Sanaa Semlali

Benoit Cormary

Maria L De Marco

Jérôme Majimel

Alix Saquet

Yannick Coppel

Mathieu Gonidec

Patrick Rosa

Glenna L Drisko

Effect of solvent on silicon nanoparticle formation and size: a mechanistic study

Silicon has emerged as the most desirable material for optics, however chemists struggle to control its morphology and dimensions for larger diametres. Here the average diametre of silicon nanoparticles is varied between 3 and 15 nm by changing the reaction solvent. Electrochemistry and NMR elucidate the role of solvent on the synthetic mechanism. Surprisingly the solvent does not stabilize the nanoparticles and there is no trend associated with chain length or open-chain versus cyclical solvent molecules. The solvents' main role is to stabilize the by-products, which prelongs the reaction lifetime.

Silicon has exceptional optical properties spanning from photoluminescence to Mie resonance, which has generated frenetic activity among physicists in recent years. 1 Silicon is an abundant and nontoxic element, thus commercial applications are expected from optically active biomedical probes to telecommunications. [1][2][3] One particularly exciting application is the prospect of forward-light scattering, which can be used for beam shaping. In pursuit of optical metamaterials, large particle diameters, that is ≥75 nm, are required. 1 It is especially difficult to grow nanoparticles larger than 10 nm using solution chemistry. The size restriction can be attributed to the sluggish particle growth kinetics compared to nucleation rate. It is wellknown that a strong reducing agent favours fast nucleation, consuming most of the precursor. 4 As small nanoparticles are typically obtained by strong reductants, 5 a gentle reductant is required to promote particle growth over nucleation. Magnesium (Mg) is a gentler reducing agent than the others commonly used, i.e. naphthalenides, alkalides, metal hydrides and Zintl salts. This synthetic protocol is similar to one published by S. Kauzlarich and coworkers, where SiCl4 was reduced by the Zyntl salt Mg2Si. 6 This metathesis reaction producing particles in glyme with a diameter between 2-5 nm, 6 where the Mg reduction reported here generates particles around 10 nm. However, Mg does not reduce SiCl4 in all solvents, 7 begging two questions: What is the reduction mechanism? And what is the solvent's role in this reaction? Scheme 1. Formation mechanism of Si NPs (blue spheres). The Mg (grey sphere) is not drawn to scale, being millimetric in size, where the silicon is nanometric. Clusters of Si are represented by a blue Y, the MgCl2 as white crystals. No one has yet studied the solvent effect on silicon nanoparticle (Si NP) size, even though the solvent changes the reactivity of reducing agents and precursors, while either stabilizing the nanoparticles or promoting their aggregation. For example, in the preparation of CdS nanoparticles, the particle size was ten times smaller in pure alcohol than in an alcohol/water mixture. 8 Here we tested six solvents, in the form of glycol ethers and heterocyclic ethers. We have chosen to use as a reductant, due to its relatively mild reactivity and because it does not pollute the silicon structure, as hydrogen can. 9 Using Mg and SiCl4, we were able to vary the mean nanoparticle size between 3-15 nm simply by changing the solvent. This reaction is relatively easy, as it occurs at room temperature and ambient pressure, using commercially available products. It is also easy to handle and poses less security risks than most other reducing agents. We have investigated the reaction mechanism using NMR and by determining the reduction potentials in the respective solvents. SiCl4 was reduced by Mg grit at room temperature and ambient pressure in the presence of THF, glyme, diglyme and tetraglyme. The reaction had little induction time, with a visible colour change occurring within a few minutes of mixing the reagents. Reduction did not occur in diethyl ether and dioxane. A previous report claims that the reaction does not occur in toluene or anisole. 7 By producing Si NPs under different conditions and studying the synthesis using solution-state NMR and electrochemistry, we propose that these nanoparticles form via the mechanism shown in Scheme 1. SiCl4 reduction is observed through a colour change, from colourless to yellow, to red, to brown, as an evolution in nanoparticle size. The SiCl4 reduction occurs at the Mg surface. We can eliminate a mechanism based on the formation of silanylmagnesium complexes, the analogue of the carbon-based Grignard reagent, a mechanism that was proposed previously. 7 Silanylmagnesium compounds cannot be produced from chloride-based silicon precursors at room temperature, and if they did form, they would cleave the glycol ether solvents. 10 At the onset of the reaction, the Mg grit is covered by a thin oxide layer. Upon SiCl4 addition, the surface of the Mg is partially scrubbed, generating a very small amount of SiO2 and a reactive magnesium surface. The SiCl4 then grows into clusters of several silicon atoms, generating MgCl2 as a byproduct. The vast majority of the silicon clusters and the MgCl2 detach from the surface of the Mg grit. However, as time progresses, the surface eventually becomes encumbered with a skin of salt and Si NPs. Elemental analysis of pieces of Mg grit at the end of the reaction in different solvents was conducted using a scanning electron microscope (SEM) (Fig. S1, S2 ESI). The surface was covered with two types of zones: One rich in silicon and one rich in both magnesium and chloride. This skin passivates the Mg surface and prevents further reaction, even if SiCl4 still exists in the solution, terminating nanoparticle growth. The solvent does not intervene directly in the reaction. We believe that the solvent's main effect on the final size of the Si NPs is due to its capacity to solvate MgCl2 and Si species, and its physical properties (e.g. viscosity, reduction potential of the SiCl4, polarity and density). NMR and electrochemistry studies provide evidence supporting this By simply changing the solvent from one glycol ether to another, the Si NP size distribution is significantly altered. Fig. 1 shows representative TEM images and histograms of the size distribution of Si NPs synthesized in four different glycol ethers. The results are not immediately logical, the trend in size is not linearly connected to the number of ether units, and oddly THF produces particles where its linear analogue, diethyl ether, does not. The smallest Si NPs were obtained in diglyme, having an average diameter of 6 ± 2 nm. Si NPs synthesized in THF have a size centred around 8 ± 8 nm. The synthesis in glyme offers a large range of Si NP sizes: from 5 to 60 nm; however half of the NPs have a diameter between 5 and 10 nm. In tetraglyme, a size maximum distribution occurs at 15 nm, with a substantial quantity of particles being larger than 20 nm. Perhaps the substantially larger size observed in tetraglyme is due to its viscosity, slowing NP diffusion away from the reactive surface. Pure, crystalline Si NPs are produced by this room temperature method, as confirmed with high resolution transmission electron microscopy (HRTEM) and Raman spectroscopy (Fig. 2). The Si NPs contain diffraction planes, visible using HRTEM (Fig. 2a). The d-spacing for the planes in this orientation were 0.19 nm, which correspond to the {220} hkl plane for the diamond cubic structure. The selected area electron diffraction (SAED) pattern indexes to the [-111] zone axis of cubic silicon, the expected crystallographic structure. Si nanoparticles with a chlorinated surface become charged by observation under the electron beam and thus leave the TEM grid after ~6 s of exposure time. This short window of observation did not allow us to do EDX spectra of the silicon nanoparticles. However, we performed electron diffraction on particles of all sizes and consistently found that the particles were silicon. The Raman spectrum of the Si NPs possesses a peak at 510 cm -1 that corresponds to crystalline silicon. 11 No other peaks were observed, indicating the purity of the sample. Thus, crystalline silicon can be successfully produced using this room temperature reaction. Analytic electrochemistry was used to probe the observed trend in Si NP size as a function of solvent (Fig. 3). The reductive potential of the Mg fluctuated rather dramatically according to the solvent used. The starting potential (Et=0) indicates the role of the solvent on the electrode. The starting potentials measured in glycol ethers are similar to the potential observed for a Mg electrode in DMF with NBu4Cl as the electrolyte salt. 12 Upon SiCl4 addition, an initial oxidation occurs. The oxidation is likely due to the reaction of SiCl4 with the passivating oxide layer present at the surface of the Mg electrode. This initial oxidation is observed in all of the solvents studied. In the case of dioxane, once the oxidation is complete, the silicon precursor no longer reacts with Mg. The starting potential in dioxane is 1.4 V/SCE. Although dioxane shows the lowest potential in the presence of Mg and absence of SiCl4, after addition of SiCl4, dioxane shows the highest potential of the set and does not form nanoparticles. In the other solvents studied, the reaction was spontaneous. After SiCl4 injection, there are different reduction kinetics. Within 200 s, the potential stabilizes at a lower value. The final potential value ranged from -1.6 to -2.0 V/SCE, and is probably the standard potential of the MgCl2/Mg couple. Upon cycling the Mg electrode in the presence of SiCl4, in all solvents except dioxane (Fig. S3, ESI) the surface became highly resistive and eventually passivated. Thus, no further reaction between Mg and SiCl4 was possible. The passivation results from the side product of the reduction, which is probably the magnesium chloride salt. 13 In dioxane, the Mg surface remained unchanged (Fig. S3b,ESI). When platinum was used as the working electrode in place of Mg, the SiCl4 was not reduced (Fig. S4,ESI). Thus, platinum does not intervene in the reaction in the same way as Mg does. The electrochemistry confirms what was observed synthetically, no nanoparticles are formed when dioxane is employed. Moreover, the reduction in the other solvents is spontaneous on the surface of the Mg. While differences in the redox potential of Mg/MgCl2 appear to induce noticeable differences in the final sizes of Si NPs in each solvent, the relationship between solvent and reductant is not univocal: diglyme yielded smaller sizes than THF and glyme while having a higher potential. It is likely that other factors, such as viscosity, come into play. NMR experiments help shed light upon the interaction between the solvent and the nanoparticles and show which species are present in solution. In situ growth NMR studies were performed without agitation and as a result, the reduction occurred much slower than the reactions taking place in a Schlenk flask. When monitoring the reaction over time using 29 Si NMR spectroscopy, at the onset only SiCl4 is present (δ -18 ppm, Fig. S5 ESI). After 4 h, peaks appeared at 0 and 30 ppm. The peak at 0 ppm later disappears, indicating that it is a relatively stable intermediate species, which is then consumed. At longer times, a peak is still observed at 30 ppm, but with a strongly increased width. F. Meyer-Wegner, et al. have published the chemical shifts for Si in SinCl2n+2 species of n=1-5. 14 These species can only be detected if they are free in solution, species bound to the Mg grit cannot be observed. The peaks that we observe do not correspond to those reported by Meyer-Wegner and coworkers, thus we believe that they correspond to clusters larger than 5 silicon atoms. Alternatively, however, these downfield shifts may correspond to SiCl2 silylene adducts: examples of ternary amine adducts are reported in the literature in cleavage reactions of chlorosilanes in the presence of reductants, so solvent adducts in our likewise reducing conditions could be a possibility. There is little evolution after 6 h, indicating that the Mg surface is no longer active. The SiCl4 peak at -18 ppm is consistently present. Another peak appears at -128 ppm after 4 h of reaction, which then shifts to -140 ppm by 6 h. After 46h, two new peaks appear at -39 and -46 ppm, which we cannot explain at this time. The peaks are only present when THF is used as a solvent, thus it is perhaps due to a degraded THF molecule attacking a silicon chloride species, as silicates can appear in this region of the spectrum. After a second addition of Mg, these three peaks disappear, which is probably because the THF becomes so crosslinked that it is no longer sufficiently mobile to be detected by liquid NMR. The solution had gellified by the end of the reaction. The mechanism is further clarified using 35 Cl NMR. The consumption of the SiCl4 precursor is monitored using 35 Cl NMR (Fig. 4). It is possible to observe the 35 Cl signal due to the tetrahedral geometry of the chloride, which averages to zero the quadrupolar coupling. Chlorides occurring in other geometries are not observable. At the beginning of the experiment, SiCl4 is added to 4 equivalents of Mg grit in deuterated THF. The intensity of the Cl peak initially decreases at a linear rate, and then stabilizes. Twenty% of the initial concentration of SiCl4 is consumed after 5.5 h. At this point, a fresh quantity of Mg (4 equivalents) is added. The SiCl4 reacts with the fresh Mg and stabilizes by 8 h. By this time, 62% of the SiCl4 has been consumed. For comparison, in the case of a reaction with just one single addition of 10 equivalents of Mg grit, the SiCl4 concentration decreases by only 35%. The second addition of Mg yields a faster consumption of the silicon precursor. The same behaviour is observed in the case of 2 successive additions of 2 equivalents of Mg (Fig. S6,ESI). These results illustrate that a successive addition of reductant is more efficient than a single addition. The NMR experiments confirm the electrochemistry results: the reaction occurs rapidly until the Mg surface becomes passivated and the reaction stops. 1 H NMR studies show no visible species, with the exception of THF (data not shown). THF shows a minor chemical shift over time, probably due to its polymerization. The NMR studies show no interaction between the solvent and the growing Si NPs, indicating that the solvent molecules are not in fast exchange between a surface-bound and solvated state, and thus it is unlikely that the solvent is behaving as a ligand as has been observed for other semiconductor nanoparticles. 15 Solvent does not seem to stabilize the formed Si NPs. Thus, the changing particle size as a function of solvent must be mainly due to its effect on the reduction potential of SiCl4 and its capacity to solvate MgCl2 and Si species. Physicists challenge chemists to develop silicon particles, but the difficulty of working with silicon leads to slow progress. In pursuit of a bottom-up chemistry synthesis of Si NPs large enough to support a Mie resonance, it is important to examine each reaction parameter separately: the silicon precursor, the reducing agent, the solvent and the reaction conditions. Here, solvent was the subject of study, where tetraglyme proved to produce nanoparticles much larger on average than in other glycol ethers. The Mg reduction of SiCl4 was studied because Mg is a gentle reducing agent, which should favour growth over nucleation. In all cases, the Si NPs obtained were cubic crystalline silicon. The Mg donates its electrons without polluting the silicon structure. Electrochemistry and NMR experiments indicate that the reaction occurs quickly and spontaneously on the Mg surface. Mg reduction did provide larger nanoparticle sizes, however polydispersity in nanoparticle size was large. Thus, Mg may not be the right reductant for solution-based routes to optically resonant silicon nanoparticles. Although silicon-based optical metamaterials still cannot be achieved using solution chemistry, this work shows that solvent needs to be considered in order to maximize nanoparticle size.

Fig. 1

 1 Fig. 1 Bright field TEM and particle size distribution of Si NPs synthesized in (a) THF, (b) diglyme, (c) glyme, and (d) tetraglyme. Scale bars represent 100 nm.

Fig. 2

 2 Fig. 2 (a) HRTEM (scale bar = 5 nm), (b) filtered HRTEM image, (c) SAED, and (d) Raman spectroscopy of a nanoparticle produced in glyme.

Fig. 3

 3 Fig. 3 Chronopotentiometry measurement using a Mg electrode of the equilibrium potential in various solvents before and after adding SiCl4.

Fig. 4

 4 Fig. 4 35 Cl NMR of SiCl4 and Mg grit in THF-D8 at different time intervals and with 4 equivalents of Mg.

|Sanaa Semlali, Benoit Cormary, Maria De Marco, Jérôme Majimel, Alix Saquet, et al.. Effect of solvent on silicon nanoparticle formation and size: a mechanistic study. Nanoscale, Royal Society of Chemistry, 2019, 11 (11), pp.4696-4700. 〈10.1039/C9NR00619B〉. 〈hal-02071674〉

Conflicts of interest

There are no conflicts to declare.

Notes and references

SS, BC, MLDM and GLD were supported by the LabEx AMADEus (ANR-10-LABX-42). They and MG acknowledge the framework of IdEx Bordeaux (ANR-10-IDEX-03-02), i.e. the d'Avenir program of the French government managed by the Agence Nationale de la Recherche. PLACAMAT (UMS 3626CNRS-Bordeaux University) provided technical support with imaging. FEGSEM and EDX instrumentation was facilitated by the Institut des Matériaux de Paris Centre (IMPC FR2482) and was funded by Sorbonne Université, CNRS and by the C'Nano projects of the Région Ile-de-France. We thank David Montero for performing the FEGSEM and EDX analysis.

TOC graphic

Silicon nanoparticle size is adjusted by simply changing the reaction solvent. The role of solvent on nanoparticle formation is investigated.


N Raymond

J Royer

ABSENCE OF EMBEDDED EIGENVALUES FOR TRANSLATIONALLY INVARIANT MAGNETIC LAPLACIANS

Translationnally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of point spectrum under various assumptions on the behavior of the magnetic field.

Context and results

1.1. Translationally invariant magnetic Laplacians. This paper is devoted to the description of the point spectrum of translationally invariant magnetic Laplacians in two dimensions. Here the magnetic field B is assumed to be a smooth enough function that only depends on its first variable. More precisely, we assume that @px, yq P R 2 , Bpx, yq " bpxq, where b P C 1 pR, Rq. Associated with B, we may consider a vector potential A " pA 1 , A 2 q where A 1 px, yq " 0 , A 2 px, yq " apxq :" a 0 `ż x 0 bpuqdu ,

for some arbitrary a 0 . When the limits exist in R Y t˘8u, we set φ ˘" lim xÑ˘8 apxq.

(1.

2)

The magnetic Laplacian under consideration in this paper is the self-adjoint differential operator L " p´i∇ ´Aq 2 " D 2

x ``D y ´apxq ˘2 , D " ´iB ,

equipped with the domain DompL q " u P H 1 A pR 2 q : p´i∇ ´Aq 2 u P L 2 pR 2 q ( , where H 1 A pR 2 q " tu P L 2 pR 2 q : p´i∇ ´Aqu P L 2 pR 2 qu . 1.2. Context and motivation. Due to the translation invariance, it is easy to see that the spectrum of L is essential:

σpL q " σ ess pL q .

The main question addressed in this paper is to find conditions under which L has no eigenvalue. Thus, we would like to exclude the existence of pλ, ψq P r0, `8q ˆDompL q such that ψ ‰ 0 and L ψ " λψ. In order to understand how subtle this question can be, let us remark the following:

-When b is constant and non-zero, it is well-known that the spectrum is made of infinitely degenerate eigenvalues, the Landau levels:

σpL q " tp2n ´1q|b| , n ě 1u .

-When φ `or φ ´is finite, one will see in our proofs that σpL q " r0, `8q .

Thus, as noticed in the seminal paper [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF], even the nature of the essential spectrum itself strongly depends on the variations of b.

In this paper, we focus our investigation on proving the absence of point spectrum, even if, in some particular situations, our proof might also imply the absolute continuity of the spectrum. In particular, in Theorem 1.2, one will see that, if bpxq behaves like x α (with α ‰ 0 and α ą ´1) at infinity, the Landau levels structure is lost as well as the existence of eigenvalues. Theorem 1.4 is of asympotic nature: when b P L 1 pR, R `q and when the magnetic field is large, we show that the only possible eigenvalues are essentially of the order of the flux squared.

Our main results deal with cases when a is semi-bounded, semi-unbounded, and when a is bounded. They partially extend the results in [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF] (where the assumptions imply lim xÑ˘8 apxq " ˘8) by considering non-necessarily bounded magnetic fields.

More generally, this paper can be considered as an exploration of the conjecture stated in [1, Theorem 6.6 & Remark 1]. Let us recall a theorem whose proof may be deduced from the investigation in [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF] (and also [START_REF] Cycon | Schrödinger operators with application to quantum mechanics and global geometry[END_REF]Theorem 6.6] where the magnetic field is allowed to vanish).

Theorem 1.1 (Ywatsuka '85). Assume (i) either that (see (1.2)) φ ´" φ `" ´8 or φ ´" φ `" `8 , (ii) or that lim xÑ˘8 bpxq " b ˘with b ˘P Rzt0u distinct.
Then L has absolutely continuous spectrum. In particular, L has no eigenvalue.

1.3. Some relations with the literature. In [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF], the author is mainly concerned by proving the absolute continuity of the spectrum. Note that this issue is closely connected to the existence of edge currents (quantified by Mourre estimates), as explained for instance in [START_REF] Hislop | Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields[END_REF], where positive magnetic fields are considered. The reader might also want to consider -the physical considerations in [START_REF] Reijniers | Snake orbits and related magnetic edge states[END_REF], -the article [3, Sec. 2] considering translationnally invariant magnetic fields constant away from a compact, -the paper [START_REF] Hislop | Band functions in the presence of magnetic steps[END_REF] considering the dispersion curves associated with non-smooth magnetic fields, -the contribution [START_REF] Tušek | On an extension of the Iwatsuka model[END_REF] generalizing Iwatsuka's result by adding a translationnaly invariant electric potential, and also [START_REF] Exner | A geometric Iwatsuka type effect in quantum layers[END_REF] where a magnetized layer invariant by translation is considered, -the paper [START_REF] Yafaev | On spectral properties of translationally invariant magnetic Schrödinger operators[END_REF] devoted to dimension three and fields having cylindrical and longitudinal symmetries, -or [START_REF] Miranda | Spectrum of the Iwatsuka Hamiltonian at thresholds[END_REF] where various estimates of the band functions are established for increasing, positive, and bounded magnetic fields, and applied to the estimate of quantum currents. Then L has no eigenvalue.

Remark 1.3.

-By the symmetry x Þ Ñ ´x, we can easily adapt this theorem to consider behaviors in ´8. We have a similar result if ´b satisfies (1.4).

-Theorem 1.2 can be applied, for instance, to b ˘pxq " xxy ˘1 2 . In particular, the same proof will establish the absence of eigenvalues for some magnetic fields tending to `8 or to 0 at infinity. -We will see in Theorem 1.4 that, when b tends to 0 too rapidly, the absence of eigenvalues is more subtle to establish.

Our second theorem gives some results in situations where φ `and φ ´are finite but with φ " φ `´φ ´" 1 (the case φ ! ´1 would be similar). By a change of gauge (take a 0 " ş 0 ´8 bpuqdu in (1.1)), we can assume that φ ´" 0 (and hence φ `ą 0).

The problem can then be rewritten in a semiclassical framework. If we set h " φ ´1 `, b 1 pxq " hbpxq and a 1 pxq " hapxq, then we have L " φ 2 `Lh where L h " h 2 D 2

x ``hD y ´a1 pxq ˘2 .

(1.5)

Thus our purpose is now to prove the absence of eigenvalues of the operator L h with a 1 pxq "

ż x ´8 b 1 psq ds, ż `8
´8 b 1 psq ds " 1. Theorem 1.4. (i) For all h ą 0, the operator L h has no point spectrum in " 1 4 , `8˘. (ii) Assume that b 1 is of class C 1 pRq and takes positive values. Assume also that for some N ě 0 we have

b 1 pxq " |x|Ñ`8 O `|x| N ˘,
a 1 P L 1 pR ´q and pa 1 ´1q P L 1 pR `q. Let pη h q hą0 be such that η h " op|lnphq| ´6q as h Ñ 0. Then, there exists h 0 ą 0 such that for h P p0, h 0 q the operator L h has no eigenvalue smaller than η h . Remark 1.5. For example, we can apply Theorem 1.4 to b 1 pxq " 1 ? π e

´x2 . An interesting question is left open: for h small enough, can we exclude the presence of eigenvalues in the interval `ηh , 1 4 ˘? One will see in the proof that this function η h is related to the harmonic approximation. To replace, for instance, η h by 1 4 ´ε would not only suppose to find a convenient effective Hamiltonian in the harmonic approximation (what is possible via a Birkhoff normal form in dimension one, under analyticity assumptions), but also to be able to deduce from it a non-trivial behavior of each dispersion curve. Even if such a description were possible, it would still not exclude the existence of embedded eigenvalues near 1 4 in the limit h Ñ 0. 1.5. Organization of the proofs. In Section 2, we recall basic facts about the Fourier fibration of translationnaly invariant magnetic Laplacians. In particular, Proposition 2.2 provides a criterion to exclude the existence of eigenvalues as soon as no dispersion curve is constant. Even though this proposition seems to be well-known, the presence of essential spectrum for the fibered operator requires to give a careful proof. This will immediately imply Theorem 1.2. Section 3 is devoted to some facts about a parameter dependent version of the harmonic approximation which will be crucial in the proof of Theorem 1.4 (ii) and which will appear when analysing the large frequency limit of the dispersion curves. This approximation will allow us to use somehow the existence of a non-constant "center-guide dynamics" to prove the non-constant character of some dispersion curves (see Remark 5.2).

Reminders on fibered magnetic Hamiltonians

Since L commutes with the translation in y, the Fourier transform in y will play a fundamental role in our analysis. For u P L 2 pR 2 q and for almost all x P R we denote by u ξ the Fourier transform of upx, ¨q. For u P SpR 2 q it is given by

u ξ px, ξq " 1 ? 2π ż R e ´iyξ upx, yqdy .
This induces the following direct integral representation (see, for instance, [12, Section XIII.16] about such direct integrals)

L " ż ' R L ξ dξ , (2.1)
where, for all ξ P R, L ξ " D 2 x `pξ ´apxqq 2 . For all ξ P R this defines an operator on L 2 pRq with domain DompL ξ q " u P H 1 pRq : apxqu P L 2 pRq and pD 2

x `pξ ´apxqq 2 qu P L 2 pRq ( " u P H 2 pRq : pξ ´apxqq 2 u P L 2 pRq ( .

In the following proposition we gather some spectral properties of L ξ that will be useful to the spectral analysis of L . Let us emphasize here that, in [7, Assumption (B)], the assumption on b implies that σ ess pL ξ q " H. This will not always be the case in this paper (see Figure 1 where the bottom of the essential spectrum is represented as a function of ξ).

Proposition 2.1. The operator L ξ is self-adjoint and non-negative for all ξ P R. The family pL ξ q ξPR is analytic of type (A). Let ξ P R.

(i) We have

σpL ξ q Ă " inf xPR `ξ ´apxq ˘2, ` 8¯.
(ii) We have

σ ess pL ξ q " " min `pξ ´φ´q 2 , pξ ´φ`q 2 ˘, `8˘.
In particular, when |φ ´| " |φ `| " `8, σ ess pL ξ q " H. (iii) If φ ˘P R we assume that pξ ´apxqq 2 ´pξ ´φ˘q 2 P L 1 pR ˘q. Then the operator L ξ has no embedded eigenvalue in σ ess pL ξ q. (iv) The eigenvalues of L ξ are simple and depend analytically on ξ.

Proof. The first statements are standard. For (ii), if φ ´and φ `are infinite then L ξ has a compact resolvent by the Riesz-Fréchet-Kolmogorov Theorem. If φ ´and φ `are finite then L ξ is a relatively compact perturbation of D 2

x `V pxq where V pxq " pξ ´φ´q 2 for x ď 0 and V pxq " pξ ´φ`q 2 for x ą 0. We conclude with the Weyl Theorem. If φ ´P R and φ `" `8 we conclude similarly by considering V pxq " pξ ´φ´q 2 if x ď 0 and V pxq " maxppξ ´φ´q 2 , pξ ´apxqq 2 q if x ą 0. The other cases are similar.

For (iii) we use Lemma A.1. If φ `P R then for λ ě pξ ´φ`q 2 we apply the lemma with ω 2 " λ ´pξ ´φ`q 2 and w " pξ ´apxqq 2 ´pξ ´φ`q 2 P L 1 pR `q. This proves that λ is not an eigenvalue. Similarly, if φ ´is finite then L ξ has no eigenvalue λ ě pξ ´φ´q 2 . Let us briefly recall why the eigenvalues of L ξ are simple. Assume that u and v are eigenfunctions of L ξ associated with the same eigenvalue λ. Letting W " u 1 u 1 2 ´u1

1 u 2 , we easily get W 1 " 0, so that W is constant. Since u 1 and u 2 belong to the domain, we get that W is integrable, and thus that W " 0. This shows that the family pu 1 , u 2 q is not free. Combining the simplicity of the eigenvalues and the analyticity of the family, we finally get the analyticity of the eigenvalues.

Let ξ P R. If L ξ has eigenvalues (necessarily simple and under the essential spectrum, according to Proposition 2.1), we label them by increasing order

pλ k pξqq 1ďkďN ξ , with λ k pξq ă λ k`1 pξq, 1 ď k ă N ξ , for some N ξ P N Y t`8u.
When σ ess pL ξ q " H, the following proposition can be found in [12, Theorem XIII.86]. In this paper, the essential spectrum will not be empty in general.

Proposition 2.2. Let λ P R and

Σ λ " tξ P R : λ R σ ess pL ξ qu . (2.2)
If λ is an eigenvalue of L , then there exists n P N ˚and a connected component I of Σ λ such that L ξ has at least n eigenvalues for all ξ P I and @ξ P I, λ n pξq " λ.

Proof. Let λ P R and u P DompL qzt0u be such that L u " λu. For almost all ξ P R, we have L ξ u ξ " λu ξ . Consider Ξ " tξ P R : u ξ ‰ 0u . In particular, λ is an eigenvalue of L ξ for all ξ P Ξ, and hence, with Proposition 2.1, Ξ Ă Σ λ . Moreover, Ξ has positive Lebesgue measure, so there exist a connected component I of Σ λ and a compact K Ă I such that K X Ξ has positive measure. Then there exists ξ 0 P K X Ξ such that rξ 0 ´ε, ξ 0 `εs X Ξ has positive measure for all ε ą 0. Since ξ 0 P Ξ, λ is an eigenvalue of L ξ 0 , so there exists n P N such that L ξ 0 has at least n eigenvalues and λ n pξ 0 q " λ. By simplicity of the eigenvalues and continuity with respect to ξ, together with the non-negativeness of L ξ (so that the eigenvalues cannot escape to ´8), there exists ε ą 0 such that σpL ξ q X rλ ´ε, λ `εs " tλ n pξqu for all ξ P rξ 0 ´ε, ξ 0 `εs. Since λ n is analytic and λ n pξq " λ on a subset of rξ 0 ´ε, ξ 0 `εs of positive measure, we have λ n pξq " λ for all ξ P rξ 0 ´ε, ξ 0 `εs.

Assume by contradiction that there exists ξ P I such that ξ ą ξ 0 and L ξ does not have n eigenvalues. Let ξ 1 " sup tξ P I : L ξ has at least n eigenvaluesu P I.

By analycity we have λ n pξq " λ for all ξ P rξ 0 , ξ 1 q. Moreover rξ 0 , ξ 1 s is a compact subset of Σ λ , so λ ă inf ξPrξ 0 ,ξ 1 s inf σ ess pL ξ q. By continuity of the spectrum of L ξ around ξ " ξ 1 we obtain that L ξ has at least n eigenvalues for ξ on some neighborhood of ξ 1 , which gives a contradiction. Then L ξ has at least n eigenvalues for all ξ P I with ξ ě ξ 0 . The case ξ ď ξ 0 is similar. Then λ n is defined on the whole interval I and, by analycity, we have λ n pξq " λ for all ξ P I.

Note that, with these properties in hand, we can easily deduce Theorem 1.1 (i): the essential spectrum of L ξ is empty so if λ P R is an eigenvalue of L there exists n P N ˚such that λ n pξq " λ for all ξ P R, which is impossible since the bottom of the spectrum of L ξ goes to `8 when ξ Ñ ˘8.

We can also easily prove the first statement of Theorem 1.4:

Proof of Theorem 1.4.(i). Note that, here, h ą 0 is fixed (and we may assume that h " 1).

Assume by contradiction that λ ě 1 4 is an eigenvalue of L . By Proposition 2.1 we have Σ λ " Rz "

´?λ, 1 `?λ ‰ . Since a is bounded, we have inf σpL ξ q Ý ÝÝÝ Ñ ξÑ˘8 `8.

Then Proposition 2.2 gives a contradiction.

We cannot use the same argument when a is surjective (since then we have inf xPR pξ ´apxqq 2 " 0 for all ξ P R) or when a is bounded and λ ă 1 4 (because Σ λ has also a bounded connected component, see Figure 1).

To go further, we will use the harmonic approximation to estimate the eigenvalues of L ξ .

Harmonic approximation for moderately small eigenvalues

In this section, we prove a parameter dependent version of the classical harmonic approximation (see for instance [START_REF] Simon | Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions[END_REF][START_REF] Helffer | Semi-Classical Analysis for the Srödinger Operator and Applications[END_REF]). The main interest of Theorem 3.1 below is that we consider eigenvalues which are "not too small" (in particular, much larger than the low lying eigenvalues, which are of order Ophq).

Without this version of the harmonic approximation, one would only be able to prove the absence of eigenvalues below Ch in Theorem 1.4. We consider a family pV θ q θPΘ of continuous and real-valued potentials on R which satisfies the following properties.

(i) We can write V θ psq " s 2 v 2 θ `s3 w θ psq where, for some v ´, v `, C w , N ą 0, we have @θ P Θ, @s P R, v ´ă v θ ă v `and |w θ psq| ď C w s N .

(3.1)

In particular, there exists ε 0 ą 0 such that @θ P Θ, @s P r´ε 0 , ε 0 s, V θ psq ě v ´s2 2 .

(ii) There exists c 8 ą 0 such that for θ P Θ and s P Rzr´ε 0 , ε 0 s we have

V θ psq ą c 8 (3.2)
Then, for h P p0, 1s and θ P Θ, we consider the operator L h,θ " h 2 D 2 s `Vθ psq, with domain DompL h,θ q " u P H 2 pRq : V θ u P L 2 pRq ( . We recall that, for h P p0, 1s, the spectrum of the operator h 2 D s `vθ s 2 is given by the sequence of simple eigenvalues p2n ´1qhv θ , n P N ˚. We prove that for h small enough the bottom of the spectrum L h,θ is given by simple eigenvalues close to those of this harmonic oscillator.

For θ P Θ and h ą 0 we denote by 0 ă λ 1 ph, θq ď λ 2 ph, θq ď ¨¨ẗ he eigenvalues of L h,θ under the essential spectrum, and we consider a corresponding orthonormal family pψ k,h,θ q of eigenvectors. Then for E P p0, inf σ ess pL θ,h qq we denote by NpE, h, θq the number of eigenvalues of L h,θ (counted with multiplicities) smaller than E: NpE, h, θq " max tn P N ˚: λ n ph, θq ď Eu .

For θ P Θ, h ą 0 and n P N ˚we set

E n ph, θq " spanpψ k,h,θ q 1ďkďn .
We consider a family pη h q hą0 of positive numbers such that

η h " hÑ0 o ˆ1 |lnphq| 6
˙.

Theorem 3.1. There exist h 0 ą 0 such that for θ P Θ and h P p0, h 0 s we have η h ă inf σ ess pL h,θ q and

Npη h , h, θq ě η h 4v `h ´1 . (3.3)
Moreover, there exists a function εphq converging to 0 as h Ñ 0 such that, for all n P t1, . . . , Npη h , h, θqu, The proof of Theorem 3.1 relies on the classical Agmon Formula (see for instance [START_REF] Raymond | Bound States of the Magnetic Schrödinger Operator[END_REF]Prop. 4.7]): Proposition 3.3. Let Φ be a real-valued, Lipschitzian and bounded function on R. Then for h ą 0, θ P Θ and u P DompL h,θ q we have ż

R ˇˇhDpe Φ uq ˇˇ2 dσ `żR `Vθ ´h2 |Φ 1 | 2 ˘e2Φ |u| 2 dσ " Re L h,θ u, e 2Φ u .
In particular, if pλ, uq is an eigenpair of L h,θ then

ż R ˇˇhDpe Φ uq ˇˇ2 dσ `żR `Vθ ´h2 |Φ 1 | 2 ´λ˘e 2Φ |u| 2 dσ " 0.
On the other hand, the following lemma is an easy consequence of Proposition 4.4 in [START_REF] Raymond | Bound States of the Magnetic Schrödinger Operator[END_REF], where we check that the rest is estimated uniformly in θ P Θ.

Lemma 3.4.

There exists h 0 ą 0 such that for all θ P Θ and h P p0, h 0 q we have inf σpL h,θ q ě hv θ 2 .

The following result about the uniform exponential decay of the eigenfunctions has its own interest: Proof. There exist κ P p0, 1q and c E ą 0 such that for all θ P Θ and s P R we have

V θ psq ě min `cE s 2 , p1 `2κqE ˘. (3.6)
Then we set

γ " v ´?κ 2 ą 0,
where v ´is given by (3.1). Let θ P Θ and h P p0, h 0 q. Let pλ, ψq be an eigenpair of L h,θ with λ ď E. For ε ą 0 and s P R we set Then we consider c 1 , . . . , c n P C such that

Φ ε psq " min ˆγ|s| ? λ , 1
ψ " n ÿ j"1 c j ψ j,h,θ .
By the triangle inequality and Proposition 3.5 we have

› › |s| 3 2 xsy N 2 ψ › › L 2 p|s|ěαph,θqq ď n ÿ j"1 |c j | › › |s| 3 2 xsy N 2 ψ j,h,θ › › L 2 p|s|ěαph,θqq À e ´γαph,θq 2 ? λnph,θq n ÿ j"1 |c j | › › ›e γ|s| ? λnph,θq ψ j,h,θ › › › L 2 p|s|ěαph,θqq À h n ÿ j"1 |c j | À h ? n. (3.10)
With (3.9) and (3.10) we get, for some c ą 0 independant of θ, h or n,

xL h,θ ψ, ψy ě xph 2 D 2 s `v2 θ s 2 qψ, ψy ´c ´λn ph, θq 3 2 |lnphq| 3 `nh 2 ¯.
This, with (3.7) and the min-max Theorem, implies that λ n ph, θq ě p2n ´1qhv θ ´c ´λn ph, θq

3 2 |lnphq| 3 `nh 2 ¯. (3.11)
In particular, if h 0 was chosen small enough, there exists C ą 0 such that, for h P p0, h 0 s, θ P Θ and n ď Npη h , h, θq, nh ď Cλ n ph, θq .

Then (3.11) yields

p2n ´1qhv θ ´λn ph, θq ď ε 1 phqλ n ph, θq , (3.13)
where ε 1 phq " c ´λn ph, θq

1 2 |lnphq| 3 `Ch ¯ÝÝÑ hÑ0 0 .
For n P N ˚we denote by f n the n-th Hermite function. It solves on R `D2 σ `σ2 ´p2n ´1q ˘fn pσq " 0 .

Then for h ą 0, θ P Θ and n P N ˚we set

f n,h,θ : s Þ Ñ h ´1 4 v 1 4 θ f n ´h´1 2 v 1 2 θ s ¯.
We have }f n,h,θ } " 1 and `h2 D 2 s `s2 v 2 θ ´p2n ´1qhv θ ˘fn,h,θ psq " 0 . For f in spanpf j,h,θ q 1ďjďn with }f } 2 L 2 pRq " 1 we have

p2n ´1qhv θ ě ph 2 D s `v2 θ s 2 qf, f ě L h,θ f, f ´Cw ż R |s| 3 s N |f | 2 ds.
Following the same lines as above we obtain, for some C 2 ą 0,

C w ż R |s| 3 s N |f | 2 ds ď ρpn, hq :" C 2 `pnhq 3 2 |lnphq| 3 `nh 2 ˘.
If n P N ˚is not greater than η h {p4v `hq we have

ρpn, hq η h ď C 2 p4v `q 3 2 η 1 2 h |lnphq| 3 `h 4v `Ý ÝÑ hÑ0 0.
Hence, if h 0 is small enough, then for h P p0, h 0 s, θ P Θ and n ď η h {p4v `hq we have L h,θ f, f ď p2n ´1qv θ h `ρpn, hq ď η h . By the min-max Theorem this implies λ n ph, θq ď η h , and (3.3) is proved.

On the other hand for n ď Npη h , h, θq we have

λ n ph, θq ´p2n ´1qhv θ ď λ n ph, θqε 2 phq, (3.14)
where, by (3.12),

ε 2 phq :" sup nďN pη h ,h,θq ρpn, hq λ 2 pn, hq ď C 2 C 3 2 η 1 2 h |lnphq| 3 `C2 Ch ÝÝÑ hÑ0 0.
Then (3.4) follows from (3.13) and (3.14).

Absence of embedded eigenvalues with transverse confinement

In this section, we prove Theorem 1.2. Since φ `" `8, we observe that if φ ´" `8, we can apply Theorem 1.1. Thus, we can restrict our attention to the cases φ ´P R and φ ´" ´8. The proof relies on the following asymptotics for the eigenvalues: Proposition 4.1. Assume that (1.4) holds (for any α ą ´1) and that φ ´P r´8, `8q. Let n P N ˚. Then for ξ large enough the operator L ξ has at least n eigenvalues and its n-th eigenvalue λ n pξq satisfies

λ n pξq " ξÑ`8 p2n ´1qc 1 c ´α 1`α 0 ξ α 1`α `opξ α 1`α q ,
where c 0 " c 1 {p1 `αq.

Proof. There exists x 0 ě 1 such that for x ě x 0 we have

a 1 pxq " bpxq ě c 1 x α 2 . (4.1)
In particular, a is increasing on rx 0 , `8q. Since a has a limit in r´8, `8q at ´8 we can assume, by choosing x 0 larger if necessary, that apx 0 q ą apxq for all x P p´8, x 0 q. We set ξ 0 " ap2x 0 q. Then for ξ ě ξ 0 there is a unique x ξ P R such that apx ξ q " ξ. Since apxq "

ż x 0 bpuq du " xÑ`8 c 0 x α`1 , it satisfies x ξ " ξÑ`8 pc ´1 0 ξq 1 1`α .
Let ξ ě ξ 0 . For v P L 2 pRq and s P R, we set

pU ξ vqpsq " x 1 2 ξ v `xξ p1 `sq ˘.
U ξ is a unitary operator on L 2 pRq and

U ξ L ξ U ´1 ξ " x ´2 ξ D 2 s ``ξ ´apx ξ p1 `sq ˘2 " ξ 2 " h 2 ξ D 2 s `Vξ psq ‰ , (4.2)
where h ξ " pξx ξ q ´1 and V ξ psq " `1 ´ξ´1 apx ξ p1 `sq ˘2 .

V ξ takes non-negative values and has a unique zero at s " 0. By the Taylor formula,

a `xξ p1 `sq ˘" ξ `sx ξ a 1 px ξ q `s2 x 2 ξ ż 1 0 p1 ´τ qa 2 `p1 `τ sqx ξ ˘dτ , (4.3)
so we can write V ξ psq " s 2 v 2 ξ `s3 w ξ psq , where, by using (1.4),

v ξ " ξ ´1x ξ a 1 px ξ q Ñ ξÑ`8 c 1 c 0
and |w ξ psq| ď Cxsy maxp2α´1,α´1q , for some C ą 0 independent of ξ and s.

Let us now consider the coercivity property away from the minimum. Let ε P `0, 1 2 ˘. Let ξ ě ξ 0 . For s ě ε we have by the Mean Value Theorem and (4.1)

apx ξ p1 `sqq ´apx ξ q ě c 1 sx α`1 ξ 2 ě c ε ξ,
for some c ε ą 0, and hence V ξ psq ě c 2 ε . Similarly, if s ď ´ε we have apx ξ q ´apx ξ p1 `sqq ě apx ξ q ´apx ξ p1 ´εqq ě

c 1 εx ξ 2 ´xξ 2 ¯α ,
and we conclude similarly. In any case we obtain c 8 ą 0 such that for ξ ě ξ 0 and |s| ě ε we have V ξ psq ě c 8 . With all these properties we can apply Theorem 3.1. We obtain that, for all n P N ˚, there exists ξ 0 ě 0 such that for ξ ě ξ 0 the operator h 2 ξ D 2 s `Vξ psq has at least n eigenvalues and its n-th eigenvalue λn pξq satisfies λn pξq "

ξÑ`8 p2n ´1qh ξ v ξ " ξÑ`8 p2n ´1qc 1 c ´α 1`α 0 ξ ´1 1`α ´1 .
The asymptotic behavior of λ n pξq follows since, by (4.2), we have λ n pξq " ξ 2 λn pξq.

Now we can prove Theorem 1.2.

Proof of Theorem 1.2. Let λ ě 0 and assume by contradiction that λ is an eigenvalue of L . Consider the case φ ´" ´8. Then, for all ξ P R the spectrum of L ξ is purely discrete. By Proposition 2.2, there exists n P N ˚such that λ n pξq " λ for all ξ P R. This gives a contradiction with Proposition 4.1.

Consider now the case φ ´P R. Then, we have Σ λ " Rzrφ ´´? λ, φ ´`? λs and we consider its two connected components in order to apply Proposition 2.2.

-By Proposition 4.1, λ cannot be an eigenvalue of L ξ for all ξ ą φ ´`? λ.

-Since a is bounded from below, Proposition 2.1 gives lim ξÑ´8 inf σpL ξ q " `8, so λ cannot be an eigenvalue of L ξ for all ξ ă φ ´´? λ. This is a contradiction. Remark 4.2. These arguments also imply Theorem 1.1 (ii). Since b ˘‰ 0, we are in a situation where σ ess pL ξ q is empty for all ξ P R, so if L has an eigenvalue there exists n P N such that λ n pξq does not depend on ξ. This gives a contradiction since, by Proposition 4.1, we should have lim ξÑ˘8 λ n pξq " p2n ´1qb ˘.

Moderately small eigenvalues without transverse confinement

In this section we prove the second statement of Theorem 1.4. We recall that b 1 , a 1 and the operators L h , h ą 0, were defined before the statement of Theorem 1.4.

For θ P R, we let L h,θ " h 2 D 2 x `pθ ´a1 pxqq 2 . Then L h is the direct integral of L h,θ , θ P R, as in (2.1).

Proof of Theorem 1.4.(ii). Since b 1 takes positive values, a 1 is an increasing bijection from R to p0, 1q. For θ P p0, 1q we set x θ " a ´1 1 pθq. Then for s P R we set V θ psq " pθ ´a1 px θ `sqq 2 . This defines a nonnegative valued potential, 0 is the unique solution of V θ p0q " 0 and V 2 θ p0q " 2b 1 px θ q 2 ą 0, so V θ has a unique non-degenerate minimum at 0 (and this minimum is not attained at infinity).

Let J be a compact interval of R on which b is not constant and Θ " apJq. As in (4.3) we write a 1 px θ `sq " θ `sb 1 px θ q `s2 Ipθ, sq, where Ipθ, sq " ż 1 0 p1 ´τ qb 1 1 px θ `sτ q dτ. This gives V θ psq " s 2 b 1 px θ q 2 `s3 `2b 1 px θ qIpθ, sq `sIpθ, sq 2 ˘.

Since b 1 is continuous and takes postive values, there exist v ´, v `ą 0 such that v ´ă b 1 px θ q ă v `for all θ P Θ. On the other hand, since b 1 1 grows at most polynomially, this is also the case for Ipθ, ¨q, uniformly in θ P Θ. Thus, we can apply Theorem 3.1. By (3.5) there exist h 0 ą 0 and ε : R ˚Ñ R ˚going to 0 at 0 such that for θ P Θ, h P p0, h 0 q and n ď Npη h , h, θq we have |λ n ph, θq ´p2n ´1qhb 1 px θ q| ď εphqp2n ´1qh.

(5.1) Let x 1 , x 2 P J be such that b 1 px 1 q ‰ b 1 px 2 q. We set θ 1 " a 1 px 1 q, θ 2 " a 1 px 2 q. Choosing h 0 smaller if necessary, we can assume that for all h P p0, h 0 q we have |b 1 px 1 q ´b1 px 2 q| ą εphq.

(5.2)

Now assume by contradiction that there exist h P p0, h 0 q and λ P r0, η h s such that λ is an eigenvalue of L h . We necessarily have λ P " 0, 1 4 ˘. Then, with Σ λ defined as in (2.2), we have Σ λ " `´8, ´?λ ˘Y `?λ, 1 ´?λ ˘Y `1 `?λ, `8˘.

As in the proof of the first statement of Theorem 1.4 we see that λ cannot be an eigenvalue of L h,θ for all θ P p´8, ´?λq or for all θ P p1 `?λ, `8q, so by Proposition 2.2 there exists k P N ˚such that λ " λ k ph, θq for all θ P `?λ, 1 ´?λ ˘. If h 0 was chosen small enough, we have θ 1 , θ 2 P `?λ, 1 ´?λ ˘, so λ k ph, θ 1 q " λ " λ k ph, θ 2 q, which gives a contradiction with (5.1) and (5.2).

Remark 5.1. Note that (5.1) describes the dispersion curves on the interval p ? λ, 1? λq, see Figure 1. The eigenvalues under consideration here are far below the "peak" of the essential spectrum.

Remark 5.2. The function Θ Q θ Þ Ñ b 1 px θ q is nothing but an effective Hamiltonian which emerges from the semiclassical limit. In the semiclassical spectral theory of the magnetic Laplacian, this effective Hamiltonian appears, for instance, in [START_REF] Raymond | Geometry and spectrum in 2D magnetic wells[END_REF]Theorem 1.1]. With this interpretation, the function θ Þ Ñ x θ corresponds to a parametrization of the "characteristic manifold" of the magnetic Laplacian.

Then we have V 1 pxq " iΩV pxq `MpxqV pxq where V " P ´1U, and M " P ´1 ˆ0 0 w 0 ˙P P L 1 pR `q .

The Duhamel Formula gives, for all x ě 0, This establishes the existence of a and b. Since they are necessarily unique, the proof is complete.

1

 1

1. 4 .Theorem 1 . 2 . 8 c 1 x

 41281 Main results. Let us now state our main theorems. In the first result we generalize Theorem 1.1.(ii) by considering situations where φ `" `8 and φ ´P R Y t´8u. Assume that φ ´exists as an element of R Y t˘8u and that for some α P p´1, 0q Y p0, `8q and c 1 , C ą 0 we have bpxq " xÑ`α and |b 1 pxq| ď C x α´1 .(1.4)

4 Figure 1 .

 41 Figure 1. Bottom of the essential spectrum as a function of ξ

. 4 )

 4 |λ n ph, θq ´p2n ´1qhv θ | ď εphqλ n ph, θq . (3Remark 3.2. From (3.4) we obtain that for h small enough we have λ n ph, θq À p2n ´1qhv θ , so with a possibly different function ε we can rewrite (3.4) as |λ n ph, θq ´p2n ´1qhv θ | ď εphqp2n ´1qh. (3.5)

Proposition 3 . 5 .

 35 Let h 0 ą 0 be as in Lemma 3.4. For E P ´0, lim inf |x|Ñ`8 inf θPΘ V θ pxq there exist γ ą 0 and C ą 0 such that for h P p0, h 0 q, θ P Θ and an eigenpair pλ, ψq of L h,θ with λ ď E we have ż R e 2γ|s|{ ? λ |ψ| 2 ds ď C}ψ} 2 L 2 pRq .

V pxq " e iΩx V p0q `ż x 0 e 8 xe 8 e

 088 iΩpx´sq MpsqV psqds . ds , and hence, by the Gronwall Lemma, }V pxq} ď }V p0q} e ş x 0 }M psq}ds . This proves that V is bounded. Thus, by (A.1) we can setA " lim xÑ`8e ´iΩx V pxq.The Duhamel Formula now givesV pxq " e iΩx A ´ż`iΩpx´sq MpsqV psqds " xÑ`iΩx A `op1q. (A.2)It remains to multiply by P to conclude. If ω " 0 then we proceed similarly, without change of basis, and using the fact that exp

  2Φε |ψ| 2 ds ď `1 `κ´1 pc `R2 `1 `κq ˘e2γR 2 Proof of Theorem 3.1. There exists h 0 P p0, 1s such that Lemma 3.4 holds and for all h P p0, h 0 s and θ P Θ we haveη h ď c 8 2 ă c 8 ď inf σ ess pL h,θ q,where c 8 is given by (3.2). Let h P p0, h 0 s and n P t1, . . . , Npη h , h, θqu. For ψ P E n ph, θq with }ψ} L 2 pRq " 1 we have xL h,θ ψ, ψy ď λ n ph, θq .

	and hence
	ż					ż
						|s|ăR ? λ	|ψ| 2 ds.
	It only remains to let ε go to 0 to conclude.
	Now we can prove Theorem 3.1.
						(3.7)
	On the other hand, by (3.1),
	xL h,θ ψ, ψy ě xph 2 D 2 s	`v2
						˙.
	Proposition 3.3 gives				ε 2 |lnphq| 3 . 3	(3.9)
		ż R	ˆVθ psq	´h2 γ 2 λ	´λ˙e 2Φε |ψ| 2 ds ď 0 .
	ż
	|s|ěR ? λ There exists c `ą 0 such that 0 ď V θ psq ď c `s2 for all θ P Θ and |s| ď R pV θ psq ´p1 `κqλqe 2Φε |ψ| 2 ds ď ´ż|s|ăR ? pV θ psq ´p1 `κqλqe 2Φε |ψ| 2 ds . λ ? E, so with (3.6) we have
	ż					ż
	κλ	|s|ěR ? λ	e 2Φε |ψ| 2 ds ď λpc `R2 `1 `κq	|s|ăR ? λ	e 2Φε |ψ| 2 ds ,

By Lemma 3.4 we have λ ě hv 2 , so ż R `Vθ psq ´p1 `κqλ ˘e2Φε |ψ| 2 ds ď 0 .

We choose R ą 0 so large that c E R 2 ´p1 `κq ě κ. Then we write R e θ s 2 qψ, ψy ´Cw ż R |s| 3 xsy N |ψ| 2 ds . (3.8) Let γ be given by Proposition 3.5 for E " c 8 {2. We set α n ph, θq " 2 γ a λ n ph, θq |lnphq| . Since α n ph, θq is bounded uniformly in θ P Θ, h P p0, h 0 s and n ď Npη h , h, θq, we have ż |s|ďαnph,θq |s| 3 xsy N |ψ| 2 ds À λ n ph, θq

Acknowledgments. This work has been supported by the CIMI Labex, Toulouse, France, under grant ANR-11-LABX-0040-CIMI. N. Raymond is deeply grateful to the Mittag-Leffler Institute where part of this work was completed.

Appendix A.

The following lemma is very classical and originally appears in [START_REF] Jost | On the scattering of a particle by a static potential[END_REF].

Lemma A.1. Let ω ě 0 and w P L 1 pR `q. Let ψ P C 2 pR `q be such that ´ψ2 ´ω2 ψ `wψ " 0 .

There exists a unique pa, bq P C 2 such that ψpxq "

xÑ`8 ae iωx `be ´iωx `op1q, ω ą 0, ψpxq "

xÑ`8

a `bx `op1q, ω " 0.

In particular, if ψ P L 2 pR `q, then ψ " 0.

Proof. We first assume that ω ą 0. For x ě 0 we set Upxq " pψpxq, ψ 1 pxqq ⊺ . Then U P C 1 pR `q and