f , W_{αβ}(Q)Qg_{n}⟩
and ⟨f , W_{αβ}(Q)g⟩ = lim_{n→∞}⟨f , W_{αβ}(Q)g_{n}⟩ .

By the previous computation,

⟨W_{αβ}(Q)Qf , iP g_{n}⟩ = ⟨W_{αβ}(Q)f , if_{1}⟩ + o(1)
+ wkα ∫_{ℝ^{d}} κ(n^{−1}|x|)(1 − κ(|x|))^{3}|x|^{1−β+|α−1|−(d+ϵ)} · sin^{2}(k|x|^{α}) dx ,

as n → ∞. By the monotone convergence theorem, the above integrals tend to

(3.7)  ∫_{ℝ^{d}}(1 − κ(|x|))^{3}|x|^{1−β+|α−1|−(d+ϵ)} · sin^{2}(k|x|^{α}) dx ,

as n → ∞. By Lemma C.7, the integral (3.7) is infinite. If (3.1) would hold true with H replaced by W_{αβ}(Q), the sequence

(⟨f, [W_{αβ}(Q), iA]g_{n}⟩)_{n}

would converge. Therefore the integral (3.7) would be finite, by (3.6). Contradiction. Thus H ∉ C^{1}(A). □ In Fig. 3, we summarised the above results. Note that the results of [FHHH2, FH] on the absence of positive eigenvalue apply the blue region. Keeping A as conjugate operator, we could try to apply another version of Mourre commutator method, namely the one that relies on “local regularity” (see [Sa]). Let us recall this type of regularity. Remember that a bounded operator T belongs to C^{1}(A) if the map t ↦ exp(itA)T exp(−itA) is strongly C^{1} (cf. Appendix B). We say that such an operator T belongs to C^{1,u}(A) if the previous map is norm C^{1}. Let I be an open subset of ℝ. We say that H ∈ C_{I}^{1}(A) (resp. H ∈ C_{I}^{1,u}(A)) if, for any function φ ∈ C_{c}^{∞}(ℝ; ℂ) with support in I,

0  1  2

1/2

α

1 β

[figure] [figure]

red

red red blue blue blue

blue

Figure 3. H ∈ C^{1,1}(A) in the blue region; H ∉ C^{1}(A) in the red region.

φ(H) ∈ C^{1}(A) (resp. C^{1,u}(A)). The Mourre theory with “local regularity” requires some C_{I}^{1+0}(A) regularity, that is stronger than the C_{I}^{1,u}(A), to prove the LAP inside I. In our situation, we focus on open, relatively compact interval I ⊂]0; +∞[ and denote by \overline{I} the closure of I. We first recall a result in [GJ2].

Proposition 3.4. [GJ2]. Assume Assumption 1.1 with w ≠ 0, α = β = 1, and Ṽ_{sr} = V_{c} = 0. Then, for any open interval I ⊂ \overline{I} ⊂]0; +∞[, H ∉ C_{I}^{1,u}(A).

Remark 3.5. Note that, in the framework of Proposition 3.4, H ∈ C^{1}(A). This implies (cf. [GJ2]) that, for any open interval I ⊂ \overline{I} ⊂]0; +∞[, H ∈ C_{I}^{1}(A). But, since the C_{I}^{1+0}(A) regularity is not available, the Mourre theory with conjugate operator A, that is developped in [Sa], cannot apply. We believe that Proposition 3.4 still holds true for nonzero Ṽ_{sr} and V_{c}.

Proposition 3.6. Assume Assumption 1.1 with V_{c} = Ṽ_{sr} = 0, w ≠ 0, α = 1, β ∈]1/2; 1[, and ρ_{lr} > 1/2. Then, for any open interval I ⊂ \overline{I} ⊂]0; +∞[, H ∉ C_{I}^{1}(A).

Remark 3.7. By Proposition 3.6, the Mourre theory with local regularity w.r.t. the conjugate operator A cannot be applied to recover Theorem 1.8 in the region V ∩ {(1; β); 0 < β < 1}. The proof of Proposition 3.6 below is close to the one of Proposition 3.4 in [GJ2]. Since H ∉ C^{1}(A), we need however to be a little bit more careful.

Proof of Proposition 3.6. We proceed by contradiction. Assume that, for some open interval interval I ⊂ \overline{I} ⊂]0; +∞[, H ∈ C_{I}^{1}(A). Then, for all φ ∈ C_{c}^{∞}(ℝ; ℂ) with support in I, φ(H) ∈ C^{1}(A), by definition. Take such a function φ. Since H_{0} ∈ C^{1}(A), φ(H_{0}) ∈ C^{1}(A). Therefore, the form [φ(H) − φ(H_{0}), iA] extends to a bounded form on L^{2}(ℝ^{d}). We shall show that, for some bounded operator B and B′ on L^{2}(ℝ^{d}), the form B[φ(H) − φ(H_{0}), iA]B′ coincides, modulo a bounded form on L^{2}(ℝ^{d}), with the form associated to a pseudodifferential operator c^{w} w.r.t. the metric g_{0} (cf. (A.2)), the symbol of which, c, is not bounded. By (A.5), c^{w} is not bounded and we arrive at the desired contradiction. Let f, g be functions in the Schwartz space 𝒮 (ℝ^{d}; ℂ) on ℝ^{d}. We write

⟨f , Cg⟩ := ⟨f , [φ(H) − φ(H_{0}), iA]g⟩
= ⟨(φ(H)^{∗} − φ(H_{0})^{∗})f , iAg⟩ − ⟨Af , i(φ(H) − φ(H_{0}))g⟩ .

Now, we use (C.5) with k = 0 and the resolvent formula to get

⟨f , Cg⟩ = ∫_{ℂ} ∂z̄φ^{ℂ}(z){⟨(z̄ − H)^{−1}V (Q)(z̄ − H_{0})^{−1}f , iAg⟩
− ⟨Af , i(z − H)^{−1}V (Q)(z − H_{0})^{−1}g⟩}dz ∧ dz̄.

Recall that V = V_{sr} + W with W = V_{lr} + W_{1β}. Using (C.12), we can find a bounded operator B_{1} such that

⟨f , (C − B_{1})g⟩ = ∫_{ℂ} ∂z̄φ^{ℂ}(z){⟨(z̄ − H)^{−1}W (Q)(z̄ − H_{0})^{−1}f , iAg⟩

− ⟨Af , i(z − H)^{−1}W (Q)(z − H_{0})^{−1}g⟩}dz ∧ dz̄.

Using again the resolvent formula and (C.12) and the fact that 2β_{lr} > 1, we can find another bounded operator B_{2} such that

⟨f , (C − B_{2})g⟩ = ∫_{ℂ} ∂z̄φ^{ℂ}(z){⟨(z̄ − H_{0})^{−1}W (Q)(z̄ − H_{0})^{−1}f , iAg⟩

− ⟨Af , i(z − H_{0})^{−1}W (Q)(z − H_{0})^{−1}g⟩}dz ∧ dz̄. (3.8)

Since the form [V_{lr}(Q), iA] is bounded from H^{2} to H^{−2}, H_{1} := H_{0} +V_{lr}(Q) has the C^{1}(A) regularity. Therefore, we can redo the above computation with H replaced by H_{1} to see that the contribution of V_{lr} in (3.8) is actually bounded. Thus, for some bounded operator B_{3},

⟨f , (C − B_{3})g⟩ = ∫_{ℂ} ∂z̄φ^{ℂ}(z){⟨(z̄ − H_{0})^{−1}W_{1β}(Q)(z̄ − H_{0})^{−1}f , iAg⟩

− ⟨Af , i(z − H_{0})^{−1}W_{1β}(Q)(z − H_{0})^{−1}g⟩}dz ∧ dz̄.

Recall that W_{1β} = w(2i)^{−1}(e_{+} − e_{−}), where e_{±} = e_{±}^{α} is given by (2.1) with α = 1. Let χ_{β} : [0; +∞[⟶ ℝ be a smooth function such that χ_{β} = 0 near 0 and χ_{β}(t) = t^{−β} when t belongs to the support of 1 − κ. Thus, ⟨f , (C − B_{3})g⟩ is

= −\frac{w}{2} ∑_{σ∈\{±1\}} σ ∫_{ℂ} ∂z̄φ^{ℂ}(z)\{⟨e_{σ}(Q)(z̄ − H_{0})^{−1}f , χ_{β}(|Q|)(z − H_{0})^{−1}Ag⟩

− ⟨χ_{β}(|Q|)(z̄ − H_{0})^{−1}Af , e_{σ}(Q)(z − H_{0})^{−1}g⟩}dz ∧ dz̄.

Now, we use the arguments of the proof of Lemma 5.5 in [GJ2] to find a symbol b ∈ S(1; g_{0}) such that, for B′ = e^{ik|Q|}, for all f, g ∈ 𝒮 (ℝ^{d}; ℂ), ⟨f , b^{w}(C − B_{3})B′g⟩ = ⟨f , c^{w}g⟩, where c is unbounded. Actually, there exist ξ ∈ ℝ^{d}, R > 0 and C > 0 such that |c(x; ξ)| ≥ C|x|^{1−β}, for |x| ≥ R. □

4. The Mourre estimate.

In this section, we establish a Mourre estimate for the operator H near appropriate positive energies. In the spirit of [FH], we deduce from it spacial decaying, polynomial bounds on the possible eigenvectors of H at that energies. Since H does not have a good regularity w.r.t. the conjugate operator A (cf. Section 3), the abstract setting of Mourre theory does not help much and we have to look more precisely at the structure of H. The properties derived in Section 2 play a key role in the result. Still working under Assumption 1.1, we shall modify, only in the case α = 1, Assumption 1.5 by requiring the following

Assumption 4.1. Let α, β > 0. Recall that β_{lr} = min(β; ρ_{lr}). Unless |α − 1| + β > 1, we take α ≥ 1 and we take β and ρ_{lr} such that β + β_{lr} > 1 or, equivalently, β > 1/2 and ρ_{lr} > 1 − β. We consider a compact interval J such that J ⊂]0; +∞[, except when α = 1 and β ∈]1/2; 1], and, in the latter case, we consider a small enough, compact interval J such that J ⊂]0; k^{2}/4[.

Remark 4.2. Assumption 4.1 is identical to Assumption 1.5, except for the change of the name of the interval and for the smallness requirement when α = 1 and β ∈]1/2; 1]. We actually need to work in a slightly larger interval J than the interval I considered in Theorem 1.8. In the case α = 1 and β ∈]1/2; 1], the smallness of J (and thus of the above I) is the one that matches the smallness required in Proposition 2.1. It depends only on the distance of the middle point of J to k^{2}/4.

As pointed out in Section 3, the form [H, A] does not extend to a bounded form from H^{2} to H^{−2} for a certain range of the parameters α and β. Thus, given a function θ ∈ C_{c}^{∞}(ℝ; ℂ), we do not know a priori if the forms θ(H)[H, iA]θ(H) and θ(H)[H, iA]′θ(H) extend to a bounded one on L^{2}. Recall that [H, iA]′ is defined in (3.2). Nethertheless these two forms are well defined and coincide on D(⟨Q⟩), by Lemma 3.1. By Section 3 again, we know that the difficulty is concentrate in the contribution of the oscillating potential W_{αβ}, namely (3.6). Thanks to the interaction between the oscillations and the kinetic operator, we are able to show the following

Proposition 4.3. Under Assumptions 1.1 and 4.1, let θ ∈ C_{c}^{∞}(ℝ; ℝ) with support inside J̊, the interior of J, the form θ(H)[W_{αβ}(Q), iA]θ(H) extends to a bounded form on L^{2}(ℝ^{d}) that is associated to a compact operator.

Remark 4.4. In dimension d = 1 with α = 1, the result still holds true if the function θ is supported inside ]0; +∞[\{k^{2}/4}.

Our proof of Proposition 4.3 relies on Propositions 2.1, 2.4, and on the following Lemma 4.5. Assume Assumptions 1.1 and 1.5 satisfied. Let θ ∈ C_{c}^{∞}(ℝ; ℂ). Then ⟨Q⟩^{β_{lr}}(θ(H)−θ(H_{0})) and ⟨Q⟩^{β_{lr}}P (θ(H)−θ(H_{0})) are bounded on L^{2}(ℝ^{d}).

Proof. See Lemma C.5. □ Proof of Proposition 4.3. It suffices to study the form θ(H)[W_{αβ}(Q), iA]′θ(H), where [W_{αβ}(Q), iA]′ is defined in (3.6). Consider first the case where |α − 1| + β > 1. By Remark 1.6, the form [W_{αβ}(Q), iA]′ is of one of the types [V_{lr}(Q), iA]′, (3.3), and (3.5). It is thus compact from H^{2} to H^{−2}. Since ⟨P ⟩^{2}θ(H) is bounded, the form θ(H)[W_{αβ}(Q), iA]′θ(H) extends to a bounded one on L^{2}(ℝ^{d}), that is associated to a compact operator on L^{2}(ℝ^{d}). We assume now that |α − 1| + β ≤ 1. Since β > 0, the form θ(H)W_{αβ}(Q)θ(H) extends to a bounded form associated to a compact operator. We study the form (f, g) ↦ ⟨P θ(H)f , W_{αβ}(Q)Qθ(H)g⟩, the remainding term being treated in a similar way. We write this form as

θ(H)P · QW_{αβ}(Q)θ(H) = (θ(H) − θ(H_{0}))P · QW_{αβ}(Q)(θ(H) − θ(H_{0}))

+ (θ(H) − θ(H_{0}))P · QW_{αβ}(Q)θ(H_{0})

+ θ(H_{0})P · QW_{αβ}(Q)(θ(H) − θ(H_{0})) (4.1)

+ θ(H_{0})P · QW_{αβ}(Q)θ(H_{0}) .

Using Lemma 4.5 and the fact that β + β_{lr} − 1 > 0, we see that the first three terms on the r.h.s. of (4.1) extends to a compact operator. So does also the last term, by Proposition 2.1 with ϵ = 1 − β, if α = 1, and by Proposition 2.4 with p = 1 − β, if α > 1. □ Now, we are in position to prove the Mourre estimate.

Proposition 4.6. Under Assumptions 1.1 and 4.1, let θ ∈ C_{c}^{∞}(ℝ, ℝ) with support inside the interior J̊ of the interval J. Denote by c > 0 the infimum of J. Then the form θ(H)[H, iA]θ(H) extends to a bounded one on L^{2}(ℝ^{d}) and there exists a compact operator K on L^{2}(ℝ^{d}) such that

(4.2)  θ(H)[H, iA]θ(H) ≥ 2c θ(H)^{2} + K .

Proof. Let K_{0} be the operator associated with the form

θ(H)[V_{sr}(Q), iA]θ(H) + θ(H)[(v · ∇Ṽ_{sr})(Q), iA]′θ(H)
+ θ(H)[V_{lr}(Q), iA]θ(H) + θ(H)[V_{c}(Q), iA]θ(H)
+ θ(H)[W_{αβ}(Q), iA]θ(H) .

It is compact by Section 3 and Proposition 4.3. Thus, as forms,

θ(H)[H, iA]θ(H) = θ(H)[H_{0}, iA]θ(H) + K_{0} .

Since [H_{0}, iA] = 2H_{0}, the form

(θ(H) − θ(H_{0}))[H_{0}, iA]θ(H) + θ(H_{0})[H_{0}, iA](θ(H) − θ(H_{0}))

is associated to a compact operator K_{1}, by Lemma 4.5, and

θ(H)[H, iA]θ(H) = θ(H_{0})[H_{0}, iA]θ(H_{0}) + K_{0} + K_{1}
≥ 2c θ(H_{0})^{2} + K_{0} + K_{1}
≥ 2c θ(H)^{2} + K_{0} + K_{1} + K_{3} ,

with compact K_{3} = 2c(θ(H_{0})^{2} − θ(H)^{2}). □

5. Polynomial bounds on possible eigenfunctions with positive energy.

In this section, we shall show a polynomially decaying bound on the possible eigenfunctions of H with positive energy. Because of the oscillating behaviour of the potential W_{αβ}, the corresponding result in [FH] does not apply (cf. Section 3) but it turns out that one can adapt the arguments from [FH] to the present situation. We note further that the abstract results in [Ca, CGH] cannot be applied here because of the lack of regularity w.r.t. the generator of dilations (cf. Section 3).

Proposition 5.1. Under Assumptions 1.1 and 4.1, let E ∈ J̊ and ψ ∈ D(H) such that Hψ = Eψ. Then, for all λ ≥ 0, ψ ∈ D(⟨Q⟩^{λ}) and ∇ψ ∈ D(⟨Q⟩^{λ}).

Corollary 5.2. Under Assumptions 1.1 and 4.1, for E ∈ J̊, Ker(H − E) ⊂ D(A).

Proof. Let ψ ∈ Ker(H − E). By Proposition 5.1, ∇ψ ∈ D(⟨Q⟩) thus ψ ∈ D(A). □ Proof of Proposition 5.1. We take a function θ ∈ C_{c}^{∞}(ℝ; ℝ) with support inside J̊ such that θ(E) = 1. By Proposition 4.6, the Mourre estimate (4.2) holds true. Now we follow the beginning of the proof of Theorem 2.1 in [FH], making appropriate adaptations. For λ ≥ 0 and ϵ > 0, we consider the function F : ℝ^{d} → ℝ defined by F (x) = λ ln(⟨x⟩(1 + ϵ⟨x⟩)^{−1}). For all x ∈ ℝ^{d}, ∇F (x) = g(x)x with g(x) = λ⟨x⟩^{−2}(1 + ϵ⟨x⟩)^{−1}. Let H(F) be the operator defined on the domain D(H(F)) := D(H_{0}) = H^{2}(ℝ^{d}) by

(5.1) H(F) = e^{F (Q)}He^{−F (Q)} = H −|∇F |^{2}(Q)+(iP ·∇F (Q)+∇F (Q)·iP) .

Setting ψ_{F} = e^{F (Q)}ψ, one has ψ_{F} ∈ D(H_{0}), H(F)ψ_{F} = Eψ_{F}, and ⟨ψ_{F} , Hψ_{F} ⟩ = ⟨ψ_{F} , (|∇F |^{2}(Q) + E)ψ_{F} ⟩. Note that, since e^{F} does not contain decay in ⟨x⟩, we a priori need some argument to give a meaning to ⟨ψ_{F} , [H, iA]ψ_{F} ⟩ when β < 1, because of the contribution of W_{αβ} in (3.2). Let χ ∈ C_{c}^{∞}(ℝ; ℝ) with χ = 1 near 0 and, for R ≥ 1, let χ_{R}(t) = χ(t/R). To replace Equation (2.9) in [FH], we claim that

(5.2) lim_{R→+∞}⟨χ_{R}(⟨Q⟩)ψ_{F} , [H , iA]χ_{R}(⟨Q⟩)ψ_{F} ⟩ = −4 · ∥g(Q)^{1/2}Aψ_{F} ∥^{2}
+ ⟨ψ_{F} , G(Q)ψ_{F} ⟩ ,

where G : ℝ^{d} ∋ x ↦ ((x·∇)^{2}g)(x)−(x·∇|∇F |^{2})(x). Notice that χ_{R}(⟨Q⟩)ψ_{F} ∈ D(⟨Q⟩⟨P ⟩), so the bracket on the l.h.s. of (5.2) is well defined. Since, for x ∈ ℝ^{d}, |g(x)| ≤ λ⟨x⟩^{−1} and |G(x)| = O(⟨x⟩^{−2}), so is the r.h.s. By a direct computation,

2ℜ⟨Aχ_{R}(⟨Q⟩)ψ_{F} , i(H(F) − E)χ_{R}(⟨Q⟩)ψ_{F} ⟩
= −⟨χ_{R}(⟨Q⟩)ψ_{F} , [H , iA]χ_{R}(⟨Q⟩)ψ_{F} ⟩ − 4 · ∥g(Q)^{1/2}Aχ_{R}(⟨Q⟩)ψ_{F} ∥^{2}

+ ⟨χ_{R}(⟨Q⟩)ψ_{F} , G(Q)χ_{R}(⟨Q⟩)ψ_{F} ⟩ . (5.3)

Note that the commutator [H(F) , χ_{R}(Q)]_{∘} is well-defined since χ_{R}(Q) preserves the domain of H(F). Furthermore [H(F) , χ_{R}(Q)]_{∘} = [H_{0}(F) , χ_{R}(Q)]_{∘}, where H_{0}(F) = e^{F (Q)}H_{0}e^{−F (Q)} is a pseudodifferential operator. Notice that the l.h.s of (5.3) is given by

2ℜ⟨Aχ_{R}(Q)ψ_{F} , i[H(F) , χ_{R}(Q)]_{∘}ψ_{F} ⟩ .

Using an explicit expression for the commutator and the fact that the family of functions x ↦ ⟨x⟩χ′_{R}(⟨x⟩) is bounded, uniformly w.r.t. R, and converges pointwise to 0, as R → +∞, we apply the the dominated convergence theorem to see that the l.h.s. of (5.3) tends to 0 and that the last two terms in (5.3) converge to the r.h.s. of (5.2). Thus the limit in (5.2) exists and (5.2) holds true. Next we claim that

lim_{R→+∞}⟨χ_{R}(⟨Q⟩)ψ_{F} , [H , iA]χ_{R}(⟨Q⟩)ψ_{F} ⟩ = ⟨θ(H)ψ_{F} , [H , iA]θ(H)ψ_{F} ⟩

+ ⟨ψ_{F} , (K_{1}B_{1,ϵ} + B_{2,ϵ}K_{2})ψ_{F} ⟩ , (5.4)

where, on L^{2}(ℝ^{d}), K_{1}, K_{2} are ϵ-independent compact operators and B_{1,ϵ}, B_{2,ϵ} are bounded operators satisfying ∥B_{1,ϵ}∥ + ∥B_{2,ϵ}∥ = O(ϵ^{0}). Notice that, by Proposition 4.6, the first term on the r.h.s of (5.4) is well defined and equal to

lim_{R→+∞}⟨θ(H)χ_{R}(⟨Q⟩)ψ_{F} , [H , iA]θ(H)χ_{R}(⟨Q⟩)ψ_{F} ⟩ .

Writing each χ_{R}(⟨Q⟩)ψ_{F} as χ_{R}(⟨Q⟩)ψ_{F} = θ(H)χ_{R}(Q)+(1−θ(H))χ_{R}(⟨Q⟩)ψ_{F}, we split ⟨χ_{R}(⟨Q⟩)ψ_{F} , [H , iA]χ_{R}(⟨Q⟩)ψ_{F} ⟩ into four terms, one of them tending to the first term on the r.h.s of (5.4). We focus on the others. Since (1 − θ(H))ψ = 0,

(1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} = − [θ(H), χ_{R}(⟨Q⟩)]_{∘}ψ_{F}

− χ_{R}(⟨Q⟩)[θ(H), e^{F (Q)}]_{∘}ψ , (5.5)

P (1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} = − P [θ(H), χ_{R}(⟨Q⟩)]_{∘}ψ_{F}
− [P, χ_{R}(⟨Q⟩)][θ(H), e^{F (Q)}]_{∘}ψ

− χ_{R}(⟨Q⟩)P [θ(H), e^{F (Q)}]_{∘}ψ . (5.6)

Lemma 5.3. Recall that β_{lr} = min(β; ρ_{lr}) ≤ 1. For intergers 1 ≤ i, j ≤ d, let τ(P) = 1, or τ(P) = P_{i}, or τ(P) = P_{i}P_{j}.

(1) For σ ∈ [0; 1], the operators

⟨Q⟩^{1−σ}τ(P)[θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}⟨Q⟩^{σ}

are bounded on L^{2}(ℝ^{d}), uniformly w.r.t. ϵ ∈]0; 1].

(2) For R ≥ 1, the operators

⟨Q⟩^{1−β_{lr}}τ(P)[θ(H), χ_{R}(⟨Q⟩)]_{∘}

are bounded on L^{2}(ℝ^{d}) and their norm are O(R^{−β_{lr}}).

Proof. For the result (2), see the proof of Lemma C.6. Let us prove (1). Making use of Helffer-Sjöstrand formula (C.5) and of (C.12), for H′ = H, we can show by induction that, for all j ∈ ℕ^{∗},

(5.7)  ⟨Q⟩^{1−σ} · ad_{⟨Q⟩}^{j}(θ(H)) · ⟨Q⟩^{σ}

is bounded on L^{2}(ℝ^{d}). Note that the function e^{F} can be written as φ_{ϵ}(⟨·⟩), where φ_{ϵ} stays in a bounded set in S^{λ}, when ϵ varies in ]0; 1]. Since θ(H) ∈ C^{∞}(⟨Q⟩) (cf. Lemma 3.1), we can apply Propositions C.3 with B = θ(H) and k > λ + 1. By (5.7), the first terms are all bounded on L^{2}(ℝ^{d}). Let us focus on the last one, that contains an integral. Exploiting (C.2) with ℓ = k + 1, (C.3), (C.7), (5.7), and the fact that φ_{ϵ}(⟨·⟩) is bounded below by 1/2 for ϵ ∈]0; 1], we see that the last term is also bounded on L^{2}(ℝ^{d}). □ Proof of Proposition 5.1 continued. Using Lemma 5.3 and (5.5), we get that

lim_{R→+∞}⟨θ(H)χ_{R}(⟨Q⟩)ψ_{F} , P · QW_{αβ}(Q)(1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} ⟩

=  − ⟨Kψ_{F} , W_{αβ}(Q)⟨Q⟩^{β}Q⟨Q⟩^{−1} · ⟨Q⟩[θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}ψ_{F} ⟩

where K is an ϵ-independent vector of compact operators and the bounded operator acting on the right ψ_{F} is uniformly bounded w.r.t. ϵ. Similarly, using Lemma 5.3 and (5.6), we see that

lim_{R→+∞}⟨θ(H)χ_{R}(⟨Q⟩)ψ_{F} , W_{αβ}(Q)Q · P (1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} ⟩

=  − ⟨K′ψ_{F} , W_{αβ}(Q)⟨Q⟩^{β}Q⟨Q⟩^{−1} · ⟨Q⟩P [θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}ψ_{F} ⟩

with K′ compact and an uniformly bounded operator acting on the right ψ_{F}. Using again (5.5) and (5.6), we also get

lim_{R→+∞}⟨(1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} , W_{αβ}(Q)Q · P (1 − θ(H))χ_{R}(⟨Q⟩)ψ_{F} ⟩

=  ⟨⟨Q⟩^{−β/2}[θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}⟨Q⟩^{β/2}⟨P ⟩K′′ψ_{F} ,
W_{αβ}(Q)⟨Q⟩^{β}Q⟨Q⟩^{−β/2}P [θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}ψ_{F} ⟩

with compact K′′ = ⟨P ⟩^{−1}⟨Q⟩^{−β/2} and uniformly bounded operators acting on the right ψ_{F} and on K′′ψ_{F}. In a similar way, we can treat the last term in the contribution of [W_{αβ}(Q), iA]′ and the contribution of the forms [H_{0}, iA]′, [V_{lr}(Q), iA]′, [V_{sr}(Q), iA]′, [V_{c}(Q), iA]′, and [(v · ∇Ṽ_{sr})(Q), iA]′ (cf. (3.2), (3.3), (3.4), (3.5)). This ends the proof of (5.4), yielding, together with (5.2),

(5.8)⟨θ(H)ψ_{F} , [H , iA]θ(H)ψ_{F} ⟩ = −4 · ∥g(Q)^{1/2}Aψ_{F} ∥^{2} + ⟨ψ_{F} , G(Q)ψ_{F} ⟩

− ⟨ψ_{F} , (K_{1}B_{1,ϵ} + B_{2,ϵ}K_{2})ψ_{F} ⟩ .

Assume that, for some λ > 0, ψ ∉ D(⟨Q⟩^{λ}). We define Ψ_{ϵ} = ∥ψ_{F} ∥^{−1}ψ_{F}. As in [FH], (H_{0} + 1)Ψ_{ϵ} and thus Ψ_{ϵ} both go to 0, weakly in L^{2}(ℝ^{d}), as ϵ → 0. Therefore ∥K_{1}Ψ_{ϵ}∥+∥K_{2}Ψ_{ϵ}∥ → 0, as ϵ → 0. Since G(Q)(H_{0} +1)^{−1} is compact, ∥G(Q)Ψ_{ϵ}∥ → 0. Since (1 − θ(H))ψ = 0,

(1 − θ(H))Ψ_{ϵ} = [θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}⟨Q⟩⟨Q⟩^{−1}(H_{0} + 1)^{−1}(H_{0} + 1)Ψ_{ϵ} .

Since [θ(H), e^{F (Q)}]_{∘}e^{−F (Q)}⟨Q⟩ is uniformly bounded w.r.t. ϵ, by Lemma 5.3, and ⟨Q⟩^{−1}(H_{0} + 1)^{−1} is compact, the weak convergence to 0 of (H_{0} + 1)Ψ_{ϵ} implies the norm convergence to 0 of (1−θ(H))Ψ_{ϵ}. Thus lim_{ϵ→0} ∥θ(H)Ψ_{ϵ}∥ = 1. Dividing by ∥ψ_{F} ∥^{2} in (5.8) and then taking the “lim inf_{ϵ→0}”, we get

lim inf_{ϵ→0}⟨θ(H)Ψ_{ϵ} , [H , iA]θ(H)Ψ_{ϵ}⟩ = −4 · lim inf_{ϵ→0}∥g(Q)^{1/2}AΨ_{ϵ}∥^{2} ≤ 0 .

Now, we apply the Mourre estimate (4.2) to Ψ_{ϵ}, yielding

lim inf_{ϵ→0}⟨θ(H)Ψ_{ϵ} , [H , iA]θ(H)Ψ_{ϵ}⟩ ≥ 2c lim inf_{ϵ→0} ∥θ(H)Ψ_{ϵ}∥^{2} + 0 = 2c > 0

and a contradiction. Therefore ψ ∈ D(⟨Q⟩^{λ}), for all λ > 0. Take λ > 0. Since V (Q) is H_{0}-bounded with relative bound 0, we can find, for any δ ∈]0; 1[, some C_{δ} > 0 such that, for all ϵ > 0,

|⟨ψ_{F} , V (Q)ψ_{F} ⟩| ≤ δ⟨ψ_{F} , H_{0}ψ_{F} ⟩ + C ∥ψ_{F} ∥^{2} = δ∥∇ψ_{F} ∥^{2} + C ∥ψ_{F} ∥^{2} .

Using the equality ⟨ψ_{F} , Hψ_{F} ⟩ = ⟨ψ_{F} , (|∇F |^{2}(Q)+E)ψ_{F} ⟩, we can find C′, C′′ > 0 such that, for all ϵ > 0,

(5.9)  ∥∇ψ_{F} ∥^{2} ≤ C′∥ψ_{F} ∥^{2} ≤ C′∥⟨Q⟩^{λ}ψ∥^{2} =: (C′′)^{2} .

Now, ∇ψ_{F} = (∇F)(Q)ψ_{F} + e^{F (Q)}∇ψ, yielding, for all ϵ > 0,

∥e^{F (Q)}∇ψ∥ ≤ C′′ + ∥ψ_{F} ∥ ≤ C′′ + ∥⟨Q⟩^{λ}ψ∥ .

This shows that ∇ψ belongs to D(⟨Q⟩^{λ}). □

6. Local finitness of the point spectrum.

In the usual Mourre theory, one easily deduces from a Mourre estimate on some compact interval J the finitness of the point spectrum in any compact interval I ⊂ J̊, the interior of J, thanks to the virial Theorem. In the present situation, for some values of the parameters α and β, we do not have the required regularity of H w.r.t. A (cf. Section 3) to apply the abstract virial Theorem. But, thanks to Corollary 5.2, we are able to get it in a trivial way.

Proposition 6.1. Under Assumptions 1.1 and 4.1, let E ∈ J̊ and ψ ∈ D(H) such that Hψ = Eψ. Then ⟨ψ, [H, A]ψ⟩ = 0.

Proof. Since ψ ∈ D(A) by Corollary 5.2, ⟨ψ, [H, A]ψ⟩ is well defined and

⟨ψ , [H, A]ψ⟩ = ⟨Hψ , Aψ⟩ − ⟨Aψ , Hψ⟩ = 0 ,

because E is real and A is self-adjoint. □ Now, the Mourre estimate in Proposition 4.6 gives the

Corollary 6.2. Under Assumptions 1.1 and 4.1, for any compact interval I ⊂ J̊, the point spectrum of H inside I is finite (counted with multiplicity).

Proof. One can follow the usual proof. See [ABG] p. 295 or [Mo], for instance.

□

Thanks to Corollaries 5.2 and 6.2, we are able to prove the following regularity result. The precise definition of the mentioned regularity is given in Appendix B.

Corollary 6.3. Under Assumptions 1.1 and 4.1, for any θ ∈ C_{c}^{∞}(ℝ; ℂ) with support included in J̊, θ(H)Π ∈ C^{1}(A) and θ(H)Π ∈ C^{∞}(⟨Q⟩).

Proof. For ψ ∈ D(A), the projector ⟨ψ, ·⟩ψ belongs to C^{1}(A) since the form, defined on D(A)^{2} by

(φ_{1}; φ_{2}) ↦ ⟨φ_{1}, [⟨ψ, ·⟩ψ , A]φ_{2}⟩ = \overline{⟨ψ, φ_{1}⟩}⟨Aψ, φ_{2}⟩ − ⟨ψ, φ_{2}⟩ ⟨φ_{1}, Aψ⟩ ,

extends to a bounded one. By Corollary 6.2, the point spectrum of H inside the support of θ is some {λ_{1}; ⋯; λ_{n}} and there exist ψ_{1}, ⋯, ψ_{n} ∈ D(H) such that Hψ_{j} = λ_{j}ψ_{j}, for all j. By Corollary 5.2, ψ_{j} ∈ D(A), for all j. Since

(6.1)  θ(H) Π = ∑_{j=1}^{n} θ(λ_{j}) ⟨ψ_{j}, ·⟩ψ_{j} ,

θ(H)Π ∈ C^{1}(A). Similarly, we show θ(H)Π ∈ C^{∞}(⟨Q⟩) using (6.1) and Proposition 5.1. □

7. Exponential bounds on possible eigenfunctions with positive energy.

In this section, unless |α − 1| + β > 1, we impose α > 1. We consider positive energies and show that, a possible eigenfunction of H, associated to such energies, must satisfy some exponential bound in the L^{2}-norm. The result and the proof are almost identical to Theorem 2.1 in [FH] and its proof. We only change some argument to take into account the influence of our oscillating potential. We try to explain in Remark 7.2 below why we do not treat here the case α = 1. However, we have some information at high energy in the case α = β = 1 (see Remark 7.3).

Proposition 7.1. Under Assumptions 1.1 and 1.5 with α > 1 when |α − 1| + β ≤ 1, let E > 0 and ψ ∈ D(H) such that Hψ = Eψ. Let

r = sup{t^{2} + E; t ∈ [0; +∞[ and e^{t⟨Q⟩}ψ ∈ L^{2}(ℝ^{d})} ≥ E .

Then r = +∞.

Proof. We exactly follow the lines of the last part of Theorem 2.1 in [FH], except for one important argument and some details. Just after formula (2.35) in [FH], the authors use the boundedness of (H_{0} + 1)^{−1}[H, iA](H_{0} + 1)^{−1} to show that the l.h.s. of this formula (2.35) is bounded w.r.t. λ. Here we cannot do so (the previous form is actually unbounded, by Section 3) but provide another argument (see (7.4)) to get the same conclusion. For completeness, we recall the main lines of this last part of the proof of Theorem 2.1 in [FH]. Assume that the result is false. Then r is finite. By Proposition 4.6, the Mourre estimate (4.2) holds true for any θ ∈ C_{c}^{∞}(ℝ) with small enough support around r. Let us take such a function θ that is also identically 1 on some open interval I′ centered at r. If r = E, let r_{0} = r = E, else let r_{0} < r such that r_{0} ∈ I′. We set r_{0} = t_{0}^{2} + E with t_{0} ≥ 0. We take t_{1} > 0 such that r_{1} := (t_{0} + t_{1})^{2} + E > r and r_{1} ∈ I′. We may assume that t_{1} ≤ 1. For λ ≥ 0, let F : ℝ^{d} ⟶ ℝ be defined by F (x) = t_{0}⟨x⟩+λ ln(1+t_{1}λ^{−1}⟨x⟩). By the definition of r, we know that ⟨Q⟩^{λ}e^{t_{0}⟨Q⟩}ψ ∈ L^{2}(ℝ^{d}) (if r = E i.e. t_{0} = 0, this follows from Proposition 5.1). Thus ψ belongs to the domain of the multiplication operator e^{F (Q)}. We define ψ_{F} = e^{F (Q)}ψ and Ψ_{λ} = ∥ψ_{F} ∥^{−1}ψ_{F}. By the end of the proof of Proposition 5.1, we can show that ∇ψ_{F} belongs to the domain of ⟨Q⟩. Thus ψ_{F} ∈ D(A), therefore the expectation value ⟨ψ_{F} , [H, iA]ψ_{F} ⟩ is well defined, and a direct computation gives

(7.1)  ⟨ψ_{F} , [H, iA]ψ_{F} ⟩ = −4 · ∥g(Q)^{1/2}Aψ_{F} ∥^{2} + ⟨ψ_{F} , G(Q)ψ_{F} ⟩ ,

where g is defined by F (x) = g(x)x and G(x) = ((Q.P)^{2}g)(x)−(Q.P |∇F |^{2})(x). Uniformly w.r.t. λ ≥ 1, |∇F (x)| = O(⟨x⟩^{0}) and the matrix norm |(∇ ⊗ ∇)F (x)| = O(⟨x⟩^{−1}). Notice that e^{(t_{0}+t_{1})⟨Q⟩}ψ ∉ L^{2}(ℝ^{d}). As in [FH], we can show that λ ↦ Ψ_{λ}, λ ↦ ∇Ψ_{λ}, and λ ↦ H_{0}Ψ_{λ} are bounded for the L^{2}(ℝ^{d})- norm and tend to 0 weakly in L^{2}(ℝ^{d}), as λ → +∞. This implies, in particular, that, for any δ > 0,

(7.2)  lim_{λ→+∞}∥⟨Q⟩^{−δ}Ψ_{λ}∥ = 0 and lim_{λ→+∞}∥⟨Q⟩^{−δ}∇Ψ_{λ}∥ = 0 .

Since |G(x)| = O(⟨x⟩^{−1}) + t_{1}(t_{0} + t_{1}), uniformly w.r.t. λ ≥ 1, we derive from (7.1) and (7.2) that

(7.3)  lim sup_{λ→+∞} ⟨Ψ_{λ}, [H, iA]Ψ_{λ}⟩ ≤ t_{1}(t_{0} + t_{1}) .

Now, we claim that

(7.4)  sup_{λ≥1} |⟨Ψ_{λ}, [H, iA]Ψ_{λ}⟩| < +∞ .

Thanks to (7.4), we can follow the arguments of [FH] to get the desired contradiction for small enough t_{1}. We are left with the proof of (7.4). The form ⟨P ⟩^{−2}[H − W_{αβ}(Q), iA]⟨P ⟩^{−2} extends to a bounded one, by Section 3. Since the family (⟨P ⟩^{2}Ψ_{λ})_{λ≥1} is bounded, so is also (|⟨Ψ_{λ}, [H − W_{αβ}(Q), iA]Ψ_{λ}⟩|))_{λ≥1}. In the case |α − 1| + β > 1, the form ⟨P ⟩^{−2}[H, iA]⟨P ⟩^{−2} also extends to a bounded one, by Section 3 and Remark 1.6. Thus we get the bound (7.4).

Now assume that |α − 1| + β ≤ 1 and α > 1. In this case, the form (f, g) ↦ ⟨W_{αβ}(Q)f, iAg⟩ is not bounded from H^{2} to H^{−2} (cf. Section 3). To get the result, we shall use the fact that ψ is localised w.r.t. H at energy E and “move” this property through the e^{F (Q)} factors appearing in (7.4). To get the boundedness of (|⟨Ψ_{λ}, [W_{αβ}(Q), iA]Ψ_{λ}⟩|)_{λ≥1}, it suffices to show

(7.5)  sup_{λ≥1} |⟨W_{αβ}(Q)Ψ_{λ}, Q · P Ψ_{λ}⟩| < +∞ .

Since V (Q) is H_{0}-compact, there exists some c_{0} > 0 such that H ≥ −c_{0}. For m > c_{0}, m + H is invertible with bounded inverse. Recall that H(F) is defined in (5.1). Let H_{0}(F) = e^{F (Q)}H_{0}e^{−F (Q)}. Since |∇F (x)| = O(⟨x⟩^{0}), uniformly w.r.t. λ ≥ 1, we can find m > 0 large enough such that, for all λ ≥ 1, m+H(F) and m + H_{0}(F) are invertible with uniformly bounded inverse. Moreover, we see that V (Q)(m + H(F))^{−1} and V (Q)(m + H_{0}(F))^{−1} are uniformly bounded. For λ ≥ 1, F stays in a bounded set of the symbol class S(1; g) (see Appendix A for details). Thus, by pseudodifferential calculus, ⟨P ⟩^{2}(m + H_{0}(F))^{−1} is uniformly bounded. By the resolvent formula, so is also ⟨P ⟩^{2}(m + H(F))^{−1}. Since H_{0} ∈ C^{1}(⟨Q⟩) and H ∈ C^{1}(⟨Q⟩) by Lemma 3.1, since F is smooth, H_{0}(F) ∈ C^{1}(⟨Q⟩) and H(F) ∈ C^{1}(⟨Q⟩). Using Propositions C.3 and C.4, we see that, for ϵ ∈ [0; 1], ⟨Q⟩^{ϵ}(m + H_{0}(F))^{−1}⟨Q⟩^{−ϵ} and ⟨Q⟩^{ϵ}(m + H(F))^{−1}⟨Q⟩^{−ϵ} are bounded, uniformly w.r.t. λ ≥ 1. For ℓ ∈ ℕ, we can write ψ = (m + E)^{ℓ}(m + H)^{−ℓ}ψ. By a direct computation,

e^{F (Q)}(m + H)^{−1}e^{−F (Q)} = (m + H(F))^{−1} .

Thus, for ℓ_{1}, ℓ_{2} ∈ ℕ,

⟨W_{αβ}(Q)Ψ_{λ}, Q · P Ψ_{λ}⟩
(7.6) = (m + E)^{ℓ_{1}+ℓ_{2}}⟨QW_{αβ}(Q)(m + H(F))^{−ℓ_{1}}Ψ_{λ} , P (m + H(F))^{−ℓ_{2}}Ψ_{λ}⟩ .

In (7.6), we write

(m+H(F))^{−ℓ_{1}} = ((m+H_{0}(F))^{−1} − (m+H_{0}(F))^{−1}V (Q)(m+H(F))^{−1})^{ℓ_{1}} ,

(m+H(F))^{−ℓ_{2}} = ((m+H_{0}(F))^{−1} + (m+H(F))^{−1}V (Q)(m+H_{0}(F))^{−1})^{ℓ_{2}} ,

and expand the products. The expansion contains, up to the factor (m + E)^{ℓ_{1}+ℓ_{2}}, terms of the form

(7.7)  ⟨QW_{αβ}(Q)(m + H_{0}(F))^{−1}V (Q)(m + H(F))^{−1}B_{1}Ψ_{λ} , B_{2}Ψ_{λ}⟩ ,

where B_{1} and B_{2} are uniformly bounded operators. By Assumption 1.5, ⟨Q⟩^{1−β−β_{lr}} is bounded. For W = V_{sr}, W = V_{lr}, and W = W_{αβ}, ⟨Q⟩^{β_{lr}}W (Q) is bounded. Since, by the resolvent formula,

⟨Q⟩^{β_{lr}}V_{c}(Q)(m + H(F))^{−1}
= ⟨Q⟩^{β_{lr}}χ_{c}(Q)V_{c}(Q)⟨P ⟩^{−2}⟨P ⟩^{2}(m + H_{0}(F))^{−1}
− ⟨Q⟩^{β_{lr}}χ_{c}(Q)V_{c}(Q)⟨P ⟩^{−2}⟨P ⟩^{2}(m + H_{0}(F))^{−1}V (m + H(F))^{−1} ,

the operator ⟨Q⟩^{β_{lr}}V_{c}(Q)(m + H(F))^{−1} is uniformly bounded. Furthermore,

⟨Q⟩^{β_{lr}}(m + H_{0}(F))^{−1}(v · ∇Ṽ_{sr})(Q)(m + H(F))^{−1}

= ⟨Q⟩^{β_{lr}}(m + H_{0}(F))^{−1}(v(Q) · iP)⟨Q⟩^{−β_{lr}} · ⟨Q⟩^{β_{lr}} Ṽ_{sr}(Q)(m + H(F))^{−1}

− ⟨Q⟩^{β_{lr}}(m + H_{0}(F))^{−1}⟨Q⟩^{−β_{lr}} · ⟨Q⟩^{β_{lr}} Ṽ_{sr}(Q) · (v(Q) · iP)(m + H(F))^{−1} ,

so it is also uniformly bounded. Therefore all the terms of the form (7.7) are bounded, uniformly w.r.t. λ ≥ 1. Up to the factor (m + E)^{ℓ_{1}+ℓ_{2}}, the previous expansion contains also terms of the form

(7.8) ⟨QW_{αβ}(Q)B′_{1}Ψ_{λ} , P (m + H(F))^{−1}V (Q)(m + H_{0}(F))^{−1}B′_{2}Ψ_{λ}⟩ ,

for uniformly bounded operators B′_{1} and B′_{2}. We note that ⟨Q⟩^{β_{lr}}P ⟨Q⟩^{−β_{lr}}⟨P ⟩^{−1} is bounded and that ⟨Q⟩^{β_{lr}}⟨P ⟩^{1}(m + H(F))^{−1}⟨Q⟩^{−β_{lr}} is uniformly bounded, use again the above arguments to conclude that all the terms of the form (7.8) are bounded functions of λ. We are left with the term

(m + E)^{ℓ_{1}+ℓ_{2}}⟨QW_{αβ}(Q)(m + H_{0}(F))^{−ℓ_{1}}Ψ_{λ} , P (m + H_{0}(F))^{−ℓ_{2}}Ψ_{λ}⟩ .

By pseudodifferential calculus,

⟨P ⟩^{2ℓ_{1}}(m + H_{0}(F))^{−ℓ_{1}} and ⟨P ⟩^{2ℓ_{2}−1}P (m + H_{0}(F))^{−ℓ_{2}}

are uniformly bounded. Thus, by Proposition 2.4, this last term is bounded, if we choose ℓ_{1} and ℓ_{2} large enough. This proves (7.5) and therefore (7.4). □

Remark 7.2. In the second part of the above proof, we used the assumption α > 1 to get (7.5). Indeed, we managed to move a ”localisation” (m + H)^{−ℓ} through the multiplication operator e^{F (Q)}, creating in this way the factors ⟨P ⟩^{−ℓ_{1}} and ⟨P ⟩^{−ℓ_{2}}. Then we applied Proposition 2.4 that only holds true for α > 1 (see Remark 2.5). In the case α = 1, it is natural to try to move an appropriate localisation θ(H) through e^{F (Q)} and then use Proposition 2.1. We do not know how to bound the operator e^{F (Q)}θ(H)e^{−F (Q)} uniformly w.r.t. λ, when θ is smooth and compactly supported. Formally, e^{F (Q)}θ(H)e^{−F (Q)} = θ(H(F)) where H(F) = e^{F (Q)}He^{−F (Q)}, but the latter is not self-adjoint (see (5.1)).

Remark 7.3. In the case α = β = 1, the Mourre estimate is valid at high energy, say on any compact interval included in some [a; +∞[ with a > 0 (cf. the proof of Proposition 4.6). Take an energy E > a and ψ ∈ D(H) such that Hψ = Eψ. The proof of Theorem 2.1 in [FH] works in this situation and yields the conclusion of Proposition 7.1, namely r = +∞.

8. Eigenfunctions cannot satisfy unlimited exponential bounds.

In this section, we work under Assumption 1.1 with |α − 1| + β > 1 or with β ≥ 1/2 and |α − 1| + β ≤ 1, but, in contrast to Section 7, we impose some lower bound on the form [V (Q), iA]. Again, we study the states ψ ∈ D(H) such that Hψ = Eψ, for some E ∈ ℝ, but also assume that ψ belongs to the domain of the multiplication operator e^{γ⟨Q⟩}, for all γ ≥ 0. We shall show that such ψ must be zero. Our proof is inspired by the corresponding result in [FHHH2] (see also Theorem 4.18 in [CFKS]). In fact, when |α − 1| + β > 1, we just apply [FHHH2]. Our new contribution concerns the case where α > 1, 1 ≥ β ≥ 1/2, and α + β ≤ 2. In that case, that is not covered by the result in [FHHH2], we still arrive at the same conclusion using an appropriate bound on the contribution of the oscillating potential W_{αβ} to the commutator form [H, iA]. This provides in particular a proof of Theorem 1.14. Under Assumption 1.1, we demand, unless |α − 1| + β > 1, that β ≥ 1/2. We require further, as in [FHHH2], that the form [(V_{c} + v · ∇Ṽ_{sr})(Q), iA] is H_{0}-form-lower-bounded with relative bound less than 2. Precisely, we demand that

∃ϵ_{c} > 0 , ∃λ_{c} > 0 ; ∀φ ∈ D(H) ∩ D(A) , (8.1)

⟨φ, [(V_{c} + v · ∇Ṽ_{sr})(Q), iA]φ⟩ ≥ (ϵ_{c} − 2)⟨φ, H_{0}φ⟩ − λ_{c}∥φ∥^{2} .

We shall need the following known

Lemma 8.1. Under the previous assumptions,

∀δ ∈]0; 1[ , ∃μ_{δ} > 0 ; ∀φ ∈ D(H) ∩ D(A) ,

⟨φ, H_{0}φ⟩ ≥ δ⟨φ, Hφ⟩ − μ_{δ}∥φ∥^{2} . (8.2)

∀ϵ > 0 , ∃λ_{ϵ} > 0 ; ∀φ ∈ D(H) ∩ D(A) ,

⟨φ, [H − W_{αβ}(Q), iA]φ⟩ ≥ (ϵ_{c} − ϵ)⟨φ, H_{0}φ⟩ − λ_{ϵ}∥φ∥^{2} . (8.3)

Proof. Since V (Q) is H_{0}-compact, it is H_{0}-bounded with relative bound 0. This implies (8.2) (see [K]). Recall that the form [V_{sr}(Q) + V_{lr}(Q), iA] is compact from H^{1} to H^{−1} (cf . (cf. (3.3), (3.4), (3.5)). Thus it is H_{0} form bounded with relative bound 0. Take ϵ > 0. There exists μ_{ϵ} > 0 such that, for all φ ∈ D(H) ∩ D(A),

|⟨φ, [V_{sr}(Q) + V_{lr}(Q), iA]φ⟩| ≤ ϵ⟨φ, H_{0}φ⟩ + μ_{ϵ}∥φ∥^{2} .

Therefore, for such φ, the l.h.s. of (8.3) is

≥ (2 − ϵ + ϵ_{c} − 2)⟨φ, H_{0}φ⟩ − (λ_{c} + μ_{ϵ})∥φ∥^{2} ,

by (8.1). This yieds (8.3) with λ_{ϵ} = λ_{c} + μ_{ϵ}. □ As in Section 7, we shall use a conjugaison by an appropriate e^{F (Q)}. For γ > 0, let F : ℝ^{d} ⟶ ℝ be the smooth function defined by F (x) = γ⟨x⟩. Setting g(x) = γ⟨x⟩^{−1}, ∇F (x) = g(x)x and

(8.4)  |∇F (x)|^{2} = γ^{2}(1 − ⟨x⟩^{−2}) .

A direct computation gives

(8.5)  ((Q.P)^{2}g)(x) = γ⟨x⟩^{−1}(1 − ⟨x⟩^{−2})(1 − 3⟨x⟩^{−2})
(8.6) −((Q.P)(|∇F |^{2}))(x) = −2γ^{2}⟨x⟩^{−2}(1 − ⟨x⟩^{−2}) ≤ 0 .

Proposition 8.2. Assume Assumption 1.1 and (8.1). Unless |α − 1| + β > 1, take β ≥ 1/2. Unless α + β ≤ 2 or β ≥ 1/2, take |w| small enough. Let ψ ∈ D(H) and E ∈ ℝ such that Hψ = Eψ. Assume further that, for all γ ≥ 0, ψ belongs to the domain of the multiplication operator e^{γ⟨Q⟩}. Then ψ = 0.

Remark 8.3. Note that Proposition 8.2 applies under (8.1) and Assumptions 1.1 and 1.5. In particular, the case α = 1 is allowed.

Proof of Proposition 8.2. First of all, we focus on the cases where a result in [FHHH2] applies. Assume that β > 1 or α < β ≤ 1. We use Remark 1.6 to derive, thanks to (8.3), the following property: for any ϵ > 0, there exists λ_{ϵ} > 0, such that, for all φ ∈ D(H) ∩ D(A),

(8.7)  ⟨φ, [V, iA]φ⟩ ≥ (ϵ_{c} − ϵ)⟨φ, H_{0}φ⟩ − λ_{ϵ}∥φ∥^{2} .

Therefore [FHHH2] applies. Assume now that α + β > 2 and β ≤ 1. Again by Remark 1.6, we know that the form [W_{αβ}(Q), iA] extends to a bounded one from H^{2} to H^{−2}. Thus, for |w| small enough, (8.7) still holds true and [FHHH2] applies. Now, we treat the last case: |α − 1| + β ≤ 1 and β ≥ 1/2. We always consider γ ≥ 1. By assumption, ψ belongs to the domain of the multiplication operator e^{F (Q)}. Setting ψ_{F} = e^{F (Q)}ψ, we claim that

(8.8)  |⟨ψ_{F} , [W_{αβ}(Q), iA]ψ_{F} ⟩| ≤ ∥g(Q)^{1/2}Aψ_{F} ∥^{2} + |w|^{2}γ^{−1}∥ψ_{F} ∥^{2} .

¿From the definition of the form [W_{αβ}(Q), iA], we observe that

|⟨ψ_{F} , [W_{αβ}(Q), iA]ψ_{F} ⟩| ≤ 2|w| · ∥g(Q)^{1/2}Aψ_{F} ∥ · ∥g(Q)^{−1/2}⟨Q⟩^{−β}ψ_{F} ∥
≤ 2|w| · ∥g(Q)^{1/2}Aψ_{F} ∥ · γ^{−1/2} · ∥⟨Q⟩^{1/2−β}ψ_{F} ∥
≤ 2 · ∥g(Q)^{1/2}Aψ_{F} ∥ · γ^{−1/2}|w| · ∥ψ_{F} ∥

since we assumed that β ≥ 1/2. Now (8.8) follows from the use of the inequality 2ab ≤ a^{2} + b^{2}, for a, b ≥ 0. Now, we essentially follows the argument in the proof of Theorem 4.18 in [CFKS] and prove the result by contradiction. Assume that ψ ≠ 0. Let ψ_{F} = e^{F (Q)}ψ. The formula (7.1) is valid with the new function F. As in the proof of Proposition 5.1, we also have

(8.9)  ⟨ψ_{F} , Hψ_{F} ⟩ = ⟨ψ_{F} , (|∇F |^{2}(Q) + E)ψ_{F} ⟩ .

Combining (7.1) and (8.8), we get, for γ ≥ 1,

⟨ψ_{F} , [H − W_{αβ}(Q), iA]ψ_{F} ⟩ ≤ −3 · ∥g(Q)^{1/2}Aψ_{F} ∥^{2} + |w|^{2}γ^{−1}∥ψ_{F} ∥^{2}
+ ⟨ψ_{F} , G(Q)ψ_{F} ⟩
(8.10) ⟨ψ_{F} , [H − W_{αβ}(Q), iA]ψ_{F} ⟩ ≤ ⟨ψ_{F} , G(Q)ψ_{F} ⟩ + |w|^{2}γ^{−1}∥ψ_{F} ∥^{2} ,

where G(Q) = (Q.P)^{2}g − (Q.P)(|∇F |^{2}). Next we deduce from (8.3) and (8.2) in Lemma 8.1, and (8.9), that, for all δ ∈]0; ϵ_{c}[, there exist some ρ_{δ}, ρ′_{δ} > 0 such that, for all γ ≥ 1,

⟨ψ_{F} , [H − W_{αβ}(Q), iA]ψ_{F} ⟩ ≥ δ⟨ψ_{F} , H_{0}ψ_{F} ⟩ − ρ_{δ}∥ψ_{F} ∥^{2}
≥ 2^{−1}δ(⟨ψ_{F} , Hψ_{F} ⟩ − 2μ_{1/2}) − ρ_{δ}∥ψ_{F} ∥^{2}
≥ 2^{−1}δ⟨ψ_{F} , (H − E)ψ_{F} ⟩ − ρ′_{δ}∥ψ_{F} ∥^{2}
(8.11)  ≥ 2^{−1}δ⟨ψ_{F} , |∇F |^{2}(Q)ψ_{F} ⟩ − ρ′_{δ}∥ψ_{F} ∥^{2} .

In view of (8.4), we introduce the function f : [0; +∞[→ [0; +∞[ given by

(8.12)  f(γ) = ⟨ψ_{F} , (1 − ⟨Q⟩^{−2})ψ_{F} ⟩ = γ^{−2}⟨ψ_{F} , |∇F |^{2}(Q)ψ_{F} ⟩ .

Since ψ ≠ 0, we can find ϵ > 0 such that ∥1I_{|·|≥2ϵ}(Q)ψ∥ > 0. For all γ ≥ 0,

\frac{∥1I_{|·|≤ϵ}(Q)e^{γ⟨Q⟩}ψ∥^{2}}{∥e^{γ⟨Q⟩}ψ∥^{2}} ≤ \frac{e^{2γ⟨ϵ⟩}∥1I_{|·|≤ϵ}(Q)ψ∥^{2}}{e^{2γ⟨2ϵ⟩}∥1I_{|·|≥2ϵ}(Q)ψ∥^{2}} ≤ e^{2γ(⟨ϵ⟩−⟨2ϵ⟩)} \frac{∥ψ∥^{2}}{∥1I_{|·|≥2ϵ}(Q)ψ∥^{2}}

and

f(γ) ≥ (1 − ⟨ϵ⟩^{−2})∥1I_{|·|≥ϵ}(Q)ψ_{F} ∥^{2}
≥ (1 − ⟨ϵ⟩^{−2}) · (∥ψ_{F} ∥^{2} − ∥1I_{|·|≤ϵ}(Q)e^{γ⟨Q⟩}ψ∥^{2})
≥ (1 − ⟨ϵ⟩^{−2}) · ∥ψ_{F} ∥^{2} · (1 − C_{ϵ}e^{2γ(⟨ϵ⟩−⟨2ϵ⟩)}) ,

where C_{ϵ} := ∥ψ∥^{2} · ∥1I_{|·|≥2ϵ}(Q)ψ∥^{−2}. Thus, there exist C > 0 and Γ ≥ 1 such that, for γ ≥ Γ,

(8.13)  f(γ) ≥ C ∥ψ_{F} ∥^{2} ≥ C ∥ψ∥^{2} > 0 .

We derive from (8.10) and (8.11), thanks to (8.12) and (8.6), that, for all γ ≥ 1,

2^{−1}δγ^{2}f(γ) − (ρ′_{δ} + |w|^{2}γ^{−1})∥ψ_{F} ∥^{2}

≤ ⟨ψ_{F} , G(Q)ψ_{F} ⟩ ≤ ⟨ψ_{F} , ((Q.P)^{2}g)(Q)ψ_{F} ⟩ .

By (8.5), ((Q.P)^{2}g)(x) ≤ γ(1 − ⟨x⟩^{−2}), for all x ∈ ℝ^{d}, yielding, for all γ ≥ Γ,

2^{−1}δγ^{2}f(γ) − (ρ′_{δ} + |w|^{2}γ^{−1})∥ψ_{F} ∥^{2} ≤ γf(γ)
and  (2^{−1}δγ^{2} − γ − (ρ′_{δ} + |w|^{2}γ^{−1})C^{−1}) · f(γ) ≤ 0 ,

by (8.13). We get a contradiction for γ large enough. □

9. LAP at suitable energies.

In this section, we prove the limiting absorption principle for H for appropriate energy regions. As already pointed out in [GJ2] and in Section 3, one cannot use the usual Mourre theory w.r.t. the generator of dilations A, since the Hamiltonian is not regular enough w.r.t. A. For the same reason, one cannot follow the lines in [Gé]. As explained in Remark 3.3, we were not able to apply the “weighted Mourre theory” developed in [GJ2], which is inspired by [Gé] and is a kind of “localised” Putnam argument. Instead, we follow the more complicated path introduced in [GJ1].

To prepare our result, we need some notation. For δ > 0 and y ∈ ℝ^{d}, we set

(9.1)  g_{δ}(y) = (2 − ⟨y⟩^{−δ})⟨y⟩^{−1}y .

Let χ ∈ C_{c}^{∞}(ℝ) with χ(t) = 1 if and only if |t| ≤ 1 and suppχ ⊂ [−2; 2]. Let χ̃ = 1 − χ. For R ≥ 1 and t ∈ ℝ, we set χ_{R}(t) = χ(t/R) and χ̃_{R}(t) = χ̃(t/R). We also set g_{δ,R}(y) = χ̃_{R}(⟨y⟩)^{2}g_{δ}(y). Recall that we set β_{lr} = min(ρ_{lr}, β). First, we show a kind of weighted Mourre estimate at infinity for the position operators Q (meaning for large |Q|), which can be seen as an energy localised (i.e. localised in H) Putnam positivity, that is also localised in |Q| at infinity. It should be compared with Section 2 in [La1].

Proposition 9.1. Assume Assumption 1.1. Under Assumption 4.1, take any compact interval I′ ⊂ J̊, the interior of J. Let δ be a small enough positive number (depending only on the potential) and s = (1+δ)/2. There exist c_{1} > 0 and R_{1} > 1 such that, for R ≥ R_{1}, there exists a bounded, self-adjoint operator B_{R} such that, for f ∈ L^{2}(ℝ^{d}) with E_{I′}(H)f = f, we have the estimate:

⟨f , [H, iB_{R}] f⟩ ≥ c_{1} ∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2} − O(R^{−γ})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥

− O(R^{−γ−1}) , (9.2)

with γ = 1 − δ > 1/2, if |α − 1| + β > 1, else γ = β − δ > 1/2. Here

B_{R} = g_{δ,R}(Q) · P + P · g_{δ,R}(Q) .

The ”O” terms in the estimate can be chosen independent of f when f stays in a bounded set for the norm ∥⟨Q⟩^{−s} · ∥.

Remark 9.2. In fact, we can give a precise upper bound on δ in Proposition 9.1. We demand that δ < min(β; ρ_{sr}; ρ′_{lr}; 1/2). In the case where α ≥ 1 and α+β ≤ 2, we know that β + β_{lr} > 1 and β > 1/2, by Assumption 4.1, and we further require that δ < min(β + β_{lr} − 1; β − 1/2). Denoting by c the infimum of J, one can take c_{1} = δc/2 in (9.2).

Proof. We choose δ according to Remark 9.2. We take f satisfying E_{I′}(H)f = f and belonging to some fix bounded set for the norm ∥⟨Q⟩^{−s} · ∥. Let θ ∈ C_{c}^{∞}(ℝ; ℝ) such that θ = 1 on I′ and supp θ ⊂ J̊. We have θ(H)f = f. Take R_{1} large enough such that, for R ≥ R_{1}, χ̃_{R}V_{c} = 0. In particular,

⟨f, [V_{c}(Q), iB_{R}]f⟩ = 2⟨V_{c}(Q)f, g_{δ,R}(Q) · iP f⟩ + 2⟨g_{δ,R}(Q) · iP f, V_{c}(Q)f⟩
= 0.

The other contributions of the potential are given by

⟨f , [V_{lr}(Q), iB_{R}]f⟩ = −⟨f , (g_{δ,R} · ∇V_{lr})(Q)f⟩
⟨f , [V_{sr}(Q), iB_{R}]f⟩ = 2ℜ⟨iP f , V_{sr}(Q)g_{δ,R}(Q)f⟩ ,
⟨f , [W_{αβ}(Q), iB_{R}]f⟩ = 2ℜ⟨iP f , W_{αβ}(Q)g_{δ,R}(Q)f⟩ ,

and

⟨f , [(v · ∇Ṽ_{sr})(Q), iB_{R}]f⟩ = 2ℜ⟨iP f , (v · ∇Ṽ_{sr})(Q)g_{δ,R}(Q)f⟩
= 2ℜ⟨iP f , [v(Q) · iP , Ṽ_{sr}(Q)]g_{δ,R}(Q)f⟩
= 2ℜ⟨(P · v(Q))P f , Ṽ_{sr}(Q)g_{δ,R}(Q)f⟩
− 2ℜ⟨Ṽ_{sr}(Q)P f , (v(Q) · P)g_{δ,R}(Q)f⟩
(9.3)  = 2ℜ⟨(v(Q) · P)P f , Ṽ_{sr}(Q)g_{δ,R}(Q)f⟩
+ 2ℜ⟨(∇ · v)(Q)P f , iṼ_{sr}(Q)g_{δ,R}(Q)f⟩
− 2ℜ⟨Ṽ_{sr}(Q)P f , g_{δ,R}(Q)(v(Q) · P)f⟩
+ 2ℜ⟨Ṽ_{sr}(Q)P f , i(v · ∇g_{δ,R})(Q)f⟩ .

Note that the term ⟨f , (g_{δ,R} · ∇V_{lr})(Q)f⟩ is O(R^{−ϵ})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}, for ϵ = ρ′_{lr} − δ > 0. We shall evaluate the size of the other terms. To this end, we shall repeatedly make use of Lemma C.5, of Lemma C.6 and of the fact that the term ∥⟨Q⟩^{−s}f∥ stays in a bounded region, for the considered f. Note that those lemmata follow from the regularity of H w.r.t. ⟨Q⟩. Writing

⟨(V_{sr}g_{δ,R})(Q)f , iP f⟩
= ⟨(V_{sr}g_{δ,R})(Q)f , iP θ(H)f⟩
= ⟨⟨Q⟩^{s}(V_{sr}g_{δ})(Q)χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f , [χ̃_{R}(⟨Q⟩) , iP θ(H)]⟨Q⟩^{s} · ⟨Q⟩^{−s}f⟩

+ ⟨⟨Q⟩^{2s}(V_{sr}g_{δ})(Q)χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f , ⟨Q⟩^{−s}iP θ(H)⟨Q⟩^{s} · χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f⟩ ,

the first term is O(R^{δ−1−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥ and the second term is at most of size O(R^{δ−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}, by Lemma C.6. Notice that O(R^{δ−1−ρ_{sr}}) = O(R^{−γ}) and that δ − ρ_{sr} < 0. Using that

χ̃_{R}(⟨Q⟩)P f = χ̃_{R}(⟨Q⟩)P θ(H)f
(9.4)  = [χ̃_{R}(⟨Q⟩) , P θ(H)]⟨Q⟩^{s} · ⟨Q⟩^{−s}f
+ ⟨Q⟩^{s} · ⟨Q⟩^{−s}P θ(H)⟨Q⟩^{s} · χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f ,

we see that the second term in (9.3) is

O(R^{δ−1−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥ + O(R^{δ−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}

and the fourth term is even better. For the third term, we use (9.4) twice to see that it is

O(R^{δ−2−ρ_{sr}}) + O(R^{δ−1−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥

+O(R^{δ−ρ_{sr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2} .

Note that O(R^{δ−2−ρ_{sr}}) = O(R^{−γ−1}). To evaluate the contribution of W_{αβ}, we use Remark 1.6. If 1 < β, then we can treat this contribution as the one of V_{sr}. If β ≤ 1 and α < β, then it is treated as the one of V_{lr}. If β ≤ 1 and α + β > 2, we follow the above treatment of the contribution of v · ∇Ṽ_{sr}. Thus, we are left with the case α ≥ 1 ≥ β and α+β ≤ 2. By Assymption 4.1, β +β_{lr} > 1 and, by Remark 9.2, β +β_{lr} > 1+δ. Shortening W_{αβ}(Q) as W_{αβ}, we write

⟨W_{αβ}g_{δ,R}(Q)f , iP f⟩
= ⟨χ̃_{R}(⟨Q⟩)^{2}W_{αβ}g_{δ}(Q)θ(H)f , iP θ(H)f⟩
= ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H)χ̃_{R}(⟨Q⟩)f , iP θ(H)χ̃_{R}(⟨Q⟩)f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H)χ̃_{R}(⟨Q⟩)f , [χ̃_{R}(⟨Q⟩), iP θ(H)]f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)[χ̃_{R}(⟨Q⟩), θ(H)]f , iP θ(H)χ̃_{R}(⟨Q⟩)f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)[χ̃_{R}(⟨Q⟩), θ(H)]f , [χ̃_{R}(⟨Q⟩), iP θ(H)]f⟩ .

The second and third terms are O(R^{δ−β})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥ and the last term is O(R^{δ−1−β}), by Lemma C.6. We now focus on the first term. We write

⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H)χ̃_{R}(⟨Q⟩)f , iP θ(H)χ̃_{R}(⟨Q⟩)f⟩ (9.5)

= ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H_{0})χ̃_{R}(⟨Q⟩)f , iP θ(H_{0})χ̃_{R}(⟨Q⟩)f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)(θ(H) − θ(H_{0}))χ̃_{R}(⟨Q⟩)f , iP θ(H_{0})χ̃_{R}(⟨Q⟩)f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H_{0})χ̃_{R}(⟨Q⟩)f , iP (θ(H) − θ(H_{0}))χ̃_{R}(⟨Q⟩)f⟩
+ ⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)(θ(H) − θ(H_{0}))χ̃_{R}(⟨Q⟩)f ,
iP (θ(H) − θ(H_{0}))χ̃_{R}(⟨Q⟩)f⟩ .

By Lemma C.5, the second and third terms on the r.h.s. of (9.5) are at most of size O(R^{δ+1−β−β_{lr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}, whereas the fourth one is seen to be O(R^{δ+1−β−2β_{lr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}. We write the first one as

⟨χ̃_{R/2}(⟨Q⟩)W_{αβ}g_{δ}(Q)θ(H_{0})χ̃_{R}(⟨Q⟩)f , iP θ(H_{0})χ̃_{R}(⟨Q⟩)f⟩
= ⟨W_{αβ}[χ̃_{R/2}(⟨Q⟩)g_{δ}(Q), θ(H_{0})]χ̃_{R}(⟨Q⟩)f , iP θ(H_{0})χ̃_{R}(⟨Q⟩)f⟩
+ ⟨W_{αβ}θ(H_{0})g_{δ}(Q)χ̃_{R}(⟨Q⟩)f , iP θ(H_{0})χ̃_{R}(⟨Q⟩)f⟩ .

By the above arguments, the first term on the r.h.s is

O(R^{δ−β})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}.

So is also the last term by Propositions 2.1 and 2.4. We are left with the contribution of H_{0} in the l.h.s. of (9.2). A direct computation gives [H_{0}, iB_{R}] = P^{T} · G_{δ,R} · P − h_{δ,R} where the entries of the d × d-matrix valued function G_{δ,R} on ℝ^{d} are given by

∂_{k}(χ̃_{R}(⟨·⟩)^{2}(g_{δ})_{j})(y) .

Here g_{δ}(y) = ((g_{δ})_{1}(y), ⋯, (g_{δ})_{d}(y))^{T} and T denotes the transposition. The real valued function h_{δ,R} on ℝ^{d} is given by

h_{δ,R}(y) = ∑_{1≤j,k≤d} ∂_{kkj}^{3}(χ̃_{R}(⟨·⟩)^{2}(g_{δ})_{j})(y) .

The contribution of h_{δ,R} to (9.2) is seen to be O(R^{δ−2}) = O(R^{−γ−1}). Since

∂_{k}(χ̃_{R}(⟨·⟩)^{2}(g_{δ})_{j})(y)

= χ̃_{R}(⟨y⟩)^{2}∂_{k}((g_{δ})_{j})(y) + 2(2 − ⟨y⟩^{−δ})χ̃_{R}(⟨y⟩)χ̃′_{R}(⟨y⟩)\frac{y_{j}y_{k}}{⟨y⟩^{2}} ,

2(2 − ⟨·⟩^{−δ})χ̃_{R}(⟨·⟩)χ̃′_{R}(⟨·⟩) ≥ 0, and the matrix (y_{j}y_{k}⟨y⟩^{−2})_{1≤j,k≤d} is nonnegative,

⟨f , P^{T} · G_{δ,R} · P f⟩ ≥ ⟨f , P^{T} · χ̃_{R}(⟨Q⟩)^{2}G_{δ}(Q) · P f⟩

where the entries of the d × d-matrix valued function G_{δ} on ℝ^{d} are given by ∂_{k}((g_{δ})_{j})(y). For y ∈ ℝ^{d}, G_{δ}(y) is the sum of two nonnegative matrices, namely

G_{δ}(y) = \frac{(2 − ⟨y⟩^{−δ})}{⟨y⟩}(δ_{jk} − \frac{y_{j}y_{k}}{⟨y⟩^{2}} )_{1≤j,k≤d} + \frac{δ}{⟨y⟩^{1+δ}} (\frac{y_{j}y_{k}}{⟨y⟩^{2}} )_{1≤j,k≤d}
≥ \frac{δ}{⟨y⟩^{1+δ}} (δ_{jk} − \frac{y_{j}y_{k}}{⟨y⟩^{2}} )_{1≤j,k≤d} + \frac{δ}{⟨y⟩^{1+δ}} (\frac{y_{j}y_{k}}{⟨y⟩^{2}} )_{1≤j,k≤d}
≥ \frac{δ}{⟨y⟩^{1+δ}} I_{d} ,

where I_{d} is the d × d identity matrix. This yields

⟨f , P^{T} · G_{δ,R} · P f⟩ ≥ δ⟨f , P^{T} · χ̃_{R}(⟨Q⟩)^{2}⟨Q⟩^{−2s}P f⟩ .

We write

⟨f , P^{T} · χ̃_{R}(⟨Q⟩)^{2}⟨Q⟩^{−2s}P f⟩
= ⟨θ(H)f , P^{T} · χ̃_{R}(⟨Q⟩)^{2}⟨Q⟩^{−2s}P θ(H)f⟩
= ⟨f , [θ(H)P^{T} , χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}] · [χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}, P θ(H)] f⟩
+ ⟨f , [θ(H)P^{T} , χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}] · P θ(H)χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s} f⟩
+ ⟨χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f , θ(H)P^{T} · [χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}, P θ(H)] f⟩
+ ⟨χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f , θ(H)H_{0}θ(H) χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f⟩ .

By Lemma C.6, the first term is O(R^{−2}) = O(R^{−γ−1}), the second and third ones are O(R^{−1})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥, thus also O(R^{−γ})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥. Writing H_{0} = H − V in the last term and using the fact that θ(H)V ⟨Q⟩^{β_{lr}} is bounded, this last term is

≥ c∥θ(H)χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2} − O(R^{−β_{lr}})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2} ,

where c is the infimum of J. Now, we write

∥θ(H)χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2}

= ⟨[θ(H), χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}]f , [θ(H), χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}]f⟩

+ ⟨[θ(H), χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}]f , χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f⟩

+ ⟨χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f , [θ(H), χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}]f⟩

+ ∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥^{2} .

By Lemma C.6 again, the first term is O(R^{−2}) = O(R^{−γ−1}), the second and third ones are O(R^{−1})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥ = O(R^{−γ})∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f∥. Gathering all the previous estimates and taking R_{1} large enough, we get (9.2) with c_{1} = δc/2. □ Now we are in position to prove our first main result, namely Theorem 1.8. To this end, we use the characterization of the LAP in terms of so called “special sequences”, that was introduced in [GJ1]. Proof of Theorem 1.8. Without loss of generality, the length of I may be assumed small enough. In particular, we can find a compact interval J satisfying Assumption 4.1 such that I ⊂ J̊. Since the validity of (1.4) for some s > 1/2 implies the validity of (1.4) for any s′ ≥ s, we may choose s > 1/2 as close to 1/2 as we want. Let θ ∈ C_{c}^{∞}(ℝ; ℝ) such that θ = 1 on I and supp θ ⊂ J̊. By Proposition 4.6, (4.2) holds true. Multiplying each term on both sides by χ(H)Π^{⊥}, with χθ = χ, and shriking the size of the support of χ so much that ∥Kχ(H)Π^{⊥}∥ ≤ c, we get (4.2) with 2c replaced by c, θ(H) replaced by χ(H)Π^{⊥}, and K = 0. This can be done with the requirement that χ = 1 on a small compact interval. Therefore we may assume that, for I, J, and θ, as above, we have the following strict, projected Mourre estimate

(9.6)  Π^{⊥}θ(H)[H, iA]θ(H)Π^{⊥} ≥ c θ(H)^{2}Π^{⊥} .

Recall that θ(H) ∈ C^{∞}(⟨Q⟩) and θ(H)Π ∈ C^{∞}(⟨Q⟩), by Lemma 3.1 and by Corollary 6.3, respectively. Thus θ(H)Π^{⊥} ∈ C^{∞}(⟨Q⟩) and we can apply Proposition 3.2 in [GJ2]. Therefore the LAP (1.4) is equivalent to the following statement: Take a sequence (f_{n}, z_{n})_{n∈ℕ} such that, for all n, z_{n} ∈ ℂ, ℜz_{n} ∈ I, ℑz_{n} ≠ 0, f_{n} ∈ D(H), Π^{⊥}f_{n} = f_{n}, θ(H)f_{n} = f_{n}, and (H − z_{n})f_{n} ∈ D(⟨Q⟩^{s}). Assume further that ℑz_{n} → 0, ∥⟨Q⟩^{s}(H − z_{n})f_{n}∥ → 0, and that (∥⟨Q⟩^{−s}f_{n}∥)_{n∈ℕ} converges to some real number η. Then η = 0. We shall prove this statement. Let us consider such a sequence (f_{n}, z_{n})_{n∈ℕ}. Take R ≥ 1. Notice that χ_{R}(⟨Q⟩)f_{n} actually belongs to D(H) ∩ D(⟨Q⟩). Note also that the operator A^{⊥} := Π^{⊥}θ(H)Aθ(H)Π^{⊥} is well-defined on D(⟨Q⟩), since P θ(H) is bounded and preserves, together with θ(H) and Π^{⊥}, the set D(⟨Q⟩). Since H commutes with θ(H)Π^{⊥}, we derive from (9.6) applied to χ_{R}(⟨Q⟩)f_{n} that

(9.7)  ⟨χ_{R}(⟨Q⟩)f_{n} , [H, iA^{⊥}]χ_{R}(⟨Q⟩)f_{n}⟩ ≥ c ∥θ(H)Π^{⊥}χ_{R}(⟨Q⟩)f_{n}∥^{2} .

Since θ(H)Π^{⊥} is smooth w.r.t. ⟨Q⟩,

θ(H)Π^{⊥}χ_{R}(⟨Q⟩)f_{n} = χ_{R}(⟨Q⟩)f_{n} + [θ(H)Π^{⊥}, χ_{R}(⟨Q⟩)]⟨Q⟩^{s} · ⟨Q⟩^{−s}f_{n}
= χ_{R}(⟨Q⟩)f_{n} + O(R^{s−1}) ,

thanks to Lemma C.6. The above O(R^{s−1}) and the following ”O” are all independent of n. Inserting this information in (9.7), we get

⟨χ_{R}(⟨Q⟩)f_{n} , [H, iA^{⊥}]χ_{R}(⟨Q⟩)f_{n}⟩ ≥ c ∥χ_{R}(⟨Q⟩)f_{n}∥^{2}

+ O(R^{s−1})∥χ_{R}(⟨Q⟩)f_{n}∥ + O(R^{2s−2}) . (9.8)

Now, we need information on the f_{n} for ”large ⟨Q⟩”. Let I′ a compact interval such that supp θ ⊂ I′ ⊂ J̊. Since f_{n} = θ(H)f_{n} and E_{I′}θ = θ, E_{I′}f_{n} = E_{I′}(H)θ(H)f_{n} = θ(H)f_{n} = f_{n}. Furthermore, the sequence (∥⟨Q⟩^{−s}f_{n}∥)_{n} is bounded since it converges to η, by assumption. Therefore we can apply Proposition 9.1 to f = f_{n} (choosing s close enough to 1/2, requiring in particular that s < γ), yielding (9.2) with f replaced by f_{n} and with n-independent ”O′s”. As in [GJ1] (cf. Corollary 3.2), we deduce from this that, for R ≥ R_{1},

(9.9) lim sup_{n}∥χ̃_{R}(⟨Q⟩)⟨Q⟩^{−s}f_{n}∥ = O(R^{−γ}) + O(R^{−(γ+1)/2}) = O(R^{−γ}) .

We rewrite the l.h.s of (9.8) as

⟨χ_{R}(⟨Q⟩)f_{n} , [H, iA^{⊥}]χ_{R}(⟨Q⟩)f_{n}⟩
= ⟨f_{n} , [H, iχ_{R}(⟨Q⟩)A^{⊥}χ_{R}(⟨Q⟩)]f_{n}⟩
+ 2ℜ⟨[H, χ_{R}(⟨Q⟩)]f_{n} , iA^{⊥}χ_{R}(⟨Q⟩)f_{n}⟩ .

Since, as form,

[H, χ_{R}(⟨Q⟩)] = [H_{0}, χ_{R}(⟨Q⟩)]_{∘} =

= −2∇ · (∇(χ_{R}(⟨·⟩)))(Q) + (Δ(χ_{R}(⟨·⟩)))(Q) ,

and since ⟨Q⟩^{−1}∇A^{⊥} is bounded, we obtain, using (9.9),

2ℜ⟨[H, χ_{R}(⟨Q⟩)]f_{n} , iA^{⊥}χ_{R}(⟨Q⟩)f_{n}⟩ = O(R^{s−γ})∥χ_{R}(⟨Q⟩)f_{n}∥ .

Therefore (9.8) yields

⟨f_{n} , [H, iχ_{R}(⟨Q⟩)A^{⊥}χ_{R}(⟨Q⟩)]f_{n}⟩ ≥ c ∥χ_{R}(⟨Q⟩)f_{n}∥^{2} + O(R^{2s−2})

+ O(R^{s−γ})∥χ_{R}(⟨Q⟩)f_{n}∥ . (9.10)

Expanding the commutator as in [GJ1] (cf. Proposition 2.15), we see that

(9.11)  lim_{n}⟨f_{n} , [H, iχ_{R}(⟨Q⟩)A^{⊥}χ_{R}(⟨Q⟩)]f_{n}⟩ = 0 .

Using (9.11) in (9.10), we deduce that

lim sup_{n}∥χ_{R}(⟨Q⟩)f_{n}∥ = O(R^{s−γ}) ,

with s − γ < 0. It follows from this and (9.9) that η = 0. □

10. Symbol-like long range potentials.

This section is devoted to the Proof of Theorem 1.18. Let H_{1} be the self-adjoint operator H_{0} + V_{lr}(Q) on D(H_{0}). Thanks to the assumption on V_{lr}, H_{1} is actually the Weyl quantization p^{w} of the symbol p ∈ S(⟨ξ⟩^{2}, g) defined by p(x; ξ) = |ξ|^{2} + V_{lr}(x) (see Appendix A for details). Now we redo the proofs of Theorems 1.8 and 1.14, replacing H_{0} by H_{1} at some appropriate places. More precisely, we perform this replacement exactly when the original proofs use the “decay” in ⟨Q⟩ of θ(H) − θ(H_{0}).

First, we claim that the last statement in Proposition 2.1 is valid if H_{0} is replaced by H_{1}. Indeed we can follow the proof of Lemma 4.3 in [GJ2] and arrive at (4.7) with θ(|ξ|^{2})θ(|ξ ∓kx̂|^{2}) replaced by θ(|ξ|^{2} +V_{lr}(x))θ(|ξ ∓kx̂|^{2} +V_{lr}(x)). Since the latter also vanishes for small enough support of θ, we conclude as in [GJ2]. For any ℓ ≥ 0 and any θ ∈ C_{c}^{∞}ℝ; ℂ), ⟨P ⟩^{ℓ}θ(H_{1}) is bounded by pseudodifferential calculus (cf. Appendix A). Therefore, the last statement in Proposition 2.4 holds true with H_{0} is replaced by H_{1}. We can check that the result in Lemma 4.5 holds true with H_{0} replaced by H_{1}. Thus, performing the same replacement in (4.1), we get the result of Proposition 4.3. We derive the Mourre estimate of Proposition 4.6 by the same proof. Also with the same proofs, we get the results of Proposition 5.1, Corollary 5.2, Proposition 6.1, Corollary 6.2, and Corollary 6.3. In the proof of Proposition 7.1, we modify the argument leading to the bound (7.5). Again, we replace H_{0} by H_{1}. We notice that H_{1}(F) = e^{F (Q)}H_{1}e^{−F (Q)} is also a pseudodifferential operator in C^{1}(⟨Q⟩) such that, for ϵ ∈ [0; 1], the operator ⟨Q⟩^{ϵ}(m + H_{1}(F))^{−1}⟨Q⟩^{−ϵ} is bounded, uniformly w.r.t. λ ≥ 1. Then, we can follow the end of the proof of Proposition 7.1 with β_{lr} replaced by β, H_{0}(F) by H_{1}(F), and V by V − V_{lr}. Next, we redo the proof of Proposition 8.2 without change. In the proof of Proposition 9.1, we only change the treatment of (9.5) in the following way. We can check that the results in Lemma C.5 are valid with H_{0} replaced by H_{1} and β_{lr} by β. Concerning the first term on the r.h.s of (9.5), we only need to point out that ⟨P ⟩^{ℓ}θ(H_{1}) is bounded for any ℓ, by pseudodifferential calculus. We thus obtain the result of Proposition 9.1. Finally, we recover the result of Theorem 1.8 by the same proof. □

Appendix A. Standard pseudodifferential calculus.

In this appendix, we briefly review some basic facts about pseudodifferential calculus. We refer to [Hö][Chapters 18.1, 18.4, 18.5, and 18.6] for a traditional study of the subject but also to [Bea, Bo1, Bo2, BC, Le] for a modern and powerful version. Denote by 𝒮 (M) the Schwartz space on the space M and by F the Fourier transform on ℝ^{d} given by

Fu(ξ) := (2π)^{−d} ∫_{ℝ^{d}} e^{−ix·ξ}u(x) dx ,

for ξ ∈ ℝ^{d} and u ∈ 𝒮 (ℝ^{d}). For test functions u, v ∈ 𝒮 (ℝ^{d}), let Ω(u, v) and Ω′(u, v) be the functions in 𝒮 (ℝ^{2d}) defined by

Ω(u, v)(x, ξ) := \overline{v}(x)Fu(ξ)e^{ix·ξ} ,
Ω′(u, v)(x, ξ) := (2π)^{−d} ∫_{ℝ^{d}} u(x − y/2)\overline{v}(x + y/2)e^{−iy·ξ} dy ,

respectively. Given a distribution b ∈ 𝒮′(T^{∗}ℝ^{d}), the formal quantities

(2π)^{−d} ∫_{ℝ^{3d}} e^{i(x−y)·ξ}b(x, ξ)v(x)u(y) dxdydξ ,
(2π)^{−d} ∫_{ℝ^{3d}} e^{i(x−y)·ξ}b((x + y)/2, ξ)u(x)u(y) dxdydξ

are defined by the duality brackets ⟨b, Ω(u, v)⟩ and ⟨b, Ω′(u, v)⟩, respectively. They define continuous operators from 𝒮 (ℝ^{d}) to 𝒮′(ℝ^{d}) that we denote by Op b(x, D_{x}) and b^{w}(x, D_{x}) respectively. Sometimes we simply write Op b and b^{w}, respectively. Choosing on the phase space T^{∗}ℝ^{d} a metric g and a weight function m with appropriate properties (cf., admissible metric and weight in [Le]), let S(m, g) be the space of smooth functions on T^{∗}ℝ^{d} such that, for all k ∈ ℕ, there exists c_{k} > 0 so that, for all x^{∗} = (x, ξ) ∈ T^{∗}ℝ^{d}, all (t_{1}, ⋯, t_{k}) ∈ (T^{∗}ℝ^{d})^{k},

(A.1)  |a^{(k)}(x^{∗}) · (t_{1}, ⋯, t_{k})| ≤ c_{k}m(x^{∗})g_{x^{∗}}(t_{1})^{1/2} ⋯ g_{x^{∗}}(t_{k})^{1/2} .

Here, a^{(k)} denotes the k-th derivative of the function a. We equip the vector space S(m, g) with the semi-norms ∥ · ∥_{ℓ,S(m,g)} defined by max_{0≤k≤ℓ} c_{k}, where the c_{k} are the best constants in (A.1). S(m, g) is a Fréchet space. The space of operators Op b(x, D_{x}) (resp. b^{w}(x, D_{x})) when b ∈ S(m, g) has nice properties (cf., [Hö, Le]). Defining x^{∗} = (x, ξ) ∈ T^{∗}ℝ^{d}, we stick here to the following metrics

(A.2)  (g_{0})_{x^{∗}} := \frac{dx^{2}}{⟨x⟩^{2}} + \frac{dξ^{2}}{⟨ξ⟩^{2}} and (g_{α})_{x^{∗}} := \frac{dx^{2}}{⟨x⟩^{2(1−α)}} + \frac{dξ^{2}}{⟨ξ⟩^{2}} ,

for 0 < α < 1, and to weights of the form, for p, q ∈ ℝ,

m(x^{∗}) := ⟨x⟩^{p}⟨ξ⟩^{q}. (A.3)

The gain of the calculus associated to each metric in (A.2) is given respectively by

h_{0}(x^{∗}) := ⟨x⟩^{−1}⟨ξ⟩^{−1} and h_{α}(x^{∗}) = ⟨x⟩^{1−α}⟨ξ⟩^{−1}. (A.4)

Take weights m_{1}, m_{2} as in (A.3), let g be g_{0} or g_{α}, and denote by h the gain of g̃. For any a ∈ S(m_{1}, g) and b ∈ S(m_{2}, g), there are a symbol a#_{r}b ∈ S(m_{1}m_{2}, g) and a symbol a#b ∈ S(m_{1}m_{2}, g) such that Op aOp b = Op (a#_{r}b) and a^{w}b^{w} = (a#b)^{w}. The maps (a, b) ↦ a#_{r}b and (a, b) ↦ a#b are continuous and so are also (a, b) ↦ a#_{r}b − ab ∈ S(m_{1}m_{2}h, g) and (a, b) ↦ a#b − ab ∈ S(m_{1}m_{2}h, g). If a ∈ S(m_{1}, g), there exists c ∈ S(m_{1}, g) such that a^{w} = Op c. The maps a ↦ c and a ↦ c − a ∈ S(m_{1}m_{2}h, g) are continuous. If a ∈ S(⟨ξ⟩^{m}, g) for m ∈ ℕ, a^{w} and Op a are bounded from H^{m}(ℝ^{d}) to L^{2}(ℝ^{d}) and the corresponding operator norms are controlled above by some appropriate semi-norm of a in S(⟨ξ⟩^{m}, g). In particular, they are bounded on L^{2}(ℝ^{d}), if a ∈ S(1, g). Futhermore, for a ∈ S(m, g),

(A.5)  Op a is bounded ⟺ a^{w} is bounded ⟺ a ∈ S(1, g) .

For a ∈ S(1, g),

(A.6)  Op a is compact ⟺ a^{w} is compact ⟺ lim_{|x^{∗}|→∞} a(x^{∗}) = 0 .

Finally, we recall the following result on some smooth functional calculus for pseudodifferential operators associated to some admissible metric g. This result is essentially contained in [Bo1] (see [GJ2, Le], for details). We also use it for g = g_{0} or g = g_{α}. For ρ ∈ ℝ, we denote by S^{ρ} the set of smooth functions φ : ℝ ⟶ ℂ such that, for all k ∈ ℕ, sup_{t∈ℝ}⟨t⟩^{k−ρ}|∂_{t}^{k}φ(t)| < ∞. If we take a real symbol a ∈ S(m, g), then the operator a^{w} is self-adjoint on the domain D(a^{w}) = {u ∈ L^{2}(ℝ_{x}^{d}); a^{w}u ∈ L^{2}(ℝ_{x}^{d})}. In particular, the operator φ(a^{w}) is well defined by the functional calculus if φ is a borelean function on ℝ. We assume that m ≥ 1. A real symbol a ∈ S(m, g) is said elliptic if (i − a)^{−1} belongs to S(m^{−1}, g). Recall that h denotes the gain of the symbolic calculus in S(m, g).

Theorem A.1. Let m ≥ 1 and a ∈ S(m, g) be real and elliptic. Take a function φ ∈ S^{ρ}. Then φ(a) ∈ S(m^{ρ}, g) and there exists b ∈ S(hm^{ρ}, g) such that

(A.7)  φ(a^{w}(x, D)) = (φ(a))^{w}(x, D) + b^{w}(x, D).

Appendix B. Regularity w.r.t. an operator.

For sake of completness, we recall here important facts on the regularity w.r.t. to a self-adjoint operator. Further details can be found in [ABG, DG, GJ2, GGM, GGé]. Let ℋ be a complex Hilbert space. The scalar product ⟨·, ·⟩ in ℋ is right linear and ∥ · ∥ denotes the corresponding norm and also the norm in B(ℋ ), the space of bounded operators on ℋ . Let M be a self-adjoint operator in ℋ . Let T be a closed operator in ℋ . The form [T, M] is defined on (D(M) ∩ D(T)) × (D(M) ∩ D(T)) by

(B.1)  ⟨f , [T, M]g⟩ := ⟨T^{∗}f , Mg⟩ − ⟨Mf , T g⟩ .

If T is a bounded operator on ℋ and k ∈ ℕ, we say that T ∈ C^{k}(M) if, for all f ∈ ℋ , the map ℝ ∋ t ↦ e^{itM}T e^{−itM}f ∈ ℋ has the usual C^{k} regularity. The following characterization is available.

Proposition B.1. [ABG, p. 250]. Let T ∈ B(ℋ ). Are equivalent:

(1) T ∈ C^{1}(M).

(2) The form [T, M] defined on D(M) × D(M) extends to a bounded form on ℋ × ℋ associated to a bounded operator denoted by ad_{M}^{1}(T) := [T, M]_{∘}.

(3) T preserves D(M) and the operator T M − MT, defined on D(M), extends to a bounded operator on ℋ .

It immediately follows that T ∈ C^{k}(M) if and only if the iterated commutators ad_{M}^{p}(T) := [ad_{M}^{p−1}(T), M]_{∘} are bounded for p ≤ k. It turns out that T ∈ C^{k}(M) if and only if, for a z outside σ(T), the spectrum of T, (T − z)^{−1} ∈ C^{k}(M). Now, let N be a self-adjoint operator in ℋ . It is natural to say that N ∈ C^{k}(M) if (N − z)^{−1} ∈ C^{k}(M) for some z ∉ σ(N). In that case, (N − z)^{−1} ∈ C^{k}(M), for all z ∉ ℝ. Lemma 6.2.9 and Theorem 6.2.10 in [ABG] gives the following characterization of this regularity:

Theorem B.2. [ABG, p. 251]. Let M and N be two self-adjoint operators in the Hilbert space ℋ . For z ∉ σ(N), set R(z) := (N − z)^{−1}. The following points are equivalent:

(1) N ∈ C^{1}(M).

(2) For one (then for all) z ∉ σ(N), there is a finite c such that

|⟨Mf, R(z)f⟩ − ⟨R(z̄)f, Mf⟩| ≤ c∥f∥^{2}, for all f ∈ D(M). (B.2)

(3) a. There is a finite c such that for all f ∈ D(M) ∩ D(N):

(B.3)  |⟨Mf, Nf⟩ − ⟨Nf, Mf⟩| ≤ c(∥Nf∥^{2} + ∥f∥^{2}).

b. The set {f ∈ D(M); R(z)f ∈ D(M) and R(z̄)f ∈ D(M)} is a core for M, for some (then for all) z ∉ σ(N).

Note that the condition (3.b) could be uneasy to check, see [GGé]. We mention [GM][Lemma A.2] to overcome this subtlety. Note that (B.2) yields that the commutator [M, R(z)] extends to a bounded operator, in the form sense. We shall denote the extension by [M, R(z)]_{∘}. In the same way, from (B.3), the commutator [N, M] extends to a unique element of B(D(N), D(N)^{∗}) denoted by [N, M]_{∘}. Moreover, if N ∈ C^{1}(M) and z ∉ σ(N),

[M, (N − z)^{−1}]_{∘} = \underbrace{(N − z)^{−1}}_{ℋ ←D(N)^{∗}}  \underbrace{[N, M]_{∘}}_{D(N)^{∗}←D(N)} \underbrace{(N − z)^{−1}}_{D(N)←ℋ}. (B.4)

Here we used the Riesz lemma to identify ℋ with its anti-dual ℋ ^{∗}. It turns out that an easier characterization is available if the domain of N is conserved under the action of the unitary group generated by M.

Theorem B.3. [ABG, p. 258]. Let M and N be two self-adjoint operators in the Hilbert space ℋ such that e^{itM}D(N) ⊂ D(N), for all t ∈ ℝ. Then N ∈ C^{1}(M) if and only if (B.3) holds true.

Appendix C. Commutator expansions.

In this appendix, we recall known results on functional calculus and on commutator expansions. Details can be found in [DG, GJ1, GJ2, Mø]. We then apply these results to get several facts used in the main part of the text. We make use of pseudodifferential calculus (cf. Appendix A) and of the regularity w.r.t. an operator, recalled in Appendix B. As in Appendix A, we consider, for ρ ∈ ℝ, the set S^{ρ} of functions φ ∈ C^{∞}(ℝ; ℂ) such that

∀k ∈ ℕ, C_{k}(φ) := sup_{t∈ℝ} ⟨t⟩^{−ρ+k}|∂_{t}^{k}φ(t)| < ∞. (C.1)

Equipped with the semi-norms defined by (C.1), S^{ρ} is a Fréchet space. We recall the following result from [DG] on almost analytic extension.

Proposition C.1. [DG]. Let φ ∈ S^{ρ} with ρ ∈ ℝ. There is a smooth function φ^{ℂ} : ℂ → ℂ, called an almost analytic extension of φ, such that, for all l ∈ ℕ,

(C.2)  φ^{ℂ}|_{ℝ} = φ, |∂z̄φ^{ℂ}(z)| ≤ c_{1}⟨Re(z)⟩^{ρ−1−l}|Im(z)|^{l} ,

supp φ^{ℂ} ⊂ {x + iy; |y| ≤ c_{2}⟨x⟩}, (C.3)

φ^{ℂ}(x + iy) = 0, if x ∉ supp φ, (C.4)

for constants c_{1}, c_{2} only depending on the semi-norms (C.1) of φ in S^{ρ}.

Next we recall Helffer-Sjöstrand’s functional calculus (cf., [HeS, DG]). As in Appendix B, we consider a self-adjoint operator M acting in some complex Hilbert space ℋ . For ρ < 0, k ∈ ℕ, and φ ∈ S^{ρ}, the bounded operators (∂^{k}φ)(M) can be recovered by

(∂^{k}φ)(M) = \frac{i(k!)}{2π} ∫_{ℂ} ∂z̄φ^{ℂ}(z)(z − M)^{−1−k}dz ∧ dz̄, (C.5)

where the integral exists in the norm topology, by (C.2) with l = 1. For ρ ≥ 0, we rely on the following approximation:

Proposition C.2. [GJ1]. Let ρ ≥ 0 and φ ∈ S^{ρ}. Let χ ∈ C_{c}^{∞}(ℝ) with χ = 1 near 0 and 0 ≤ χ ≤ 1, and, for R > 0, let χ_{R}(t) = χ(t/R). For f ∈ D(⟨M⟩^{ρ}) and k ∈ ℕ, there exists

(∂^{k}φ)(M)f = lim_{R→+∞} \frac{i}{2π} ∫_{ℂ} ∂z̄(φχ_{R})^{ℂ}(z)(z − M)^{−1−k}f dz ∧ dz̄. (C.6)

The r.h.s. converges for the norm in ℋ . It is independent of the choise of χ.

Notice that, for some c > 0 and s ∈ [0; 1], there exists some C > 0 such that, for all z = x + iy ∈ {a + ib | 0 < |b| ≤ c⟨a⟩} (like in (C.3)),

∥⟨M⟩^{s}(M − z)^{−1}∥ ≤ C⟨x⟩^{s} · |y|^{−1}. (C.7)

Observing that the self-adjointness assumption on B is useless, we pick from [GJ1] the following result in two parts.

Proposition C.3. [BG, DG, GJ1, Mø]. Let k ∈ ℕ^{∗}, ρ < k, φ ∈ S^{ρ}, and B be a bounded operator on ℋ such that B ∈ C^{k}(M). As forms on D(⟨M⟩^{k−1}) × D(⟨M⟩^{k−1}),

(C.8) [φ(M), B] = ∑_{j=1}^{k−1} \frac{1}{j!}(∂^{j}φ)(M)ad_{M}^{j}(B)

+ \frac{i}{2π} ∫_{ℂ} ∂z̄φ^{ℂ}(z)(z − M)^{−k}ad_{M}^{k}(B)(z − M)^{−1}dz ∧ dz̄. (C.9)

In particular, if ρ ≤ 1, then B ∈ C^{1}(φ(M)).

The rest of the previous expansion is estimated in Proposition C.4. [BG, GJ1, Mø]. Let B be a bounded operator on ℋ such that B ∈ C^{k}(M). Let φ ∈ S^{ρ}, with ρ < k. Let I_{k}(φ) be the rest of the development of order k (C.8) of [φ(M), B], namely (C.9). Let s, s′ ≥ 0 such that s′ < 1, s < k, and ρ + s + s′ < k. Then, for φ staying in a bounded subset of S^{ρ}, ⟨M⟩^{s}I_{k}(φ)⟨M⟩^{s′} is bounded and there exists a M and φ independent constant C > 0 such that ∥⟨M⟩^{s}I_{k}(φ)⟨M⟩^{s′}∥ ≤ C∥ad_{M}^{k}(B)∥.

Now, we show a serie of results needed in the main text. Most of them are more or less known. We provide proofs for completeness. Proof of Lemma 3.1. The assumptions 1.1 and 1.5 are not required for the proof of (1). We note that (1 + H_{0})^{−1} = a^{w} and ⟨Q⟩ = b^{w}, where a(x, ξ) = (1 + |ξ|^{2})^{−1} and b(x, ξ) = ⟨x⟩, Since a ∈ S(⟨ξ⟩^{−2}, g_{0}) and b ∈ S(⟨x⟩, g_{0}), where the metric g_{0} defined in (A.2), the form [(1 + H_{0})^{−1}, ⟨Q⟩] is associated to c^{w} with c ∈ S(h⟨ξ⟩^{−2}⟨x⟩, g_{0}) = S(⟨ξ⟩^{−3}, g_{0}), by pseudodifferential calculus. Since S(⟨ξ⟩^{−3}, g_{0}) ⊂ S(1, g_{0}), the form [(1 + H_{0})^{−1}, ⟨Q⟩] extends to bounded one on L^{2}(ℝ^{d}). Similarly, we can show that the iterated commutators ad_{⟨Q⟩}^{p}((1 + H_{0})^{−1}) all extend to bounded operator on L^{2}(ℝ^{d}). By the comment just after Proposition B.1, (1+H_{0})^{−1} ∈ C^{∞}(⟨Q⟩) and H_{0} ∈ C^{∞}(⟨Q⟩), by definition. Since ⟨P ⟩ = d^{w} with d(x, ξ) = (1 + |ξ|^{2})^{1/2}, we can follow the same lines to prove that ⟨P ⟩^{−1} ∈ C^{∞}(⟨Q⟩) and thus ⟨P ⟩ ∈ C^{∞}(⟨Q⟩). Similarly, P_{i}, P_{i}P_{j}, ⟨P ⟩^{2} ∈ C^{∞}(⟨Q⟩). Since the form [⟨P ⟩, ⟨Q⟩] is associated to bounded pseudodifferential operator, we see that D(⟨Q⟩⟨P ⟩) = D(⟨P ⟩⟨Q⟩). By a direct computation, we see that the group e^{it⟨Q⟩} (for t ∈ ℝ) preserves the Sobolev space H^{2}(ℝ^{d}), which is the domain of H. Furthermore the form [H, ⟨Q⟩] coïncide on D(H) ∩ D(⟨Q⟩) with [H_{0}, ⟨Q⟩]. The latter is associated, by pseudodifferential calculus, to a pseudodifferential operator that is bounded from H^{1}(ℝ^{d}) to L^{2}(ℝ^{d}). By Theorem B.3, H ∈ C^{1}(⟨Q⟩) and, for z ∈ ℂ \ ℝ,

(C.10)  [(z − H)^{−1}, ⟨Q⟩]_{∘} = (z − H)^{−1} [H, ⟨Q⟩]_{∘}(z − H)^{−1} .

On D(⟨Q⟩) × D(⟨Q⟩), we can write the form [[(z − H)^{−1}, ⟨Q⟩]_{∘}] as

[(z − H)^{−1}, ⟨Q⟩] [H, ⟨Q⟩]_{∘}(z − H)^{−1} + (z − H)^{−1}[H, ⟨Q⟩]_{∘} [(z − H)^{−1}, ⟨Q⟩]

+ (z − H)^{−1} [[H, ⟨Q⟩]_{∘} , ⟨Q⟩] (z − H)^{−1} .

Since [[H, ⟨Q⟩]_{0}, ⟨Q⟩] = [[H_{0}, ⟨Q⟩]_{0}, ⟨Q⟩] is associated to a bounded pseudodifferential operator, H ∈ C^{2}(⟨Q⟩) by Proposition B.1. Now we conclude the proof of (2) by induction, making use of (C.10) and the fact that the form ad_{⟨Q⟩}^{p}(H) = ad_{⟨Q⟩}^{p}(H_{0}) extends to a bounded one, if p ≥ 2. Let N = H or H_{0}. For z ∈ ℂ \ ℝ, we have (C.10) with H replaced by N, thanks to (1) and (2). Using the resolvent equality for the difference (z − N)^{−1} − (i − N)^{−1}, we see that

(C.11)  ∥[(z − N)^{−1}, ⟨Q⟩]_{∘}∥ ≤ C(1 + \frac{⟨ℜz⟩}{|ℑz|}) .

where C only depends on the operator norm of [N, ⟨Q⟩]_{∘}. Now we use (C.5) with φ = θ to express the form [θ(H), ⟨Q⟩] and see that it extends to a bounded one, thanks to (C.11). This shows that θ(N) ∈ C^{1}(⟨Q⟩). In a similar way, we can show by induction that θ(N) ∈ C^{∞}(⟨Q⟩). The above arguments actually show that P_{i}[θ(N), ⟨Q⟩]_{∘} is a bounded operator on L^{2}(ℝ^{d}). So is also [P_{i}θ(N), ⟨Q⟩]_{∘} and, since P_{i}θ(N) is bounded, P_{i}θ(N) ∈ C^{1}(⟨Q⟩). Again we can derive by induction that P_{i}θ(N) ∈ C^{∞}(⟨Q⟩). Similarly we can verify that P_{i}P_{j}θ(N) ∈ C^{∞}(⟨Q⟩). Note that θ(H)D(⟨Q⟩) ⊂ D(H) = D(H_{0}). Let z ∈ ℂ \ ℝ. By (2), (z − H)^{−1} preserves D(⟨Q⟩) and, on D(⟨Q⟩),

⟨Q⟩ (z − H)^{−1} = (z − H)^{−1} ⟨Q⟩ + [⟨Q⟩, (z − H)^{−1}]_{∘} .

Thus ⟨Q⟩ (z − H)^{−1}⟨Q⟩^{−1} is bounded and

⟨Q⟩ (z − H)^{−1}⟨Q⟩^{−1} = (z − H)^{−1} + [⟨Q⟩, (z − H)^{−1}]_{∘} ⟨Q⟩^{−1} .

By (C.10), we see that ⟨P ⟩⟨Q⟩ (z − H)^{−1}⟨Q⟩^{−1}

= ⟨P ⟩(z − H)^{−1} + ⟨P ⟩(z − H)^{−1} [⟨Q⟩, H]_{∘} (z − H)^{−1}⟨Q⟩^{−1}

is bounded and, for some z-independent C′ > 0,

∥⟨P ⟩⟨Q⟩ (z − H)^{−1}⟨Q⟩^{−1}∥ ≤ \frac{C′}{|ℑz|}(1 + \frac{⟨ℜz⟩}{|ℑz|}) .

Therefore, ⟨P ⟩⟨Q⟩ θ(H)⟨Q⟩^{−1} is bounded, by (C.5) with k = 0. This implies that θ(H)D(⟨Q⟩) ⊂ D(⟨P ⟩⟨Q⟩). □

Lemma C.5. Assume Assumptions 1.1 and 1.5. For integers 1 ≤ i, j ≤ d, let the operator τ(P) be either 1, or P_{i}, or P_{i}P_{j}. Then, for any θ ∈ C_{c}^{∞}(ℝ; ℂ) and any σ ≥ 0, ⟨Q⟩^{β_{lr}−σ}τ(P)(θ(H) − θ(H_{0}))⟨Q⟩^{σ}, ⟨Q⟩^{−σ}τ(P)θ(H)⟨Q⟩^{σ}, and ⟨Q⟩^{−σ}τ(P)θ(H_{0})⟨Q⟩^{σ} are bounded on L^{2}(ℝ^{d}).

Proof. We first note that, for δ ∈ [−1; 1], the form [H, ⟨Q⟩^{δ}] = [H_{0}, ⟨Q⟩^{δ}] extends to a bounded one from H^{1}(ℝ^{d}) to L^{2}(ℝ^{d}). Thus, as in the previous proof (the one of Lemma 3.1), for H′ = H and H′ = H_{0}, there exists C > 0, such that, z ∈ ℂ \ ℝ,

(C.12)  ∥⟨P ⟩^{2}⟨Q⟩^{−δ} (z − H′)^{−1}⟨Q⟩^{δ}∥ ≤ \frac{C}{|ℑz|}(1 + \frac{⟨ℜz⟩}{|ℑz|}) .

Since, for δ ∈ [0; 1], we can write

⟨Q⟩^{−1−δ}(z − H′)^{−1}⟨Q⟩^{1+δ} = ⟨Q⟩^{−δ}(z − H′)^{−1}⟨Q⟩^{δ}

+ ⟨Q⟩^{−1−δ}(z − H′)^{−1}[H′, ⟨Q⟩]_{∘}(z − H′)^{−1}⟨Q⟩^{δ}

with [H′, ⟨Q⟩]_{∘} = [H_{0}, ⟨Q⟩]_{∘}, (C.12) implies (C.12) with δ replaced by δ +1. By induction, we get (C.12) for all δ ≥ 0. For δ ∈ [−1; 0], we can similarly show (C.12) with δ replaced by δ − 1 and then, by induction, (C.12) for all δ ≤ 0. For z ∈ ℂ \ ℝ,

⟨Q⟩^{β_{lr}−σ}V_{c}(Q)(z − H_{0})^{−1}⟨Q⟩^{σ}
= ⟨Q⟩^{β_{lr}}χ_{c}(Q) · V_{c}(Q)⟨P ⟩^{−2} · ⟨P ⟩^{2}⟨Q⟩^{−σ}(z − H_{0})^{−1}⟨Q⟩^{σ}

and, for W = W_{αβ} + V_{lr} + V_{sr},

⟨Q⟩^{β_{lr}−σ}W (Q)(z − H_{0})^{−1}⟨Q⟩^{σ} = ⟨Q⟩^{β_{lr}}W (Q) ⟨Q⟩^{−σ}(z − H_{0})^{−1}⟨Q⟩^{σ} ,

and, using iP · v(Q) = (∇ · v)(Q) + v(Q) · iP,

⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}(v · ∇Ṽ_{sr})(Q)(z − H_{0})^{−1}⟨Q⟩^{σ}

= ⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}(v(Q) · iP )Ṽ_{sr}(Q)(z − H_{0})^{−1}⟨Q⟩^{σ}

− ⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}Ṽ_{sr}(Q)(v(Q) · iP )(z − H_{0})^{−1}⟨Q⟩^{σ}

= ⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}iP ⟨Q⟩^{σ−β_{lr}} · v(Q)⟨Q⟩^{β_{lr}} Ṽ_{sr}(Q)⟨Q⟩^{−σ}(z − H_{0})^{−1}⟨Q⟩^{σ}

− ⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}⟨Q⟩^{σ−β_{lr}}(∇ · v)(Q)⟨Q⟩^{β_{lr}} Ṽ_{sr}(Q)⟨Q⟩^{−σ}(z − H_{0})^{−1}⟨Q⟩^{σ}

− ⟨Q⟩^{β_{lr}−σ}(z − H)^{−1}⟨Q⟩^{σ−β_{lr}}⟨Q⟩^{β_{lr}} Ṽ_{sr}(Q)v(Q) · ⟨Q⟩^{−σ}iP (z − H_{0})^{−1}⟨Q⟩^{σ} .

By (C.12) for H′ = H and δ = σ − β_{lr}, (C.12) for H′ = H_{0} and δ = σ, and by the resolvent formula, we see that the operator

⟨P ⟩^{2}⟨Q⟩^{β_{lr}−σ}((z − H)^{−1} − (z − H_{0})^{−1})⟨Q⟩^{σ}

is bounded and its norm is dominated by some z-independent C′ times the r.h.s. of (C.12) squared. Now, we use (C.5) with k = 0 to get the boundedness of ⟨P ⟩^{2}⟨Q⟩^{β_{lr}−σ}(θ(H) − θ(H_{0}))⟨Q⟩^{σ}. This shows the desired result for the first considered operator. The result for the last two operators follows from (C.12) and (C.5) with k = 0. □

Lemma C.6. Assume Assumptions 1.1 and 1.5 satisfied. Let θ ∈ C_{c}^{∞}(ℝ; ℂ). Let χ ∈ C_{c}^{∞}(ℝ; ℝ) with χ = 1 near 0 and, for R ≥ 1, let χ_{R}(t) = χ(t/R) and χ̃_{R}(t) = 1 − χ_{R}(t). Let τ(P) be either 1, or P_{i}, or P_{i}P_{j}, for 1 ≤ i, j ≤ d.

(1) For σ ∈ [0; 1[ and ϵ ≥ 0, the operators

⟨Q⟩^{σ−ϵ}[τ(P)θ(H), χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} and ⟨Q⟩^{σ−ϵ}τ(P)[θ(H), χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ}

are bounded on L^{2}(ℝ^{d}) and their norm are O(R^{2σ−1−ϵ}).

(2) The operators

⟨Q⟩^{1−β_{lr}}[τ(P)θ(H), χ_{R}(⟨Q⟩)]_{∘} and ⟨Q⟩^{1−β_{lr}}τ(P)[θ(H), χ_{R}(⟨Q⟩)]_{∘}

are bounded on L^{2}(ℝ^{d}) and their norm are O(R^{−β_{lr}}).

Proof. We only prove (1). The proof of (2) is similar since [θ(H), χ_{R}(⟨Q⟩)] = −[θ(H), χ̃_{R}(⟨Q⟩)] and [τ(P)θ(H), χ_{R}(⟨Q⟩)] = −[τ(P)θ(H), χ̃_{R}(⟨Q⟩)]. Note that, on D(⟨Q⟩^{σ}),

⟨Q⟩^{σ−ϵ}[τ(P)θ(H), χ_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} = ⟨Q⟩^{σ−ϵ}[τ(P), χ_{R}(⟨Q⟩)]_{∘}θ(H)⟨Q⟩^{σ}
+ ⟨Q⟩^{σ−ϵ}τ(P)[θ(H), χ_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} ,

where [θ(H), χ_{R}(⟨Q⟩)]_{∘} is explicit and satisfies

∥⟨Q⟩^{σ−ϵ}[θ(H), χ_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ}∥ = O(R^{2σ−1−ϵ}) .

Thus, it suffices to study the second operator in (1). The form [H, χ_{R}(⟨Q⟩)] = [H_{0}, χ_{R}(⟨Q⟩)] extends to a bounded one from H^{1}(ℝ^{d}) to L^{2}(ℝ^{d}). Furthermore,

[H, χ̃_{R}(⟨Q⟩)]_{∘} = [H_{0}, χ̃_{R}(⟨Q⟩)]_{∘} = −χ′_{R}(⟨Q⟩)⟨Q⟩^{−1}Q · P + B_{R} ,

with bounded B_{R} such that ∥B_{R}∥ = O(R^{−2}). Using the proofs of Lemma 3.1 and of Lemma C.5, we get, for z ∈ ℂ \ ℝ, the operator

⟨Q⟩^{σ−ϵ}[(z − H)^{−1}, χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} = −⟨Q⟩^{σ−ϵ}(z − H)^{−1}[H, χ̃_{R}(⟨Q⟩)]_{∘}(z − H)^{−1}⟨Q⟩^{σ}

is bounded and, there exist C > 0 such that, for all z ∈ ℂ \ ℝ and all R ≥ 1,

∥⟨Q⟩^{σ−ϵ}[(z − H)^{−1}, χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ}∥ ≤ R^{2σ−1−ϵ} \frac{C}{|ℑz|^{2}} (1 + \frac{⟨ℜz⟩}{|ℑz|})^{2} .

Using (C.5) with k = 0, we get the boundedness of ⟨Q⟩^{σ−ϵ}[θ(H), χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} and the desired upper bound on its norm. Similarly, we can treat the operators ⟨Q⟩^{σ−ϵ}P_{i}[θ(H), χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ} and ⟨Q⟩^{σ−ϵ}P_{i}P_{j}[θ(H), χ̃_{R}(⟨Q⟩)]_{∘}⟨Q⟩^{σ}. □

Lemma C.7. Let (α; β) such that |α−1|+β < 1. Let ϵ ∈]2|α−1|; 1−β+|α−1|]. Then the integral (3.7) is infinite.

Proof. Denote by I this integral (3.7). Note that its integrand is nonnegative. Using spherical coordinates,

I = c_{d} ∫_{0}^{+∞}(1 − κ(r))^{3}r^{1−β+|α−1|−(d+ϵ)+d−1} sin^{2}(kr^{α}) dr

where c_{d} > 0 is the mesure of the unit sphere in ℝ^{d}. For n ∈ ℕ, let

a_{n} = \frac{1}{k} (\frac{π}{2} − \frac{π}{4} + 2nπ) and b_{n} = a_{n} + \frac{π}{2k} .

For r ∈ [a_{n}^{1/α}; b_{n}^{1/α}], sin^{2}(kr^{α}) ≥ 1/2. Let N be a large enough integer such that, for n ≥ N, a_{n}^{1/α} lies outside the support of κ(| · |). Thus,

\frac{2I}{c_{d}} ≥ ∑_{n=N}^{∞} ∫_{a_{n}^{1/α}}^{b_{n}^{1/α}} r^{−β+|α−1|−ϵ} dr .

The general term in the above serie is bounded below by c·n^{α^{−1}(1−β+|α−1|−ϵ)−1}, for some c > 0. By assumption, 1 − β + |α − 1| − ϵ ≥ 0, therefore the serie diverges, showing that I is infinite. □

Appendix D. Strongly oscillating term.

In this section, we focus on the case α > 1 and prove the key result on oscillations, namely Proposition 2.4. To this end, we recall the following well-known result.

Lemma D.1. Schur’s lemma. Let (n; m) ∈ (ℕ^{∗})^{2}. Let K : ℝ^{n} × ℝ^{m} ⟶ ℂ be a measurable function such that, there exists C > 0 such that

sup_{x∈ℝ^{n}} ∫_{ℝ^{m}} |K(x; y)| dy ≤ C and sup_{y∈ℝ^{m}} ∫_{ℝ^{n}} |K(x; y)| dx ≤ C .

Then the operator A : L^{2}(ℝ^{m}) ⟶ L^{2}(ℝ^{n}), that maps f ∈ L^{2}(ℝ^{m}) to the function

x ↦ ∫_{ℝ^{m}} K(x; y) · f(y) dy ,

is well-defined, bounded and its operator norm is bounded above by C.

Proof of Proposition 2.4. Recall that, by (2.1), denoting 1 − κ by χ,

e_{±}^{α}(Q) = (1 − κ(|Q|))e^{±ik|Q|^{α}} = χ(|Q|)e^{±ik|Q|^{α}} ,

where κ ∈ C_{c}^{∞}(ℝ; ℝ) is identically 1 near 0. Note that, for ϵ, δ > 0, ⟨Q⟩^{−ϵ}⟨P ⟩^{−δ} is compact on L^{2}(ℝ^{d}; ℂ). By pseudodifferential calculus (or commutator expansions, cf. [GJ1]), ⟨Q⟩^{−ϵ}⟨P ⟩^{−ℓ}⟨Q⟩^{ϵ} is bounded on L^{2}(ℝ^{d}; ℂ) for any ℓ ≥ 0. Thus, the desired result follows from the boundedness on L^{2}(ℝ^{d}; ℂ) for all p ≥ 0 of ⟨P ⟩^{−ℓ_{1}}⟨Q⟩^{p}e_{±}^{α}(Q)⟨P ⟩^{−ℓ_{2}}, for appropriate ℓ_{1} and ℓ_{2}. Given p, we seek for ℓ_{1}, ℓ_{2} ≥ 0 and C > 0 such that, for all function f ∈ 𝒮 (ℝ^{d}; ℂ), the Schwartz space on ℝ^{d},

∥⟨P ⟩^{−ℓ_{1}}⟨Q⟩^{p}e_{±}^{α}(Q)⟨P ⟩^{−ℓ_{2}}f∥^{2}

= ⟨⟨P ⟩^{−ℓ_{2}}f , ⟨Q⟩^{p}e_{∓}^{α}(Q)⟨P ⟩^{−2ℓ_{1}}⟨Q⟩^{p}e_{±}^{α}(Q)⟨P ⟩^{−ℓ_{2}}f⟩

is bounded above by C∥f∥^{2}. Given f ∈ 𝒮 (ℝ^{d}; ℂ), we set g = ⟨P ⟩^{−ℓ_{2}}f ∈ 𝒮 (ℝ^{d}; ℂ) and write

f_{1}(x) := (⟨Q⟩^{p}e_{∓}^{α}(Q)⟨P ⟩^{−2ℓ_{1}}⟨Q⟩^{p}e_{±}^{α}(Q)g)(x)
(D.1)  = (2π)^{−d} ∫_{ℝ^{2d}} e^{iφ_{α,±}(x;y;ξ)}⟨x⟩^{p}χ(x)⟨ξ⟩^{−2ℓ_{1}}⟨y⟩^{p}χ(y)g(y) dy dξ ,

where φ_{α,±}(x; y; ξ) = (x − y) · ξ ∓ k(|x|^{α} − |y|^{α}) and the integral converges absolutely, if ℓ_{1} > d/2. Take δ ∈]0; 1/2[ and τ ∈ C_{c}^{∞}(ℝ) such that τ(t) = 1 if |t| ≤ 1 − 2δ and τ(t) = 0 if |t| ≥ 1 − δ. On the support of (x; y) ↦ χ(x)χ(y)τ(|x − y| · |x|^{−1}), |x − y| ≤ (1 − δ)|x|. In particular, on this support, 0 does not belong the segment [x; y] and, for all t ∈ [0; 1],

(D.2)  (2 − δ)|x| ≥ |tx + (1 − t)y| ≥ |x| − (1 − t)|y − x| ≥ δ|x| .

We write f_{1}(x) = f_{2}(x) + f_{3}(x) where f_{2} (resp. f_{3}) is given by (D.1) with g(y) replaced by (1−τ(|x−y|·|x|^{−1}))g(y) (resp. τ(|x−y|·|x|^{−1})g(y)). On the support of the function (x; y) ↦ χ(x)χ(y)(1−τ(|x−y|·|x|^{−1})), |x−y| ≥ (1−2δ)|x| > 0 and |x − y| ≥ C_{δ}|y|, for some (x; y)-independent, positive constant C_{δ}. Since

(L_{x,y,D_{ξ}} −1)e^{i(x−y)·ξ∓ik(|x|^{α}−|y|^{α})} = 0 for L_{x,y,D_{ξ}} = |x−y|^{−2}(x−y)·D_{ξ} , we get, by integration by parts, that, for all n ∈ ℕ,

f_{2}(x) = (2π)^{−d} ∫ e^{iφ_{α,±}(x;y;ξ)}⟨x⟩^{p}χ(x)⟨y⟩^{p}χ(y)g(y)(1 − τ(|x − y|⟨x⟩^{−1}))
· (L_{x,y,D_{ξ}}^{∗})^{n}(⟨ξ⟩^{−2ℓ_{1}}) dydξ .

Choosing n large enough, we can apply Lemma D.1 to show that the map f ↦ f_{2} is bounded on L^{2}(ℝ^{d}). On the support of the function (x; y) ↦ χ(x)χ(y)τ(|x − y| · |x|^{−1}), we can write φ_{α,±}(x; y; ξ) = (x − y) · (ξ ∓ kw_{α}(x; y)) where

w_{α}(x; y) = α ∫_{0}^{1} |tx + (1 − t)y|^{α−2}(tx + (1 − t)y) dt .

Setting, for j ∈ {0; 1},

λ_{j} = ∫_{0}^{1} |tx + (1 − t)y|^{α−2} t^{j} dt ,

λ_{0} ≥ λ_{1} > 0 and α^{−1}w_{α}(x; y) = λ_{1}x+(λ_{0}−λ_{1})y = λ_{0}((λ_{1}/λ_{0})x+(1−λ_{1}/λ_{0})y). By (D.2),

λ_{0} ≥ λ_{1} ≥ 2^{−1}(δ|x|)^{α−2}

and |w_{α}(x; y)| ≥ αλ_{0}δ|x|. Furthermore |w_{α}(x; y)| ≤ α((2 − δ)|x|)^{α−1}, thus

2^{−1}δ^{α−1} ≤ α^{−1}|x|^{1−α} · |w_{α}(x; y)| ≤ (2 − δ)^{α−1} , (D.3)

2^{−1}δ^{α−1}(2 − δ)^{1−α} ≤ α^{−1}|y|^{1−α} · |w_{α}(x; y)| ≤ δ^{1−α}(2 − δ)^{α−1} . (D.4)

In the integral defining f_{3}, we make the change of variables ξ ↦ η = ξ ∓ kw_{α}(x; y) and obtain

f_{3}(x) = (2π)^{−d} ∫ e^{i(x−y)·η}⟨x⟩^{p}χ(x)⟨y⟩^{p}χ(y)g(y)τ(|x − y| · |x|^{−1})

· ⟨η ± kw_{α}(x; y)⟩^{−2ℓ_{1}} dydη . (D.5)

We write f_{3}(x) = f_{4}(x) + f_{5}(x) where f_{4} (resp. f_{5}) is given by (D.5) with g(y) replaced by τ(|η|·|kw_{α}(x; y)|^{−1})g(y) (resp. (1−τ(|η|·|kw_{α}(x; y)|^{−1}))g(y)). On the support of the integrand of f_{4}, |η| ≤ (1 − δ)|kw_{α}(x; y)| which implies that |η ± kw_{α}(x; y)| ≥ δ|kw_{α}(x; y)|. Take ℓ_{1} > (α − 1)^{−1}(p + d). By (D.3), (D.4), and Lemma D.1, the map f ↦ f_{4} is bounded on L^{2}(ℝ^{d}). On the support of the integrand of f_{5}, |η| ≥ (1 − 2δ)|kw_{α}(x; y)| > 0. Since

M_{η,D_{x}}e^{i(x−y)·η} = e^{i(x−y)·η} = −M_{η,D_{y}}e^{i(x−y)·η} for M_{η,D_{z}} = |η|^{−2}η · D_{z} ,

we get, by integration by parts, that, for all n ∈ ℕ,

⟨g , f_{4}⟩ = (2π)^{−d} ∫ e^{i(x−y)·η}( − M_{η,D_{x}}^{∗}M_{η,D_{y}}^{∗})^{n}[⟨x⟩^{p}χ(x)\overline{g(x)}⟨y⟩^{p}χ(y)g(y)

· τ(|x − y| · |x|^{−1})(1 − τ(|η| · |kw_{α}(x; y)|^{−1}))] dxdydη .

Choosing the integer n such that n(α − 1) > p + d, using (D.3) and (D.4), we can apply Lemma D.1 to get some f-independent constant C_{0} > 0 such that

|⟨g , f_{4}⟩| ≤ C_{0} sup_{0≤|γ|≤n}(∥g∥^{2} + ∥P^{γ}g∥^{2}) .

Now the r.h.s. is bounded above by C∥f∥^{2} if ℓ_{2} is greater than 1 plus the integer part of (α − 1)^{−1}(p + d). □

References

[A] S. Agmon: Lower bounds for solutions of Schrödinger equations. J. Analyse Math. 23, 1970, 1-25.

[ABG] W.O. Amrein, A. Boutet de Monvel, V. Georgescu: C_{0}-groups, commutator methods and spectral theory of N-body hamiltonians., Birkhäuser 1996.

[Bea] R. Beals: Weighted distribution spaces and pseudodifferential operators, J. Analyse Math. 39 (1981), 131–187.

[Ben] M. Ben-Artzi: Smooth Spectral Theory, Operator Theory: Advances and Applications, 111 (2010), 119-182.

[BD] M. Ben-Artzi, A. Devinatz: Spectral and scattering theory for the adiabatic oscillator and related potentials. J. Maths. Phys. 111 (1979), p. 594-607.

[Bo1] J-M. Bony: Caractérisation des opérateurs pseudo-différentiels. Séminaire ÉDP (1996-1997), exposé numéro XXIII.

[Bo2] J-M. Bony: Sur l’ingalité de Fefferman-Phong. Séminaire ÉDP (1998-1999), exposé numéro III.

[BC] J-M. Bony, J-Y. Chemin: Espaces fonctionnels associés au calcul de Weyl-Hörmander. Bulletin de la S.M.F., tome 122, numéro 1 (1994), p. 77-118.

[BG] N. Boussaid, S. Golénia: Limiting absorption principle for some long range pertubations of Dirac systems at threshold energies. Comm. Math. Phys. 299, 677-708 (2010).

[Ca] L. Cattaneo: Mourre’s inequality and Embedded bounded states, Bull. Sci Math. 129, Issue 7, 591–614, 2005.

[CGH] L. Cattaneo, G. M. Graf, W. Hunziker: A general resonance theory based on Mourre’s inequality, Ann. H. Poincaré 7 (2006), 3, p. 583-601.

[Co] M. Combescure: Spectral and scattering theory for a class of strongly oscillating potentials, Comm. Math. Phys. 73, 43-62 (1980).

[CG] M. Combescure, J. Ginibre: Spectral and scattering theory for the Schrdinger operator with strongly oscillating potentials, Ann. IHP, section A, tome 24, n^{o} 1 (1976), p. 17-30.

[CHM] J. Cruz-Sampedro, I. Herbst, R. Martinez-Avendaño: Pertubations of the Wigner-Von Neumann potential leaving the embedded eigenvalue fixed. Ann. H. Poincaré 3 (2002) 331-345.

[CFKS] H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon: Schrödinger operators with applications to quantum mechanics and global geometry. Springer Verlag 1987.

[DG] J. Dereziński, C. Gérard: Scattering theory of classical and quantum N-particle systems. Springer-Verlag 1997.

[DMR] A. Devinatz, R. Moeckel, P. Rejto: A limiting absorption principle for Schrödinger operators with Von Neumann-Wigner potentials. Int. Eq. and Op. Theory, vol. 14 (1991).

[DR1] A. Devinatz, P. Rejto: A limiting absorption principle for Schrödinger operators with oscillating potentials. Part I. J. Diff. Eq., 49, pp. 29-84 (1983).

[DR2] A. Devinatz, P. Rejto: A limiting absorption principle for Schrödinger operators with oscillating potentials. Part II. J. Diff. Eq., 49, pp. 85-104 (1983).

[FH] R.G. Froese, I. Herbst: Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators. Comm. Math. Phys. 87, 429-447 (1982).

[FHHH1] R.G. Froese, I. Herbst, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof: L^{2}-Exponential Lower Bounds to solutions of the Schrödinger Equation. Comm. Math. Phys. 87, 265-286 (1982).

[FHHH2] R.G. Froese, I. Herbst, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof : On the absence of positive eigenvalues for one-body Schrödinger operators. J. Analyse Math. 41, 272-284, (1982).

[GGé] V. Georgescu, C. Gérard: On the Virial Theorem in Quantum Me-

chanics, Comm. Math. Phys. 208, 275–281, (1999).

[GGM] V. Georgescu, C. Gérard, J.S. Møller: Commutators, C_{0}-semigroups and resolvent estimates, J. Funct. Anal. 216, no 2, p. 303-361, 2004.

[GM] V. Georgescu, M. Mantoiu: On the spectral theory of singular Dirac type Hamiltonians, J. Operator Theory 46 (2001), no. 2, 289321.

[Gé] C. Gérard: A proof of the abstract limiting absorption principle by

energy estimates, J. Funct. Anal. 254 (2008), no 11, 2707-2724.

[GJ1] S. Golénia, Th. Jecko: A new look at Mourre’s commutator theory, Compl. Ana. Op. Theory, Vol. 1, No. 3, p. 399-422, August 2007. Also available on ArXiv: “arXiv:math/0607275 [math.SP] ”.

[GJ2] S. Golénia, Th. Jecko: Weighted Mourre’s commutator theory, application to Schrödinger operators with oscillating potential, J. Oper. Theory 70:1 (2013), 109-144. Also available on ArXiv: “arXiv:1012.0705 [math.SP] ”.

[GM] S. Golénia, S. Moroianu: Spectral analysis of magnetic Laplacians on conformally cusp manifolds., Ann. H. Poincaré 9 (2008), no. 1, 131–179.

[HeS] B. Helffer, J. Sjöstrand: Opérateurs de Schrödinger avec champs magnétiques faibles et constants. Exposé No. XII, Séminaire EDP, février 1989, Ecole Polytechnique.

[Hö] L. Hörmander: The analysis of linear partial differential operators

III., Springer-Verlag Berlin Heidelberg 1983.

[HuS] W. Hunziker, I.M. Sigal: The quantum N-body problem, J. Math. Phys. 41 (6), 3448–3510, 2000.

[IJ] A.D. Ionescu, D. Jerison: On the absence of positive eigenvalues of Schrödinger operators with rough potentials, GAFA, Geom. Funct. Anal. 13, 2003, 1029-1081.

[K] A. Kiselev: Imbedded singular continuous spectrum for Schrödinger operators, J. Amer. Math. Soc. 18 (2005), no. 3, 571603.

[JMP] A. Jensen, E. Mourre, P. Perry: Multiple commutator estimates and resolvent smoothness in quantum scattering theory. Ann. Inst. H. Poincaré vol. 41, no 2, 1984, p. 207-225.

[K] T. Kato: Pertubation theory for linear operators. Springer-Verlag 1995.

[La1] R.B. Lavine: Commutators and scattering theory. I. Repulsive interactions. Comm. Math. Phys. 20, 301-323, (1971).

[La2] R.B. Lavine: Commutators and scattering theory. II. A class of one body problems. Indiana Univ. Math. J., Vol. 21, No. 7, 643-656, (1972).

[Le] N. Lerner: Metrics on phase space and non-selfadjoint pseudodifferential operators. Birkhaüser 2010.

[Man] M. Mandich: The limiting absorption principle for the discrete Wigner-Von Neumann operator. Preprint. Available on “http://arxiv.org/abs/1605.00879” and “https://hal.archives-ouvertes.fr/hal-01297239”.

[Mar] A. Martin: On the limiting absorption principle for Schrödinger Hamiltonians. Preprint. Available on “http://arxiv.org/abs/1510.03543” and “https://hal.archives-ouvertes.fr/hal-01214523”.

[MU] K. Mochizuki, J. Uchiyama: Radiation conditions and spectral theory for 2-body Schrödinger operators with “oscillating” long-range potentials I: the principle of limiting absorption. J. Math. Kyoto Univ. 18, 2, 377–408, 1978.

[Mø] J.S. Møller: An abstract radiation condition and applications to N-

body systems. Rev. Math. Phys. 12 (2000), no. 5, 767803.

[Mo] E. Mourre: Absence of singular continuous spectrum for certain selfadjoint operators. Comm. in Math. Phys. 78, 391–408, 1981.

[N] S. Nakamura: A remark on the Mourre theory for two body Schrödinger operators. J. Spectral Theory 4, (2015), No. 3, 613–619.

[RS4] M. Reed, B. Simon: Methods of Modern Mathematical Physics, Tome IV: Analysis of operators. Academic Press.

[Rem] C. Remling: The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials, Comm. Math. Phys. 193 (1998), 151–170.

[ReT1] P. Rejto, M. Taboada: A limiting absorption principle for Schrödinger operators with generalized Von Neumann-Wigner potentials I. Construction of approximate phase. J. Math. Anal. and Appl. 208, p. 85–108 (1997).

[ReT2] P. Rejto, M. Taboada: A limiting absorption principle for Schrödinger operators with generalized Von Neumann-Wigner potentials II. The proof. J. Math. Anal. and Appl. 208, p. 311-336 (1997).

[Sa] J. Sahbani: The conjugate operator method for locally regular Hamiltonians. J. Oper. Theory 38, No. 2, 297–322 (1997).

[Sc] L. Schwartz: Théorie des distributions. Hermann, Paris 1966.

[Si] B. Simon: On positive eigenvalues of one body Schrödinger operators. Comm. Pure Appl. Math. 22, 531–538 (1969).

Th. Jecko AGM, UMR 8088 du CNRS Université de Cergy Pontoise Site de Saint Martin 2, rue Adolphe Chauvin F-95302 Cergy-Pontoise cedex France A. Mbarek AGM, UMR 8088 du CNRS Université de Cergy Pontoise Site de Saint Martin, 2, rue Adolphe Chauvin F-95302 Cergy-Pontoise cedex France