Mathilde Payen

María Pardos De La Gándara

Aurélie Cointe

Alix Massiot

Philippe Bidet

François-Xavier Weill

Stéphane Bonacorsi
email: stephane.bonacorsi@aphp.fr

Stéphane Pr

Bonacorsi

Diagnostic challenge of gastrointestinal infection due to lactose-fermenting Salmonella enterica subsp. enterica serovar 4,5:i

Keywords: Monophasic-Salmonella, Salmonella Typhimurium, Lac operon, Lactose fermenting Salmonella, C8-esterase Case

Here, we describe a case of a non-typhoidal Salmonella disease caused by a Salmonella enterica serovar 4,5:i:-(monophasic Salmonella Typhimurium) which acquired a Lac operon. This lactose-fermenting bacterium presents a major challenge for phenotypical detection of Salmonella. Only specific agar plates or molecular techniques allow reliable detection.

department for aqueous diarrhea with fever up to 39.5°C that had started three days before. No history of travel was noted and no family members were reported ill. A stool culture was then prescribed to search for Salmonella, Shigella, Campylobacter and Yersinia.

Twenty-four and 48 hours later no enteric pathogen was isolated. In particular no suspicious colonies were observed either on Hektoen agar (Biomérieux, Marcyl'Étoile) or on C8-esterase agar (ChromID Salmonella®, Biomérieux), two media conventionally used for the detection of Salmonella (Figure 1). On Hektoen medium, Salmonella appear as blue-green colonies due to their inability to ferment lactose, and also generally have a black iron sulfite precipitate due to the reduction of thiosulfate in H 2 S, while lactose-fermenting Enterobacteria grow yellow to red, due to the pH change caused by the lactose fermentation. On C8-esterase agar, Salmonella are expected to be pink to mauve due to C8-esterase activity, compared to white when there is no C8-estearase activity. Figure 1 : Atypical Salmonella enterica serovar 4,5:i:-isolate appearing (A) as lactose-fermenting and H 2 S non-producing colonies on Hektoen enteric agar medium and (B) as C8 esterase non-producing on chromogenic Salmonella medium (Biomérieux). In order to search for other pathogens, a multiplex PCR (FilmArray GI panel, Biomérieux®) was performed on the stool and it detected Salmonella spp. as the sole pathogen. A bacterial identification performed by MALDI-TOF® technique (Brucker, Wissembourg, France) on the predominant lactose-fermenting (Lac+), H 2 S negative (H 2 S-), and C8-esterase negative (C8-) colonies, confirmed the presence of Salmonella spp. We then used three chromogenic agars: CHROMagar Salmonella®, CHROMagar Salmonella Plus® (CHROMagar, Paris, France) and BBL CHROMagar Salmonella® (Becton Dickinson, Le Pont-de-Claix, France) and only the CHROMagar Salmonella Plus® plate enabled the detection of the Lac+ Salmonella isolate (Figure 2). No antibiotic resistance genes were detected using the Resfinder tool (https://cge.cbs.dtu.dk/services/ResFinder/). The genomic analysis allowed the identification of a Lac operon, 100% identical to the chromosomal Lac operon of Enterobacter hormaechei, as well as an IncHI1 plasmid also related to E. hormaechei plasmid (CP010380.1). Genes encoding the C8-esterase were complete.

Complementary C8-esterase spot test (MUCAP® test, Biomérieux) confirmed the C8-esterase activity of this isolate.

Salmonella are Gram-negative, rod-shaped bacteria, belonging to the Enterobacteriaceae family. The Salmonella genus comprises two species, S. bongori and S. enterica. The S. enterica species is itself classically subdivided in 6 subspecies, including S. enterica subsp. enterica, a food-borne pathogen isolated from numerous warm-blooded animals [START_REF] Havelaar | World Health Organization Global Estimates and Regional Comparisons of the Burden of Foodborne Disease in 2010[END_REF]. This subspecies contains many serovars which can be classified in 2 groups depending on clinical symptoms: typhoidal serovars and non-typhoidal serovars.

Salmonella enterica serovar Typhimurium (hereafter referred as to S. Typhimurium) and its monophasic variant are non-typhoidal serovars responsible of gastrointestinal disease in humans. Non-typhoidal Salmonella infection is characterised by fever, diarrhoea, abdominal pain and sometimes vomiting. Most of the time, this gastroenteritis is uncomplicated in immunocompetent patients. However, bacteraemia can occur in vulnerable patients and in this case antibiotics can be prescribed.

In the laboratory, the Salmonella genus is traditionally identified on Hektoen agar thanks to its biochemical properties leading to Lac-and H 2 S+ colonies. Lactose nonfermentation is a key point in rapid Enterobacteriaceae screening and Salmonella identification. Only S. enterica subspecies diarizonae and arizonae are known to be naturally Lac+ with respectively approximately 75% and 25% of their serotypes being Lac+ [START_REF] Grimont | Antigenic Formulae of the Salmonella Serovars. 9th[END_REF]. Since the 90s, detection of S. enterica subsp. enterica in human stools has been improved by detecting the C8-esterase activity, specific to this Salmonella subspecies [START_REF] Cooke | A novel chromogenic ester agar medium for detection of Salmonellae[END_REF][START_REF] Gaillot | Comparison of CHROMagar Salmonella medium and hektoen enteric agar for isolation of salmonellae from stool samples[END_REF].

Lac+ S. Typhimurium human isolates were first reported during a prolonged outbreak in São Paulo, Brazil in the early 1970s [START_REF] Falcão | Unusual Enterobacteriaceae: lactose-positive Salmonella typhimurium which is endemic in São Paulo, Brazil[END_REF]. Six other outbreaks involving Lac+ but C8+ (referring to spot test not to chromogenic agars) S. Typhimurium were also described in US cattle in 2000 (McDonough et al., 2000).

According to this latter study, the black iron sulfite precipitate is dissolved due to acid production in the media during lactose hydrolysis, explaining the H2S-phenotype.

Other authors postulated that color intensity on chromogenic C8-esterase agar might also be affected by the acidic environment [START_REF] Cooke | A novel chromogenic ester agar medium for detection of Salmonellae[END_REF].

A recent study described the Lac operon genetic support in 13 Lac+ S. enterica belonging to different non-Typhimurium serovars. The Lac operon was found either on a plasmid similar to one carried by other Enterobacteriaceae, or on the chromosome, flanked by IS elements [START_REF] Leonard | Acquisition of the lac operon by Salmonella enterica[END_REF].

Due to the difficult detection of Lac+ Salmonella on conventional agar media, it is possible that such isolates may not be so rare. Querying the full Enterobase database for the presence of the Lac operon [START_REF] Luhmann | BlastFrost: Fast querying of 100,000s of bacterial genomes in Bifrost graphs[END_REF] might allow accurate estimation of the proportion of the Lac+ S. enterica subsp. enterica isolates that have been detected, sequenced and submitted to this genomic database.

In our laboratory, this Lac+ Salmonella isolate represented a major challenge and we believe that without a molecular approachhere used in a second step to search for other pathogensthe diagnostic would have been missed. Multiplex molecular panels are increasingly replacing conventional methods for the detection of enteric pathogens from stool samples. Our study did not aim to validate this approach.

Instead, we recommend the systematic use of an agar medium able to detect Lac+ Salmonella, like CHROMagar Salmonella Plus®, in case of negative reflex conventional cultures for Salmonella. The systematic use of an agar able to detect Lac+ Salmonella in clinical laboratories that only rely upon culture-based methods might not be indicated given the apparent rarity of Lac+ Salmonella isolates and the lack of specificity data with these media. They can, however, be used in a second line if conventional media do not show Lac-colonies, whereas there is a strong suspicion of salmonellosis.

Declarations of interest: none

Figure 2 :

 2 Figure2: The atypical Salmonella enterica serovar 4;5:i:-isolate appears (A) nondetectable on CHROMagar Salmonella® (blue instead of pink), (B) detectable on CHROMagar Salmonella Plus® (pink color expected of Salmonella) and (C) nondetectable on BBL CHROMagar® Salmonella (blue instead of pink).


J Ér

Émy Martin

Karel Pravda-Starov
email: karel.pravda-starov@univ-rennes1.fr

Jérémy Martin
email: jeremy.martin@univ-rennes1.fr

Karel Pravda

GEOMETRIC CONDITIONS FOR THE EXACT CONTROLLABILITY OF FRACTIONAL FREE AND HARMONIC SCHR ÖDINGER EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 93B05. Key words and phrases. Exact controllability, Schrödinger equation, harmonic oscillator

We provide necessary and sufficient geometric conditions for the exact controllability of the one-dimensional fractional free and fractional harmonic Schrödinger equations. The necessary and sufficient condition for the exact controllability of fractional free Schrödinger equations is derived from the Logvinenko-Sereda theorem and its quantitative version established by Kovrijkine, whereas the one for the exact controllability of fractional harmonic Schrödinger equations is deduced from an infinite dimensional version of the Hautus test for Hermite functions and the Plancherel-Rotach formula.

Introduction

The purpose of this article is to provide necessary and sufficient geometric conditions on control subsets to ensure exact controllability of fractional free Schrödinger equations and fractional harmonic Schrödinger equations. The notion of exact controllability is defined as follows: Definition 1.1 (Exact controllability). Let B be a closed operator on L 2 (R d ) which is the infinitesimal generator of a strongly continuous semigroup (e -tB ) t≥0 on L 2 (R d ), T > 0 and ω be a measurable subset of R d . The equation

(1.1) (∂ t + B)f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
is said to be exactly controllable from the control set ω in time T > 0 if, for any initial datum f 0 ∈ L 2 (R d ) and any final datum f T ∈ L 2 (R d ), there exists u ∈ L 2 ((0, T ) × R d ), supported in (0, T ) × ω, such that the mild (or semigroup) solution of (1.1) satisfies f (T, •) = f T . The equation (1.1) is said to be exactly controllable from the control set ω if there exists T > 0 such that this system is exactly controllable from ω in time T > 0.

By the Hilbert uniqueness method, see [START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], the exact controllability of the evolution equation (1.1) is equivalent to the exact observability of the adjoint system

(1.2) (∂ t + B * )g(t, x) = 0 , x ∈ R d , t > 0 , g| t=0 = g 0 ∈ L 2 (R d ),
where B * denotes the L 2 (R d )-adjoint of the operator B. The notion of exact observability is defined as follows: Definition 1.2 (Exact observability). Let T > 0 and ω be a measurable subset of R d . The evolution equation (1.2) is said to be exactly observable from the control set ω in time T > 0 if there exists a positive constant C T > 0 such that, for any initial datum g 0 ∈ L 2 (R d ), the mild (or semigroup) solution of (1.2) satisfies

R d |g 0 (x)| 2 dx ≤ C T T 0 ω |g(t, x)| 2 dx dt .
In this paper, we study evolution equations associated to skew-selfadjoint operators. More precisely, we consider evolution equations

(1.3) ∂ t f (t, x) + i(Af )(t, x) = 0 , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
where the selfadjoint operators A on L 2 (R d ) generating a strongly continuous group (e itA ) t∈R on L 2 (R d ) is specifically given by a fractional Laplacian A = (-∆ x ) s or a fractional harmonic oscillator A = (-∆ x + |x| 2 ) s , with s > 0.

Our starting point is the following result proved by Miller. This author shows in [START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]Corollary 2.17] that the observability of the system (1.3) is equivalent to the following spectral estimates: Proposition 1.3 (Miller [11,Corollary 2.17]). Let A be a selfadjoint operator on L 2 (R d ), which is the infinitesimal generator of a strongly continuous group (e itA ) t∈R on L 2 (R d ). If the evolution equation (1.3) is exactly observable from a measurable subset ω ⊂ R d at time T > 0 then there exist some positive constants k > 0 and D > 0 such that

(1.4) ∀λ ∈ R, ∀f ∈ 1l {|A-λ|≤ √ D} L 2 (R d ) , f L 2 (R d ) ≤ √ k f L 2 (ω) .
Conversely, when the spectral estimates (1.4) hold for some k > 0 and D > 0, then the system (1.3) is exactly observable from ω at any time

T > π 1 + k D .
The operators 1l {|A-λ|≤ √ D} appearing in (1.4) are defined by functional calculus. It is important to point out that the constants D and k do not depend on λ. We recall the proof given by Miller in Appendix (Section 5.2) since the minimal time obtained by Miller in [START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]Corollary 2.17] is different to ours.

When the operator A = (-∆ x ) s , with s > 0, is a fractional Laplacian, the family (1l {|A-λ|≤ √ D} ) λ∈R is a family of frequency cutoff operators on bounded frequency subsets and for all D > 0 and λ ∈ R,

(1.5) 1l {|A-λ|≤ √ D} L 2 (R d ) = f ∈ L 2 (R d ) : supp f ⊂ ξ ∈ R d , ||ξ| 2s -λ| ≤ √ D ,
where | • | denotes the Euclidean norm on R d . The spectral estimates (1.4) are in this case related to the notion of annihilating pairs:

Definition 1.4 (Annihilating pairs). Let S, Σ be two measurable subsets of R d .

-The pair (S, Σ) is said to be a weak annihilating pair if the only function

f ∈ L 2 (R d ) with supp f ⊂ S and supp f ⊂ Σ is zero f = 0.
-The pair (S, Σ) is said to be a strong annihilating pair if there exists a positive constant

C = C(S, Σ) > 0 such that for all f ∈ L 2 (R d ), R d |f (x)| 2 dx ≤ C R d \S |f (x)| 2 dx + R d \Σ | f (ξ)| 2 dξ .
It can be readily checked that a pair (S, Σ) is a strong annihilating one if and only if there exists a positive constant C = C(S, Σ) > 0 such that

(1.6) ∀f ∈ L 2 (R d ), supp f ⊂ Σ, f L 2 (R d ) ≤ C f L 2 (R d \S) .
We deduce from the Hilbert uniqueness method, Proposition 1.3 and (1.5) that the exact controllability of the fractional free Schrödinger equation

∂ t f (t, x) + i(-∆ x ) s f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
with s > 0, from a measurable control set ω holds if and only if its complement R d \ ω forms a strong annihilating pair with the frequency set

C D (λ) = ξ ∈ R d : ||ξ| 2s -λ| ≤ √ D ,
for some D > 0 and all λ ∈ R with an uniform constant C with respect to λ in (1.6).

An exhaustive description of annihilating pairs is for now out of reach. However, the Logvinenko-Sereda Theorem [START_REF] Logvinenko | Equivalent norms in spaces of entire functions of exponential type[END_REF] gives a complete description of all support sets forming a strong annihilating pair with a given bounded frequency set:

Theorem 1.5 (Logvinenko-Sereda [START_REF] Logvinenko | Equivalent norms in spaces of entire functions of exponential type[END_REF]). Let S, Σ ⊂ R d be measurable subsets with Σ bounded. Denoting S = R d \ S, the following assertions are equivalent:

-The pair (S, Σ) is a strong annihilating pair -The subset S is thick, that is, there exists a cube K ⊂ R d with sides parallel to coordinate axes and a positive constant

0 < γ ≤ 1 such that ∀x ∈ R d , |(K + x) ∩ S| ≥ γ|K| > 0,
where |A| denotes the Lebesgue measure of the measurable set A.

Notice that if the support set S forms a strong annihilating pair with some bounded frequency set Σ, then S forms a strong annihilating pair with any bounded frequency set. In order to be able to use the Logvinenko-Sereda theorem in control theory, it is essential to understand how the constant C(S, Σ) appearing in (1.6) depends on the bounded frequency set Σ. In [START_REF] Kovrijkine | Some results related to the Logvinenko-Sereda Theorem[END_REF]Theorem 3], Kovrijkine adresses this question by establishing the following quantitative version of the Logvinenko-Sereda Theorem: Theorem 1.6 (Kovrijkine [START_REF] Kovrijkine | Some results related to the Logvinenko-Sereda Theorem[END_REF]Theorem 3]). Let ω ⊂ R d be a measurable subset γ-thick at scale L > 0, that is, satisfying

(1.7) ∀x ∈ R d , |([0, L] d + x) ∩ ω| ≥ γL d > 0, with 0 < γ ≤ 1.
There exists a universal positive constant C > 0 independent on the dimension d ≥ 1 such that for all f ∈ L 2 (R d ) satisfying supp f ⊂ J, with J a cube with sides of length b parallel to coordinate axes,

(1.8) f L 2 (R d ) ≤ c(γ, d, L, b) f L 2 (ω) , with c(γ, d, L, b) = C d γ Cd(Lb+1)
.

This result is extended to the case of frequency supports covered by the union of finitely many parallepipeds in [START_REF] Kovrijkine | Some results related to the Logvinenko-Sereda Theorem[END_REF]Theorem 4]. In this work, we only use this result in the onedimensional setting:

Theorem 1.7 (Kovrijkine [8, Theorem 2]). Let ω ⊂ R be a measurable subset γ-thick at scale L > 0, that is, satisfying ∀x ∈ R, |([0, L] + x) ∩ ω| ≥ γL > 0, with 0 < γ ≤ 1. There exists a universal positive constant C > 0 such that for all f ∈ L 2 (R) satisfying supp f ⊂ m k=1 J k , with (J k ) 1≤k≤m a finite family of intervals with length |J k | = b for all 1 ≤ k ≤ m, (1.9) f L 2 (R) ≤ c(γ, m, L, b) f L 2 (ω) , with c(γ, m, L, b) = C γ Lb C γ m +m-1 2 .
Notice that the constant c(γ, m, L, b) depends only on the number of intervals and their length but not on their locations. Thanks to this explicit dependence of the constant with respect to the length of the intervals in (1.9), Egidi and Veselić [START_REF] Egidi | Sharp geometric condition for null-controllability of the heat equation on R d and consistent estimates on the control cost[END_REF]; and Whang, Whang, Zhang and Zhang [START_REF] Wang | Observable set, observability, interpolation inequality and spectral inequality for the heat equation in R d[END_REF] have independently established that the heat equation

(1.10) (∂ t -∆ x )f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
is null-controllable in any positive time T > 0 from a measurable control set ω ⊂ R d if and only if the control subset ω is thick in R d . In the recent work [START_REF] Beauchard | Spectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations[END_REF], Beauchard, Jaming and Pravda-Starov prove that this thickness condition is also sufficient for the null-controllability of the harmonic heat equation

(∂ t + (-∆ x + |x| 2 ))f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ).
More generally, the result of [START_REF] Beauchard | Spectral inequality for finite combinations of Hermite functions and null-controllability of hypoelliptic quadratic equations[END_REF]Theorem 2.2] shows that the thickness condition is a sufficient condition for the null-controllability of a large class of hypoelliptic quadratic equations.

When the operator A appearing in the evolution equation (1.3) has a compact resolvent, the result of Proposition 1.3 is specified further as the following infinite dimensional version of the Hautus test: Proposition 1.8 (Miller [11,Corollary 2.18]). Let A be a selfadjoint operator on L 2 (R d ), which is the infinitesimal generator of a strongly continuous group (e itA ) t∈R on L 2 (R d ). When the operator A has a compact resolvent with a spectral gap γ > 0 in the following sense: |λ -µ| ≥ γ for all distinct eigenvalues λ and µ, the evolution equation (1.3) is exactly observable at some time T > 0 from a mesurable subset ω if and only if

∃δ > 0, for all eigenvector ψ of A, ψ L 2 (R d ) ≤ δ ψ L 2 (ω) .
When the spectral gap condition holds, the observability of the evolution equation (1.3) then depends only on some properties of the eigenvectors of A. Notice in particular that this spectral gap condition holds for the one-dimensional harmonic oscillator

H = -∆ x + x 2 ,
as well as the fractional harmonic oscillators H s , when s ≥ 1, since the spectra of these operators are given by σ (-∆ x + x 2 ) s = {(2n + 1) s : n ∈ N}.

In this work, we aim at studying necessary and sufficient geometric conditions on the control subsets ω to ensure exact controllability of the free fractional Schrödinger equations

∂ t f (t, x) + i(-∆ x ) s f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
and the fractional harmonic Schrödinger equations

(1.11) ∂ t f (t, x) + i(-∆ x + |x| 2 ) s f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
with s > 0. The first result contained in this paper (Corollary 2.7) shows that the thickness condition defined in Theorem 1.5 is a necessary and sufficient geometric condition for the exact controllability of the fractional free Schrödinger equations when the fractional parameter satisfies s ≥ 1 2 . In the multidimensional setting, the thickness condition is always a necessary condition for exact controllability when the fractional parameter satisfies s > 0 (Theorem 2.5). A sufficient geometric condition to ensure the exact controllability from a neighborhood of the control subset is also given in the multidimensional case when s ≥ 1 2 (Proposition 2.9). These results are based on the Logvinenko-Sereda Theorem, its quantitative versions proved by Kovrijkine and new uncertainty principles recently established by Green, Jaye and Mitkovski [START_REF] Green | Uncertainty principles associated to sets satisfying the geometric control condition[END_REF].

Regarding the fractional harmonic Schrödinger equations, we prove that the following geometric condition

(1.12) ∀A ∈ O(R d ), lim inf R 1 →+∞ ... lim inf R d →+∞ |A(ω) ∩ [-R 1 , R 1 ] × ... × [-R d , R d ]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| > 0,
turns out to be necessary for exact controllability in any dimension d ≥ 1 (Theorem 2.3) and sufficient in the one-dimensional setting (Theorem 2.2). The proof of this result is based on the infinite dimensional version of the Hautus test given by Proposition 1.8 applied with Hermite functions together with the Rotach-Plancherel formula for Hermite polynomials. This necessary condition enables us to derive a negative result of exact controllability from a cone in dimension 2. In the case of the one-dimensional harmonic Schrödinger equation, we establish that exact observability from a control subset satisfying (1.12) holds at time T = π.

Statements of the main results

2.1. Hautus test and exact controllability of fractional harmonic Schrödinger equations. In this section, we state the results related to the exact controllability of fractional harmonic Schrödinger equations. We recall that the exact controllability of the evolution equations (1.11) is equivalent to the observability of the adjoint systems

(2.1) ∂ t f (t, x) -i(-∆ x + |x| 2 ) s f (t, x) = 0 , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ).
The observability of (2.1) is deduced thanks to Proposition 1.8 by an infinite dimensional Hautus test on the Hermite functions seen as eigenvectors of the fractional harmonic oscillator.

Hermite functions are normalized L 2 (R)-functions defined as

(2.2) ∀n ≥ 0, ψ n (x) = 1 π 1 4 2 n 2 √ n! e -x 2 2 H n (x),
where H n stands for the Hermite polynomial of degree n ≥ 0, see e.g. [START_REF] Szegö | Orthogonal polynomials[END_REF], formula (5.5.1). The family (ψ n ) n≥0 defines a Hilbert basis of L 2 (R). They satisfy the following infinite dimensional Hautus test:

Proposition 2.1. Let ω be a measurable subset of R and (ψ n ) n≥0 be the Hermite basis on L 2 (R) defined in (2.2). The following assertions are equivalent:

(i) lim inf R→+∞ |ω ∩ [-R, R]| |[-R, R]| > 0, (ii) ∃c > 0, ∀n ≥ 0, 1 = ψ n L 2 (R) ≤ c ψ n L 2 (ω) , (iii) lim inf n→+∞ ψ n L 2 (ω) > 0.
The proof of Proposition 2.1 is given in Section 3 and is based on the Plancherel-Rotach formula. We readily deduce from Proposition 1.8 and 2.1 the following result of exact controllability in the one-dimensional case: Theorem 2.2. Let s ≥ 1 and ω be a measurable set in R. The two following assertions are equivalent:

(i) The fractional harmonic Schrödinger equation

∂ t f (t, x) + i(-∆ x + x 2 ) s f (t, x) = u(t, x)1l ω (x) , x ∈ R, t > 0, f | t=0 = f 0 ∈ L 2 (R),
is exactly controllable from the set ω for some positive time T > 0, (ii) The control set obeys the following geometric condition

lim inf R→+∞ |ω ∩ [-R, R]| |[-R, R]| > 0.
If one of the two above conditions holds in the non fractional case s = 1, then the harmonic Schrödinger equation is exactly controllable from the control subset ω in time T = π.

The equivalence between (i) and (ii) directly follows from Propositions 1.8 and 2.1, since the Hermite basis (ψ n ) n≥0 is a Hilbert basis of eigenvectors for the fractional harmonic oscillator H s associated to the eigenvalues ((2n + 1) s ) n≥0 that satisfy the spectral gap condition when s ≥ 1. Let us now check that when exact controllability holds for some positive time T 0 > 0 then it necessary holds in time T = π. To that end, we just notice that the function

t ∈ R → e it(-∆x+x 2 ) f 0 2 L 2 (ω) , is π-periodic since ∀t ∈ R, ∀f 0 ∈ L 2 (R), e i(t+π)(-∆x+x 2 ) f 0 = n∈N e i(t+π)(2n+1) f 0 , ψ n L 2 (R) ψ n = -e it(-∆x+x 2 ) f 0 .
By recalling that exact controllability at time T 0 > 0 is equivalent to the observability of the evolution equation

∂ t f (t, x) -i(-∆ x + x 2 )f (t, x) = 0, x ∈ R, t > 0, f | t=0 = f 0 ∈ L 2 (R), at time T 0 > 0, that is, ∃C T 0 > 0, ∀f 0 ∈ L 2 (R), f 0 2 L 2 (R) ≤ C T 0 T 0 0 e it(-∆x+x 2 ) f 0 2 L 2 (ω) dt.
The π-periodicity property readily implies observability and exact controllability at time T = π, since for all f 0 ∈ L 2 (R),

f 0 2 L 2 (R) ≤ C T 0 T 0 0 e it(-∆x+x 2 ) f 0 2 L 2 (ω) dt ≤ C T 0 ( T 0 π +1)π 0 e it(-∆x+x 2 ) f 0 2 L 2 (ω) dt ≤ C T 0 T 0 π k=0 (k+1)π kπ e it(-∆x+x 2 ) f 0 2 L 2 (ω) dt ≤ C T 0 T 0 π + 1 π 0 e it(-∆x+x 2 ) f 0 2 L 2 (ω) dt,
where • denotes the floor function. Theorem 2.2 provides a necessary and sufficient geometric condition on the control subset ω to ensure exact controllability of fractional harmonic Schrödinger equations in the one-dimensional setting. In higher dimensions d ≥ 2, it is natural to investigate if the geometric condition

(2.3) lim inf R→+∞ |ω ∩ B d (0, R)| |B d (0, R)| > 0,
where B d (0, R) denotes the Euclidean ball in R d centered at 0 with radius R, turns out to be sufficient to ensure exact controllability. This is actually not the case in general as the next result shows that condition (2.3) is not sufficient for the exact controllability of the harmonic Schrödinger equation posed on R 2 . The stronger necessary geometric condition for exact controllability given by the following proposition proves in particular that the exact controllability of the harmonic Schrödinger equation posed on R 2 cannot hold from the cone (2.4)

ω δ = (r cos θ, r sin θ) : r ≥ 0, |θ| ≤ π 2 -δ ⊂ R 2 , with 0 < δ ≤ π 2 ,
Cone ω δ δ δ as the control subset ω δ satisfies condition (2.3), but fails to satisfy the following necessary geometric condition lim inf

R 1 →+∞ lim inf R 2 →+∞ |ω ∩ [-R 1 , R 1 ] × [-R 2 , R 2 ]| |[-R 1 , R 1 ] × [-R 2 , R 2 ]| > 0,
given by the following result:

Theorem 2.3. Let s ≥ 1 and ω be a measurable subset of R d with d ≥ 1. If the fractional harmonic Schrödinger equation

(2.5) ∂ t f (t, x) + i(-∆ x + |x| 2 ) s f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
is exactly controllable from ω for some positive time T > 0, then the control subset ω does obey the following geometric condition

(2.6) ∀A ∈ O(R d ), lim inf R 1 →+∞ ... lim inf R d →+∞ |A(ω) ∩ [-R 1 , R 1 ] × ... × [-R d , R d ]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| > 0,
where O(R d ) stands for the orthogonal group in dimension d.

This result shows that the exact controllability of the fractional harmonic Schrödinger equations requires the control subset to be distributed in any space direction. The sufficiency of condition (2.6) for exact controllability is an open question in dimension d ≥ 2. The proof of Theorem 2.3 is given in Section 4. An example of sufficient geometric condition is given by the following proposition, which is deduced from Theorem 2.2:

Example 2.4. Let d ≥ 1, s ≥ 1 and ω be a measurable subset of R satisfying lim inf R→+∞ |ω ∩ [-R, R]| |[-R, R]| > 0.
The evolution equation

(2.7) ∂ t f (t, x) + i(-∆ x + |x| 2 ) s f (t, x) = u(t, x)1l ω×R d-1 (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ), is exactly controllable from ω × R d-1 for some positive time T > 0.
Proof. We first notice that the spectral gap condition in Proposition 1.8 is satisfied by the fractional harmonic operator (-∆ x + |x| 2 ) s , since its spectrum is given by

σ (-∆ x + |x| 2 ) s = (2n + d) s : n ∈ N .
Let N ∈ N and f be an eigenvector associated to the eigenvalue (2N + d) s . There exists a family of complex numbers (C α ) α∈N d ,|α|=N such that

f = α∈N d , |α|=N C α Ψ α , with ∀α = (α 1 , ..., α d ) = (α 1 , α ) ∈ N d , ∀x = (x 1 , ..., x d ) = (x 1 , x ) ∈ R d , Ψ α (x) = d i=1 ψ α i (x i ).
We notice that

f 2 L 2 (ω×R d-1 ) = ω R d-1 α∈N d , |α|=N C α Ψ α (x 1 , x ) 2 dx dx 1 = ω R d-1 α ∈N d-1 , |α |≤N C N -|α |,α ψ N -|α | (x 1 )Ψ α (x ) 2 dx dx 1 = ω α ∈N d-1 , |α |≤N |C N -|α |,α | 2 |ψ N -|α | (x 1 )| 2 dx 1 = α ∈N d-1 , |α |≤N |C N -|α |,α | 2 ψ N -|α | 2 L 2 (ω) .
By using Proposition 2.1,

∃c > 0, ∀n ≥ 0, 1 = ψ n L 2 (R) ≤ c ψ n L 2 (ω) ,
we deduce that

(2.8) f 2 L 2 (ω×R d-1 ) ≥ 1 c 2 α ∈N d-1 , |α |≤N |C N -|α |,α | 2 = 1 c 2 α∈N d , |α|=N |C α | 2 = 1 c 2 f 2 L 2 (R d ) .
Thanks to Proposition 1.8, we deduce from (2.8) the observability of the adjoint system

∂ t g(t, x) -i(-∆ x + |x| 2 ) s g(t, x) = 0 , x ∈ R d , t > 0, g| t=0 = g 0 ∈ L 2 (R d ),
and then the exact controllability of the evolution equation (2.7).

The above example ensures the exact controllability of fractional harmonic Schrödinger equations from the half space R + × R d-1 . In particular, it appears that the cone (2.4) is a limit case for exact controllability.

2.2. Exact controllability of fractional free Schrödinger equations. In this section, we study the exact controllability of the fractional free Schrödinger equation (2.9)

(∂ t + i(-∆ x ) s )f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
with s > 0, from a measurable control subset ω ⊂ R d . By the Hilbert uniqueness method, the exact controllability of the evolution equation (2.9) is equivalent to the observability of the adjoint system

(2.10) (∂ t -i(-∆ x ) s )g(t, x) = 0, x ∈ R d , t > 0, g| t=0 = g 0 ∈ L 2 (R d ).
We first notice that the Logvinenko-Sereda Theorem recalled in Theorem 1.5 shows that the thickness condition is a necessary geometric condition for the exact controllability of the fractional Schrödinger equation (2.9) in any dimension d ≥ 1: Theorem 2.5. Let s > 0, ω be a measurable subset of R d and T > 0. If the fractional free Schrödinger equation (2.9) is exactly controllable from ω in some time T > 0, then the control subset ω is a thick set.

Proof. The assumptions of Theorem 2.5 imply that the system (2.10) is exactly observable from ω for the time T > 0. We deduce from Proposition 1.3 that there exist some positive constants k > 0 and D > 0 such that for all λ ∈ R and

f ∈ L 2 (R d ), (2.11) supp f ⊂ ξ ∈ R d : ||ξ| 2s -λ| ≤ √ D =⇒ f L 2 (R d ) ≤ √ k f L 2 (ω) .
While taking λ = 0, the assertion (2.11) implies in particular that the sets R d \ ω and the Euclidean ball

B d (0, D 1 4s
) in R d make a strong annihilating pair. Since B d (0, D

) is a bounded set, the Logvinenko-Sereda Theorem (Theorem 1.5) ensures that the control subset ω must be thick.

In the one-dimensional setting, the quantitative version of the Logvinenko-Sereda Theorem established by Kovrijkine (Theorem 1.6) together with Proposition 1.3 show that the thickness condition is also sufficient for the exact controllability of the fractional free Schrödinger equations (2.9) when s ≥ 1 2 :

Theorem 2.6. If s ≥ 1 2 and ω is a thick measurable subset of R, then there exists a positive constant T 0 > 0 such that for any time T > T 0 , the fractional free Schrödinger equation (2.9) is exactly controllable from ω in time T .

Proof. Let ω ⊂ R be a measurable subset γ-thick at scale L > 0. Let D > 0, λ ∈ R and f ∈ L 2 (R) such that supp f ⊂ ξ ∈ R : ||ξ| 2s -λ| ≤ √ D .
If |λ| > √ D + 1, the Fourier transform f is supported in the union of two intervals supp f ⊂ -(

√ D + |λ|) 1 2s , -(- √ D + |λ|) 1 2s ∪ (- √ D + |λ|) 1 2s , ( √ D + |λ|) 1 2s
and

-( √ D + |λ|) 1 2s , -(- √ D + |λ|) 1 2s
= (-

√ D + |λ|) 1 2s , ( √ D + |λ|) 1 2s ≤ √ D s , since s ≥ 1 2 and |λ| ≥ √ D + 1.
It follows from Theorem 1.7 that there exists a universal positive constant C > 0 such that (2.12)

f L 2 (R) ≤ C γ L √ D s ( C γ ) 2 + 3 2 f L 2 (ω) .
On the other hand, we notice that supp f ⊂ 0, 2

√ D + 1 1 2s
when |λ| ≤ √ D + 1. We deduce from Theorem 1.6 that there exists a universal positive constant C > 0 such that

(2.13) f L 2 (R) ≤ C γ C(1+L(2 √ D+1) 1 2s ) f L 2 (ω) .
By setting

d = max C γ L √ D s ( C γ ) 2 + 3 2 , C γ C(1+L(2 √ D+1) 1 2s ) 2
, it follows from (2.12) and (2.13) that for all λ ∈ R and f ∈ L 2 (R) with supp f ⊂ ξ ∈ R :

||ξ| 2s -λ| ≤ √ D , f L 2 (R) ≤ √ d f L 2 (ω)
. Theorem 2.6 is then a direct consequence of the Hilbert uniqueness method and Proposition 1.3.

We deduce the following necessary and sufficient geometric condition for the exact controllability of the fractional free Schrödinger equations: Corollary 2.7. Let s ≥ 1 2 and ω be a measurable subset of R. The one-dimensional fractional free Schrödinger equation

(∂ t + i(-∆ x ) s )f (t, x) = u(t, x)1l ω (x) , x ∈ R, t > 0, f | t=0 = f 0 ∈ L 2 (R),
is exactly controllable from ω if and only if ω is thick.

It is interesting to point out that the sharp geometric condition for the exact controllability of the free Schrödinger equation is actually the same as the one for the nullcontrollability of the free heat equation (1.10), even if the behavior of these two equations are very different. This thickness condition is also a necessary and sufficient condition for the null-controllability of the fractional heat equations as showed by Alphonse and Bernier [1, Remark 1.13], when the fractional parameter satisfies s > 1 2 . In a recent work [START_REF] Green | Uncertainty principles associated to sets satisfying the geometric control condition[END_REF] , Green, Jaye and Mitkovski prove new uncertainty principles. These results can be seen as a generalization of the Logvinenko-Sereda Theorem and its quantitative versions given by Kovrijkine. They establish in particular the following uncertainty principle: Theorem 2.8 (Green, Jaye & Mitkovski [5, Corollary 3]). Let ω be a measurable subset of R d satisfying the following one-dimensional geometric control condition: there exist some positive constants L > 0 and 0 < γ ≤ 1 such that for all straight lines D ⊂ R d and for all line segment S ⊂ D of length L,

H(S ∩ ω) ≥ γL,
where H denotes the one-dimensional Lebesgue measure of D. Then, for all δ > 0, β > 0, there exists C > 0 such that for all R > 0 and f

∈ L 2 (R d ), supp f ⊂ ξ ∈ R d : R -β ≤ |ξ| ≤ R + β =⇒ f L 2 (R d ) ≤ C f L 2 (U δ (ω)) ,
where U δ (ω) denotes the δ-neighborhood of ω defined by

U δ (ω) = {x ∈ R d : d(x, ω) < δ}.
This result together with Proposition 1.3 and the Hilbert uniqueness method allow to derive the following sufficient geometric condition on the control subset ω to ensure the exact controllability of the fractional free Schrödinger equation (2.9): Proposition 2.9. Let d ≥ 1, s ≥ 1 2 and ω be a measurable subset of R d satisfying that there exist some positive constants L > 0 and 0 < γ ≤ 1 such that for all straight lines D ⊂ R d and for all line segment S ⊂ D of length L,

(2.14) H(S ∩ ω) ≥ γL,
where H denotes the one-dimensional Lebesgue measure of D. Then, for all δ > 0, the fractional free Schrödinger equation

(∂ t + i(-∆ x ) s )f (t, x) = u(t, x)1l ω (x) , x ∈ R d , t > 0, f | t=0 = f 0 ∈ L 2 (R d ),
is exactly controllable from the control subset U δ (ω).

Notice that any measurable subset ω ⊂ R d satisfying the condition (2.14) is a thick subset in the sense of (1.7). The condition (2.14) is therefore stronger than the thickness condition.

Addendum. After the completion of this work, we find out the recent preprint [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF], which points out the same necessary and sufficient geometric condition on control subsets for the exact controllability of the one-dimensional free and harmonic Schrödinger equations. The authors of [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF] also derive a necessary and sufficient geometric condition for the exact controllability of the Schrödinger equation associated to the one-dimensional operator P = -∆ x + V (x), with V (x) = x 2m , when m ≥ 2, which is not discussed in this work. On the other hand, the fractional case is not discussed in [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF] and the multidimensional results contained in this work are also not covered by the results of [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF]. The proofs given in the two papers are also different. Regarding the free Schrödinger equation, the proof of [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF] given only in the non-fractional case s = 1 makes the use of explicit computations of the Fourier transform of Gaussians which cannot be directly used in the fractional case. This proof does not make explicit the condition s ≥ 1 2 found out in Corollary 2.7 to obtain the necessary and sufficient geometric condition for exact controllability. Regarding the harmonic Schrödinger equation, the proof of [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF] is much more involved since it is given in the general case when m ≥ 1 with no explicit simplification in the harmonic case m = 1. The use of the Plancherel-Rotach formula in this work allows to obtain more directly the result in the harmonic case and accounts for the fact that the present paper is much shorter than the preprint [START_REF] Huang | Observable sets, potentials and Schrödinger equations[END_REF].

Proof of Proposition 2.1

This section is devoted to the proof of Proposition 2.1. The assertions (ii) and (iii) are clearly equivalent. Indeed, (ii) implies (iii) by passing to the limit inferior lim inf

n→+∞ ψ n L 2 (ω) ≥ 1 c > 0.
On the other hand, we readily deduce from formula (2.2) that for any measurable set ω ⊂ R of positive measure |ω| > 0,

(3.1) ∀n ≥ 0, ψ n L 2 (ω) > 0,
since H n is a polynomial of degree n. Then, assertion (ii) directly follows from (iii) and (3.1). It is therefore sufficient to prove the equivalence of assertions (i) and (iii).

As a preliminary step, we begin by establishing that

(3.2) ∀0 < ε < π 2 , lim n→+∞ ψ n L 2 (Iε,n) = 1 - 2ε π ,
where I ε,n stands for the open interval (-

√ 2n + 1 cos ε, √ 2n + 1 cos ε). Let 0 < ε < π 2 .
The Plancherel-Rotach formula for Hermite polynomials [START_REF] Szegö | Orthogonal polynomials[END_REF] (Theorem 8.22.9) provides that for all ε ≤ θ ≤ π -ε,

(3.3) e -x 2 2 H n (x) = 2 n 2 + 1 4 √ n! (πn) 1 4 √ sin θ sin n 2 + 1 4 sin(2θ) -2θ + 3π 4 + R n (x) ,
with x = √ 2n + 1 cos θ, where the remainder term satisfies

(3.4) ∃C > 0, ∀n ≥ 0, ∀y ∈ [- √ 2n + 1 cos ε, √ 2n + 1 cos ε], |R n (y)| ≤ C n + 1 .
It follows from (2.2) and (3.3) that the Hermite function

ψ n (x) = 1 π 1 4 2 n 2 e -x 2 2 H n (x), ψ n L 2 (R) = 1, n ≥ 0,
satisfies that for all ε ≤ θ ≤ π -ε and n ≥ 1,

(3.5) ψ n (x) = 2 1 4
√ πn

1 4 √ sin θ sin n 2 + 1 4 sin(2θ) -2θ + 3π 4 + R n (x) ,
with x = √ 2n + 1 cos θ. We obtain from (3.5) that for all n ≥ 1,

(3.6) F n L 2 (Iε,n) -G n L 2 (Iε,n) ≤ ψ n L 2 (Iε,n) ≤ F n L 2 (Iε,n) + G n L 2 (Iε,n) ,
where the two functions F n and G n are defined for all ε < θ < π -ε,

(3.7) F n (x) = 2 1 4
√ πn

1 4 √ sin θ sin n 2 + 1 4 sin(2θ) -2θ + 3π 4 and (3.8) G n (x) = 2 1 4
√ πn

1 4 √ sin θ R n (x),
with x = √ 2n + 1 cos θ. We deduce from (3.4) that for all n ≥ 1,

G n 2 L 2 (Iε,n) = 2(2n + 1) π √ n π-ε ε |R n ( √ 2n + 1 cos θ)| 2 dθ ≤ C 2 2(2n + 1)(π -2ε) π √ n(n + 1) 2 , implying that (3.9) lim n→+∞ G n L 2 (Iε,n) = 0.
On the other hand, we deduce from (3.7) that for all n ≥ 1,

(3.10) |F n (x)| 2 = 1 √ 2nπ sin θ 1 -cos n + 1 2 sin(2θ) -2θ + 3π 2 .
It follows from (3.10) that for all n ≥ 1, (3.11)

Iε,n |F n (x)| 2 dx = Iε,n √ 2n + 1 √ 2nπ (2n + 1) -x 2 dx - √ 2n + 1 √ 2nπ π-ε ε cos n + 1 2 sin(2θ) -2θ + 3π 2 dθ, since sin 2 θ = 1 -x 2 2n+1 , when x = √ 2n + 1 cos θ.
We have for all n ≥ 1, (3.12)

Iε,n √ 2n + 1 √ 2nπ (2n + 1) -x 2 dx = 1 π 2n + 1 2n cos ε -cos ε dy 1 -y 2 = 2 π 2n + 1 2n arcsin(cos ε) = 2 π 2n + 1 2n π 2 -arccos(cos ε) = 2n + 1 2n 1 - 2ε π , since ∀ -1 ≤ x ≤ 1, arccos x + arcsin x = π 2 .
It follows from (3.12) that

(3.13) lim n→+∞ Iε,n √ 2n + 1 √ 2nπ (2n + 1) -x 2 dx = 1 - 2ε π .
On the other hand, we have for all n ≥ 1, (3.14)

√ 2n + 1 √ 2nπ π-ε ε cos n + 1 2 sin(2θ) -2θ + 3π 2 dθ = √ 2n + 1 √ 2nπ ϕ(π-ε) ϕ(ε) 1 ϕ (ϕ -1 (t)) cos n + 1 2 t - 3π 2 dt,
where

ϕ(θ) = 2θ -sin(2θ) is a increasing C ∞ -diffeomorphism from [ε, π -ε] to [ϕ(ε), ϕ(π -ε)] = [2ε -sin(2ε), 2π -2ε + sin(2ε)].
An integration by parts shows that for all n ≥ 1,

ϕ(π-ε) ϕ(ε) 1 ϕ (ϕ -1 (t)) cos n + 1 2 t - 3π 2 dt = sin((n + 1 2 )t -3π 2 ) ϕ (ϕ -1 (t))(n + 1 2 ) ϕ(π-ε) ϕ(ε) - 1 n + 1 2 ϕ(π-ε) ϕ(ε) d dt 1 ϕ (ϕ -1 (t)) sin n + 1 2 t - 3π 2 dt, implying that (3.15) ϕ(π-ε) ϕ(ε) 1 ϕ (ϕ -1 (t)) cos n + 1 2 t - 3π 2 dt ≤ C 0 n + 1 2 (2 + 2π -4ε + 2 sin(2ε)), with 0 < C 0 = sup t∈[2ε-sin(2ε),2π-2ε+sin(2ε)] 1 ϕ (ϕ -1 (t)) + d dt 1 ϕ (ϕ -1 (t)) < +∞.
We deduce from (3.14)

F n L 2 (Iε,n) = 1 - 2ε π .
Thanks to (3.6), (3.9) and (3.17), we obtain that (3.2) holds.

Let ω be a measurable subset of R verifying

(3.18) δ = lim inf n→+∞ ψ n L 2 (ω) > 0. Let 0 < ε 0 < π 2 satisfying 0 < ε 0 < π 2 δ 2 .
By using that ψ n L 2 (R) = 1, we observe that for all 0 < ε < π 2 and n ≥ 0,

(3.19) ψ n 2 L 2 (ω) = ψ n 2 L 2 (ω∩Iε,n) + ψ n 2 L 2 (ω∩(R\Iε,n)) ≤ ψ n 2 L 2 (ω∩Iε,n) + ψ n 2 L 2 (R\Iε,n) = ψ n 2 L 2 (ω∩Iε,n) + 1 -ψ n 2 L 2 (Iε,n) . It follows from (3.2), (3.18) and (3.19) that for all 0 < ε ≤ ε 0 , (3.20) lim inf n→+∞ ψ n L 2 (ω∩Iε,n) > 0.
On the other hand, we deduce from (3.5), (3.7) and (3.8) that for all n ≥ 1 and 0

< ε < π 2 , (3.21) F n L 2 (ωε,n) -G n L 2 (ωε,n) ≤ ψ n L 2 (ωε,n) ≤ F n L 2 (ωε,n) + G n L 2 ( ωε,n) , with (3.22)
ω ε,n = ω ∩ I ε,n . By using that G n L 2 (ωε,n) ≤ G n L 2 (Iε,n) , it follows from (3.9) that (3.23) ∀0 < ε < π 2 , lim n→+∞ G n L 2 (ωε,n) = 0.
We deduce from (3.10) and (3.22) that for all n ≥ 1, (3.24)

ωε 0 ,n |F n (x)| 2 dx ≤ ωε 0 ,n 2 √ 2n + 1 √ 2nπ (2n + 1) -x 2 dx ≤ ωε 0 ,n √ 2 √ nπ √ 1 -cos 2 ε 0 dx ≤ √ 2 π sin ε 0 |ω ∩ (- √ 2n + 1, √ 2n + 1)| √ n .
By using (3.20), (3.21), (3.22), (3.23) and (3.24), it follows that

(3.25) lim inf n→+∞ |ω ∩ (- √ 2n + 1, √ 2n + 1)| √ n ≥ lim inf n→+∞ π sin ε 0 √ 2 ωε 0 ,n |F n (x)| 2 dx = lim inf n→+∞ π sin ε 0 √ 2 ωε 0 ,n |ψ n (x)| 2 dx > 0.
We readily obtain from (3.25) that

(3.26) lim inf n→+∞ |ω ∩ [- √ 2n + 1, √ 2n + 1]| √ n > 0.
Since the floor function satisfies

∀R > 0, R ≤ R < R + 1, it follows that for all R ≥ 2, (3.27) √ 2n + 1 ≤ R < √ 2n + 3, with n = n(R) = R 2 -1 2 ∈ N. We obtain that for all R ≥ 2, (3.28) |ω ∩ [-R, R]| |[-R, R]| ≥ |ω ∩ [- √ 2n + 1, √ 2n + 1]| 2R = √ n 2R |ω ∩ [- √ 2n + 1, √ 2n + 1]| √ n .
Notice that (3.27) implies that

(3.29) lim R→+∞ √ n 2R = 1 2 √ 2 .
It follows from (3.26), (3.28) and (3.29) that lim inf

R→+∞ |ω ∩ [-R, R]| |[-R, R]| ≥ 1 2 √ 2 lim inf n→+∞ |ω ∩ [- √ 2n + 1, √ 2n + 1]| √ n > 0.
It proves that assertion (ii) implies assertion (i).

Conversely, let ω be a measurable subset of R verifying lim inf

R→+∞ |ω ∩ [-R, R]| |[-R, R]| > 0 and define δ = lim inf n→+∞ |ω ∩ [- √ 2n + 1, √ 2n + 1]| √ n > 0.
In order to prove (iii), it follows from (3.22) that it is actually sufficient to show that

∃0 < ε < π 2 , lim inf n→+∞ ψ n L 2 (ωε,n) > 0.
According to (3.21) and (3.23), it is then sufficient to show that

(3.30) ∃0 < ε < π 2 , lim inf n→+∞ F n L 2 (ωε,n) > 0.
While using the substitution rule with x = √ 2n + 1 cos θ, we deduce from (3.7) and (3.22) that for all 0 < ε < π 2 and n ≥ 1,

(3.31) ωε,n |F n (x)| 2 dx = 1 π 4n + 2 n ωε,n sin 2 n 2 + 1 4 sin(2θ) -2θ + 3π 4 dθ, with (3.32) ωε,n = arccos ω ε,n √ 2n + 1 ⊂ (ε, π -ε).
According to (3.30) and (3.31), it is then sufficient to show that

(3.33) ∃0 < ε < π 2 , lim inf n→+∞ ωε,n sin 2 n 2 + 1 4 sin(2θ) -2θ + 3π 4 dθ > 0. Let 0 < ε 0 < π 2 verifying 0 < 2 √ 2(1 -cos ε 0 ) < δ.
By noticing from (3.22) that for all 0 < ε < π 2 and n ≥ 1,

|ω ε,n | √ n ≥ |ω ∩ [- √ 2n + 1, √ 2n + 1]| √ n - 2 √ 2n + 1 √ n (1 -cos ε), we deduce that lim inf n→+∞ |ω ε 0 ,n | √ n ≥ δ -2 √ 2(1 -cos ε 0 ) > 0.
It follows that there exist some positive constants c 0 > 0 and n 0 ≥ 1 such that

(3.34) ∀n ≥ n 0 , |ω ε 0 ,n | ≥ c 0 √ n.
On the other hand, we notice from (3.32) and (3.34) that for all n ≥ n 0 ,

(3.35) |ω ε 0 ,n | = ωε 0 ,n |κ n (x)|dx = ωε 0 ,n dx (2n + 1) -x 2 ≥ |ω ε 0 ,n | √ 2n + 1 ≥ c 0 n 2n + 1 , with κ n (x) = arccos x √
2n+1 . It follows from (3.35) that there exists a positive constant c 1 > 0 such that

(3.36) ∀n ≥ n 0 , |ω ε 0 ,n | ≥ c 1 > 0.
By using anew the substitution rule with t = 2θ -sin(2θ), we observe that (3.37)

ωε 0 ,n sin 2 n 2 + 1 4 sin(2θ) -2θ + 3π 4 dθ = ϕ(ωε 0 ,n) 1 ϕ (ϕ -1 (t)) sin 2 n 2 + 1 4 t - 3π 4 dt ≥ c 2 ϕ(ωε 0 ,n) sin 2 n 2 + 1 4 t - 3π 4 dt, with 0 < c 2 = inf t∈[2ε 0 -sin(2ε 0 ),2π-2ε 0 +sin(2ε 0 )] 1 ϕ (ϕ -1 (t)) < +∞, where ϕ(θ) = 2θ -sin(2θ) is the increasing C ∞ -diffeomorphism from [ε 0 , π -ε 0 ] to [2ε 0 - sin(2ε 0 ),
sin 2 n 2 + 1 4 t - 3π 4 dt > 0,
where according to (3.36), the subset ϕ(ω ε 0 ,n ) ⊂ (2ε 0 -sin(2ε 0 ), 2π-2ε 0 +sin(2ε 0 )) satisfies for all n ≥ n 0 ,

(3.39) |ϕ(ω ε 0 ,n )| = ωε 0 ,n |ϕ (x)|dx ≥ inf [ε 0 ,π-ε 0 ] |ϕ | |ω ε 0 ,n | ≥ c 3 , with 0 < c 3 = c 1 inf [ε 0 ,π-ε 0 ] |ϕ | < +∞.
By using anew the substitution rule with x = ( n 2 + 1 4 )t -3π 4 , we observe that (3.40)

ϕ(ωε 0 ,n) sin 2 n 2 + 1 4 t - 3π 4 dt = 4 2n + 1 Ωε 0 ,n sin 2 x dx,
where according to (3.32) and (3.39), the subset

Ω ε 0 ,n = n 2 + 1 4 ϕ(ω ε 0 ,n ) - 3π 4 ⊂ n 2 + 1 4 ϕ(ε 0 ) - 3π 4 , n 2 + 1 4 ϕ(π -ε 0 ) - 3π 4 , satisfies (3.41) ∃c 4 > 0, ∀n ≥ n 0 , |Ω ε 0 ,n | ≥ c 4 n.
According to (3.38) Let k ≥ 1 be an integer depending only on ε 0 such that

(3.43) ∀n ≥ n 0 , Ω ε 0 ,n ⊂ -knπ + π 2 , knπ + π 2 .
We denote N n = #I n the cardinality of the following set (3.44)

I n = i ∈ [-kn, kn -1] ∩ Z : Ω ε 0 ,n ∩ iπ + π 2 , π(i + 1) + π 2 ≥ c 4 2k(π + 1)
,

with n ≥ n 0 . It follows from (3.43) and (3.44) that for all n ≥ n 0 ,

|Ω ε 0 ,n | = kn-1 i=-kn Ω ε 0 ,n ∩ iπ + π 2 , (i + 1)π + π 2 (3.45) = i∈In Ω ε 0 ,n ∩ iπ + π 2 , (i + 1)π + π 2 + i / ∈In, i∈[-kn,kn-1]∩Z Ω ε 0 ,n ∩ iπ + π 2 , (i + 1)π + π 2 ≤ πN n + 2kn c 4 2k(π + 1) = πN n + nc 4 π + 1 .
We deduce from (3.41) and (3.45) that

(3.46) ∀n ≥ n 0 , N n ≥ nc 4 π + 1 .
Let us now prove that the estimate (3.42) holds. By using the π-periodicity of the function x → sin 2 x and (3.43), we obtain that for all n ≥ n 0 ,

1 n Ωε 0 ,n sin 2 x dx = 1 n kn-1 i=-kn Ωε 0 ,n∩[iπ+ π 2 ,(i+1)π+ π 2 ]
sin 2 x dx (3.47)

≥ 1 n i∈In (Ωε 0 ,n-(i+1)π)∩[- π 2 , π 2 ]
sin 2 x dx.

An application of Lemma

m i,n = (Ω ε 0 ,n -(i + 1)π) ∩ - π 2 , π 2 = Ω ε 0 ,n ∩ iπ + π 2 , (i + 1)π + π 2 ,
for all n ≥ n 0 and i ∈ [-kn, kn -1] ∩ Z. This establishes (3.42) and concludes that assertion (i) implies assertion (iii). This ends the proof of Proposition 2.1.

Proof of Theorem 2.3

This section is devoted to the proof of Theorem 2.3. For any α = (α 1 , ..., α d ) ∈ (N\{0}) d and ε ≥ 0, we consider the parallelepipeds

C α,ε = d j=1 [-2α j + 1 cos ε, 2α j + 1 cos ε]. Let 0 < ε < π 2 .
We deduce from the one-dimensional Plancherel-Rotach formula (3.5) that for all α ∈ (N \ {0}) d , 0 < ε < π 2 and x ∈ C α,ε , (4.1)

Ψ α (x) = F α (x) + G α (x), with (4.2) F α (x) = 2 d 4 π d 2 α 1 4 d i=1 sin α i 2 + 1 4 (sin(2θ i ) -2θ i ) + 3π 4 √ sin θ i and (4.3) G α (x) = 2 d 4 π d 2 α 1 4 d i=1 √ sin θ i R α (x), with x = ( √ 2α 1 + 1 cos θ 1 , ..., √ 2α d + 1 cos θ d ) ∈ C α,ε . The remainder terms R α satisfy (4.4) ∃C ε > 0, ∀α ∈ (N \ {0}) d , ∀x ∈ C α,ε , |R α (x)| ≤ C ε d i=1 1 α i + 1
.

By using the preliminary step in the proof of Proposition 2.1, we obtain from (3.2) that for all 0 < ε < π 2 , lim

α 1 →+∞ ... lim α d →+∞ Ψ α 2 L 2 (Cα,ε) (4.5)
= lim

α 1 →+∞ ... lim α d →+∞ d j=1 ψ α j 2 L 2 ([- √ 2α j +1 cos ε, √ 2α j +1 cos ε]) = d j=1 lim α j →+∞ ψ α j 2 L 2 ([- √ 2α j +1 cos ε, √ 2α j +1 cos ε]) = 1 - 2ε π d 2 .
Let us assume that the fractional harmonic Schrödinger equation (2.5) with s ≥ 1 is exactly controllable from a measurable subset ω ⊂ R d for some time T > 0. The spectral gap condition in Proposition 1.8 is satisfied by the fractional harmonic oscillator (-∆ x + |x| 2 ) s since its spectrum is given by

σ (-∆ x + |x| 2 ) s = (2n + d) s : n ∈ N .
We can deduce from the infinite dimensional Hautus test [START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]Corollary 2.18] recalled in Proposition 1.8 that there exists δ > 0 such that

(4.6) ∀α ∈ N d , δ = δ Ψ α L 2 (R d ) ≤ Ψ α L 2 (ω) . Let 0 < ε 0 < π 2 such that δ -1 -1 - 2ε 0 π d 2 > 0.
By writing for all α ∈ N d ,

Ψ α 2 L 2 (ω) = Ψ α 2 L 2 (ω∩Cα,ε 0 ) + Ψ α 2 L 2 (ω\Cα,ε 0 ) ≤ Ψ α 2 L 2 (ω∩Cα,ε 0 ) + 1 -Ψ α 2 L 2 (Cα,ε 0 ) ,
we deduce from (4.5) and (4.6) that lim inf

α 1 →+∞ ... lim inf α d →+∞ Ψ α L 2 (ω∩Cα,ε 0 ) ≥ δ 2 -1 -1 - 2ε 0 π d 2 > 0.
It follows from (4.3) and (4.4) that

∀α ∈ (N \ {0}) d , G α 2 L 2 (Cα,ε 0 ) ≤ C 2 ε 0 2 d 2 α 1 2 d i=1 1 α i + 1 2 d i=1 √ 2α i + 1 1 - 2ε 0 π d and (4.7) lim sup α 1 →+∞ ... lim sup α d →+∞ G α L 2 (Cα,ε 0 ) = 0.
We can therefore deduce from (4.1) and (4.7) that lim inf

α 1 →+∞ ... lim inf α d →+∞ F α L 2 (ω∩Cα,ε 0 ) (4.8) ≥ lim inf α 1 →+∞ ... lim inf α d →+∞ Ψ α L 2 (ω∩Cα,ε 0 ) -G α L 2 (ω∩Cα,ε 0 ) ≥ lim inf α 1 →+∞ ... lim inf α d →+∞ Ψ α L 2 (ω∩Cα,ε 0 ) -G α L 2 (Cα,ε 0 ) ≥ lim inf α 1 →+∞ ... lim inf α d →+∞ Ψ α L 2 (ω∩Cα,ε 0 ) > 0.
On the other hand, we notice from (4.2) that for all α ∈

(N \ {0}) d , ∀x ∈ C α,ε 0 , |F α (x)| 2 ≤ 2 d 2 π d α 1 2 d i=1 √ 2α i + 1 2α i + 1 -x 2 i
and this implies that for all α ∈ (N \ {0}) d , (4.9)

F α 2 L 2 (ω∩Cα,ε 0 ) ≤ 2 d 2 π d α 1 2 ω∩Cα,ε 0 d i=1 √ 2α i + 1 2α i + 1 -x 2 i dx ≤ 2 d 2 π d sin d ε 0 |ω ∩ [- √ 2α 1 + 1, √ 2α 1 + 1] × ... × [- √ 2α d + 1, √ 2α d + 1]| α 1 2
.

We deduce from (4.8) and (4.9) that (4.10) lim inf

α 1 →+∞ ... lim inf α d →+∞ |ω ∩ [- √ 2α 1 + 1, √ 2α 1 + 1] × ... × [- √ 2α d + 1, √ 2α d + 1]| |[- √ 2α 1 + 1, √ 2α 1 + 1] × ... × [- √ 2α d + 1, √ 2α d + 1]| > 0.
It follows from (3.27) that for all (R 1 , ..

., R d ) ∈ [2, +∞) d , (4.11) ∀1 ≤ i ≤ d, √ 2n i + 1 ≤ R i < √ 2n i + 3,
where

n i = n i (R i ) = R 2 i -1 2 ∈ N for all 1 ≤ i ≤ d. We obtain that |ω ∩ [-R 1 , R 1 ] × ... × [-R d , R d ]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| (4.12) ≥ |ω ∩ [- √ 2n 1 + 1, √ 2n 1 + 1] × ... × [- √ 2n d + 1, √ 2n d + 1]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| = d i=1 √ 2n i + 1 R i |ω ∩ [- √ 2n 1 + 1, √ 2n 1 + 1] × ... × [- √ 2n d + 1, √ 2n d + 1]| |[- √ 2n 1 + 1, √ 2n 1 + 1] × ... × [- √ 2n d + 1, √ 2n d + 1]| .
We observe that (4.11) implies that (4.13)

∀1 ≤ i ≤ d, lim R i →+∞ √ 2n i + 1 R i = 1.
It follows from (4.10), (4.12) and (4.13) that lim inf

R 1 →+∞ ... lim inf R d →+∞ |ω ∩ [-R 1 , R 1 ] × ... × [-R d , R d ]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| ≥ lim inf α 1 →+∞ ... lim inf α d →+∞ |ω ∩ [- √ 2α 1 + 1, √ 2α 1 + 1] × ... × [- √ 2α d + 1, √ 2α d + 1]| |[- √ 2α 1 + 1, √ 2α 1 + 1] × ... × [- √ 2α d + 1, √ 2α d + 1]| > 0. It establishes (2.6) with A = I d . Let A ∈ O(R d ).
By using the invariance of the Weyl quantization under affine symplectic transformations (see e.g. [6, Theorem 18.5.9]), we notice that

(4.14) R A • H = H • R A , with H = -∆ x +|x| 2 , where R A is the unitary operator on L 2 (R d ) defined by R A f = f (A •) for f ∈ L 2 (R d ). The identity (4.14) implies in particular that (R A Ψ α ) α∈N d is a Hilbert basis of L 2 (R d ) satisfying ∀α ∈ N d , H(R A Ψ α ) = (2|α| + d)R A Ψ α , when A ∈ O(R d )
. By using that the operator

R A : Span (Ψ α ) α∈N d ,|α|=N → Span (Ψ α ) α∈N d ,|α|=N ,
is a unitary operator for any N ∈ N, we notice that for all s ≥ 1 and

f ∈ D(H s ), (R A • H s )f = (R A • H s ) α∈N d f, R A -1 Ψ α L 2 (R d ) R A -1 Ψ α = R A α∈N d (2|α| + d) s f, R A -1 Ψ α L 2 (R d ) R A -1 Ψ α = α∈N d (2|α| + d) s R A f, Ψ α L 2 (R d ) Ψ α = (H s • R A )f, that is (4.15) ∀A ∈ O(R d ), ∀s ≥ 1, R A • H s = H s • R A . Let f 0 , f T ∈ L 2 (R d ). By assumption, there exists a control function u ∈ L 2 ([0, T ] × R d ) supported in R + × ω such that the semigroup solution g : R + × R d → R of the system (2.5) associated to the initial condition f 0 (A •) satisfies g(T, •) = f T (A •). We deduce from (4.15) that the function f defined on R + × R d by ∀t ∈ R + , ∀x ∈ R d , f (t, x) = g(t, A -1 x),
is the semigroup solution of (2.5) associated to the initial condition f 0 and the control function v(t, x) = u(t, A -1 x) ∈ L 2 ([0, T ] × R d ) supported in R + × A(ω); and satisfies f (T, •) = f T (•). It implies that the harmonic Schrödinger equation (2.5) is exactly controllable from the control set A(ω). By using the first part of this proof with ω replaced by A(ω), we obtain that lim inf

R 1 →+∞ ... lim inf R d →+∞ |A(ω) ∩ [-R 1 , R 1 ] × ... × [-R d , R d ]| |[-R 1 , R 1 ] × ... × [-R d , R d ]| > 0.
This ends the proof of Theorem 2.3.

Appendix

This appendix contains the proof of two instrumental results. The first part is devoted to the proof of a technical lemma used in the proof of Proposition 2.1. The purpose of the second part is to recall the proof of Miller's result stated in Proposition 1.3. 5.2. Spectral inequalities and exact controllability. This section is devoted to recall the proof of Miller's result [START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]Corollary 2.17] stated in Proposition 1.3 which provides necessary and sufficient spectral estimates for the observability of system (1.3) to hold. The proof of Proposition 1.3 is based on another Miller's result [START_REF] Miller | Resolvent conditions for the control of unitary groups and their approximations[END_REF]Theorem 2.4] characterizing the observability with resolvent conditions which is recalled below and whose proof is omitted: Proposition 5.2 (Miller [11,Theorem 2.4]). Let (A, D(A)) be a selfadjoint operator on L 2 (R d ), which is the infinitesimal generator of a strongly continuous group (e itA ) t∈R on L 2 (R d ). The system (1.3) is exactly observable from a measurable subset ω ⊂ R d if and only if there exist some positive constants M > 0 and m > 0 such that (5.3) ∀f ∈ D(A), ∀λ ∈ R,

f 2 L 2 (R d ) ≤ M (A -λ)f 2 L 2 (R d ) + m f 2 L 2 (ω) .
When condition (5.3) is satisfied, exact observability holds in any time T > π √ M .

We consider the system (1.3). If this system is exactly observable from a measurable subset ω ⊂ R d at time T > 0, Proposition 5.2 proves that there exist some positive constants M > 0 and m > 0 such that the resolvent estimate (5. Conversely, let us assume that there exist some positive constants D > 0 and k > 0 such that the spectral estimates (1.4) holds. Let λ ∈ R and f ∈ D(A). With f λ = 1l {|A-λ|≤ √ D} f and f ⊥ = f -f λ , we deduce from (1.4) and the functional calculus that for all λ ∈ R and f ∈ D(A),

(5.6) f 2 L 2 (R d ) = f λ 2 L 2 (R d ) + f ⊥ 2 L 2 (R d ) ≤ k f λ 2 L 2 (ω) + 1 D (A -λ)f ⊥ 2 L 2 (R d ) ≤ k f λ 2 L 2 (ω) + 1 D (A -λ)f 2 L 2 (R d ) .
Let T > π 1+k D and ε > 0 such that

T 2 > π 2 1 + (1 + ε 2 )k D .
The functional calculus shows that for all λ ∈ R and f ∈ D(A),

(5.7) f λ 2 L 2 (ω) = f -f ⊥ 2 L 2 (ω) ≤ (1 + ε -2 ) f 2 L 2 (ω) + (1 + ε 2 ) f ⊥ 2 L 2 (ω) ≤ (1 + ε -2 ) f 2 L 2 (ω) + (1 + ε 2 ) f ⊥ 2 L 2 (R d ) ≤ (1 + ε -2 ) f 2 L 2 (ω) + 1 + ε 2 D (A -λ)f ⊥ 2 L 2 (R d ) ≤ (1 + ε -2 ) f 2 L 2 (ω) + 1 + ε 2 D (A -λ)f 2 L 2 (R d ) .
It follows from (5.6) and (5.7) that ∀λ ∈ R, ∀f ∈ D(A),

f 2 L 2 (R d ) ≤ k(1 + ε -2 ) f 2 L 2 (ω) + 1 + k(1 + ε 2 ) D (A -λ)f 2 L 2 (R d ) .
We can deduce from Proposition 5.2 that the system (1.3) is exactly observable from ω at time T .

5. 1 . 1 : 5 . 1 . 2 -δ 2 sin 2

 1151222 A technical lemma. The following technical result is used in the proof of Proposition 2.Lemma If A is a measurable subset of [-π 2 , π2] and δ = |A| denotes its Lebesgue measure then A sin 2 x dx ≥ δ x dx.

  3) holds. Let λ ∈ R, 0 < D < 1 M and f ∈ 1l {|A-λ|≤ √ D} D(A) . The functional calculus shows that (5.4) (A -λ)f L 2 (R d ) ≤ √ D f L 2 (R d ) .It follows from (5.3) and (5.4) that(5.5) ∀λ ∈ R, ∀f ∈ 1l {|A-λ|≤ √ D} D(A) , f 2 L 2 (R d ) ≤ M D f 2 L 2 (R d ) + m f 2 L 2 (ω) , that is ∀λ ∈ R, ∀f ∈ 1l {|A-λ|≤ √ D} D(A) , f 2 L 2 (R d ) ≤ m 1 -M D f 2 L 2 (ω) .It establishes the spectral estimates (1.4) for all f ∈ 1l {|A-λ|≤ √ D} L 2 (R d ) since the domain D(A) is dense in L 2 (R d ).