Gimenez-Arnau Elena

Stingeni Luca

Foti Caterina

Aerts Olivier

Gonçalo Margarida

Giménez-Arnau Ana
email: anamariagimenezarnau@gmail.com

Isabel A Ogueta

Johanna Brared Christensson

Elena Giménez-Arnau

Richard Brans

Mark Wilkinson

Luca Stingeni

Caterina Foti

Olivier Aerts

Cecilia Svedman

Margarida Gonçalo

Ana M Giménez-Arnau

M Gonçalo With Abbvie

Leo

Lilly

Limonene and linalool hydroperoxides review: pros and cons for routine patch testing

Keywords: allergic contact dermatitis, hydroperoxides, limonene, linalool, patch test

Limonene and linalool are among the most common fragrance terpenes used in products of everyday life. They are pre-haptens forming hydroperoxides (Lim-OOHs, Lin-OOHs) upon oxidation inducing frequent positive patch test reactions in patients with dermatitis. Still, they

are not yet routinely tested in Europe. This review evaluates the patch testing experience with Lim-OOHs and Lin-OOHs by answering key questions such as whether hydroperoxide patch testing is warranted, understand difficulties or challenges related to the reading and interpretation of hydroperoxide patch test results with currently available material, assessing their relevance. Studies are increasingly pointing out to high percentages of positive reactions in patients consecutively patch tested with these oxidized products. An association between a positive clinical history and a strong patch test reaction has been described, but problems with doubtful/irritant reactions have also been reported. Considering the high frequencies of relevant positive reactions, the incorporation of Lim-OOHs 0.3% and Lin-OOHs 1% in the baseline series could be discussed and is maybe justified. Since exposure, sensitization and elicitation limits of Lim-OOHs and Lin-OOHs in the products still need to be better determined, an assessment of previous exposure, possible sensitizations and reactions may help to improve the clinical assessment.

Introduction

Allergic contact dermatitis (ACD) to fragrances is common, affecting 1.1% to 2.6% of the general population in Europe [START_REF] Thyssen | The prevalence and morbidity of sensitization to fragrance mix I in the general population[END_REF][START_REF] Uter | Categorization of fragrance contact allergens for prioritization of preventive measures: clinical and experimental data and consideration of structure-activity relationships[END_REF][START_REF] Diepgen | Prevalence of fragrance contact allergy in the general population of five European countries: a cross-sectional study[END_REF] . Moreover, some studies report a prevalence that could even reach 15% in patients with a history of dermatitis [START_REF] Marks | North American Contact Dermatitis Group patch test results for the detection of delayed-type hypersensitivity to topical allergens[END_REF][START_REF] Nardelli | Frequency of and trends in fragrance allergy over a 15-year period[END_REF] .

Many fragrance materials used today belong to the chemical group of terpenes [START_REF] De Groot | Essential Oils, Part III: Chemical Composition[END_REF] and, among them, limonene (citrus scent) and linalool (lavender scent) are frequently found in multiple household (hygienic and cosmetic products) and occupational products, essential oils, natural products and in aromatherapy [START_REF] Buckley | Fragrance ingredient labelling in products on sale in the U.K[END_REF][START_REF] Rastogi | Fragrance chemicals in domestic and occupational products[END_REF][START_REF] Rastogi | The composition of fine fragrances is changing[END_REF][START_REF] Bennike | Fragrance contact allergens in 5588 cosmetic products identified through a novel smartphone application[END_REF][START_REF] Yazar | Preservatives and fragrances in selected consumer-available cosmetics and detergents[END_REF][START_REF] Bråred Christensson | Positive patch test reactions to oxidized limonene: exposure and relevance[END_REF] which come into contact with the skin daily.

Limonene and linalool are known to be pre-haptens. They oxidize upon contact with air forming hydroperoxides as primary oxidation products and these have been recognized as important contact allergens [START_REF] Matura | Oxidized citrus oil (R-limonene): A frequent skin sensitizer in Europe[END_REF][START_REF] Sköld | Studies on the autoxidation and sensitizing capacity of the fragrance chemical linalool, identifying a linalool hydroperoxide[END_REF] . If the compounds limonene and linalool are patch tested with not deliberately oxidized ("pure") form, positive patch test reactions are rarely found (Table 1), whereas the hydroperoxides are frequent causes of positive patch test reactions in patients when using standardized patch test materials for oxidized limonene and linalool developed in the last decade (Tables 2 and3 and references therein).

However, oxidized limonene and linalool are not yet routinely tested in Europe. In recent years, there has been debate about the most useful patch test concentrations to obtain a reliable result when testing oxidized limonene and oxidized linalool. In addition, although limonene and linalool are found in many consumer products, it is often difficult to identify these oxidized terpenes as the culprit ingredients causing ACD. Moreover, it is not easy to confirm the clinical relevance of these hydroperoxides is not easy to specify as these products when patch tested "as is" often induce false-negative patch test reactions, and analyses hardly detect and allow quantification of hydroperoxides in commercial products. Therefore, exposure to oxidation products, such as limonene hydroperoxides (Lim-OOHs) and linalool hydroperoxides (Lin-OOHs), remains largely elusive. Nevertheless, it has also been shown in repeated open application test (ROAT) studies that both oxidized limonene and oxidized linalool can cause ACD in sensitized patients [START_REF] Björkman | Air-oxidized linalool elicits eczema in allergic patients -a repeated open application test study[END_REF][START_REF] Bennike | Allergic contact dermatitis caused by hydroperoxides of limonene and dose-response relationship-A repeated open application test (ROAT) study[END_REF] . One possible hypothesis of Lim-OOHs and Lin-OOHs sensitization is that there is repeated exposure to the respective allergen from many sources, and even low concentrations of oxidized terpenes in each product can induce cumulative exposure capable of generating ACD in previously sensitized individuals [START_REF] Björkman | Air-oxidized linalool elicits eczema in allergic patients -a repeated open application test study[END_REF] .

The objectives of this review are to identify and try to clarify doubts regarding the need to incorporate oxidized limonene (Lim-OOHs 0.3% pet.) and oxidized linalool (Lin-OOHs 1.0% pet.) in the baseline series for patch testing as it was proposed [START_REF] Wilkinson | The European baseline series: Criteria for allergen inclusion with reference to formaldehyde releasers[END_REF] .

Methods

This document is based on a systematic search of all literature published from the first publication on the subject (1985) to 2021. The following bibliographical sources were used for the search: Pubmed, Embase. The literature searches were designed to cover the following items alone and/or combined: allergy, contact dermatitis, eczema, hydroperoxides limonene, hydroperoxides linalool, limonene, linalool, terpenes. The following MeSh terms were used: allergic contact dermatitis and (limonene or limonene-2-hydroperoxide or limonene -1hydroperoxide) and (linalool or linalool oxide) and patch tests. The manuscript was initially carried out by five recognized experts in chemistry and contact dermatitis due to oxidized terpenes. The primary document was assessed and reviewed by the European Environmental Contact Dermatitis Research Group (EECDRG) members. Then at least two rounds of discussion were performed before reaching consensus on the final document submitted.

3

Current knowledge about the sensitizing capacity and the diagnostic system

Radical mechanisms possibly involved in contact sensitization to Lim-OOHs and

Lin-OOHs

Lin-7-OOH/Lin-6-OOH and Lim-1-OOH/Lim-2-OOH hydroperoxides have been identified

as the components in the oxidation mixtures which are responsible for the sensitization potential of autoxidized linalool and R-(+)-limonene, respectively (Figure 1). Cutaneous allergens (haptens) are low molecular weight compounds that are unable to stimulate an adaptive immune response, but which become immunogenic after reacting with proteins to form stable hapten-protein conjugates that are subsequently processed and presented to the immune system. The best-known mechanism for hapten-protein interaction is the formation of covalent bonds by two-electrons mechanisms. Indeed, very often the allergen is electrophile and reacts with nucleophilic side chains of amino acids from skin proteins, such as cysteine and lysine. However, organic hydroperoxides (R-OOHs) do not fit this model and one-electron radical-mediated mechanisms are suspected to be involved [START_REF] Lepoittevin | Molecular aspects in allergic and irritant contact dermatitis[END_REF] . Indeed, sensitizing R-OOHs such as Lin-OOHs and Lim-OOHs can react through radical mechanisms that begin with the cleavage of the O-O bond of weak dissociation energy (ca.

175 kJ mol -1 ). Such cleavage is an easy process that allows the initially formed unstable alkoxyl radicals (RO • ) to be efficiently converted into longer half-life carbon-centered radicals through different known rearrangement processes. An example is shown in Figure 2 for Lin-7-OOH [START_REF] Kao | Synthesis of allylic hydroperoxides and EPR Spin-Trapping studies on the formation of radicals in iron systems as potential initiators of the sensitizing pathway[END_REF] . All these radical intermediates could potentially be reactive toward amino acids in the skin.

To understand how R-OOHs can form immunogenic structures once they penetrate into the epidermis, reactivity studies have been performed with amino acids known to be involved in radical processes. These studies were performed in solution (buffer, semi-organic) and were followed by nuclear magnetic resonance (NMR) and liquid chromatography combined to mass spectrometry (LC-MS). It was shown that carbon-radicals emitted by R-OOHs could react with amino acids such as cysteine, histidine, and tryptophan when Fe(II)/Fe(III) redox systems were used to trigger radical initiation [START_REF] Kao | Synthesis of allylic hydroperoxides and EPR Spin-Trapping studies on the formation of radicals in iron systems as potential initiators of the sensitizing pathway[END_REF][START_REF] Kao | Mechanistic studies on the reactivity of sensitizing allylic hydroperoxides: investigation of the covalent modification of amino acids by carbon-radical intermediates[END_REF] . Importantly, during these studies it was also noted that oxide-reduction processes are promoted during the reactions inducing chemical modifications of amino acids, thus demonstrating the involvement of oxidative stress.

The above mentioned studies were performed in solution and, of course, a critical step in understanding skin sensitization to these hydroperoxides is to verify if these radicals are really formed in the skin. To achieve this goal, efforts are today focused on the use of electron (Figure 3) [START_REF] Kuresepi | Potential of EPR spin-trapping to investigate in situ free radicals generation from skin allergens in reconstructed human epidermis: cumene hydroperoxide as proof of concept[END_REF][START_REF] Sahli | Understanding the skin sensitization capacity of ascaridole: a combined study of chemical reactivity and activation of the innate immune system (dendritic cells) in the epidermal environment[END_REF] . EpiSkin TM RHE (Episkin, Lyon, France) consists of a normal multi-layered culture of keratinocytes, main cell type of the epidermis playing a key role in inflammatory reactions in the skin. EpiSkin TM RHE is a very good replica of human epidermis architecture.

It is used for irritation and penetration testing of cosmetic and chemical compounds in replacement of in vivo testing [START_REF]Vitro Skin Irritation: Reconstructed Human Epidermis Test Method[END_REF] . Several oxygen-and carbon-centered radicals were initially identified in this complex environment, using cumene hydroperoxide and ascaridole as proofs of concept. Further, basic triggers for the activation in the skin of hydroperoxides derived from the autoxidation of linalool and limonene were studied using this methodology [START_REF] Kuresepi | Mechanistic insights on skin sensitization to linalool hydroperoxides: EPR evidence on radical intermediates formation in reconstructed human epidermis and 13C NMR reactivity studies with thiol residues[END_REF][START_REF] Lichter | Skin sensitization to fragrance hydroperoxides: interplay between dendritic cells, keratinocytes and free radicals[END_REF] . Experiments were conducted based on a topical application procedure to estimate real-life sensitization. Skin samples were initially loaded with DEPMPO to ensure its presence at the site of radical generation prior to application of the target compounds. The RHE were thus pre-treated with DEPMPO allowing its penetration through the stratum corneum. Then, subsequent topical exposure of DEPMPO-pre-treated RHE to different concentrations of hydroperoxides demonstrated that DEPMPO was able to trap carboncentered radicals formed in this epidermal-like skin environment, together often with the hydroxyl radical and minor alkoxyl radicals. Last but not least, no Fe(II) was used for radical initiation when RHE was used to mimic real-life cutaneous allergy situations. Therefore, it could be hypothesized that the skin reaction of hydroperoxides in the presence of oneelectron donor agents (e.g., amino and thiol groups present in amino acids, metal complexes, enzymes) may initiate electron transfer processes under radical oxidation conditions and subsequent haptenation of skin proteins.

Major drawback of these studies is the impossibility of accurately assigning the carboncentered radicals formed in the RHE among the different options resulting from radical rearrangement processes (Figure 2). To overcome this, one possibility is to carry out spintrapping studies with target compounds site-specific labelled with [START_REF] Matura | Oxidized citrus oil (R-limonene): A frequent skin sensitizer in Europe[END_REF] C at positions precursors of potentially reactive radicals. Recent EPR-ST studies with cumene hydroperoxide [START_REF] Matura | Oxidized citrus oil (R-limonene): A frequent skin sensitizer in Europe[END_REF] Csubstituted at methyl positions confirmed the trapping of methyl radicals released by decomposition of the hydroperoxide in the RHE model [START_REF] Sahli | Formation of methyl radicals derived from cumene hydroperoxide in reconstructed human epidermis: an EPR spin trapping confirmation by using 13C-substitution[END_REF] .

As a conclusion to this section, radicals formed from hydroperoxides are key intermediates that could explain the reactivity with skin proteins and, therefore, their sensitizing potential.

Moreover, now we know that these intermediates are formed in a 3D model of epidermis, under conditions closer to what may occur in vivo.

Experience from patch testing with Lim-OOHs and Lin-OOHs

Up to the present, a number of studies have been conducted to evaluate the prevalence of contact allergy to limonene and linalool, both in not deliberately oxidized and oxidized forms (Tables 1, 2 and3). In some studies, not deliberately oxidized limonene and/or linalool were tested concomitantly with the oxidized forms. Overall, very few patients reacted to not deliberately oxidized limonene or linalool (range 0%-0.88%) (Table 1 and Refs in Table 1).

Oxidized limonene has been tested in different concentrations (Table 2), especially at 3% (containing 0.3% Lim-OOHs), based on experience from a study published in 1997 [START_REF] Karlberg | Contact allergy to oxidized d-limonene among dermatitis patients[END_REF] . An early multicenter study with 2273 dermatitis patients using 3% oxidized limonene showed 2.8% positive patch test reactions [START_REF] Matura | Oxidized citrus oil (R-limonene): A frequent skin sensitizer in Europe[END_REF] .

The first multicenter study of patch testing with oxidized linalool 2% pet. (containing 0.33% Lin-OOHs) was published in 2005, showing 1.3% positive patch test reactions in 1511 patients [START_REF] Matura | Selected oxidized fragrance terpenes are common contact allergens[END_REF] . Subsequently, a dose-response study investigated whether a higher concentration of oxidized linalool could be useful to detect a higher number of allergic reactions, and a patch test concentration of oxidized linalool 6.0% (containing Lin-OOHs at 1.0%) was suggested [START_REF] Bråred Christensson | Linalool -a significant contact sensitizer after air exposure[END_REF] . Few irritant reactions were identified and there were no cases of active sensitization.

Several multicenter studies have since been conducted using oxidized limonene and oxidized linalool. In 2009, a total of 2900 consecutive dermatitis patients were evaluated in 9 centers dedicated to epicutaneous patch testing (Copenhagen, Malmo, Odense, Barcelona, London, Singapore, Melbourne, Seville and Gothenburg) [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF][START_REF] Bråred Christensson | An international multicentre study on the allergenic activity of air-oxidized R-limonene[END_REF] . The overall prevalence of positive reactions to oxidized limonene 3% pet. was 5.2% [START_REF] Bråred Christensson | An international multicentre study on the allergenic activity of air-oxidized R-limonene[END_REF] , ranging from 2.3% to 12.1% in the individual centers. There were notable differences between the northern and southern

European centers, with patch testing positive in 3.8% (Gothenburg, Malmo, London,

Copenhagen, and Odense) versus 5.5% (Seville and Barcelona). Twenty five percent of patients who had a positive patch test showed a strongly positive (++/+++) reaction and 91%

had their maximum reaction at D3-D4 readings. Concomitant reactions between R-limonene and other fragrances and/or colophonium from the baseline series were found in 42% of the oxidized limonene-positive cases and in 19.6% of the patients with doubtful (?+) reactions.

Finally, in patients negative to oxidized R-limonene, only 10.6% of patients had positive reactions to fragrances and/or colophonium.

For oxidized linalool 6% pet. (1.0% Lin-OOHs), 6.9% of 2900 tested patients showed positive patch test reactions [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF] , with frequencies ranging from 3.3% to 14.3% in the different centers [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF] . Of the patients with positive reactions, 28% were strongly positive (++/+++), 86% of them at D3-D4 readings, and 40% were also positive to other baseline series fragrances and/or colophonium, which is statistically significant different from patients with doubtful (?+) or negative reactions (19% and 10%, respectively). No delayed reactions or active sensitization was described.

These studies led to the development of standardized patch test materials of oxidized limonene 3%, with a standardized content of Lim-OOHs at 0.3% (hydroperoxides of Limonene 0.3% pet.®) and oxidized linalool 6%, with a standardized content of Lin-OOHs at 1.0% (hydroperoxides of Linalool 1.0% pet.®) which are manufactured by Chemotechnique Diagnostics AB, Vellinge, Sweden [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF][START_REF] Bråred Christensson | An international multicentre study on the allergenic activity of air-oxidized R-limonene[END_REF] . The stability of the patch test material was monitored repeatedly during the study and showed a stable content of hydroperoxides during use.

In subsequent multicenter and single-center studies, patch testing with Lim-OOHs 0.3% pet.

has shown positive patch test reactions in 1.2% to 9.4% of cases (Table 2 and references therein). Likewise, Lin-OOHs 1.0% pet. has been used in many studies, giving 3.9% to 20% positive patch test reactions (Table 3 and references therein). Interestingly, in multicenter studies, wide ranges in rates of positive reactions were found between the test centers [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF][START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] , as further discussed below.

Patch test concentrations used in these studies have been further evaluated. In Spain, the GEIDAC group (Grupo Español de Investigación de Dermatitis de Contacto y Alergia

Cutánea) conducted a study with 22 Spanish centers and 3639 consecutive patients tested.

Overall, 5.1% of the patients showed positive patch test reactions to Lim-OOHs 0.3% pet.

with frequencies ranging between 0% and 24.8% at the different test centers [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] . Similarly, Lin-OOHs 1.0% pet. gave 4.9% positive patch test reactions overall, with a frequency ranging between 0% and 13.9% [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] . The GEIDAC study also compared three concentrations of Lim-OOHs at 0.1%, 0.2%, and 0.3% pet. giving 1.4%, 3.4% and 5.1% positive patch test reactions respectively (Table 2). Concomitantly, three concentrations of Lin-OOHs at 0.25%, 0.5%, and 1.0% pet. were tested giving 1.3%, 2.9% and 4.9% positive patch test reactions respectively [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] (Table 3). Overall, 8% of patients responded to one or both hydroperoxides.

The prevalence of positive patch tests increased with increasing patch test concentrations for both terpenes. In 30% of the cases of allergy to oxidized terpenes there was a concurrent contact allergy to other fragrance markers in the baseline series but in 70% of the allergic patients there was only a contact allergy to oxidized terpenes, indicating that the patients would have gone unnoticed if they had not been tested with the oxidized terpenes patch test.

In this study, only 25% of the positive cases were simultaneously positive for both hydroperoxides, supporting the specificity of the reaction.

A British multicenter study was conducted by Wlodek et al. in 2017 [START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] , in which 4563 consecutive patients were patch tested with Lim-OOHs at 0.1%, 0.2% and 0.3% pet. and with Lin-OOHs at 0.25%, 0.5% and 1.0% pet (Tables 2 and3), with 5.3% positive patch test reactions for Lim-OOHs 0.3% pet., and 7.7% for Lin-OOHs 1.0% pet. [START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] . Positive patch test reactions were considered clinically relevant in approximately 2/3 of the patients, as they had a clear current or previous history of fragrance allergy, or dermatitis from a product labelled to contain limonene or linalool, or dermatitis from a botanical oil with limonene or linalool, or a positive ROAT to such a product. Lim-OOHs at 0.3% pet. and Lin-OOHs at 1.0% pet. were recommended to be included in the British baseline patch test series [START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] .

In a recent analysis of 5773 patients tested 2013 to 2020, significantly increasing trends in contact allergy to Lim-OOHs at 0.3% pet. and Lin-OOHs at 1.0% pet. were observed, and the patients with contact allergy to Lim-OOHs and Lin-OOHs were significantly younger than those with contact allergy to other fragrances [START_REF] Sukakul | Contact allergy to oxidized linalool and oxidized limonene: Patch testing in consecutive patients with dermatitis[END_REF] . There was also a female predominance.

Nowadays, also due to the availability of the commercial test preparations, increasing numbers of cases of sensitization to Lim-OOHs and Lin-OOHs have been described when studying contact allergy to cosmetics, e.g. on the face [START_REF] Bruusgaard-Mouritsen | Facial contact dermatitis caused by cosmetic-relevant allergens[END_REF] . In addition, early and continuous exposure to Lim-OOHs and Lin-OOHs may have an impact on children. A Belgian study demonstrated that the most frequent sensitizations in 329 children patch tested from 2010 until 2019 were to nickel sulfate and Lin-OOHs (both 16%), Lim-OOHs (13.5%) and para-phenylenediamine (10.9%) 37

Doubtful and irritant patch test reactions. Are doubtful patch test reactions clinically relevant?

Doubtful and/or irritant reactions in studies on Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. have been a matter of debate. In general, in the multicenter studies, comparatively high rates of doubtful and/or irritant reactions have been recorded at some test sites, while other test sites have had low frequencies of such reactions.

For oxidized limonene and oxidized linalool, irritation studies have been conducted showing very low irritation in the range of tested concentrations of 3% pet. and 6% pet., respectively [START_REF] Bråred Christensson | Air oxidation increases skin irritation from fragrance terpenes[END_REF] .

However, doubtful reactions to Lin-OOHs 1.0% pet. ranged from 0% to 36% in the international multicenter study 31 , and from 0% to 19% in an UK multicenter study [START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] . The corresponding ranges for Lim-OOHs 0.3% pet. were 0% to 24% in the international multicenter study 32 and 0% to 17% in the UK multicenter study [START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] . Similar figures were shown in the Spanish multicenter study [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] , where the range of doubtful reactions for Lim-OOHs 0.3% pet. was 0% to 3.6% and for Lin-OOHs 1.0% pet. was 0% to 2.8%, while the range of irritant reactions for Lim-OOHs 0.3% pet. was 0% to 3.6% and for Lin-OOHs 1.0% pet. was 0% to 7.6% [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] . In all of these multicenter studies, when Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. were tested concomitantly, centers that showed high rates doubtful or irritant reactions recorded them for both test preparations, while most centers recorded very low frequencies of doubtful/irritant reactions. Then, readings may to some extent be the result of differences in the expert interpretation and recording of patch test results or a lack of standardized methodology (e.g., administered dose) [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF][START_REF] Bråred Christensson | An international multicentre study on the allergenic activity of air-oxidized R-limonene[END_REF][START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF][START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] . Furthermore, it has been shown in dose-response studies when different concentrations of oxidized limonene and/or oxidized linalool were tested, that a proportion of the doubtful reactions at the lower concentration showed a positive reaction at the next higher concentration. In the Swedish dose-response study to oxidized linalool, 25% of the patients who reacted with doubtful reactions to oxidized linalool 4% pet.

(0.66% Lin-OOHs) had a positive reaction to 6% pet. (1.0% Lin-OOHs) [START_REF] Bråred Christensson | Linalool -a significant contact sensitizer after air exposure[END_REF] . Similarly, it was shown in the Spanish multicenter study that 33% and 39% of the doubtful reactions to Lim-OOHs 0.2% pet. and Lin-OOHs 0.5% pet. respectively, were interpreted as positive reactions at the next higher test concentration of the respective test material. A weak allergic reaction at a low concentration became positive at an increased dose. Therefore, it is important to differentiate between irritant and doubtful. The irritant reaction will not change nature but will of course become stronger. Thus, some of the doubtful reactions may, in fact, be weak positive reactions. This is supported by a ROAT study with oxidized limonene, where 2/13 (15%) subjects with doubtful patch test reactions to Lim-OOHs had positive ROAT reactions to Lim-OOHs applied in realistic doses [START_REF] Bennike | Allergic contact dermatitis caused by hydroperoxides of limonene and dose-response relationship-A repeated open application test (ROAT) study[END_REF] .

There has been discussion regarding the specificity of positive reactions to oxidized patch test materials, due to the overall high rates of reactions to these substances. Many studies have confirmed that positive reactions to the separate oxidation mixtures as well as to their major allergens are specific. In these studies, when oxidized limonene and oxidized linalool are tested concomitantly, in the entire group of positive patients, about 25% will react to both Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet., while the rest (75%) will react to one or to the other [START_REF] Bråred Christensson | Oxidized limonene and oxidized linalool -concomitant contact allergy to common fragrance terpenes[END_REF][START_REF] Audrain | Allergy to oxidized limonene and linalool is frequent in the U[END_REF] . These concomitant reactions are in accordance with the frequency of exposure to both fragrances in everyday products, and with the documented common tandem exposure of limonene and linalool in products [START_REF] Buckley | Fragrance ingredient labelling in products on sale in the U.K[END_REF][START_REF] Rastogi | Fragrance chemicals in domestic and occupational products[END_REF][START_REF] Bennike | Fragrance contact allergens in 5588 cosmetic products identified through a novel smartphone application[END_REF][START_REF] Yazar | Preservatives and fragrances in selected consumer-available cosmetics and detergents[END_REF][START_REF] Uter | Coupled exposure to ingredients of cosmetic products: I. Fragrances[END_REF] . Considering the total number of tested patients, the majority do not react to either test material, again opposing that the reactions for hydroperoxides are unspecific. Also, for the main oxidized limonene allergens, Lim-1-OOH and Lim-2-OOH (Figure 1), it has been shown that there is specificity for reactions between these two structurally similar hydroperoxides [START_REF] Bråred Christensson | Limonene hydroperoxide analogues differ in allergenic activity[END_REF] . The statistically significant difference in concomitant contact allergy to other fragrance markers between the groups showing positive, doubtful, or negative patch test reactions to the oxidized terpenes as described above, also support that the patch test materials for oxidized limonene and linalool are detecting true fragrance allergy.

The clinical relevance of a positive contact allergen can be assessed in many ways, and a common method is to interview the patient and search for exposure that can be expected to contribute to the patient´s dermatitis. The relevance assessment can be difficult when interpreting positive patch test to Lim-OOH and Lin-OOH because the suspected culprits, the terpene hydroperoxides, are not labelled or identified as such in the products, which is in fact a limitation. Nevertheless, it is very difficult to find a product that does not contain terpenes that potentially could be oxidized. In about 35-70% of patients with positive reactions to Lim-OOHs 0.3% pet., an exposure to products containing listed limonene has been assessed to be relevant for the dermatitis [START_REF] Bråred Christensson | Positive patch test reactions to oxidized limonene: exposure and relevance[END_REF][START_REF] Audrain | Allergy to oxidized limonene and linalool is frequent in the U[END_REF] . For Lin-OOHs 1.0% pet., similar assessments have been made and similar estimates of relevance have been reported [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF][START_REF] Audrain | Allergy to oxidized limonene and linalool is frequent in the U[END_REF][START_REF] Dittmar | Contact sensitization to hydroperoxides of limonene and linalool: Results of consecutive patch testing and clinical relevance[END_REF] .

The GEIDAC study [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] reported that at least 50% of the reactions were relevant. The difficulty of assessing the relevance would benefit from an in vitro or in vivo test that would be able to detect the responsible hydroperoxide in the specific product.

Sources of exposure and thresholds of elicitation

Commercial notification of the presence of fragrances is optional in some countries (such as Australia and Singapore), so their presence, both qualitative and quantitative, is unknown. In Europe, since 2005, the Cosmetics Directive determines that 26 fragrances must be declared in cosmetic products when their concentration exceeds 10 ppm in leave-on and 100 ppm in rinse-off cosmetics, respectively 44 . This mandatory declaration of content is not applicable in other parts of the world.

Limonene and linalool have been identified in up to 80% of common products (mainly personal hygiene products, cosmetics, and cleaning household products) either by their labelling or by chemical analyses [START_REF] Buckley | Fragrance ingredient labelling in products on sale in the U.K[END_REF][START_REF] Rastogi | Fragrance chemicals in domestic and occupational products[END_REF][START_REF] Bennike | Fragrance contact allergens in 5588 cosmetic products identified through a novel smartphone application[END_REF] , with limonene also being present e.g., in industrial soaps and citrus-based solvents [START_REF] Bennike | Non-mix fragrances are top sensitizers in consecutive dermatitis patients - a cross-sectional study of the 26 EU-labelled fragrance allergens[END_REF] and adhesives [START_REF] Dendooven | Concomitant positive patch test reactions in FreeStyle-allergic patients sensitized to isobornyl acrylate[END_REF][START_REF] Mowitz | Patch Testing with Isobornyl Acrylate in 16 Swedish Patients with Contact Dermatitis from Glucose Sensors and/or Insulin Pumps[END_REF][START_REF] Thomas | Limonene[END_REF] . Linalool has been found to be the fragrance with the highest exposure, based on daily cumulative exposure from many sources [START_REF] Marks | North American Contact Dermatitis Group patch test results for the detection of delayed-type hypersensitivity to topical allergens[END_REF] . Limonene and linalool were also found to be the most common tandem exposure and are frequently used together in fragranced/scented products [START_REF] Yazar | Preservatives and fragrances in selected consumer-available cosmetics and detergents[END_REF][START_REF] Uter | Coupled exposure to ingredients of cosmetic products: I. Fragrances[END_REF][START_REF] Nanyan | Fragrance Allergens in Hair Removal Cosmetic Products[END_REF] . Both are also common components of essential oils, and they are found in aromatherapy and natural products.

Limonene is a component of citrus peel oil (genus Citrus, family Rutaceae) as well as eucalyptus essential oils (leaf of Eucalyptus, family Myrtaceae), whereas linalool is a major component of lavender oil (distillate of LavandulaAngustifoli), which consists of approximately 50% linalool [START_REF] Hagvall | Lavender oil lacks natural protection against autoxidation, forming strong contact allergens on air exposure[END_REF] . It has been shown that lavender oil of natural origin will autoxidize similarly to synthetic lavender oil (made by blending the three main components linalool, linalyl acetate, and caryophyllene), and in turn, in a similar way to its separate components [START_REF] Sköld | Studies on the autoxidation and sensitizing capacity of the fragrance chemical linalool, identifying a linalool hydroperoxide[END_REF][START_REF] Hagvall | Lavender oil lacks natural protection against autoxidation, forming strong contact allergens on air exposure[END_REF][START_REF] Sköld | Autoxidation of linalyl acetate, the main component of lavender oil, creates potent contact allergens[END_REF] .

With this scenario, to establish the relevance of Lim-OOHs and Lin-OOHs positive patch tests, it would be ideal to assess the minimal levels of the oxidized forms capable of sensitizing individuals as well as eliciting contact dermatitis in sensitized individuals. To study elicitation levels, ROATs have been performed for oxidized linalool and limonene. In these studies, the daily use of a product ("cream" or "perfume") is simulated by repeated skin applications of the products containing the allergen(s) in low concentrations. In the case of oxidized linalool, individuals who earlier had been shown to be allergic in patch testing, reacted in the ROAT to a concentration as low as 0.3% oxidized linalool, which contained 560 μg/g Lin-OOHs both in perfume and cream base [START_REF] Björkman | Air-oxidized linalool elicits eczema in allergic patients -a repeated open application test study[END_REF] . In the case of oxidized limonene, allergic individuals reacted to as low as 140 μg/g in the ROAT, while three allergic patients reacted to concentrations as low as 24 μg/g Lim-OOHs in a series of patch test dilutions [START_REF] Bennike | Allergic contact dermatitis caused by hydroperoxides of limonene and dose-response relationship-A repeated open application test (ROAT) study[END_REF] .

These studies demonstrate that allergic individuals react to very low concentrations of hydroperoxides.

Isolated cases of contact dermatitis to Lim-OOHs have been reported. Some were due to occupational exposure, with a histology technician that developed a recalcitrant hand eczema after being in contact with a limonene-based solvent agent [START_REF] Foti | Occupational contact dermatitis to a limonene-based solvent in a histopathology technician[END_REF] . The patient turned out negative to patch test with limonene and the solvent agent at the appropriate concentrations, while positive to patch test with Lim-OOHs. Chemical analysis failed to detect Lim-OOHs in the solvent agent used by the patient, so the authors suggested that oxidation of limonene may occur during the handling of the product, especially in presence of oxidant stains (frequently used in histological laboratories.

Another case showed a lymphomatoid contact dermatitis due to hygiene products demonstrated by patch and provocation tests [START_REF] Gatica-Ortega | Lymphomatoid contact dermatitis caused by limonene hydroperoxides confirmed by an exposure provocation test with the involved personal hygiene products[END_REF] . Two cases of dermatitis in patients allergic to oxidized linalool (Lin-OOHs 1.0% pet.) in which the culprit products were analyzed have been reported [START_REF] Elliott | Severe intractable eyelid dermatitis probably caused by exposure to hydroperoxides of linalool in a heavily fragranced shampoo[END_REF][START_REF] Isaksson | Allergic contact dermatitis caused by oxidized linalool in a deodorant[END_REF] .

Quantification of hydroperoxides in products

Hydroperoxides are very difficult to quantify and at present, no consensual detection limit values in products have not been agreed A study from 2019 56 evaluated 104 samples of consumer products that patients suspected could trigger their symptoms, as well as products containing essential oils, including old and new samples of the same brand. Using GS-MS screening analysis, four samples (3.8%) contained >50 μg/g of at least one of the hydroperoxides. However, specifically analysis by liquid chromatography-mass spectrometry (LC-MS) methods, of the samples recovered from patch test positive patients showed levels <50 μg/g of the different hydroperoxides. In a previous study, a geometric mean of 30 μg/g was determined in 22/39 (56%) samples investigated by LC-MS analysis [START_REF] Kern | Detection of potentially skin sensitizing hydroperoxides of linalool in fragranced products[END_REF] .

New and improved methods (e.g different matrices) could help to assess even lower amounts of such hydroperoxides and elucidate the clinical relevance of positive patch tests [START_REF] Ramzi | A two-dimensional non-comprehensive reversed/normal phase high-performance liquid chromatography/tandem mass spectrometry system for determination of limonene and linalool hydroperoxides[END_REF] .

Using a two-dimensional LC-MS method, a content of 8.4 μg/g Lin-6-OOH and 5.6 μg/g Lin-7-OOH was demonstrated in the deodorant that caused eczema in the axilla of a female patient with a solitary reaction to oxidized linalool (Lin-OOHs 1.0% pet.). The use test showed clear clinical relevance [START_REF] Isaksson | Allergic contact dermatitis caused by oxidized linalool in a deodorant[END_REF][START_REF] Ramzi | A two-dimensional non-comprehensive reversed/normal phase high-performance liquid chromatography/tandem mass spectrometry system for determination of limonene and linalool hydroperoxides[END_REF] . In another case, a content of 0.2 μg/g Lin-OOHs was detected in a shampoo that caused eyelid dermatitis in a child [START_REF] Elliott | Severe intractable eyelid dermatitis probably caused by exposure to hydroperoxides of linalool in a heavily fragranced shampoo[END_REF] . A cumulative exposure could be the reason for sensitization to Lim-OOHs and Lin-OOHs.

Taken together, the results obtained so far from chemical analyses of limonene and linalool hydroperoxides in scented products indicate that the content of these hydroperoxides may cause elicitation in sensitized individuals. About 4% of the products analyzed by Natsch et al. [START_REF] Natsch | Exposure source for skin sensitizing hydroperoxides of limonene and linalool remains elusive: An analytical market surveillance[END_REF] contained >50 μg/g of either hydroperoxide. Considering the widespread use of products containing limonene and linalool, there are many opportunities to reach amounts of hydroperoxides capable of eliciting ACD in daily exposures. The ROAT, as well as previous case reports, support that very low amounts of hydroperoxides can elicit ACD.

One question raised concerns the timing of terpene oxidation. Between ordering and receiving essential oils, oxidation had already occurred at the time of delivery [START_REF] Rudbäck | Essential oils can contain allergenic hydroperoxides at eliciting levels, regardless of handling and storage[END_REF] . Moreover, essential oils oxidize in any environment (stored in the dark, in the refrigerator, etc.) [START_REF] Rudbäck | Essential oils can contain allergenic hydroperoxides at eliciting levels, regardless of handling and storage[END_REF] . Thus, oxidation of a product can occur both before formulation of the product (during production or transport from the fragrance producer), during formulation, during transport of the consumer product (e.g., shampoo or cream), during storage, and during use. The presence of antioxidants is considered a protective factor for chemical degradation, but their usefulness has been shown to last for a limited time period. In the case of limonene, it has been demonstrated that the antioxidant butylated hydroxytoluene (BHT) will prevent degradation during a certain period, but will be consumed, after which the oxidation of the terpene will occur [START_REF] Karlberg | Influence of an anti-oxidant on the formation of allergenic compounds during auto-oxidation of d-limonene[END_REF] .

Challenges in Lim-OOHs and Lin-OOHs path testing

Patch testing with standardized patch test material of oxidized limonene 3% (standardized content of Lim-OOHs at 0.3%; hydroperoxides of limonene 0.3% pet.®) and oxidized linalool 6% (standardized content of Lin-OOHs at 1.0%; hydroperoxides of linalool 1.0% pet.®) (Chemotechnique Diagnostics AB, Vellinge, Sweden [START_REF] Bråred Christensson | Air-oxidized linalool: a frequent cause of fragrance contact allergy[END_REF][START_REF] Bråred Christensson | An international multicentre study on the allergenic activity of air-oxidized R-limonene[END_REF] ) has now been performed in most large studies, showing consistent and stable results. Both selected Lim-OOH and Lin-OOH concentrations allow to better identify doubtful and irritant reactions than lower concentrations. [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF] A large part of the positive reactions has been judged relevant by the clinicians.

A great number of patients with positive patch test reactions to Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. show no concomitant reactions to other fragrances or related substances.

If Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. are not used for routine patch test, especially in patients without other fragrance allergies, a large part of the fragrance-allergic patients will not be informed of fragrance allergy, exposing them to the risk of a relapsing episodes of clinical ACD, without the possibility of secondary prevention strategies.

Therefore, Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. should routinely be tested in patients suspected of having fragrance allergy.

At present it seems that positive patch test reactions to Lim -OOHs and Lin-OOHs still do not fulfill the necessary criteria to justify their inclusion in the European baseline series, although some countries already use them in their national baseline series [START_REF] Wilkinson | The European baseline series: Criteria for allergen inclusion with reference to formaldehyde releasers[END_REF] . One challenge when performing Lim-OOHs and Lin-OOHs patch tests is the misinterpretation of positive reactions due to the considerable percentage of doubtful and irritant reactions. In cases of weak patch test reactions, it is important to rule out a false positive, irritant reaction. A ROAT may then help to distinguish true contact allergies from irritant reactions [START_REF] Bråred Christensson | Positive patch test reactions to oxidized limonene: exposure and relevance[END_REF][START_REF] Bennike | Allergic contact dermatitis caused by hydroperoxides of limonene and dose-response relationship-A repeated open application test (ROAT) study[END_REF][START_REF] Johansen | Quantitative aspects of isoeugenol contact allergy assessed by use and patch tests[END_REF] .

How to reliably assess Lim-OOHs and Lin-OOHs exposure is still difficult, and sensitization and elicitation limits of Lim-OOHs and Lin-OOHs in products remain unknown.

Finally, further patch test materials for assessing fragrance allergy need to be developed to be up to date with current exposures to fragrances. Many other fragrance terpenes are known to be pre-or prohaptens (e.g., geraniol, citronellol, linalyl acetate, eugenol) and are known to be altered by air oxidation and/or skin metabolism and patch test materials need to be adjusted for this [START_REF] Karlberg | Activation of non-sensitizing or lowsensitizing fragrance substances into potent sensitizers -prehaptens and prohaptens[END_REF] .

Conclusions

We currently recommend that Lim-OOHs 0.3% pet. and Lin-OOHs 1.0% pet. should be patch tested in all patients with suspected fragrance allergy. Additionally, it might be considered to include them in the baseline series [START_REF] Deza | Contact sensitization to limonene and linalool hydroperoxides in Spain: a GEIDAC* prospective study[END_REF][START_REF] Wlodek | Recommendation to test limonene hydroperoxides 0•3% and linalool hydroperoxides 1•0% in the British baseline patch test series[END_REF] , as positive reactions are frequently relevant as such allowing the detection of additional cases of fragrance allergy. However, the routine patch test conditions of these materials, as well as their sources of exposure should be better clarified before definitely "allowing" them in a "reference series" that should be practical/usable by "all" dermatologists, also the less experienced ones. Testing Moreover, more work should be done to develop stable and reliable methods for detection of hydroperoxides in complex consumer products to identify and confirm the actual exposure sources causing sensitization and elicitation of ACD.

It should be kept in mind that the discussion on the general interest about when and how patch testing for some common skin contact allergens should be performed transcends the boundaries of our daily practice and has a direct impact on the health of our patients and on all preventive measures designed to avoid a public and multidisciplinary problem.

Credit (Contribution Roles of authors)

Isabel A Ogueta, Johanna Brared Radical decomposition of Lin-7-OOH

  paramagnetic resonance (EPR) spectroscopy, committed to the study of reactive free radical species, in combination with the spin trapping (ST) technique. EPR-ST is nowadays the best method to identify transient radicals in chemical and biological systems 21 . Hydroperoxidederived radicals are generally very short-lived or are formed at a too low concentration to be directly detected by EPR. Then, the EPR-ST methodology is employed. EPR-ST is based on the reaction of a spin-trap with a transient radical to form a longer half-life radical adduct detectable by EPR. For many years, EPR-ST experiments were carried out in solution by activating hydroperoxides with Fe(II) to induce radical initiation. The use of spin-traps such as 5-diethoxyphosphoryl-5-methyl-1-pyrroline N-oxide (DEPMPO) and 5,5-dimethyl-1pyrroline N-oxide (DMPO) allowed the identification of different oxygen-and carboncentered free radicals derived from the target compounds 22 . These studies revealed that several intermediate reactive radicals could be possible depending on the hydroperoxide chemical structure, suggesting that different chemical modifications of immunogenic proteins could lead to sensitization. However, these in-solution studies are far from mimicking real-life scenarios. With the aim to produce the in vivo model, a new EPR-ST methodology has been developed to investigate in situ the formation and behaviour of hydroperoxide-derived free radicals in a 3D model of reconstructed human epidermis (RHE)

  simultaneously lower concentrations, such as Lim-OOHs 0.2% and Lin-OOHs 0.5%, may help in interpretation of patch test results, discovering contact allergies. Moreover, weak (+) reactions should always be interpreted with caution as they may still, in fact, represent irritant reaction. Relevance assessment should include an in-depth evaluation of exposure to limonene and linalool in different types of products and thus a comprehensive exposure history. Therefore, the development of patient questionnaires in which exposure and skin problems are assessed prior to testing may allow for better evaluation of clinical relevance[START_REF] Bråred Christensson | Positive patch test reactions to oxidized limonene: exposure and relevance[END_REF] .

Figure 1 .

 1 Figure 1. Chemical structures of linalool, R-(+)-limonene and corresponding sensitizing hydroperoxides.

Figure 2 .

 2 Figure 2. Radical decomposition of Lin-7-OOH.

Figure

  Figure3. EPR-ST methodology developed in RHE EpiSkin TM and applied here to Lin-OOHs using DEPMPO as spin-trap. Experimental EPR spectra are analyzed by means of computer simulation using labmade scripts based on Easyspin toolbox under Matlab (Mathworks) environment. It is shown the EPR experimental spectrum (Exp) of Lin-OOHs (10mM)/DEPMPO (250 mM) in RHE, together with computer simulation (Sim) and deconvolution affording spectra of adducts representing trapping of hydroxy, carbon and alkoxy radicals formed in the epidermis[START_REF] Kuresepi | Mechanistic insights on skin sensitization to linalool hydroperoxides: EPR evidence on radical intermediates formation in reconstructed human epidermis and 13C NMR reactivity studies with thiol residues[END_REF] .

13 *

 13 Patch test material of oxidized limonene containing Lim-OOHs at 0.1% ** Patch test material of oxidized limonene containing Lim-OOHs at 0.2% *** Patch test material of oxidized limonene containing Lim-OOHs at 0.3%

*Figure 1

 1 Figure 1

Figure 2

 2 Figure 2

Table 1 .

 1 3. EPR-ST methodology developed in RHE EpiSkinTM and applied here to Lin-OOHs using DEPMPO as spin-trap. Experimental EPR spectra are analyzed by means of computer simulation using labmade scripts based on Easyspin toolbox under Matlab (Mathworks) environment. It is shown the EPR experimental spectrum (Exp) of Lin-OOHs (10mM)/DEPMPO (250 mM) in RHE, together with computer simulation (Sim) and deconvolution affording spectra of adducts representing trapping of hydroxy, carbon and alkoxy radicals formed in the epidermis[START_REF] Kuresepi | Mechanistic insights on skin sensitization to linalool hydroperoxides: EPR evidence on radical intermediates formation in reconstructed human epidermis and 13C NMR reactivity studies with thiol residues[END_REF] . Prevalence of positive reactions to linalool and limonene ("pure" or not intentionally oxidized / not deliberately oxidized) when patch tested at different concentrations in various studies

		Test	Number	Positive	Study
		Concentration	tested	cases (%)
		(%)
	Limonene	10	4731	0.2	Audrain et al. (2014) 40
		2	2396	0.1	Schnuch et al. (2007) 63
		2	320	0	van Oosten et al. (2009) 64
		2	1241	0.88	Uter et al. (2010) 65
		2	1200	0	Santucci et al. (1987) 66
	Linalool	30	179	0	de Groot et al. (1985) 67
		20	1825	0.2	de Groot et al. (2000) 68
		10	320	0.6	van Oosten et al. (2009) 64
		10	985	0.16	Uter et al. (2010) 65
		10	2401	0.2	Schnuch et al. (2007) 63
		10	4731	0.3	Audrain et al. (2014) 40

Table 2 .

 2 Prevalence of positive reactions to oxidized R-limonene/Lim-OOHs when patch tested at different concentrations in various studies.

	Test	Number	Positive	Doubtful	Irritative	Study
	Concentra	tested	cases %	cases %	cases %
	tion of
	Lim-
	OOHs (%)
	0.1*	3639	1.4	0.7	0.3	Deza et al. (2017) 33
		4563	1.3	1.2	0.6	Wlodek et al. (2017) 34
	0.2**	3639	3.4	0.6	0.8	Deza et al. (2017) 33
		4563	3.2	2.1	0.9	Wlodek et al. (2017) 34
	0.3***	5773	5.1	7.9	0.1	Sukakul et al. (2021) 35
		821	9.4	17.2	0.9	Dittmar & Schuttelaar
						(2019) 43
		2084	4.3	1.4	1.4	Ung et al. (2018) 69
		103	8	--	--	Nath et al. (2017) 70
		3639	5.1	0.4	1.5	Deza et al. (2017) 33
		4563	5.3	2.4	2	Wlodek et al. (2017) 34
		6004 63	2.5	13.7	5.8	Bennike et al. (2016,
		3843 64				2017) 45,71
		1292	5	4.3	9.8	Sabroe et al. (2016) 72

Table 3 .

 3 Prevalence of positive reactions to oxidized linalool/Lin-OOHs when patch tested at different concentrations in various studies.

	Test	Number	Positi	Doubtf	Irritativ	Study
	Concentra-	tested	ve	ul cases	e cases
	tion of Lin-		cases	(%)	(%)
	OOHs (%)		(%)
	0.25*	3639	1.3	0.9	0.2	Deza et al. (2017) 33
		4563	2.5	2.9	1.1	Wlodek et al. (2017) 34
	0.33% **	1693	0.83	1.9	0	Christensson et al. (2010) 30
	0.33%**	1511	1.3	--	--	Matura et al. (2005) 29
	0.5***	3639	2.9	0.8	1.2	Deza et al. (2017) 33
		4563	5.1	3.4	2	Wlodek et al. (2017) 34
	0.66%****	2075	3.2	5.1	0.34	Christensson et al. (2010) 30
	1.0% *****	5773	7.0	10.1	0.2	Sukakul et al. (2021) 35
		821	11.7	21.9	1.9	Dittmar & Schuttelaar (2019) 43
		2084	7.4	1.5	1.5	Ung et al. (2018) 69
		3639	4.9	0.5	1.9	Deza et al. (2017) 33
		4563	7.7	2.9	3.9	Wlodek et al. (2017) 34
		103	20	--	--	Nath et al. (2017) 70
		6004	3.9	20.9	7.2	Bennike et al. (2016, 2017) 45,71
		4731	5.9	--	5.9	Audrain et al. (2014) 40
		2900	6.9	9.2	1.3	Christensson et al. (2012) 31
		1725	5.3	3.4	0.23	Christensson et al. (2010) 30
		1292	9.8	6.6	13.6	Sabroe et al. (2016) 72
	1.8%*****	1004	7.2	7.3	0.7	Christensson et al. (2010) 30
	*

Conflict of Interest


Nicolas Raymond

Éric Soccorsi

MAGNETIC QUANTUM CURRENTS IN THE PRESENCE OF A NEUMANN WALL

We consider the Schrödinger operator with constant transverse magnetic field on a half-plane, endowed with Neumann boundary conditions. We study the low energy currents flowing along the boundary and we establish a Limiting Absorption Principle for the magnetic Neumann Laplacian under perturbation of an electric potential.

Context and motivation

1.1. Definition of the main operator. In this paper, we consider the Hamiltonian with constant magnetic field on the half-plane R 2 + = {(x, y) ∈ R 2 : x > 0} and Neumann boundary condition:

L h := D 2 x + (hD y -x) 2 , D = -i∂ , h > 0 , (1.1)
acting on the domain

Dom (L h ) = ψ ∈ L 2 (R 2 + ) : D x ψ ∈ L 2 (R 2 + ) , (hD y -x)ψ ∈ L 2 (R 2 + ) , D 2
x + (hD y -x) 2 ψ ∈ L 2 (R 2 + ) and ∀y ∈ R , ∂ x ψ(0, y) = 0 .

This operator appears in many contexts. For instance, it plays a major role in the study of surface superconductivity, see [START_REF] Fournais | Spectral methods in surface superconductivity[END_REF]. It also acquired a life of its own over the years, see [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]. The present paper is concerned with the semiclassical spectral analysis of the operator L h , that is to say that we mostly focus on spectral results for the operator L h as h → 0. It might seem surprising at first sight that y is the only variable affected by the semiclassical parameter h in this model, but this can be explained as follows. By performing the rescaling

x = h -1 2 x , y = h 1 2 y ,
in (1.1), we notice that the operator L h is unitarily equivalent to

B -1 D 2
x + (D y -Bx) 2 , where B = h -1 , which shows that the semiclassical limit corresponds to the large magnetic field regime. Such partially semiclassical scaling has already been used by numerous authors in spectral theory of magnetic Schrödinger operators, as, e.g., in [START_REF] Bonnaillie-Noël | Magnetic WKB constructions[END_REF][START_REF] Bonnaillie-Noël | Purely magnetic tunneling effect in two dimensions[END_REF] and in [START_REF] Raymond | Absence of embedded eigenvalues for translationally invariant magnetic Laplacians[END_REF], where magnetic models à la Iwatsuka (see [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF]) are considered. This paper is also fitted in the semiclassical framework in order to give a lighter formulation of the results and to use the power of the theory of pseudo-differential operators.

1.2. A short bibliography. Magnetic quantum Hall currents have attracted a lot of attention from the mathematical community over the last decades, see, e.g., [START_REF] Combes | Edge states for quantum Hall Hamiltonians[END_REF][START_REF] De Bièvre | Propagating edge states for a magnetic Hamiltonian[END_REF][START_REF] Exner | Magnetic transport in a straight parabolic channel[END_REF][START_REF] Ferrari | Extended edge states in finite Hall systems[END_REF][START_REF] Fröhlich | On the extended nature of edge states of quantum Hall Hamiltonians[END_REF][START_REF] Hislop | Edge currents for Quantum Hall systems, II. Two-edge, bounded and unbounded geometries[END_REF][START_REF] Hislop | Edge currents for quantum Hall systems, I. One-edge, unbounded geometries[END_REF][START_REF] Macris | On edge states in semi-infinite quantum Hall systems[END_REF], this list being non exhaustive. Quantum Hall devices can be modeled by a magnetic Laplacian describing the planar motion of an electron submitted to a constant transverse magnetic field. The electron is confined to unbounded regions of the plane by potential barriers or Dirichlet boundary conditions. The edge, be it the edge of the electrostatic confining potential or the Dirichlet boundary of the spatial domain, creates edge currents, whereas for Iwatsuka Hamiltonians (see [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF]), quantum currents are generated by changes in the strength of the magnetic field, see, e.g., [START_REF] Dombrowski | Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators[END_REF][START_REF] Hislop | Edge states induced by Iwatsuka Hamiltonians with positive magnetic fields[END_REF]. Some authors have also observed that the role of the "edge" could be played by a discontinuity of the magnetic field, see [START_REF] Hislop | Band functions in the presence of magnetic steps[END_REF] or [START_REF] Assaad | Lowest energy band function for magnetic steps[END_REF] where the dispersion curves are studied in detail.

In translationally invariant unbounded straight-edge geometries such as the halfplane or the infinite strip, the unperturbed quantum Hall Hamiltonian is fibered and edge currents occur at energies associated with fibers whose group velocity is non-zero, see [START_REF] Combes | Edge states for quantum Hall Hamiltonians[END_REF][START_REF] De Bièvre | Propagating edge states for a magnetic Hamiltonian[END_REF][START_REF] Fröhlich | On the extended nature of edge states of quantum Hall Hamiltonians[END_REF][START_REF] Hislop | Edge currents for Quantum Hall systems, II. Two-edge, bounded and unbounded geometries[END_REF][START_REF] Hislop | Edge currents for quantum Hall systems, I. One-edge, unbounded geometries[END_REF][START_REF] Macris | On edge states in semi-infinite quantum Hall systems[END_REF]. In this case, the existence of edge currents can be linked to the absolutely continuous spectral nature of the spectrum through the use of Mourre's positive commutator method, see, e.g. [START_REF] Briet | Mourre estimates for a 2D quantum Hamiltonian on strip-like domains[END_REF][START_REF] Briet | Spectral properties of a magnetic quantum Hamiltonian on a strip[END_REF], or also [START_REF] Raymond | Absence of embedded eigenvalues for translationally invariant magnetic Laplacians[END_REF] (where the absolute continuity is established by elementary means for Iwatsuka Hamiltonians). Similarly, limiting absorption for the quantum Hall Hamiltonian in R 2 + at all energies except Landau levels, was derived in [START_REF] Popoff | Limiting absorption principle for the magnetic Dirichlet Laplacian in a half-plane[END_REF] from the monotonicity of the dispersion relations.

Notice that quantum Hall currents in two-edge geometries such as the infinite strip, propagate in opposite directions along the left and right edges of the band, respectively, see [START_REF] Hislop | Edge currents for quantum Hall systems, I. One-edge, unbounded geometries[END_REF]. This can be understood from the fact that, unlike for the case of one-edge geometries, the dispersion functions of the corresponding quantum Hamiltonian are no longer monotonic, see [START_REF] Hislop | Edge currents for Quantum Hall systems, II. Two-edge, bounded and unbounded geometries[END_REF][START_REF] Hislop | Edge currents for quantum Hall systems, I. One-edge, unbounded geometries[END_REF]. Moreover, these functions being "symmetric" about their minimum value, the net current flowing across any line orthogonal to the strip, is zero.

This picture is quite reminiscent of the one of the present paper where none of the dispersion functions of the Neumann Laplacian in R 2 + is monotonic. Nevertheless, in contrast to the quantum Hall Hamiltonian in the band, the dispersion relations of the Neumann Laplacian are not symmetric about their minimum, which motivates for a closer look into its transport and spectral properties. 1.3. Basic results on low energy currents. In this section, we describe some elementary results about the low energy currents associated with the Hamiltonian L h . Due to the translational invariance of this operator in the y-direction, we shall see that its discrete spectrum is empty. 1.3.1. Fiber decomposition. Let us define the (unitary) semiclassical Fourier transform with respect to y as

F h ϕ(x, ξ) = 1 √
2πh R e -iξy/h ϕ(x, y)dy .

Then, L h being invariant in the y-direction, we have the direct integral decomposition

F h L h F -1 h = ⊕ R ξ dξ ,
where for all ξ ∈ R, ξ is the Neumann realization in L 2 (R + ) of the differential operator D 2

x + (ξ -x) 2 . The operator ξ is sometimes referred as the "de Gennes operator". Since x → (ξ-x) 2 is unbounded as x goes to infinity, ξ has a compact resolvent and we denote by µ j (ξ) its j-th eigenvalue, j ∈ N := {1, 2, . . .}. The well-known properties of this operator are recalled in the following proposition (see the original paper [START_REF] Dauge | Eigenvalues variation. I. Neumann problem for Sturm-Liouville operators[END_REF], and [14, Section 3.2], [START_REF] Raymond | Bound states of the magnetic Schrödinger operator[END_REF]Section 2.4]).

Proposition 1.1. For all j ∈ N, it holds true that:

i. ξ → µ j (ξ) is analytic on R and has a unique minimum denoted by ξ j-1 , which is non-degenerate. ii. µ j decreases on (-∞, ξ j-1 ) and increases on (ξ j-1 , +∞). iii. Θ j-1 = ξ 2 j-1 , where Θ j-1 := µ j (ξ j-1 ). iv. lim ξ→+∞ µ j (ξ) = 2j -1 and lim ξ→-∞ µ j (ξ) = +∞. v. Θ 0 ∈ (0, 1) and Θ j-1 ∈ (2j -3, 2j -1) for j 2.

Since the µ j 's are non-constant by Proposition 1.1.iv, the spectrum of L h is purely absolutely continuous (see [START_REF] Iwatsuka | Examples of absolutely continuous Schrödinger operators in magnetic fields[END_REF]). Moreover we have

sp(L h ) = [Θ 0 , +∞) , see [26, Theorem XIII.85].
1.3.2. Basic properties of low energy magnetic quantum currents. Let us define the current operator in the direction y by

J h := [L h , iy] = 2h(hD y -x) ,
and the current operator with energy concentration in I by

J h,I := 1 I (L h )J h 1 I (L h ) .
The operator J h,I is bounded on E h,I := range(1 I (L h )) and symmetric.

Put λ min (h, I) := inf sp (J h,I ) , λ max (h, I) := sup sp (J h,I ) . We recall from the min-max principle that

λ min (h, I) = inf ψ∈E h,I ψ =0 J h,I ψ, ψ ψ 2 L 2 (R 2 + ) , λ max (h, I) = sup ψ∈E h,I ψ =0 J h,I ψ, ψ ψ 2 L 2 (R 2 + )
.

Then, the spectral radius of J h,I , given by

ρ (J h,I ) = max (|λ min (h, I)|, |λ max (h, I)|) ,
is the maximal strength of the (absolute value of the) current carried by the energy window I.

Definition 1.2. Let e ∈ [Θ 0 , Θ 1 ) and consider the equation

µ 1 (ξ) = e . (1.2)
i. When e ∈ (Θ 0 , 1), the equation (1.2) has two solutions µ -1 -(e) < ξ 0 < µ -1 + (e). ii. When e = Θ 0 , the equation (1.2) has only one solution µ -1 ± (Θ 0 ) = ξ 0 . iii. When e ∈ (1, Θ 1 ), the equation (1.2) has one solution µ -1 -(e) < ξ 0 and, by convention, we set µ -1 + (e) = +∞.

For all e ∈ [Θ 0 , Θ 1 ), we define the algebraic current as

c(e) = µ 1 (µ -1 -(e)) + µ 1 (µ -1 + (e)) , (1.3)
where we have set µ 1 (+∞) = 0.

In the sequel, it will be assumed at some stage of the analysis that the following conjecture (which is supported by numerical simulations carried out by M. P. Sundqvist with Mathematica) is verified.

Conjecture 1.3. We have µ

(3) 1 (ξ 0 ) < 0. Lemma 1.4. Assume that Conjecture 1.3 is true. Then, there exists e ∈ (Θ 0 , 1) such that for all e ∈ (Θ 0 , e ), we have c(e) < 0.

For e ∈ [Θ 0 , Θ 1 ), we set

I δ = [e -δ, e + δ] .
Proposition 1.5. Assume that R + h → δ(h) tends to 0 as h ↓ 0. Then, we have

h -1 λ min (I δ , h) = µ 1 (µ -1 -(e)) + O(δ(h)) , h -1 λ max (I δ , h) = µ 1 (µ -1 + (e)) + O(δ(h)) .
In particular, the spectral radius reads

ρ h -1 J I δ ,h = max |µ 1 (µ -1 -(e))|, |µ 1 (µ -1 + (e))| + o(1)
. Assume in addition that Conjecture 1.3 holds. Then, for all e ∈ (Θ 0 , e ) the current of maximal strength is negative provided h is sufficiently small, i.e., we have ρ

(J I δ ,h ) = |λ min (I δ , h)|.
Moreover, the low energy states are localized in the vicinity of the Neumann boundary x = 0, as can be seen from the following Agmon type estimate.

Proposition 1.6. Put J e := (-∞, e) , e < 1 . Then, for all h > 0, all K > 0 and all ψ ∈ E Je,h , we have

R 2 + e Kx |ψ(x, y)| 2 dxdy C ψ 2 L 2 (R 2 + ) ,
for some constant C > 0 depending only on e and K.

1.4. Mourre estimate and applications. When the perturbation V is switched off, the presence of a positive edge current with energy concentration in I is, by definition of the current operator J h,I , tied to the existence of a positive local commutator estimate (or Mourre estimate) for the operator L h in the same energy interval. This motivates for a closer look into the problem of designing a Mourre inequality for the perturbed magnetic Laplacian L h,V := D 2

x + (hD y -x) 2 + h γ V (x, y, h) , for some γ > 0 and some suitably smooth (in a sense that will be made precise further) bounded perturbation V . Such an estimate will prove useful for characterizing the spectrum of L h,V in the corresponding energy interval. We are more specifically interested in energy intervals lying in the vicinity of Θ 0 , in which the unperturbed current J h,I tends to vanish. For that purpose we introduce the following spectral window located above Θ 0 , whose size depends on h > 0:

I := [e -δ, e + δ] , e := Θ 0 + ah α ∈ (Θ 0 , 1) , δ := bh β , where 0 α < 1 , β > 2α , 0 < b < a .
When α = 0, we choose a ∈ (0, 1 -Θ 0 ).

1.4.1.
Statement of the Mourre estimate. In view of designing a Mourre estimate, i.e., a positive commutator estimate for the perturbed operator L h,V on the "sliding" energy window I, where the strength of current is very weak, we pick a smooth compactly supported function

f h satisfying ∀ξ ∈ (µ -1 ∓ (e ± h α ), µ -1 ∓ (e ∓ h α )) , f h (ξ) = ∓1 , where µ -1 ± (E) denotes the largest/smallest solution of the equation µ(ξ) = E. We can take f h ∈ S α 2 (1)
, where, as in [27, Section 4.4], we write for any (order) function g, S γ (g

) := {ψ ∈ C ∞ (R) : ∀α ∈ N , ∃C α > 0 , |∂ α ψ| C α h -γ|α| g} .
Then, the function (y, η) → yf h (η) being in S α 2 ( y ), we set

A h := yf h (hD y ) + f h (hD y )y = Op w h (yf h (η))
, where we recall that the Weyl quantization is defined by

Op w h a ψ(x) = 1 2πh R 2 e iη(x-y)/h a x + y 2 , η ψ(y)dydη .
For the sake of notational simplicity, we drop the dependence on h and write f instead of f h in the sequel.

Theorem 1.7 (Mourre estimate). Assume that V (x, •, h) ∈ S( y -1 ) uniformly in (x, h) and pick γ β so large that γ > 1 + 2α. Then, there exist c0 > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ), we have:

∀φ ∈ range 1 I (L h,V ) , [L h,V , iA h ]φ, φ c0 h 1+α φ 2 .
1.4.2. Application. One of the main consequences of the above Mourre estimate is the following Limiting Absorption Principle (LAP) for the operator L h,V in I, and subsequently the absolute continuity of its spectrum in I. This is detailled in Section 3.

Corollary 1.8. Let V and γ be the same as in Theorem 1.7. Then, there exist h 0 > 0 and C > 0 such that for all h ∈ (0, h 0 ) and all uniformly (w.r.t. h) bounded operator C h such that C h A h and A h C h are also uniformly bounded, we have for all

z ∈ I × R \ {0}, C h (L h,V -z) -1 C h Ch min(-2+3α,-3/2+α,-1-α) .
In particular, it holds true that:

(i) For all z ∈ I × R \ {0}, y -1 (L h,V -z) -1 y -1
Ch min(-2+3α,-3/2+α,-1-α) .

(ii) The spectrum of L h,V lying in I is absolutely continuous.

Remark 1.9. For all α ∈ (0, 1), it follows from Corollary 1.8 that the spectrum of the operator L h,V is purely absolutely continuous spectrum in (Θ 0 + ah α , 1).

1.5. Structure of the article. The paper is organized as follows. In Section 2 we present basic considerations on low energy magnetic quantum currents and we prove Propositions 1.5 and 1.6. We explain in Section 3 how to derive the LAP by revisiting the Mourre theory and estimating carefully the involved constants (what is crucial since the spectral window of interest also depends on h). Finally, in Section 4, we give the proof of Theorem 1.7 and we establish Corollary 1.8 with the aid of Section 3.

Basic properties of low energy currents

2.1. Proof of Lemma 1.4. We have

µ 1 (ξ) = µ 1 (ξ 0 ) + µ 1 (ξ 0 ) 2 (ξ -ξ 0 ) 2 + µ (3) 1 (ξ 0 ) 6 (ξ -ξ 0 ) 3 + r(ξ -ξ 0 ) ,
from the analyticity of µ 1 at ξ 0 , where r is an analytic function at 0 such that r(p) = p→0 O(p 4 ). Since µ 1 (ξ 0 ) > 0, the Morse lemma then yields that the equation

µ 1 (ξ) = µ 1 (ξ 0 ) + ε
has two distinct solutions, provided that ε > 0 is sufficiently small. Putting ξ = ξ 0 + p in the above equation, this amounts to saying that there exist two solutions

p -(ε) < p + (ε) such that p ± (ε) = ± √ ε 2 µ 1 (ξ 0 ) + u ± (ε) , where u ± (ε) = o( √ ε)
. Namely, we get through standard computations that

u ± (ε) = - µ (3) 1 (ξ 0 ) 6µ 1 (ξ 0 ) ε + o(ε) .
As a consequence we have

µ 1 (ξ 0 + p -(ε)) + µ 1 (ξ 0 + p + (ε)) = 5µ (3) 1 (ξ 0 ) 6µ 1 (ξ 0 ) ε + o(ε) ,
which is negative whenever ε is sufficiently small.

2.2.

Proof of Proposition 1.5.

Preliminaries.

Lemma 2.1. For all ψ ∈ E h,I δ , it holds true that

F h ψ(x, ξ) = 1 I δ (µ 1 (ξ)) F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) u 1 (x, ξ) , (x, ξ) ∈ R 2 +
, where u j (•, ξ) denotes the normalized eigenfunction associated with µ j (ξ).

Proof. Since 1 I δ (L h )ψ = ψ, we have

F h 1 I δ (L h )F -1 h F h ψ = 1 I δ (F h L h F -1 h )F h ψ = F h ψ . Plugging this into the following decomposition of F h ψ(•, ξ) on the Hilbertian basis (u j (•, ξ)) j 1 of L 2 (R+), F h ψ(x, ξ) = j 1 F h ψ(•, ξ), u j (•, ξ) L 2 (R + ) u j (x, ξ) ,
and taking into account that

1 I δ (F h L h F -1 h )u j (•, ξ) = 1 I δ (µ j (ξ))u j (•, ξ), we get the desired result upon remembering that I δ ∩ (Θ 1 , +∞) = ∅.
Lemma 2.2. For all ψ ∈ E h,I δ , we have

J I δ ,h ψ, ψ = h R 1 I δ (µ 1 (ξ))µ 1 (ξ)| F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) | 2 dξ .
Proof. In light of Lemma 2.1, we have

J I δ ,h ψ, ψ = 2h R + R 1 I δ (µ 1 (ξ))(ξ -x)|u 1 (x, ξ)| 2 | F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) | 2 dxdξ ,
by the Parseval formula. Lemma 2.2 follows from this and the Feynman-Hellmann formula

µ 1 (ξ) = 2 R + (ξ -x)|u 1 (x, ξ)| 2 dx ,
upon applying Fubini's theorem.

Armed with Lemma 2.2, we are now in position to prove Proposition 1.5.

Estimation of the current.

With reference to Lemma 2.2, we have for all ψ ∈ E h,I δ ,

J I δ ,h ψ, ψ = h R 1 I δ (µ 1 (ξ))µ 1 (ξ)| F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) | 2 dξ .
Therefore, we infer from the analyticity of µ 1 in I δ that

h -1 J I δ ,h ψ, ψ = µ -1 1 (e-δ,e+δ)∩{ξ<ξ 0 } µ 1 (µ -1 -(e))| F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) | 2 dξ + µ -1 1 (e-δ,e+δ)∩{ξ>ξ 0 } µ 1 (µ -1 + (e))| F h ψ(•, ξ), u 1 (•, ξ) L 2 (R + ) | 2 dξ + O(δ) ψ 2 .
Bearing in mind that (ξ -ξ 0 )µ 1 (ξ) 0 for all ξ ∈ µ -1 1 (e -δ, e + δ), the min-max theorem then yields that

h -1 λ min (J I δ ,h ) µ (µ -1 -(e)) + O(δ) and h -1 λ max (J I δ ,h ) µ (µ -1 + (e)) + O(δ) .
The converse inequalities can be shown upon considering appropriate trial states. For instance, for λ min (J I δ ,h ), it is enough to take ψ h ∈ E h,I δ such that

F h ψ h (x, ξ) = δ -1 (2π) -1 4 u 1 (x, µ -1 -(e)))e -( ξ-µ -1 -(e) ) 2 2δ 4
1 I δ (µ(ξ)) .

2.3.

Proof of Proposition 1.6. Since e ∈ (Θ 0 , 1), we have

µ -1 1 (J e ) = (µ -1 -(e), µ -1 + (e))
and µ -1 j (J e ) = ∅, j 2.

(2.1)

Further, for ψ ∈ E Je,h we have

1 Je (µ 1 (ξ))F h ψ(•, ξ) = F h ψ(•, ξ) for all ξ ∈ µ -1 1 (J e ), and
consequently ξ F h ψ(•, ξ) = µ 1 (ξ)F h ψ(•, ξ).
by arguing as in Lemma 2.1. Next, by Agmon's theorem (see for instance [START_REF] Agmon | Bounds on exponential decay of eigenfunctions of Schrödinger operators[END_REF], [14, Section 7.2] and [24, Section 4.2]), the following localization formula holds for any

χ ∈ W 1,∞ (R + ) and all ξ ∈ µ -1 1 (J e ), Re ξ F h ψ(•, ξ), χ 2 F h ψ(•, ξ) = q ξ (χF h ψ(•, ξ)) -χ F h ψ(•, ξ) 2 L 2 (R + )
, where q ξ denotes the quadratic form associated with the operator ξ . It follows that

R + (ξ -x) 2 χ(x) 2 -χ (x) 2 -χ(x) 2 µ 1 (ξ) |F h ψ(x, ξ)| 2 dx = -(χF h ψ) (•, ξ) 2 L 2 (R + ) 0 . Therefore, since (ξ -x) 2 x 2
2 -ξ 2 and µ 1 (ξ) e, we get that

R + x 2 2 -ξ 2 -e χ(x) 2 -χ (x) 2 |F h ψ(x, ξ)| 2 dx 0 , ξ ∈ µ -1 1 (J e ).
In light of (2.1), this entails that

R + x 2 2 -C e χ(x) 2 -χ (x) 2 |F h ψ(x, ξ)| 2 dx 0 ,
where C e := max{|µ -1 -(e)|, |µ -1 + (e)|}. Thus, taking χ(x) = e Φ(x) in the above estimate, where Φ is a bounded nonnegative K-Lipschitzian function, we obtain that

R + x 2 2 -C e -Φ 2 (x) |e Φ(x) F h ψ(x, ξ)| 2 dx 0 ,
and hence that

+∞ x e,K x 2 2 -C e -K 2 |e Φ(x) F h ψ(x, ξ)| 2 dx - x e,K 0 x 2 2 -C e -K 2 |e Φ(x) F h ψ(x, ξ)| 2 dx ,
where x e,K := 2(C e + K 2 + 1) > 0. Therefore, there exists a constant C > 0, depending only on e, K and Φ L ∞ (0,x e,K ) , such that

+∞ x e,K |e Φ(x) F h ψ(x, ξ)| 2 dx C x e,K 0 |F h ψ(x, ξ)| 2 dx ,
from where we get

R + |e Φ(x) F h ψ(x, ξ)| 2 dx C F h ψ(•, ξ) 2 L 2 (R + ) ,
upon substituting C + e Φ L ∞ (0,x e,K ) for C. Now, integrating the above estimate with respect to ξ and applying the Parseval formula, we obtain that

R 2 + |e Φ(x) ψ(x, y)| 2 dxdy C ψ 2 L 2 (R 2 + ) . (2.2) Next, for n ∈ N, put Φ n (s) := K ×      s for 0 s n , 2n -s for n s 2n , 0 for s 2n .
Notice that Φ n L ∞ (0,x e,K ) = Kx e,K for all n x e,K , in such a way that (2.2) holds with Φ = Φ n , where the constant C is independent of n.

Finally, the result follows from this upon sending n to infinity and applying Fatou's lemma.

Limiting absorption revisited

In this section we build a LAP for a self-adjoint operator from a Mourre estimate. The derivation of this result is inspired by [START_REF] Mourre | Absence of singular continuous spectrum for certain selfadjoint operators[END_REF] and [8, Section 4.3] but we provide here a different approach based on coercivity estimates.

3.1. Assumptions. We consider two self-adjoint operators L and A satisfying

1 J (L )B1 J (L ) c 0 1 J (L ) , B := [L , iA ] , c 0 > 0. (3.1)
Moreover, we assume that (A) [L , A ](L + i) -1 is bounded:

∃c 1 > 0, [L , A ](L + i) -1 c 1 . (B) [[L , A ], A ](L + i) -1 is bounded: ∃c 2 > 0, [[L , A ], A ](L + i) -1 c 2 .

3.2.

Limiting absorption through coercivity estimates. Let I ⊂⊂ J. We consider ε 0 and z ∈ C such that Re z ∈ I and Im z 0. For the sake of notational simplicity we write z ∈ I × [0, +∞) in the sequel. Set

L z,ε := L -z -iεB, Dom L z,ε := Dom L .
It is apparent that the family (L z,ε ) ε∈R is analytic of type (A) in the sense of Kato. Moreover, L z,ε is bijective provided z is not on the imaginary axis. Lemma 3.1. Let M > 0. Then, there exists ε 0 > 0 such that for all ε ∈ (0, ε 0 ) and all z ∈ I × [M, +∞), the operator L z,ε is bijective and satisfies the estimate:

L -1 z,ε 4 Im z 4 M . Proof. Since L z,ε = L -Re z -iIm z -iεB, we have L z,ε u 1 2 (L -Re z)u + 1 2 Im z u -ε Bu ,
and hence

2 L z,ε u (L -Re z)u + Im z u -2c 1 ε (L + i)u .
It follows that

2 L z,ε u (L -Re z)u + Im z u -2c 1 ε( (L -Re z)u + (i + Re z)u ) ,
and consequently

2 L z,ε u (1 -2c 1 ε) (L -Re z)u + Im z -2c 1 ε max x∈I |i + x| u .
Next, choosing ε so small in the above line that ε min

1 2c 1 , max x∈I |i+x| 4c 1 , we get that 2 L z,ε u Im z 2 u ,
which shows that L z,ε is injective with closed range. Arguing as above with the adjoint operator L * z,ε = L z,-ε instead of L z,ε , we get that L * z,ε is injective as well. Therefore, L z,ε has a dense range and the conclusion follows. Lemma 3.2. Let M > 0, set B := I × [0, M ] and put

ε 1 := sup B | + i| dist(I, J c ) + 1 -1 min c 0 4c 2 1 sup J | + i| , 1 2c
1 and

ε 0 := min   ε 1 , 2 sup J | + i| c 0 sup B | +i| dist(I,J c ) + 1 1 + 4c 1 c 0 sup J | + i|   .
Then, for all ε > 0 and all z ∈ B, the following estimates hold.

(a)

∀u ∈ Dom(L ) , 1 J c (L )L z,ε u C(ε, z) (L + i)1 J c (L )u -c 1 ε sup J | + i| 1 J (L )u ,
where

C(ε, z) := 1 -c 1 ε 1 + |z + i| dist(I, J c ) 1 + |z + i| dist(I, J c ) -1 . Moreover, we have C(ε, z) 1 2 sup B | +i| dist(I,J c ) + 1 -1 for 0 < ε 1 2c 1 sup B | +i| dist(I,J c ) + 1 -1 . (b) ∀u ∈ Dom(L ) , L z,ε u D 1 (ε, z) 1 J (L )u where D 1 (ε, z) := Im z + c 0 ε -(c 1 ε) 2 sup J | +i| C(ε,z) 1 + c 1 ε C(ε,z)
.

Moreover, we have D 1 ( , z) c 0 ε 2 1 - c 2 1 ε sup J | +i| c 0 C(ε,z) c 0 ε 4 provided that 0 < ε ε 1 . (c) ∀u ∈ Dom(L ) , L z,ε u D 2 (ε, z) (L + i)1 J c (L )u , where D 2 (ε, z) = C(ε, z) 1 + c 1 εD 1 (ε, z) -1 sup J | + i| . Moreover, we have D 2 (ε, z) sup B | +i| dist(I,J c ) +1 -1 2 1+ 4c 1 c 0 sup J | +i| whenever 0 < ε ε 1 .
(d) In particular L z,ε is bijective and

(L + i)L -1 z,ε D 3 (ε, z) -1 ,
where

D 3 (ε, z) := 1 √ 2 min D 1 (ε, z) sup J | + i| , D 2 (ε, z) satisfies D 3 (ε, z) c 0 ε 4 √ 2 sup J | +i| provided that 0 < ε ε 0 . Moreover, for ε = 0, L z,ε is bijective for Im z > 0. (e) ∀u ∈ Dom(L ) , ∀ε ∈ (0, ε 2 ), | L z,ε u, u | + L z,ε u 2 c0 ε u 2 ,
where

ε 2 := min c 0 sup J | + i| 2c 2 1 C(ε, z) , 2D 2 (ε, z) 2 c 0 , ε 1 and c0 := c 0 2    1 + 2 1 + 4c 1 c 0 sup J | + i| sup B | +i| dist(I,J c ) + 1 -1 + 4c 1 c 0    -1
.

Proof. (a) We have

1 J c (L )L z,ε u 1 J c (L )(L -z)u -ε Bu 1 J c (L )(L -z)u -c 1 ε (L + i)u ,
by (A) and thus

1 J c (L )L z,ε u 1 J c (L )(L -z)u -c 1 ε sup J | + i| 1 J (L )u -c 1 ε (L + i)1 J c (L )u ,
from the orthogonal decomposition of (L + i)u. This entails that

1 J c (L )L z,ε u (1 -c 1 ε) 1 J c (L )(L -z)u -c 1 ε sup J | + i| 1 J (L )u -c 1 ε|z + i| 1 J c (L )u . (3.2)
Further, since

1 J c (L )(L -z)u dist(I, J c ) 1 J c (L )u (3.3)
and

1 J c (L )(L -z)u (L + i)1 J c (L )u -|i + z| 1 J c (L )u ,
we obtain that

1 J c (L )(L -z)u |i + z| dist(I, J c ) + 1 -1 (L + i)1 J c (L )u . ( 3
-Im L z,ε u, 1 J (L )u = Im z 1 J (L )u 2 + ε 1 J (L )B1 J (L )u, u + εRe B1 J c (L )u, 1 J (L )u Im z 1 J (L )u 2 + c 0 ε 1 J (L )u 2 -c 1 ε 1 J (L )u (L + i)1 J c (L ) (3.

5) by (A) and

(L + i)1 J c (L ) C(ε, z) -1 1 J c (L )L z,ε u + c 1 ε sup J | + i| 1 J (L )u , from (a), we get that -Im L z,ε u, 1 J (L )u Im z 1 J (L )u 2 + c 0 ε 1 J (L )u 2 -C(ε, z) -1 c 1 ε 1 J (L )u 1 J c (L )L z,ε u + c 1 ε sup J | + i| 1 J (L )u . (3.6)
An application of the Cauchy-Schwarz inequality on the left-hand-side of the above inequality then yields

1 + c 1 ε C(ε, z) L z,ε u 1 J (L )u Im z + c 0 ε - (c 1 ε) 2 sup J | + i| C(ε, z) 1 J (L )u 2 ,
which entails (b). (c) This statement follows readily from (a) and (b).

(d) We have L z,ε u 2 -1 2 min(D 1 (ε, z), D 2 (ε, z))
u by (b) and (c), hence L z,ε is injective with closed range. Since the same is true for its adjoint L z,-ε , the operator L z,ε is bijective and

L -1 z,ε √ 2 min(D 1 (ε,z),D 2 (ε,z))
. Next, we have

(L + i)1 J (L )u sup J | + i| 1 J (L )u D -1 1 (ε, z) sup J | + i| L z,ε u , from (b)
. Putting this together with (c) we obtain that

L z,ε u 1 √ 2 min D 1 (ε, z) sup J | + i| , D 2 (ε, z) (L + i)u .
(e) By combining the identity

-Im L z,ε u, u + Im L z,ε u, 1 J c (L )u = -Im L z,ε u, 1 J (L )u , with (3.6)
, we get that

-Im L z,ε u, u + Im L z,ε u, 1 J c (L )u c 0 ε - (c 1 ε) 2 sup J | + i| C(ε, z) 1 J (L )u 2 -C(ε, z) -1 c 1 ε 1 J (L )u L z,ε u ,
and consequently

| L z,ε u, u | + L z,ε u 1 J c (L )u + C(ε, z) -1 c 1 ε 1 J (L )u L z,ε u c 0 ε - (c 1 ε) 2 sup J | + i| C(ε, z) 1 J (L )u 2 .
From this, (b) and (c), it then follows that

| L z,ε u, u | + D 2 (ε, z) -1 + D 1 (ε, z) -1 C(ε, z)c 1 ε L z,ε u 2 c 0 ε - (c 1 ε) 2 sup J | + i| C(ε, z) 1 J (L )u 2 .
Next, with reference to (c) we may add L z,ε u 2 on the left-hand-side of the above estimate and D 2 (ε, z) 2 1 J c (L )u 2 on its right-hand-side. We obtain that

| L z,ε u, u | + 1 + D 2 (ε, z) -1 + D 1 (ε, z) -1 C(ε, z)c 1 ε L z,ε u 2 c 0 ε - (c 1 ε) 2 sup J | + i| C(ε, z) 1 J (L )u 2 + D 2 (ε, z) 2 1 J c (L )u 2 .
As a consequence we have for all ε ∈ (0, ε 2 ],

| L z,ε u, u | + 1 + D 2 (ε, z) -1 + D 1 (ε, z) -1 C(ε, z)c 1 ε L z,ε u 2 c 0 ε 2 u 2 .
Bearing in mind that C( , z) ∈ (0, 1), this entails that

| L z,ε u, u | +    1 + 2 1 + 4c 1 c 0 sup J | + i| sup B | +i| dist(I,J c ) + 1 -1 + 4c 1 c 0    L z,ε u 2 c 0 ε 2 u 2 ,
which yields the desired result.

Having established Lemma 3.2, we can now state the following technical result.

Lemma 3.3. For all bounded self-adjoint operator C , we have

L -1 z,ε C 1 c0 ε 1 2 ( C + C L -1 z,ε C 1 2 )
and

C L -1 z,ε 1 c0 ε 1 2 ( C + C L -1 z,ε C 1 2
) .

Proof. Taking u = L -1 z,ε C ϕ in (e), we obtain the first estimate. The second one follows from this and the fact that C L -1 z,ε is the adjoint of L -1 z,-ε C . Armed with Lemma 3.3, we are in position to prove the main result of this section. Theorem 3.4 (Limiting Absorption Principle). Let L and A fulfill the conditions of Section 3.1. Then, for any bounded self-adjoint operator C such that C A and A C are bounded, it holds true for all ε ∈ (0, min (1,

ε 0 )] that sup Im z>0 ,Re z∈I C (L -z -iεB) -1 C ≤ C,
where

C := C(ε 0 ) + (K 1 + K 2 )(1 + C(ε 0 ) 1 2 ) 1 0 dt t 1 2 + √ 2K 1 2 (K 1 + K 2 ) 1 0 | ln(t)| 1 2 t 1 2
dt,

K 1 := 2 √ c0 max( C A , A C ) , K 2 := 4 √ 2c 2 sup J | + i| c 0 C , K := (K 1 + K 2 ) C 1 + 2 sup J | + i| 1 2
√ c 0 and C(ε 0 ) is a positive constant satisfying

C(ε 0 ) 4 √ 2 sup J | + i| c 0 C 2 ε -1 0 .

Moreover, we have sup

Im z>0 ,Re z∈I

C (L -z) -1 C C .
Proof. Let us differentiate F (ε) := C L -1 z,ε C w.r.t. ε. We obtain that

F (ε) = C L -1 z,ε [L , A ]L -1 z,ε C = C L -1 z,ε [L z,ε , A ]L -1 z,ε C -εC L -1 z,ε [[L , A ], A ]L -1 z,ε C = C A L -1 z,ε C -C L -1 z,ε A C -εC L -1 z,ε [[L , A ], A ]L -1 z,ε C .
Bearing in mind that A C and C A are bounded, we refer to (B), (d) and Lemma 3.3, and deduce from the above estimate that

F (ε) C A L -1 z,ε C + A C C L -1 z,ε + c 2 ε C C L -1 z,ε (L + i)L -1 z,ε 2 √ c0 ε max( C A , A C ) + c 2 ε C D 3 (ε, z) -1 ( C + F 1 2
)

K 1 ε -1 2 + K 2 ( C + F 1 2 ) . (3.7)
Further, since F (ε)

C 2 D 3 (ε, z) -1 4 sup J | +i| c 0 C 2 ε -1 for all ε ∈ (0, ε 0 ], by (d)
, we infer from (3.7) upon possibly substituting 1 for ε 0 , that

F (ε) K 1 ε -1 2 + K 2 C 1 + ε -1 2 2 sup J | + i| 1 2 √ c 0 (K 1 + K 2 ) C 1 + 2 sup J | + i| 1 2 √ c 0 ε -1 .
Integrating the above estimate over (ε, ε 0 ) then yields

F (ε) K| ln(ε)| + F (ε 0 ) , ε ∈ (0, ε 0 ].
Plugging this into (3.7), we obtain that

F (ε) (K 1 + K 2 )ε -1 2 C + K 1 2 | ln(ε)| 1 2 + F (ε 0 ) 1 2
, which, upon integrating over (ε, ε 0 ), leads to

F (ε) ≤ F (ε 0 ) + (K 1 + K 2 ) C + F (ε 0 ) 1 2 1 0 dt t 1 2 + K 1 2 1 0 | ln t| 1 2 t 1 2
dt .

This and the estimate

F (ε 0 ) = C (L + i) -1 (L + i)L -1 z,ε 0 C ≤ (L + i)L -1 z,ε 0 C 2 ≤ 4 √ 2 sup J | + i| c 0 C 2 ε -1 0 ,
arising from (d), yield the desired result with C(ε 0 ) = F (ε 0 ) .

Mourre estimates and limiting absorption

4.1. A Mourre estimate for the unperturbed operator. Since V (x, •, h) ∈ S α 2 ( y -1 ) and A h ∈ S α 2 ( y ), the following lemma is a direct consequence of the composition theorem of pseudo-differential operators (see [START_REF] Zworski | Semiclassical analysis[END_REF]Theorem 4.18]) and the Calderon-Vaillancourt theorem (see [START_REF] Zworski | Semiclassical analysis[END_REF]Theorem 4.23]).

Lemma 4.1. The pseudo-differential operator [V (x, •, h), A h ] is bounded from L 2 (R y ) to L 2 (R y ). More precisely, there exist C > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ) and all x > 0, we have

[V (x, •, h), A h ] L 2 (L 2 (Ry)) Ch 1-α .
In particular, this operator extends to a bounded operator on L 2 (R 2 + ). Remark 4.2. The expression of the above commutator can be explicitly calculated at the cost of rather tedious computations, but it is not needed thanks to the pseudodifferential approach.

Lemma 4.3. The following commutator is well defined on Dom(L h ) and

[L h , iA h ] = 4h(hD y -x)f (hD y ) . Proof. Since [L h , iA h ] = [(hD y -x) 2 , iA h ] = i[(hD y -x) 2 , yf (hD y ) + f (hD y )y] = i[(hD y -x) 2 , y]f (hD y ) + ihf (hD y )[(hD y -x) 2 , y], (4.1)
we get the desired result upon recalling that [(hD y -x) 2 , y] = -2ih(hD y -x).

For d := bh α , put J := [e -d, e + d]. Since β > 2α, we have 0 < d < δ whenever h ∈ (0, 1), and consequently I is a proper subset of J: I ⊂ J. Proposition 4.4. There exist c0 > 0 and h 0 > 0 such that for all h ∈ (0, h 0 ), we have

1 J (L h )[L h , iA h ]1 J (L h ) c0 h 1+α > 0 .
Proof. By the Parseval formula, we get from Lemma 4.3 that for all u ∈ Ran(1

J (L h )), [L h , iA h ]u, u L 2 (R + ×R) = 4h (η -x)f (η)F h u, F h u L 2 (R + ×Rη) .
Moreover, analogously to Lemma 2.1 we have

F h u(x, η) = 1 µ 1 (η)∈J U (η)u 1 (x, η) , where U (η) = F h u(•, η), u 1 (•, η) L 2 (R + ) ,
in such a way that

[L h , iA h ]u, u L 2 (R + ×R) = 4h R + R (η -x)f (η)1 µ 1 (η)∈J |U (η)| 2 |u 1 (x, η)| 2 dηdx .
Therefore, applying the Feynman-Hellmann formula, we obtain that

[L h , iA h ]u, u L 2 (R + ×R) = 2h µ 1 (η)∈J f (η)µ 1 (η)|U (η)| 2 dη . Moreover, we have µ 1 (η)∈J f (η)µ 1 (η)|U (η)| 2 dη µ 1 (η)∈J |µ 1 (η)||U (η)| 2 dη
from the definition of f , and consequently

[L h , iA h ]u, u L 2 (R + ×R) c0 h 1+α µ 1 (η)∈J |U (η)| 2 dη c0 h 1+α u 2 ,
from the quadratic behavior of µ 1 at its minimum, expressed in Proposition 1.1.iii.

One of the benefits of a Mourre estimate is its stability under perturbation. Having established a Mourre inequality for the unperturbed operator L h in the above proposition, we turn now to extending this estimate to the case of the perturbed operator L h + V . 4.2. The case of the perturbed operator. Since the proof of a Mourre estimate for the perturbed operator essentially boils down to the existence of a Mourre inequality for unperturbed operator, we preliminarily establish the following property of the spectral decomposition associated with L h . 4.2.1. Spectral decomposition associated with L h . Lemma 4.5. Let φ ∈ range 1 I (L h,V ). Then, φ decomposes as

φ = φ 1 + φ 2 , φ 1 = 1 J (L h )φ , φ 2 = 1 R\J (L h )φ ,
and we have

φ 2 c h φ , c h = d -1 (δ + h γ V ∞ ) . Proof. Since φ 2 = 1 R\J (L h )1 I (L h,V )φ, we have φ 2 = (L h -e) -1 1 R\J (L h )(L h -e)1 I (L h,V )φ = (L h -e) -1 1 R\J (L h )(L h,V -e -h γ V )1 I (L h,V )φ , which immediately yields that φ 2 d -1 (δ + h γ V ∞ ) φ . 4.2.2. Proof of Theorem 1.7. For all φ ∈ range 1 I (L h,V ), we have [L h,V , iA h ]φ, φ = [L h , iA h ]φ, φ + h γ [V, iA h ]φ, φ . (4.2)
The second term on the right-hand-side of the above line can be estimated with the aid of Lemma 4.1: since γ -α + 1 > 1 + α, we get that

h γ | [V, iA h ]φ, φ | h γ [V, iA h ] L (L 2 (R 2 + )) φ 2 Ch γ-α+1 φ 2 c0 4 h 1+α φ 2 , (4.3
) provided that h is small enough. In the first term on right-hand-side of (4.2), the commutator [L h , iA h ] is acting on the state φ ∈ range 1 I (L h,V ), which decomposes according to Lemma 4.5, giving:

[L h , iA h ]φ, φ = [L h , iA h ]φ 1 , φ 1 + [L h , iA h ]φ 2 , φ 2 + 2Re [L h , iA h ]φ 1 , φ 2 .
Therefore, we have and that (hD y -x)φ 2 2 = φ 2 , L h φ 2 = φ 2 , L h φ = φ 2 , (L h + h γ V )φ -φ 2 , h γ V φ

[L h , iA h ]φ, φ c0 h 1+α φ 1 2 -| [L h , iA h ]φ 2 , φ 2 | -2| [L h , iA h ]φ 1 ,
(e + δ + h γ V ∞ ) φ 2 φ ,
we get that

[L h , iA h ]φ, φ c0 h 1+α φ 1 2 -4h(e + δ + h γ V ∞ ) 1 2 f ∞ φ 2 3 2 φ 1 2 -8h(e + d) 1
2 f ∞ φ 1 φ 2 . From this and Lemma 4.5 it then follows that

[L h , iA h ]φ, φ c0 h 1+α -8(e + d + h γ V ∞ ) 1 2 f ∞ (1 + c 1 2 h ) + c h h α c h h φ 2 .
Thus, bearing in mind that d = h α in such a way that c h (which scales like h β-α ) can be made arbitrarily small relative to h α (as we have β > 2α) in the asymptotic regime h ↓ 0, we infer from the above estimate that

[L h , iA h ]φ, φ 3c 0 4 h 1+α φ 2 ,
whenever h is sufficiently small. Putting this together with (4.2)-( 4.3), we get the result of Theorem 1.7 upon replacing c0 2 by c0 . 4.3. Proof of Corollary 1.8. By Theorem 1.7, the self-adjoint operator L h,V satisfies a Mourre estimate of type (3.1) on the interval I, associated with the local commutator A h and a constant c 0 of size h 1+α . Next, since f (hD y ) is a bounded operator by definition of f , Lemmas 4.1 and 4.3 ensure us that the pair (L h,V , A h ) fulfills the assumption (A) of Section 3.1 with a constant c 1 of order O(h 1-α ). Moreover, we have

[[L h , A h ],

A h ] = -4ih[(hD y -x)f (hD y ), yf h (hD y ) + f h (hD y )y] = -4ih -2ih(hD y -x)f (hD y )f (hD y ) -2ihf 2 (hD y ) , by a straightforward computation. We deduce from this and the boundedness of the operators f (hD y ) and f (hD y ) that [[L h , A h ], A h ](L h + i) -1 is bounded and that its norm is of order O(h 2-α ). Moreover, since V (x, •, h) ∈ S α 2 ( y -1 ) by assumption and A h ∈ S α 2 ( y ), we can check upon arguing in the same fashion as in the derivation of Lemma 4.1 that (L h,V , A h ) satisfies the assumption (B) of Section 3.1, where the constant c 2 is of order O(h 2-α ).

Therefore, with reference to Section 3, the statement of Corollary 1.8 follows directly from Theorem 3.4 and the fact that we can actually track the powers of h in Lemma 3.2 and Theorem 3.4. As a matter of fact it can be checked that (dist(I, J c )) -1 h -α , ε 1 h -1+4α , ε 0 h -1+4α and c0 h 1+α in Lemma 3.2, whereas in Theorem 3.4 we have

K 1 = O(h -1 2 -α 2 ), K 2 = O(h 1-2α ), K = O(h -1-α ), C(ε 0 ) = O(h -2+3α )
and finally C = O(h min(-2+3α,-3/2+α,-1-α) ) .

. 4 )

 4 Now, plugging (3.3) and (3.4) into (3.2), we get (a). (b) Since

  φ 2 | , by Proposition 4.4, and hence[L h , iA h ]φ, φ c0 h 1+α φ 1 2 -4h f ∞ | φ 2 (hD y -x)φ 2 -8h f ∞ | (hD y -x)φ 1 φ 2 ,from Lemma 4.3. Further, taking into account that(hD y -x)φ 1 2 φ 1 L h φ 1 (e + d) φ 1 2

Acknowledgments. Nicolas Raymond and Éric Soccorsi are deeply grateful to the CIRM where this work was completed. Éric Soccorsi is partially supported by the Agence Nationale de la Recherche (ANR) under grant ANR-17-CE40-0029.