Ann-Sofi Röj-Lindberg

Kirsti Hemmi
email: khemmi@abo.fi

The student's perspective on school mathematicsa case study

Keywords: Habits of the mathematics classroom, long-term case study, spaces for participation, student perspective

This paper is based on a long-term case study of students' experiences of reform mathematics classrooms in Finland. Specifically, it focuses on emerging school mathematical actions and activities experienced by three committed and engaged students. Students described their experiences in interviews during and ten years after lower secondary school. The paper asks why taking part in certain practices makes some students learn to like and others to dislike school mathematics, despite high attainment concerning reform-related objectives on achievement and responsibility. The case study showed the importance of making classrooms places where all students belong to communities in which mathematical knowledge is negotiated.

Introduction

In mathematics education, it is challenging to develop classroom cultures in which students maintain positive relationships towards mathematics after their first school years [START_REF] Sullivan | Perhaps the decision of some students not to engage in learning mathematics in school is deliberate[END_REF]. Although Finnish students have performed well in mathematics in international comparisons, especially in the Programme for International Student Assessment (PISA), many Finnish students' attitudes, self-confidence and engagement are poor in comparison to other countries. Based on PISA 2003, Viking Brunell (2007) found that only one third of the students in Swedish-speaking classrooms in Finland looked forward to mathematics lessons, and one half did not express any interest in mathematics whatsoever. Later, the 2015 Trends in International Mathematics and Science Study showed that Grade 4 Finnish students had a less positive attitude towards school mathematics than those in other European countries [START_REF] Mullis | TIMSS 2015 international results in mathematics[END_REF]. This was especially the case with girls, who chose to invest in subjects other than mathematics quite early on [START_REF] Metsämuuronen | Oppia ikä kaikki: Matemaattinen osaaminen toisen asteen koulutuksen lopussa 2015[END_REF]. While much research has confirmed some students' negative relationship with mathematics (cf. [START_REF] Brunell | Klimat och resultat i den finlandssvenska grundskolan: En fördjupad analys av PISA 2003[END_REF][START_REF] Lewis | Disaffection with school mathematics[END_REF], very few studies have focused on why engaged and committed students develop negative attitudes and abandon mathematics by neglecting to study advanced courses in the subject at upper secondary level. The aim of this paper was to contribute to our understanding of the reasons behind these choices.

For some decades, attempts have been made in many countries to improve/reform school mathematics to make it more meaningful for all students (Fennema & Scott Nelson, 1997). This has also been the case in Finland. This paper was situated in a reform context in a Swedish-speaking lower secondary school in Finland. The reform emerged in connection with a three-year action research project that focused on learning processes and assessment, especially those involving problem-solving and investigative project work. The national core curriculum declared that solving problems and attending to the internal logic of mathematics were the two most important principles for mathematics teaching (Finnish National Board of Education, 1994). By probing and assessing the mathematical thinking of students, teachers wanted to encourage the discovery of useful and viable mathematical ideas that they described as 'active' knowledge and to support individual students' emerging sense of control over and responsibility for learning to a greater degree than was possible through 'ordinary' classroom practices.

Few studies have investigated school mathematics from students' perspectives, and we have not come across any studies that involved students as legitimate participants in an interpretive research methodology that relates to teachers' reform work during lower secondary school and beyond. In the doctoral study on which this paper was based (Röj-Lindberg, 2017), participating students were interviewed several times from the age of 13 to approximately 25. In this paper, we aimed to answer the following question, "Why do some committed and engaged students develop a negative identity concerning mathematics during lower secondary school?"

School mathematics from the student's perspective

About thirty years ago, Alan [START_REF] Schoenfeld | When good teaching leads to bad results: The disasters of "well-taught" mathematics courses[END_REF] argued that school mathematics might unintentionally produce "disasters" even though most actions and activities in mathematics classrooms proceed as envisaged by teachers in terms of the curriculum and teaching practices. As a result of his inquiry, Schoenfeld called for broad research agendas that should aim "to understand the world from the student's point of view" (p. 165). His message was that if learning could be perceived as much more than what can be inferred from observing students in classrooms, including their mathematical performance, different forms of assessments, and so on, then the voices of students really should start to matter in school mathematics and educational research. Since then, the same type of request has been formulated repeatedly both outside and within mathematics education [START_REF] Berinderjeet | Student voice in mathematics classrooms around the world[END_REF].

Jo [START_REF] Boaler | Studying and capturing the complexity of practice: the case of the dance of agency[END_REF] defined classroom practices as the recurrent activities and norms that develop in classrooms over time. We argue here that even if researchers are strongly focused on changing classroom practices, they have tended to apply a top-down perspective, looking at students and processes of learning from the outside. Classroom practices have been studied in isolation from students themselves, though they have been conceptualized as methods of teaching1 that provide a social context that influences and allows the experiences and expectations of students. From an observer's perspective in a classroom, it may be possible to describe and conceptualize regularities within observable actions and activities. But such a stance does not take into account the notion that activities and actions in the classroom are complex when seen from the perspectives of students who are the immediate participants.

Here, we have taken reflexivity as a premise of the research and argue that students' accounts of what happens or is expected to happen in mathematics classrooms are socio-cultural in their very nature. We have accepted that social situatedness should be taken as a premise when students' perspectives are the focus of interest. Whatever else might be innate, students' expectations concerning school mathematics are not; these are formed over the years, largely through students' interactions with their parents, peer group and teachers. Based on such a premise, we interpreted the experiential accounts given by students in interviews as having a social and cultural origin. Interpretations can be created through "zooming in" [START_REF] Lerman | A moment in the zoom of a lens: Towards a discursive psychology of mathematics teaching and learning[END_REF] on the complex territory of classroom practices through accounts given by students and by looking at classroom actions and activities through students' 'eyes'. On the other hand, one must "zoom out" to view individual students' experiential accounts without losing contact with the actions and activities that constitute school mathematics in their individual classrooms.

The study was embedded in social practice theory and focused specifically on students' alignment and their development of identities and imagination in relation to classroom practices in which they took part. Learning was defined as increasing participation in classroom practices, which were understood as social processes where meanings are negotiated and students become or avoid becoming persons for whom mathematics is important [START_REF] Wenger | Communities of practice[END_REF].

Methodology

This paper was based on a case study (Röj-Lindberg, 2017) within the context of teacher-initiated action research and reform work. The main case record material (see [START_REF] Bassey | Case study research[END_REF] was semistructured interviews with four to five students from each reform classroom, in total 120 interviews with 27 students. Four to five interviews were done with each of them during the three lower secondary school years. Students were asked to describe and comment on their experiences of school mathematics, including their wishes and expectations, for instance, in relation to recent mathematics lessons. Students were also probed about their views on problem-solving activities and other reform-related activities, for instance, on being asked to explain one's thinking publicly. Interviews with three committed and engaged classmates, Joakim (male), Kristina and Nette (females), were selected for further analysis for the case study. These three students, who had experienced the same classroom context, were also invited to look back on their experiences as adults. Adult interviews were conducted when the three students had finished university and focused not only on lower secondary school classroom practices but also on mathematical practices more widely. The first author's preliminary interpretations of the lower secondary school interviews were used as inputs in the adult interviews, which were conducted to deepen the researcher's understanding of school mathematics from a student's point of view, not for fact-finding.

The analysis of the interviews, which were conducted in Swedish, was first inductive and then deepened through holistic interpretations of each interview (see [START_REF] Merriam | Qualitative research[END_REF]. The researcher's responsiveness to emic issues appearing in students' accounts was high. Important emic issues were structure and closure emerging from the lower secondary school interviews and experiences of identity work from the adult interviews (see [START_REF] Röj-Lindberg | Structure and closure of school mathematical practice: The experiences of Kristina[END_REF][START_REF] Röj-Lindberg | School mathematical practices as experiences of identity work: The learning journeys of three students[END_REF]. Emic issues were theoretically connected to Wenger's social practice theories. In the final analysis, they were connected to other theoretical constructs from this theory, such as communities of practice (applied to school mathematics), negotiation of meaning, modes of belonging and students' trajectories within a specific practice. These were applied during the interpretation of interview accounts. It is beyond the scope of this paper to give a thorough explanation of the use of these constructs. It is also impossible to give an exhaustive account of the extent to which the reform influenced activities in the classrooms (for detailed information, see [START_REF] Röj-Lindberg | Skolmatematisk praktik i förändring: En fallstudie[END_REF]). Yet, when presenting the results we connected the issues brought forth by students to relevant classroom events. The quotes have been translated into English by the first author. Results presented here do not intend to generalise but to understand school mathematics as it emerged in one reform classroom from students' perspectives.

Results

In the following section, we have introduced the students, Joakim, Kristina and Nette, and their trajectories [START_REF] Wenger | Communities of practice[END_REF] in mathematical practices. Next, we have reported two reasons that we claim to be of crucial importance to students who make negative choices concerning the role of mathematics in their lives. These reasons are neither mutually exclusive nor exhaustive. They are more like what can be seen in a kaleidoscope; each turn gives a new view of school mathematics from the student's perspective.

Student trajectories

As newcomers in the seventh grade, Joakim, Kristina and Nette were highly engaged students. Nette's engagement clearly related to an imagined future as a veterinarian, while Joakim and Kristina were generally confident about the significance of learning mathematics. During lower secondary school, they were responsible students who were evaluated by the teacher as skilful/capable in different types of assessment. They wanted to take part in actions and activities in school mathematics communities, and they appreciated the problem-solving activities that were introduced within the reform. In short, we can describe them as successful students who expressed strong alignments in relation to reform objectives. Nevertheless, their trajectories and identifications related to school mathematics were constituted very differently over the following school years, which strongly affected their futures. Joakim's trajectory continued to be structured by positive choices concerning mathematics. He passed the advanced mathematics courses he had selected in upper secondary school2 with success and chose a mathematics-intensive university program to obtain a MSc in engineering. The trajectories of Kristina and Nette differed significantly from Joakim's. Both of them chose basic mathematics courses at secondary level and neither of them chose university programs that included mathematics. Nette's dream of becoming a veterinarian turned into a claim that mathematics was not for her, and she chose a university program in psychology. Kristina described school mathematics as a more or less alienating experience, but in contrast to Nette, she continued to be very accepting. However, she dropped her plan of studying science, because she did not want to align with the workings of the mathematics classrooms she had experienced in school. Instead, she chose a university program in pedagogy.

The dominant habits of the mathematics classroom

From a student perspective, the actions and activities in a mathematics classroom are complex. A characteristic that supports participation in the case of a student at one moment in time might not be supportive for all students and might not continue to be supportive for that student over time. However, our analysis of students' interview accounts showed recurrent patterns in mathematics classrooms that we described as dominant habits. As the space of the paper does not allow us to thoroughly account for all habits, we have chosen those that most likely led to the development of non-participation [START_REF] Wenger | Communities of practice[END_REF] of Kristina and Nette in the long run. Two dominant habits that appeared in all student interviews in the case study were as follows:

(1) The "all-look-alike" structure of lessons, which contributed to the sense that mathematics was a monotonous subject and led to a kind of meaninglessness for students who did not accept nonparticipation as an adventure (see the work of imagination, Wenger, 1998, pp. 185-186):

All lessons look alike, we have theory and write down the things we are doing. The teacher asks different things. Then, we may count at our own pace from the book. The last lesson we did x times x and such. (Nette, Grade 7) I don't think there have been any changes. It just continues from year seven to year eight and then, to year nine. In my opinion, nothing has changed. It has been fairly similar all the time. Of course, they follow and teach by the book. (Nette, Grade 9)

(2) The general mode of working individually during lessons, which meant that students were mostly left by themselves to struggle for coherence regarding mathematical structures. Hence, there were limited opportunities to develop a sense of belonging to any community where mathematical knowledge was negotiated in the classroom:

We do work alone all of us. It is like that anyhow, the teacher has to come to everybody, so it is the fastest way to ask the one you sit next to, so you talk to everybody. (Kristina, Grade 9) Joakim continued to take part in communities that included the teacher and students he described as "smart". Moreover, the support he received from his teacher reminded him repeatedly of his membership in communities of mathematically-able people.

The textbook continued to be the main pedagogical artefact during the reform, and teachers continued to structure the tempo of student learning with the textbook. The habit of working alone with textbook tasks may have added to some students' sense of enjoyment. But as Kristina stated, the positive effect emerged only "when you succeed and keep up with new things". In the long run, both Kristina and Nette expressed a sense of not keeping up with the tempo. Hence, the way the textbook was used in the classroom may have contributed to their developing identities of nonparticipation.

All students who took part in the case study welcomed reform-related activities, such as investigative project work, which "did show a little what mathematics can be", as Joakim stated as an adult. However, some students considered investigative project work to have a marginal role in their learning of mathematics even if their accounts clearly described the opposite: You learn things that don't have so much to do with mathematics. I did learn about unemployment rates in Finland. But this has nothing to do with mathematics. I think that I haven't learned anything that has to with mathematics. I mean, like what we are doing in the book for (math) theory. I don't learn in that way when I do investigative work. (Kristina, Grade 9)

In general, reform-related activities had the potential to increase participation for all students. Yet, as the extract above shows, students did not always see the connection between these activities and their development of mathematical knowledge. Therefore, the activities did not necessarily enhance students' participation.

The character of spaces for participation in classroom practices

We defined learning as increasing participation in classroom practices. Thus, it is natural to consider the character of mathematical spaces where meanings were negotiated in the classroom [START_REF] Wenger | Communities of practice[END_REF]. We have attended here to the most obvious characteristics that seem to have contributed to some students becoming persons for whom mathematics was not important.

The most important aspect was that "right or wrong" classroom practices limited the possibilities for constructing mathematical knowledge for all students. Nette succinctly captured this aspect and her perspective on one of its side effects, which was that questioning might be interpreted by the teacher as a lack of responsibility for learning and not as comments from students striving for understanding:

You cannot dispute lower secondary school mathematics, because it is right-or-wrong mathematics. If you start questioning it, the teacher thinks you are trying to escape. (Nette, adult)

Within "right-or-wrong" practices, there was little room for social, playful and investigative mathematics. Aligning to the practice meant individually struggling for coherence and understanding by listening to the teacher. The struggle for understanding was accompanied by feelings of insecurity, because the act of giving a wrong answer in public was shameful. Reaching the right answers rapidly was, from the students' perspective, a sign of understanding and often meant immediate rewards, such as praise from the teacher or not being constrained by homework. The case study showed that tentativeness, wrong answers, misinterpretations or extended interpretations of textbook tasks were not considered by students to be characteristics of a space where every student could belong and participate. For some students such as Nette and Kristina, this situation contributed to the development of negative identifications towards mathematics, which strongly affected their futures.

Discussion

We argue that a main concern for reforms in mathematics education should be a critical focus on the reality of the classroom, which from students' perspectives is even more complex than interviews can reveal (see [START_REF] Berinderjeet | Student voice in mathematics classrooms around the world[END_REF]. Also, dimensions that extend the classroom level had an impact on students' relationship to mathematics. According to [START_REF] Gregg | The tensions and contradictions of the school mathematics tradition[END_REF], reformers should encourage teachers to think in new ways by implementing new 'idioms' into their stories and explanations. This is, however, exactly what teachers who were involved in this case study did; they were supported by the scholarly community to integrate new idioms based on constructivism (see, for example, Black & Atkins, 1996) into their talk about classroom practices. Idioms such as active learning, discovery processes and responsibility for knowledge construction were strongly present when teachers talked about reforming their practice. Yet, they did not oppose the epistemological hegemony of academe by asking, "What if we lift our gaze from the student as a 'constructing individual' to zoom out and include the 'socio-cultural whole of the student and ourselves?'" They attended to the practice-in-individual but not to the individual-in-practice [START_REF] Lerman | A moment in the zoom of a lens: Towards a discursive psychology of mathematics teaching and learning[END_REF]. The latter perspective confronts us with the question "What is the nature of school mathematics practices in which all students should become qualified in order to develop positive identifications and to enhance the role of mathematics in their lives?" This question must be the starting point whenever we want new curriculum ideas to change classroom culture in a direction that makes participation an inclusive experience for as many students as possible. This is an issue to which hardly any research attends. Several factors interacting in a complex manner influence students' choices and relationships to mathematics, and, therefore, it is valuable to explore some cases deeply to understand them. In this study, [START_REF] Wenger | Communities of practice[END_REF] theoretical constructs proved fruitful for revealing hidden regularities behind the development of students' negative identifications concerning mathematics. According to Wenger, learning is so much more than gaining skills and knowledge, it is a process of becoming that transforms both who the learner is and what the learner can do (p. 215). Concerning the results presented here, this a very strong statement. When students, regardless of gender or school mathematical practices, learn disengagement (e.g., [START_REF] Sullivan | Perhaps the decision of some students not to engage in learning mathematics in school is deliberate[END_REF] or explicitly refuse mathematics in school (e.g., [START_REF] Chronaki | Refusing mathematics: A discourse theory approach on the politics of identity work[END_REF], their lives as adults, and especially their possibilities to partake in the working life, might be diminished.

  When students are supposed to answer in public, I get fairly nervous. I think what if I give the wrong answer. (Nette, Grade 7) You have to listen all the time to what the teacher is saying and you get so tired. As soon as you talk just a little bit with your friend then it goes, oh, now I do not understand anything again. It is rather strenuous. You should have a better ability to concentrate. (Kristina, Grade 9)

In the literature, teaching methods have been described in different ways, for instance, as open, progressive, traditional, constructivist and discussion oriented.

The upper secondary mathematics curriculum is divided into basic and advanced level courses. Only students who select advanced courses must take mathematics in the matriculation examination.


Ugo Boscain

Eugenio Pozzoli

Mario Sigalotti

Classical and quantum controllability of a rotating 3D symmetric molecule

Keywords: Quantum control, Schr ödinger equation, rotational dynamics, symmetric molecule, bilinear control systems AMS Classification: 68Q25, 68R10, 68U05

come

Introduction

The control of molecular dynamics takes an important role in quantum physics and chemistry because of the variety of its applications, starting from well-established ones such as rotational state-selective excitation of chiral molecules ( [START_REF] Doyle | Enantiomer-specific state transfer of chiral molecules[END_REF][START_REF] Domingo | Coherent enantiomer-selective population enrichment using tailored microwave fields[END_REF]), and going further to applications in quantum information ( [START_REF] Yu | A scalable quantum computing platform using symmetric-top molecules[END_REF]). For a general overview of controlled molecular dynamics one can see, for example, [START_REF] Koch | Quantum control of molecular rotation[END_REF]. 1 CNRS, Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France (e-mail: ugo.boscain@upmc.fr). 2 Inria, Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, Paris, France (e-mail: eugenio.pozzoli@inria.fr) 3 Inria, Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions, Paris, France (e-mail: mario.sigalotti@inria.fr) [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF] Rotations can, in general, couple to vibrations in the so-called ro-vibrational states. In our mathematical analysis, however, we shall restrict ourselves to the rotational states of the molecule. Due to its discrete quantization, molecular dynamics perfectly fits the mathematical quantum control theory which has been established until now. In fact, the control of the Schr ödinger equation has attracted substantial interest in the last 15 years. See, for instance, [START_REF] Boscain | Multi-input Schr ödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF][START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schr ödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Caponigro | Exact controllability in projections of the bilinear Schr ödinger equation[END_REF][START_REF] Boscain | Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems[END_REF][START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF][START_REF] Domenico | Introduction to quantum control and dynamics[END_REF][START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF][START_REF] Altafini | Modeling and control of quantum systems: an introduction[END_REF][START_REF] Keyl | Controlling several atoms in a cavity[END_REF] for the state of the art.

The problem of controlling the rotational dynamics of a planar molecule has been analyzed in [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schr ödinger equation with application to the control of a rotating planar molecule[END_REF] by means of two orthogonally polarized electric fields; in particular, approximate controllability has been proved using non-resonant connectedness chains in the spectrum of the rotational Hamiltonian. In [START_REF] Boscain | Multi-input Schr ödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF] the approximate controllability of a linear-top controlled by three orthogonally polarized electric fields has been established. There, a new sufficient condition for controllability, called the Lie-Galerkin tracking condition, has been introduced in an abstract framework, and applied to the linear-top system.

Here, we study the symmetric-top as a generalization of the linear one, characterizing its controllability in terms of the position of its dipole moment in Theorems 6, 7, 9. While for the linear-top two quantum numbers j, m are needed to describe the motion, the main and more evident difference here is the presence of a third quantum number k, which classically represents the projection of the total angular momentum on the symmetry axis of the molecule. This should not be a surprise, since the configuration space of a linear-top is the 2-sphere S 2 , while the symmetric-top evolves in the Lie group SO(3), a three-dimensional manifold. As a matter of fact, by fixing k = 0, one recovers the linear-top as a subsystem inside the symmetric-top. It is worth mentioning that the general theory developed in [START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF][START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schr ödinger equation with application to the control of a rotating planar molecule[END_REF][START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF] is based on non-resonance conditions on the spectrum of the internal Hamiltonian. A major difficulty in studying the controllability properties of the rotational dynamics is that, even in the case of the linear-top, the spectrum of the rotational Hamiltonian has severe degeneracies at the so-called m-levels. The symmetric-top is even more degenerate, due to the additional presence of the so-called k-levels.

The Schr ödinger equation for a rotating molecule controlled by three orthogonally polarized electric fields reads i ∂ ∂t ψ(R, t) = Hψ(R, t) + of the molecule, P 1 , P 2 , P 3 are the angular momentum differential operators, and B i (R, δ) = -Rδ, e i is the interaction Hamiltonian between the dipole moment δ of the molecule and the direction e i , i = 1, 2, 3. Finally, R ∈ SO(3) is the matrix which describes the configuration of the molecule in the space. We shall study the symmetric-top, that is, the case I 1 = I 2 . In this case, closed expression for the spectrum and the eigenfunctions of H are known. The case of the asymmetric-top, which goes beyond the scope of this paper, could be tackled either by a perturbative approach or with a further computational effort. Here, the position of the dipole moment plays a decisive role: when it is neither along the symmetry axis, nor orthogonal to it, as in the above Figure (b), then approximate controllability holds, under some non-resonance conditions, as it is stated in Theorem 7. To prove it, we introduce in Section 2.1 a block-wise Lie-Galerkin condition, closely related to the Lie-Galerkin tracking condition, which is shown to provide a general controllability test for the multi-input Schr ödinger equation (Theorem 5). We then apply this result to the symmetric-top system. The control strategy is based on the excitation of the system with external fields in resonance with three families of frequencies, corresponding to internal spectral gaps. One frequency is used to overcome the m-degeneracy in the spectrum, and this step is quite similar to the proof of the linear-top's approximate controllability (Appendix A).

The other two frequencies are used in a next step to break the k-degeneracy, in a three-wave mixing scheme (Appendix B) typically used in quantum chemistry to obtain enantio-and stateselectivity for chiral molecules ( [START_REF] Giesen | Principles of enantio-selective excitation in threewave mixing spectroscopy of chiral molecules[END_REF], [START_REF] Yachmenev | Detecting chirality in molecules by linearly polarized laser fields[END_REF], [START_REF] Averbukh | Selective orientation of chiral molecules by laser fields with twisted polarization[END_REF]). The two dipole configurations to which the Theorem 7 does not apply are extremely relevant from the physical point of view. Indeed, the dipole moment of a symmetric-top lies usually along its symmetry axis (Figure (a)), and if not, for accidentally symmetric-top, it is often found in the orthogonal plane (Figure (c)). Here two different symmetries arise, implying the non-controllability of these systems, as we prove, respectively, in Theorems 6 and 9. These two conserved quantities stimulated and motivated the study of the classical dynamics of the symmetric-top, presented in the first part of the paper: the first conserved quantity, appearing in Theorem 6, corresponds to a classical observable, that is, the component of the angular momentum along the symmetry axis, and it turns out to be a prime integral also for the classical controlled dynamics, as remarked in Theorem 2. The second conserved quantity, appearing in Theorem 9, is more challenging, because it does not have a counterpart in the classical dynamics, being mainly due to the superposition of k and -k states in the quantum dynamics. We show that this position of the dipole still correspond to a controllable system for the classicaltop, while it does not for the quantum-top. Thus, the latter is an example of a system whose quantum dynamics are not controllable even though the classical dynamics are. The possible discrepancy between quantum and classical controllability has been already noticed, for example, in the harmonic oscillator dynamics ( [START_REF] Mirrahimi | Controllability of quantum harmonic oscillators[END_REF]). It should be noticed that the classical controllability of a rigid body has been analyzed in several works like [1, Section 6.4], [17, Section 4.6] using internal controls; nevertheless here we adopt as control fields three external forces, a strategy which has never been considered, as far as we know, in the previous literature.

The paper is organized as follows: in Section 1 we study the controllability of the classical Hamilton equations for a symmetric-top. The main results are summarized in Theorems 2 and 3, where we prove, respectively, the non-controllability when the dipole lies along the symmetry axis of the body and the controllability in any other case. In Section 2 we study the controllability of the Schr ödinger equation for a symmetric-top. The main controllability result is Theorem 7, where we prove the approximate controllability when the dipole is neither along the symmetry axis, nor orthogonal to it. In the two cases left, we prove the non-controllability in Theorems 6 and 9.

Classical symmetric-top molecule 1.Controllability of control-affine systems with recurrent drift

We recall in this section some useful results on the controllability properties of (finite-dimensional) control-affine systems.

Let M be an n-dimensional manifold, X 0 , X 1 , . . . , X m a family of smooth (i.e., C ∞ ) vector fields on M , U ⊂ R m a set of control values which contains a neighbourhood of the origin. We consider the controlled system

q = X 0 (q) + m i=1 u i (t)X i (q), q ∈ M, (1)
where the control functions u are taken in L ∞ (R, U ). The vector field X 0 is called the drift.

• The reachable set from q 0 ∈ M is Reach(q 0 ) :={q ∈ M | ∃ u, T s.t. the solution to (1) with q(0) = q 0 satisfies q(T ) = q}.

• The system (1) is said to be controllable if Reach(q 0 ) = M for all q 0 ∈ M .

• The family of vector fields X 0 , X 1 , . . . , X m is said to be Lie bracket generating if

dim(Lie q {X 0 , X 1 , . . . , X m }) = n
for all q ∈ M , where Lie q {X 0 , X 1 , . . . , X m } denotes the evaluation of q of the Lie algebra generated by X 0 , X 1 , . . . , X m .

• Let X be a complete vector field on M and φ t its flow at time t, t ∈ R. Then X is said to be recurrent if for every open nonempty subset V of M and every time t > 0, there exists

t > t such that φ t(V ) ∩ V = ∅.
The following is a basic result in geometric control theory (see, for example, [17, Section 4.6]).

Theorem 1. If X 0 is recurrent and the family X 0 , X 1 , . . . , X m is Lie bracket generating, then system (1) is controllable.

Using the Orbit Theorem (see, e.g., [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Chapter 5]) and assuming the analyticity of the vector fields, one proves the following result, which is useful to verify the Lie bracket generating condition.

Lemma 1.

If the family of analytic vector fields X 0 , X 1 , . . . , X m is Lie bracket generating on the complement of a sub-manifold N ⊂ M and Reach(q) ⊂ N , for all q ∈ N , then the family is Lie bracket generating on M .

Hamilton equations on a Lie group

The dynamics on a Lie group G, relative to a Hamiltonian function H on T * G ∼ = G × g * , where g is the Lie algebra of G, can be described in terms of the Hamiltonian field H, defined as a vector field on T * G by dH(V ) = ω( H, V ), V ∈ T (T * G), where ω is the canonical symplectic 2-form on T * G. We briefly describe this classical approach to dynamical systems on Lie groups (see, for instance, [START_REF] Jurdjevic | Geometric control theory[END_REF]Chapter 12]). First we identify

T (T * G) ∼ = T (G × g * ) ∼ = T G × T g * ∼ = (G × g) × (g * × g * ),
that is, an element of T (T * G) can be seen as ((g, X), (p, Y * )) where (X, Y * ) is a tangent vector with base (g, p) and X(g, p) ∈ g, Y * (g, p) ∈ g * , for all (g, p) ∈ G × g * .

If V is a vector field on T * G, we write V = (X, Y * ) as before, and f is a function on T * G, V acts as a derivation on f by the formula

(V f )(g, p) = (L g ) * ∂f ∂g (g, p) X(g, p) + Y * (g, p) ∂f ∂p (g, p)
for all (g, p) ∈ G × g * , where

∂f ∂g ∈ T * g G, ∂f ∂p ∈ T * p g * ∼ = g * * ∼ = g. Notice that V 1 = (X 1 , Y * 1 ), V 2 = (X 2 , Y * 2 ) are tangent vectors at (q, p), then we have ω(V 1 , V 2 ) = Y * 2 (X 1 ) -Y * 1 (X 2 ) -((adX 1 ) * p)X 2 .
As a consequence the expression of the Hamiltonian field

H = (X, Y * ) in terms of its defining Hamiltonian function H is      X(g, p) = ∂H ∂p (g, p) Y * (g, p) = -(L g ) * ∂H ∂g (g, p) -(adX) * p.
Thus the integral curves t → (g(t), (

p(t)) ∈ G × g * of H satisfy      dg(t) dt = (L g(t) ) * ∂H ∂p (g(t), p (t)
These are the equations of motion on G. If H is left-invariant, then ∂H ∂g = 0. Thus one obtains the so-called equations in vertical coordinates.

The classical dynamics of a molecule subject to electric fields

Since the molecule is a rigid body that can just rotate, the configuration space is the Lie group SO(3). We now apply the machinery just developed for a general Lie group to G = SO(3).

Consider the basis

A 1 =   0 0 0 0 0 -1 0 1 0   , A 2 =   0 0 1 0 0 0 -1 0 0   , A 3 =   0 -1 0 1 0 0 0 0 0   ,
of g = so(3). Let A 1 , A 2 , A 3 be the corresponding left-invariant vector fields. Denoting by e 1 , e 2 , e 3 a fixed orthonormal frame on R 3 and by a 1 , a 2 , a 3 a fixed orthonormal frame on the rigid body with the same orientation, the configuration of the molecule is identified with the unique g ∈ SO(3) such that ga i = e i , for i = 1, 2, 3.

As external forces on the system we consider the three electric fields with intensities u 1 (t), u 2 (t), u 3 (t) and directions e 1 , e 2 , e 3 . Denoting by δ the dipole of the molecule written in the moving frame, the three potential energies due to the interaction with the electric fields are

V i = -u i (t) gδ, e i , i = 1, 2, 3.
Set V := 3 i=1 V i the total potential energy. Finally let H i , for i = 1, 2, 3, be the (left-invariant) Hamiltonian function relative to the left-invariant vector field A i (that is, H i (p) := p( A i ), ∀p ∈ so(3)

* ) and we introduce the Hamiltonian of the dynamical system

H := 1 2 H 2 1 I 1 + H 2 2 I 2 + H 2 3 I 3 + V, (3)
defined on SO(3) × so(3) * , where I 1 , I 2 , I 3 are the moments of inertia of the rigid body in a moving frame made by principal axes of inertia.

Remark 2. H is not left-invariant since V = V (g).

Applying (2), the integral curves (g(t), p(t))

∈ SO(3) × so(3) * of H satisfy      dg(t) dt = 3 i=1 1 I i H i (p(t))A i (g(t)), dp(t) dt = -(L g(t) ) * (dV ) -(adΩ(p(t))) * p(t), (4)
where Ω(p) :=

3 i=1 1 I i H i (p) A i .
Now using the isomorphism so(3) * → so(3), p → P , defined by K( P , A) = p( A), where K(•, •) is the scalar product on so(3) given by K( A, B) = -1 2 tr( A B), ∀ A, B ∈ so(3), and the classical identification of Lie algebras

(so(3), [•, •]) → (R 3 , ×), A =   0 -a 3 a 2 a 3 0 -a 1 -a 2 a 1 0   → A =   a 1 a 2 a 3   ,
where × is the vector product, the Hamilton equations (4) read on SO(3

) × R 3      dg(t) dt = g(t) Ω(t), dP (t) dt = P (t) × Ω(t) + 3 i=1 u i (t)(g -1 (t)e i ) × δ. (5)
These are the Euler equations for a charged rigid body subject to three orthogonal electric fields. We shall also use the notation S(A) for the anti-symmetric matrix A. System (5) can be seen as a control-affine system with three controlled fields: by computing

Ω(t) = Ω(p(t)) = 3 i=1 1 I i H i (p(t))A i = 3 i=1 1 I i p(t)( A i )A i = 3 i=1 1 I i P i (t)A i ,
we obtain Ω(p(t)) = βP (t), where βP = (P 1 /I 2 , P 2 /I 2 , P 3 /I 3 ) t . Then (5) reads ġ Ṗ = X(g, P ) +

3 i=1 u i (t)Y i (g, P ), (g, P ) ∈ SO(3) × R 3 , (6)
where X(g, P ) := gS(βP

) P × (βP ) , Y i (g, P ) := 0 (g -1 e i ) × δ , i = 1, 2, 3. (7)
Rotating molecule dynamics can also be represented in terms of quaternions. The 3-sphere S 3 (seen as a double covering space of SO(3)) inherits a Lie group structure from the quaternions H and its elements act as rotations on R 3 . Let us now reformulate Euler equations [START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF] in this formalism.

We identify S 3 = {q 0 + iq 1 + jq 2 + kq 3 | (q 0 , q 1 , q 2 , q 3 ) ∈ R 4 , q 2 0 + q 2 1 + q 2 2 + q 2 3 = 1} ⊂ H and T 1 S 3 = R 3 = {iP 1 + jP 2 + kP 3 | (0, P 1 , P 2 , P 3 ) ∈ R 4 } ⊂ H. Thus, if q ∈ S 3 and P ∈ R 3 , q rotates P by the formula qP q as a quaternions multiplication. Via this identification, the vector product

P × Ω becames 1 2 [P, Ω] := 1 2 (P Ω -ΩP )
. Thus, we get

           dq(t) dt = q(t)Ω(t), dP (t) dt = 1 2 [P (t), Ω(t)] + u 1 (t) 2 [q(t)iq(t), δ] + u 2 (t) 2 [q(t)jq(t), δ] + u 3 (t) 2 [q(t)kq(t), δ]. (8)
This description will be useful for computations.

Non-controllability of the classical genuine symmetric-top

In most cases of physical interest, the electric dipole δ of a symmetric-top molecule lies along the symmetry axis of the molecule. If I 1 = I 2 , the symmetry axis is the third one, and we have Proof. In order to compute the equation satisfied by P 3 in (6), notice that

that δ =   0 0 δ 3   , δ 3 = 0,
P (t) × βP (t) =   P 1 (t) P 2 (t) P 3 (t)   ×   P 1 (t)/I 2 P 2 (t)/I 2 P 3 (t)/I 3   =      1 I 3 - 1 I 2 P 2 (t)P 3 (t) 1 I 2 - 1 I 3 P 1 (t)P 3 (t) 0      . Moreover, u i (t)(g -1 (t)e i )×δ = u i (t)(g -1 (t)e i )×   0 0 δ 3   =   0   .
Hence, for a genuine symmetrictop, the equation for

P 3 becomes dP 3 (t) dt = 0.
As a consequence, the controlled dynamics live in the hypersurfaces {P 3 = const} and hence system (6) is not controllable in the 6-dimensional manifold SO(3) × R 3 .

Controllability of the classical accidentally symmetric-top

In Theorem 2 we proved that P 3 is a prime integral for equations (5), using both the symmetry of the mass and the symmetry of the charge, meaning that I 1 = I 2 and δ = (0, 0, δ 3 ) t . We consider now a symmetric-top molecule with electric dipole δ not along the symmetry axis of the body,

that is, δ =   δ 1 δ 2 δ 3   , with δ 1 = 0 or δ 2 =
0. This system is usually called accidentally symmetric-top.

Theorem 3. For an accidentally symmetric-top molecule system (6) is controllable.

Proof. The drift X is recurrent, as observed in [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]Section 8.4]. Thus, by Theorem 1, to prove controllability it suffices to show that, for any (g,

P ) ∈ SO(3)×R 3 , dim Lie (g,P ) {X, Y 1 , Y 2 , Y 3 } = 6.
Actually, we will find six vector fields on SO(3) × R 3 such that their span is six-dimensional everywhere but on a hypersurface, and we will conclude by applying Lemma 1. Notice that, since

[DY i ]Y j = 0 0 0 0 (g -1 e j ) × δ = 0 0 , the control vector fields commute: [Y i , Y j ] = 0, for i, j = 1, 2, 3. Moreover, [X, Y i ](g, P ) = -gS(β[(g -1 )e i × δ]) .
Denote by Π SO(3) the projection onto the SO(3) part of the tangent bundle, that is,

Π SO(3) : T (g,P ) (SO(3) × R 3 ) = T g SO(3) × T P R 3 → T g SO(3).
Then we have

span{Π SO(3) X(g, P ), Π SO(3) [X, Y 1 ](g, P ), Π SO(3) [X, Y 2 ](g, P ), Π SO(3) [X, Y 3 ](g, P )} = gS β[{δ} ⊥ ⊕ span{P }] .
Hence, if P, δ = 0, we have

dim span{Π SO(3) X(g, P ), Π SO(3) [X, Y 1 ](g, P ), Π SO(3) [X, Y 2 ](g, P ), Π SO(3) [X, Y 3 ](g, P )} = 3. (9)
To go further in the analysis, it is convenient to use the quaternionic parametrization (8) in which every field is polynomial. We have, in coordinates q = (q 0 , q 1 , q 2 , q 3 ) ∈ S 3 , P =

(P 1 , P 2 , P 3 ) ∈ R 3 , X(q, P ) = qβP 1 2 [P, βP ] =                    -q 1 P 1 I 2 -q 2 P 2 I 2 -q 3 P 3 I 3 q 0 P 1 I 2 + q 2 P 3 I 3 -q 3 P 2 I 2 q 0 P 2 I 2 -q 1 P 3 I 3 + q 3 P 1 I 2 q 0 P 3 I 3 + q 1 P 2 I 2 -q 2 P 1 I 2 1 I 3 - 1 I 2 P 2 P 3 1 I 2 - 1 I 3 P 1 P 3 0                    , Y 1 (q, P ) = 0 1 2 [qiq, δ] =             0 0 0 0 (q 1 q 2 -q 0 q 3 )δ 3 -(q 1 q 3 + q 0 q 2 )δ (q 1 q 3 + q 0 q 2 )δ 1 - 1 2 (q 2 0 + q 2 1 -q 2 2 -q )δ 3 1 2 (q 2 0 + q 2 1 -q 2 2 -q 2 3 )δ 2 -(q 1 q 2 -q 0 q )δ 1             , Y 2 (q, P ) = 0 1 2 [qjq, δ] =             0 0 0 0 1 2 (q 2 0 -q 2 1 + q 2 2 -q 2
3 )δ 3 -(q 2 q 3 -q 0 q )δ 2 (q 2 q 3 -q 0 q 1 )δ 1 -(q 1 q 2 + q 0 q 3 )δ

(q 1 q 2 + q 0 q 3 )δ 2 - 1 2 (q 2 0 -q 2 1 + q 2 2 -q )δ 1             . Let us consider the six vector fields X, Y 1 , Y 2 , [X, Y 1 ], [X, Y 2 ], [[X, Y 1 ], Y 1 ]:
we have that a 6 × 6 minor of the matrix

(X(q, P ), Y 1 (q, P ), Y 2 (q, P ), [X, Y 1 ](q, P ), [X, Y 2 ](q, P ), [[X, Y 1 ], Y 1 ](q, P ))
has determinant equal to S(q, P ) := S 1 (q)S 2 (q)S 3 (q)S 4 (q)S 5 (P ), where

S 1 (q) := I 2 -I 3 I 2 2 I 2 3 q 1 , S 2 (q) := (-2q 1 q 2 δ 1 + 2q 0 q 3 δ 1 + q 2 0 δ 2 + q 2 1 δ 2 -(q 2 2 + q 2 3 )δ 2 ), S 3 (q) := (q 0 (-2q 2 δ 1 + 2q 1 δ 2 ) + 2q 3 (q 1 δ 1 + q 2 δ 2 ) + (q 2 0 -q 2 1 -q 2 2 + q 2 3 )δ 3 ) 2 , S 4 (q) := (-2(q 0 q 2 + q 1 q 3 )(δ 2 1 + δ 2 2 ) + ((q 2 0 + q 2 1 -q 2 2 -q 2
3 )δ 1 + 2(q 1 q 2 -q 0 q 3 )δ 2 )δ 3 ), S 5 (P ) := P 1 δ 1 + P 2 δ 2 + P 3 δ 3 = P, δ .

Hence, for all (q, P ) such that S(q, P ) = 0, dim span{X(q, P ),

Y 1 (q, P ), Y 2 (q, P ), [X, Y 1 ](q, P ), [X, Y 2 ](q, P ), [[X, Y 1 ], Y 1 ](q, P )} = 6, that is, outside the hypersurface V (S) := {(q, P ) ∈ S 3 × R 3 | S(q, P ) = 0} the family X, Y 1 , Y 2 is Lie bracket generating.
Now we are left to prove that

Reach(q, P ) ⊂ V (S), ∀(q, P ) ∈ V (S),
and then to apply Lemma 1. Let us start by considering the factor S 5 of S and notice that, for any fixed q ∈ S 3 , V (S 5 ) defines a surface inside R 3 . Denote by Π R 3 the projection onto the R 3 part of the tangent bundle, that is,

Π R 3 : T (q,P ) (S 3 × R 3 ) = T q S 3 × T P R 3 → T P R 3 = R 3 .
The vector field Π R 3 X is tangent to V (S 5 ) when

∇ P S 5 , Π R 3 X = δ, [P, βP ] = 0,
that is, if and only if

P 3 = 0 or P 2 δ 1 -P 1 δ 2 = 0. Notice that one vector between Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3 is not tangent to V (P 3 ), otherwise span{Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3 } = {P 3 = 0} =   0 0 1   ⊥ . However, span{Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3 } = δ ⊥ ,
which would imply that δ is collinear to (0, 0, 1) t , which is impossible since the molecule is accidentally symmetric. Concerning the hypersurface V (P 2 δ 1 -P 1 δ 2 ), we consider again Π R 3 X, which is tangent when ∇ P (P 2 δ 1 -P 1 δ 2 ), Π R 3 X = 0, that is, if and only if P 3 = 0 or P 1 δ 1 + P 2 δ 2 = 0. We treat the second case, being P 3 = 0 already treated. Hence, we consider the intersection

P 2 δ 1 -P 1 δ 2 = 0, P 1 δ 1 + P 2 δ 2 = 0.
The only solution of the system is P 1 = P 2 = 0, because the molecule is accidentally symmetric. Finally, when

P 1 = P 2 = 0, we consider the two-dimensional distribution span{Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3 },
which cannot be tangent to the P 3 axis.

Summarizing, if δ is not collinear to (0, 0, 1) t , we have Reach(q, P ) ⊂ V (S 5 ), ∀(q, P ) ∈ V (S 5 ).

To conclude, if (q, P ) ∈ V (S i ), i = 1, . . . , 4, then we fix P and we get a two-dimensional manifold {q ∈ S 3 | S i (q) = 0} ⊂ S 3 . Now the projections of the vector fields X,

[X, Y 1 ], [X, Y 2 ], [X, Y 3 ]
on the base part of the bundle span a three-dimensional vector space if P, δ = 0, as observed in [START_REF] Boscain | Approximate controllability, exact controllability, and conical eigenvalue intersections for quantum mechanical systems[END_REF]. So, by possibly steering P until P, δ = 0, it is possible to exit from V (S i ). This concludes the proof of the theorem.

Reachable sets of the classical genuine symmetric-top

Theorem 2 states that the hypersurfaces {P 3 = const} are invariant for the controlled motion. Next we prove that the restriction of system (5) to any such hypersurface is controllable. Theorem 4. Let I 1 = I 2 and δ = (0, 0, δ 3 ) t , δ 3 = 0. Then for (g, P ) ∈ SO(3) × R 3 , P = (P 1 , P 2 , P 3 ), one has

Reach(g, P ) = {(g, P ) ∈ SO(3) × R 3 | P 3 = P 3 }.
Proof. From Theorem 2 we know that {P 3 = const} is invariant. Since the drift X is recurrent, it suffices to prove that system ( 5) is Lie bracket generating on the 5-dimensional manifold {P 3 = const}. We recall from (9) that, if P, δ = 0, that is, if P 3 = 0, we have

dim span{Π SO(3) X(g, P ), Π SO(3) [X, Y 1 ](g, P ), Π SO(3) [X, Y 2 ](g, P ), Π SO(3) [X, Y 3 ](g, P )} = 3.
Moreover, taking the projections of the control fields on the R 3 part of the bundle Π R 3 Y i (q, P ) = (g -1 e i ) × δ we see that

dim span{Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3 } = 2 (10)
everywhere. Thus, if P 3 = 0, we have that

dim span{X(g, P ), Y 1 (g, P ), Y 2 (g, P ), Y 3 (g, P ), [X, Y 1 ](g, P ), [X, Y 2 ](g, P ), [X, Y 3 ](g, P )} = 5.
So the system is Lie bracket generating on the manifold {P 3 = const = 0}.

We are left to consider the case

P 3 = 0. Notice that Π R 3 Y 1 , Π R 3 Y 2 , Π R 3 Y 3
give a two-dimensional distribution for any value of P 3 . So we consider in the quaternionic parametrization the projections

of X, [X, Y 1 ], [X, Y 2 ], [[X, Y 1 ]
, X] on the S 3 part of the bundle and we obtain

dim span{Π S 3 X(q, P ), Π S 3 [X, Y 1 ](q, P ), Π S 3 [X, Y 2 ](q, P ), Π S 3 [[X, Y 1 ], X](q, P )} = 3,
for P 3 = 0, except when q 3 [2P 2 (q 1 q 2 -q 0 q 3 ) + P 1 (q 2 0 + q 2 1 -q 2 2 -q 2

3 )] = 0. This equation defines the union of two surfaces inside S 3 . (Notice that we can assume P 1 = 0 and P 2 = 0 because [START_REF] Caponigro | Exact controllability in projections of the bilinear Schr ödinger equation[END_REF] gives local controllability in (P 1 , P 2 )). On {q 3 = 0}, we have that Π S 3 X is tangent if and only if q 1 P 2 -q 2 P 1 = 0. On the curve γ ⊂ S 3 of equation

q 3 = 0, q 1 P 2 -q 2 P 1 = 0, we can consider the two-dimensional distribution spanned by Π S 3 [X, Y 1 ], Π S 3 [X, Y 2 ], Π S 3 [X, Y 3 ],
which is clearly not tangent to γ. Following Lemma 1, the system is Lie bracket generating also on {q 3 = 0}.

Analogously, on {2P 2 (q 1 q 2 -q 0 q 3 ) + P 1 (q 2 0 + q 2 1 -q 2 2 -q 2 3 ) = 0} we consider the vector field

Π S 3 [[[X, Y 1 ], X], Y 2 ]
which is tangent if and only if (q 0 q 2 + q 1 q 3 )(P 1 q 0 q 1 + P 2 q 0 q 2 -P 2 q 1 q 3 + P 1 q 2 q 3 ) = 0. Again, since the distribution spanned by

Π S 3 [X, Y 1 ], Π S 3 [X, Y 2 ], Π S 3 [X, Y 3 ] is two- dimensional,
we can exit from the set of equation 2P 2 (q 1 q 2 -q 0 q 3 ) + P 1 (q 2 0 + q 2 1 -q 2 2 -q 2 3 ) = 0, (q 0 q 2 + q 1 q 3 )(P 1 q 0 q 1 + P 2 q 0 q 2 -P 2 q 1 q 3 + P 1 q 2 q 3 ) = 0, whose strata have dimension at most one. Thus, applying again Lemma 1, we can conclude that the restriction of the system on the manifold {P 3 = 0} is Lie bracket generating.

Quantum symmetric-top molecule 2.1 Controllability of the multi-input Schr ödinger equation

Let ∈ N, a > 0 and U := [-a, a] . Let H be an infinite-dimensional Hilbert space with scalar product •, • (linear in the first entry and conjugate linear in the second), H, B 1 , . . . , B be (possibly unbounded) self-adjoint operators on H, with domains D(H), D(B 1 ), . . . , D(B ). We consider the controlled Schr ödinger equation

i dψ(t) dt = (H + j=1 u j (t)B j )ψ(t), ψ(t) ∈ H, u(t) ∈ U. ( 11
)
Definition 1.

• We say that the operator H satisfies (A1) if it has discrete spectrum with infinitely many distinct eigenvalues (possibly degenerate). Denote by B a Hilbert basis (φ k ) k∈N of H made of eigenvectors of H associated with the family of eigenvalues (λ k ) k∈N and let L be the set of finite linear combination of eigenstates, that is,

L = span{φ k | k ∈ N}. • We say that (H, B 1 , . . . , B , B) satisfies (A2) if φ k ∈ D(B j ) for every k ∈ N, j = 1, . . . , . • We say that (H, B 1 , . . . , B , B) satisfies (A3) if H + j=1 u j B j : L → H is essentially self-adjoint for every u ∈ U . • We say that (H, B 1 , . . . , B , B) satisfies (A) if H satisfies (A1) and
(H, B 1 , . . . , B , B) satisfies (A2) and (A3).

If (H, B 1 , . . . , B , B) satisfies (A) then, for every (u 1 , . . . , u ) ∈ U , H + j=1 u j B j generates
a subgroup e -it(H+ j=1 uj Bj ) of the group of unitary operators U (H). It is therefore possible to define the propagator Γ u T at time T of system (11) associated with a piecewise constant control law u(•) = (u 1 (•), . . . , u (•)) by composition of flows of the type e -it(H+ j=1 uj Bj ) . If, moreover, B 1 , . . . , B are bounded operators then the definition can be extended by continuity to every L ∞ control law (see [START_REF] Ball | Controllability for distributed bilinear systems[END_REF]). Definition 2. Let (H, B 1 , . . . , B , B) satisfy (A).

• Given ψ 0 , ψ 1 in the unit sphere S of H, we say that ψ 1 is reachable from ψ 0 if there exist a time T > 0 and a piecewise constant control law u : [0, T ] → U such that ψ 1 = Γ u T (ψ 0 ). We denote by Reach(ψ 0 ) the set of reachable points from ψ 0 .

• We say that [START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF] is approximately controllable if for every ψ 0 ∈ S the set Reach(ψ 0 ) is dense in S.

We introduce the notion of module-tracker (m-tracker, for brevity) that is, a system for which any given curve can be tracked up to (relative) phases. The identification up to phases of elements of H in the basis B = (φ k ) k∈N can be accomplished by the projection

M : ψ → k∈N | φ k , ψ |φ k .
Definition 3. Let (H, B 1 , . . . , B , B) satisfy (A). We say that system [START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF] is an m-tracker if, for every r ∈ N, ψ 1 , . . . , ψ r in H, Γ : [0, T ] → U (H) continuous with Γ 0 = Id H , and > 0, there exists an invertible increasing continuous function τ : [0, T ] → [0, T τ ] and a piecewise constant control

u : [0, T τ ] → U such that M( Γ t ψ k ) -M(Γ u τ (t) ψ k ) < , k = 1, . . . , r, for every t ∈ [0, T τ ].
Remark 3. We recall that if system (11) is an m-tracker, then it is also approximately controllable, as noticed in [8, Remark 2.9].

We now introduce some objects in order to state a new sufficient condition for a system to be m-tracker, based on the controllability results in [START_REF] Boscain | Multi-input Schr ödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF].

Let {I j } j∈N be a family of subsets of N such that

∪ j∈N I j = N and n j := |I j | < ∞, ∀j ∈ N,
where |I j | denotes the cardinality of I j . Consider the subspaces

M j := span{φ n | n ∈ I j } ⊂ H.
Let us denote by Σ j := {|λ l -λ l | | l, l ∈ I j } the spectral gaps in M j , and define the orthogonal projections

Π Mj : H ψ → n∈Ij φ n , ψ φ n ∈ H.
Given a linear operator Q on H we identify the linear operator Π Mj QΠ Mj preserving M j with its complex matrix representations with respect to the basis (φ n ) n∈Ij . We define

B (j) i := Π Mj B i Π Mj for every i = 1, . . . , .
We define

Ξ 0 j = {(σ, i) ∈ Σ j × {1, . . . , } | (B i ) l,k = 0 for every l ∈ N, k ∈ N \ I j such that |λ l -λ k | = σ},
and

Ξ 1 j = {(σ, i) ∈ Σ j × {1, . . . , } | (B i ) l,k = 0 for every l ∈ I j , k ∈ N \ I j such that |λ l -λ k | = σ}.
While the set Ξ 0 j is made by totally non-resonant gaps, in Ξ 1 j the resonances corresponding to pairs of eigenstates outside I j are allowed.

For every σ ≥ 0, and every square matrix M of dimension m, let

E σ (M ) = (M l,k δ σ,|λ l -λ k | ) l,k=1,...,m ,
where δ l,k is the Kronecker delta. The square matrix E σ (B

i ) of dimension n j , i = 1, . . . , , corresponds to the activation in B (j) i of the spectral gap σ ∈ Σ j : every element is 0 except the (l, k)-elements such that |λ l -λ k | = σ. Moreover, for every ξ ∈ S 1 ⊂ C, we consider the matrix operator W ξ such that

(W ξ (M )) l,k =      ξM l,k , λ l < λ k , 0, λ l = λ k , ξM l,k , λ l > λ k . (12)
Next, we consider the sets of excited modes

ν s j := {W ξ (E σ (iB (j) i )) | (σ, i) ∈ Ξ s j , σ = 0, ξ ∈ S 1 }, s = 0, 1. (13)
Notice that ν 0 j ⊂ ν 1 j ⊂ su(n j ). We denote by Lie(ν s j ) the Lie algebra generated by the matrices in ν s j , s = 0, 1, with respect to the bracket of su(n j ), and define T j as the minimal ideal of Lie(ν 1 j ) containing ν 0 j . Finally, we introduce the graph G with vertices V = {I j } j∈N and edges E = {(I j , I k ) | j, k ∈ N, I j ∩ I k = ∅}. We are now in a position to state a sufficient condition for a system to be an m-tracker, and thus, approximately controllable. Theorem 5. Assume that (A) holds true. If the graph G is connected and T j = su(n j ), for every j ∈ N, then [START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF] is an m-tracker.

Proof. We shall prove that the assumptions of the theorem imply that system [START_REF] Chambrion | Controllability of the discrete-spectrum Schr ödinger equation driven by an external field[END_REF] satisfies the Lie-Galerkin tracking condition ([8, Definition 2.7]) and hence that ( 11) is an m-tracker ([8, Theorem 2.8]).

Up to reordering the sets I j , we can assume that

I j+1 ∩ (∪ j k=1 I k ) = ∅, ∀j ∈ N. (14)
Let us fix n 0 ∈ N and consider m ∈ N such that {φ 1 , . . . , φ n0 } ⊂ m j=1 M j . Then, according to [8, Definition 2.7], we shall prove that

Lie(∪ m j=1 T j ) = su(dim( m j=1 M j )), (15)
where the set of operators T j is obtained similarly to T j , replacing the set of operators ν 0 j , ν 1 j by

{W ξ (E σ (iΠ m j=1 Mj B i Π m j=1 Mj )) | (σ, i) ∈ Ξ s j , σ = 0, ξ ∈ S 1 }, s = 0, 1.
We proceed by induction on m. For m = 1, (15) is true, since we have that Lie(T 1 ) = T 1 = su(n 1 ) = su(dim(M 1 )). Assume now that ( 15) is true for m, and consider the vertex I m+1 . Consider t, p ∈ ∪ m+1 j=1 I j and let us prove that E t,p := e t,p -e p,t is in Lie(∪ m+1 j=1 T j ), where e a,b is the matrix with entries all 0 except for the one in row a and column b, which is equal to 1 (and the indices in ∪ m+1 j=1 I j are identified with the elements of {1, . . . , dim(

m+1 j=1 M j )}).
Notice that, by definition of Ξ 0 j , Ξ 1 j , a matrix in T m+1 has the form

0 0 0 Q , Q ∈ su(n m+1 ),
where the upper-left square matrix has dimension dim( m+1 j=1 M j ) -n m+1 . Similarly, a matrix in ∪ m j=1 T j has the form

Q 0 0 0 , Q ∈ su(dim( m j=1 M j )),
where the lower-right square matrix has dimension dim(

m+1 j=1 M j ) -dim( m j=1 M j ). If t, p ∈ ∪ m
j=1 I j or t, p ∈ I m+1 the conclusion follows from the induction hypothesis and the identity T m+1 = su(n m+1 ). Let then t ∈ I m+1 \ (∪ m j=1 I j ) and p ∈ ∪ m j=1 I j . Fix, moreover, r ∈ I m+1 ∩ (∪ m j=1 I j ), whose existence is guaranteed by [START_REF] Domingo | Coherent enantiomer-selective population enrichment using tailored microwave fields[END_REF]. Again by the induction hypothesis and the identity T m+1 = su(n m+1 ), we have that E p,r and E r,t are in Lie(∪ m+1 j=1 T j ). The bracket [E p,r , E r,t ] = E p,t is therefore also in Lie(∪ m+1 j=1 T j ). By similar arguments, we deduce that every basis element of su(dim(

m+1 j=1 M j )) is in Lie(∪ m+1 j=1 T j ).

The Schr ödinger equation of a symmetric-top subject to electric fields

We recall in this section some general facts about Wigner functions and the theory of angular momentum in quantum mechanics (see, for instance, [START_REF] Biedenharn | of Encyclopedia of Mathematics and its Applications[END_REF][START_REF] Varshalovich | Quantum theory of angular momentum[END_REF][START_REF] Cook | Microwave molecular spectra[END_REF]).

We use Euler's angles (α, β, γ) ∈ [0, 2π) × [0, π] × [0, 2π) to describe the configuration space SO(3) of the molecule. More precisely, the coordinates of a vector change from the body fixed frame a 1 , a 2 , a 3 to the space fixed frame e 1 , e 2 , e 3 via three rotations

  X Y Z   = R e3 (α)R e2 (β)R e3 (γ)   x y z   =: R(α, β, γ)   x y z   (16)
where (x, y, z) t are the coordinates of the vector in the body fixed frame, (X, Y, Z) t are the coordinates of the vector in the space fixed frame and R ei

In Euler coordinates, the angular momentum operators are given by

             J 1 = i cos α cot β ∂ ∂α + i sin α ∂ ∂β -i cos α sin β ∂ ∂γ , J 2 = i sin α cot β ∂ ∂α -i cos α ∂ ∂β -i sin α sin β ∂ ∂γ , J 3 = -i ∂ ∂α . (18)
These are linear operators acting on the Hilbert space L2 (SO(3)), self-adjoint with respect to the Haar measure 1 8 dαdγ sin βdβ. Using ( 18), the self-adjoint operator P 3 := -i ∂ ∂γ can be written as P 3 = sin β cos αJ 1 + sin β sin αJ 2 + cos βJ 3 , that is,

P 3 = 3 i=1 R i3 (α, β, γ)J i ,
where R = (R ij ) 3 i,j=1 is given in [START_REF] Jurdjevic | Geometric control theory[END_REF]. In the same way we define

P 1 = 3 i=1 R i1 (α, β, γ)J i , P 2 = 3 i=1 R i2 (α, β, γ)J i .
The operators J i and P i , i = 1, 2, 3, are the angular momentum operators expressed in the fixed and in the body frame, respectively. Finally, we consider the square norm operator J 2 := J 2 1 + J 2 2 + J 2 3 = P 2 1 + P 2 2 + P 2 3 . Now, J 2 , J 3 , P 3 can be considered as the three commuting observables needed to describe the quantum motion of a molecule. Indeed, [J 2 , J 3 ] = [J 2 , P 3 ] = [J 3 , P 3 ] = 0, and hence there exists an orthonormal Hilbert basis of L 2 (SO(3)) which diagonalizes simultaneously J 2 , J 3 and P 3 . In terms of Euler coordinates, this basis is made by the so-called Wigner functions ([23, Chapter 4], [6, Section 3.8,3.9])

D j k,m (α, β, γ) := e i(mα+kγ) d j k,m (β), j ∈ N, k, m = -j, . . . , j, (19)
where

d j k,m (β) := n j,k,m sin β 2 |k-m| cos β 2 |k+m| F sin β 2 2 .
The function F sin β 2

Summarizing, the family of Wigner functions {D j k,m | j ∈ N, k, m = -j, . . . , j} forms an orthonormal Hilbert basis for L 2 (SO(3)). Moreover,

     J 2 D j k,m = j(j + 1)D j k,m , J 3 D j k,m = mD j k,m , P 3 D j k,m = kD j k,m .
Thus, m and k are the quantum numbers which correspond respectively to the projections of the angular momentum on the third axis of the fixed and the body frame. The rotational Hamiltonian of a molecule is H = 1 2

P 2 1 I 1 + P 2 2 I 2 + P 2 3 I 3
, which is seen here as a self-adjoint operator acting on the Hilbert space L 2 (SO( 3)). From now on, we impose the symmetry condition for the molecule, that is,

I 1 = I 2 , which implies that H = J 2 2I 2 + 1 2I 3 - 1 2I 2 P 2 3 . Thus, HD j k,m = j(j + 1) 2I 2 + 1 2I 3 - 1 2I 2 k 2 D j k,m =: E j k D j k,m . (20)
Hence, the Wigner functions are the eigenfunctions of H, i.e., the rotational states of the symmetric top and its eigenvalues E j k , j ∈ N, k = -j, . . . , j, are the rotational energies of the molecule. Since the eigenvalues of H do not depend on m, the energy level E j k is (2j + 1)degenerate with respect to m. This property is common to every molecule in nature: the spectrum σ(H) does not depend on m, just like in classical mechanics kinetic energy does not depend on the direction of the angular momentum. Moreover, the energy level E j k is also 2degenerate with respect to k, when k = 0. This extra degeneracy is actually a characterizing property of every symmetric molecule, which is not verified by asymmetric ones. Breaking this k-symmetry will be one important feature of our controllability analysis.

The interaction Hamiltonian between the dipole δ inside the molecule and the external electric field in the direction e i , i = 1, 2, 3, is given by the Stark effect ([12, Chapter 10])

B i (α, β, γ) = -R(α, β, γ)δ, e i ,
seen as a multiplicative self-adjoint operator acting on L 2 (SO(3)). Thus, the rotational Schr ödinger equation for a symmetric-top molecule subject to three orthogonally polarized electric fields reads

i ∂ ∂t ψ(α, β, γ; t) = Hψ(α, β, γ; t) + 3 l=1 u l (t)B l (α, β, γ)ψ(α, β, γ; t), ψ(t) ∈ L 2 (SO(3)), (21)
u(t) ∈ [-a, a] 3 , a > 0.

Non-controllability of the quantum genuine symmetric-top

We Proof. We have to show that H and B l , for l = 1, 2, 3, do not couple different levels of k, that is,

D j k,m , iHD j k ,m L 2 (SO(3)) = 0, k = k , D j k,m , iB l D j k ,m L 2 (SO(3)) = 0, k = k , l = 1, 2, 3. (22)
The first equation of ( 22) is obvious since the orthonormal basis {D j k,m } diagonalizes H. Under the genuine symmetric-top assumption, the second equation of ( 22) is also true: for l = 1 and k = k we compute

D j k,m , iB 1 D j k ,m L 2 (SO(3)) = - 2π 0 dα 2π 0 dγ π 0 dβ sin(β)D j k,m (α, β, γ)iB 1 (α, β, γ)D j k ,m (α, β, γ) = -iδ 3 ( 2π
B 1 (α, β, γ) = -   0 0 δ 3   , R -1 (α, β, γ)   1 0 0   = -δ 3 sin β cos α.
The computations for l = 2, 3 are analogous, since the multiplicative potentials B l do not depend on γ.

Remark 5. Equation ( 22) also shows that, for a genuine symmetric-top, the third component of the angular momentum P 3 commutes with B l , i.e.,

[P 3 , B l ] = 0, l = 1, 2, 3.
Thus, by Noether Theorem ([22, Theorem 8.1]), we have that ψ(t), P 3 ψ(t) is a conserved quantity, where ψ is the solution of (21).

Controllability of the quantum accidentally symmetric-top

So far we have studied the dynamics of a symmetric-top molecule with electric dipole moment along its symmetry axis and we have proven that its dynamics are trapped in the eigenspaces of P 3 . Nevertheless, for applications to molecules charged in the laboratory, or to particular molecules present in nature such as D 2 S 2 (Figure 2.6) or H 2 S 2 , it is interesting to consider also the case in which the dipole has non-zero components outside the symmetry axis: this case is called the accidentally symmetric molecule.

Under a non-resonance condition, we are going to prove that, if the dipole moment is not orthogonal to the symmetry axis of the molecule, the rotational dynamics of an accidentally symmetric-top are approximately controllable. To prove this statement, we are going to show that the hypothesis of Theorem 5 are satisfied by [START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF].

Theorem 7. Let I 1 = I 2 , I 2 I 3 / ∈ Q. If δ = (δ 1 , δ 2 , δ 3 ) t is such that δ =   0 0 δ 3   and δ =   δ 1 δ 2 0   , then the
Schrödinger equation ( 21) is an m-tracker.

Proof. First of all, one can check, for example in [12, Table 2.1], that the pairings induced by the interaction Hamiltonians satisfy

D j k,m , iB l D j k ,m = 0, (23)
when |j -j| > 1, or |k -k| > 1 or |m -m| > 1, for every l = 1, 2, 3. Equation ( 23) is the general form of the so-called selection rules.

We then define for every j ∈ N the set

I j := {ρ(l, k, m) | l = j, j + 1, k, m = -l, . . . , l} ⊂ N, where ρ : {(l, k, m) | l ∈ N, k, m = -l, .
. . , l} → N is the lexicographic ordering. The graph G whose vertices are the sets I j and whose edges are {(I j , I j ) | I j ∩ I j = ∅} = {(I j , I j+1 ) | j ∈ N} is therefore linear. In order to apply Theorem 5 we shall then consider the projection of [START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF] onto each space M j := H j ⊕ H j+1 , where H l := span{D l k,m | k, m = -l, . . . , l}. The dimension of M j is (2j + 1) 2 + (2(j + 1) + 1) 2 , and we identify su(M j ) with su((2j + 1) 2 + (2(j + 1) + 1) 2 ).

According to [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF], the three types of spectral gaps in M j which we should consider are

λ j k := |E j+1 k+1 -E j k | = j + 1 I 2 + 1 2I 3 - 1 2I 2 (2k + 1) , k = -j, . . . , j, (24)
corresponding to pairings for which both j and k move (see Figure 2),

η k := |E j k+1 -E j k | = 1 2I 3 - 1 2I 2 (2k + 1) , k = -j, . . . , j, (25)
and

σ j := |E j+1 k -E j k | = j + 1 I 2 , ( 26
)
for which, respectively, only k or j moves (see, Figures 3(a) and 3(b)).

Having introduced the spectral gaps needed in our analysis, we now classify them with respect to the sets Ξ 0 j and Ξ 1 j introduced in Section 2.1.

Lemma 2 (External resonances).

Let I 2 /I 3 / ∈ Q. Then (λ j k , l), (σ j , l) ∈ Ξ 0 j , and (η k , l) ∈ Ξ 1 j , for all k = -j, . . . , j, l = 1, 2, 3. Proof. Because of the selection rules ( 23), we only need to check if there are common spectral gaps in the spaces M j and M j for j = j. We start by proving that (λ j k , l), (σ j , l) ∈ Ξ 0 j by showing that a spectral gap of the type λ j k (respectively, σ j ) is different from any spectral gap of the type λ j k , σ j , or η k unless λ j k = λ j k and (k, j) = (k , j ) (respectively, σ j = σ j and j = j ).

Using the explicit structure of the spectrum [START_REF] Mirrahimi | Controllability of quantum harmonic oscillators[END_REF], any spectral gap of the type λ j k , σ j , or η k can be written as

q 1 I 2 + q 2 1 I 3 - 1 I 2 , q 1 , q 2 ∈ Q.
Since, moreover, 1 I2 and 1 I3 -1 I2 are Q-linearly independent, one easily deduces that, indeed, (λ j k , l), (σ j , l) ∈ Ξ 0 j . Notice that the gaps of the type η k correspond to internal pairings in the spaces H j . Henceforth, in order to prove that (η k , l) ∈ Ξ 1 j it is enough to check that η k is different to any gap of the type λ j k , σ j . This fact has already been noticed in the proof of the first part of the statement. The proof of the lemma is then concluded.

Next, we introduce the family of excited modes at the frequencies λ j k ,

F j := {E λ j k (iB l ), W i (E λ j k (iB l )) | l = 1, 2, 3, k = -j, . . . , j},
where the operators E µ and W ξ are defined in Section 2.1, and where, with a slight abuse of notation, we write B l instead of Π Mj B l Π Mj . Notice that F j ⊂ ν 0 j as it follows from Lemma 2, where ν 0 j is defined as in [START_REF] Domenico | Introduction to quantum control and dynamics[END_REF].

Denote by L j := Lie(F j ) the Lie algebra generated by the matrices in F j . Let us introduce the generalized Pauli matrices E j,k = e j,k -e k,j , F j,k = ie j,k + ie k,j , D j,k = ie j,j -ie k,k , where e j,k denotes the (2j + 1) 2 + (2(j + 1) + 1) 2 -square matrix whose entries are all zero, except the one at row j and column k, which is equal to 1. Consider again the lexicographic ordering ρ : {(l, k, m) | l = j, j + 1, k, m = -l, . . . , l} → N. By a slight abuse of notation, also set e (l,k,m),(l ,k ,m ) = e ρ(l,k,m),ρ(l ,k ,m ) . The analogous identification can be used to define E (l,k,m),(l ,k ,m ) , F (l,k,m),(l ,k ,m ) , D (l,k,m),(l ,k ,m ) . The next proposition tells us how the elements in L j look like. For a detailed proof, see Appendix A.

Proposition 1. The matrices X (j,k,m),(j+1,k+1,m) -X (j,-k,m),(j+1,-k-1,m) and X (j,k,m),(j+1,k+1,m±1) + X (j,-k,m),(j+1,-k-1,m±1) are in L j for all m = -j, . . . , j and k = -j, . . . , j, k = 0, where X ∈ {E, F }.

To break the degeneracy between k and -k which appears in the elementary matrices that we found in Proposition 1, and obtain all the elementary matrices that one needs to generate su(M j ), we need to exploit the other two types of spectral gaps that we have introduced in ( 25) and (26) (see Figure 3).

Let us introduce the family of excited modes at the frequencies σ j and η k ,

P j := {E σ j (iB l ), W i (E σ j (iB l )), E η k (iB l ), W i (E η k (iB l )) | l = 1, 2, 3, k = -j, . . . , j},
and notice that, by Lemma 2, P j ⊂ ν 1 j (cf. ( 13)). Therefore,

P j := {A, [B, C] | A, B ∈ L j , C ∈ P j } ⊂ T j ,
where we recall that T j is the minimal ideal of Lie(ν 1 j ) containing ν 0 j . The following proposition concludes the proof of Theorem 7. Proposition 2. Lie( P j ) = su(M j ).

For a proof of Proposition 2, see Appendix B.

Remark 6.

The assumption I 2 /I 3 / ∈ Q on the moments of inertia appearing in Theorem 7 is technical, and prevents the system from having both external resonances (as we saw in Lemma 2) and internal ones (as we will see in Lemma 3). Anyway, we have not proven that controllability fails if the ratio I 2 /I 3 is rational.

Reachable sets of the quantum genuine symmetric-top

In [START_REF] Teschl | Mathematical methods in quantum mechanics[END_REF] we see that, when δ = (0, 0, δ 3 ) t , transitions k → k are forbidden if k = k . Thus, if the quantum system is prepared in the initial state ψ(0) with P 3 ψ(0) = kψ(0), the wave function ψ evolves in the subspaces S k = span{D j k,m | j ∈ N, m = -j, . . . , j}. The next theorem tells us that the restriction of ( 21) to this subspace is controllable. 21) is an m-tracker in the Hilbert space S k . In particular, Reach(ψ) is dense in S k ∩ S for all ψ ∈ S k ∩ S.

Theorem 8. Let I 1 = I 2 and fix k ∈ Z. If δ =   0 0 δ 3   , δ 3 = 0, then the Schrödinger equation (
Proof. For every integer j ≥ |k|, let I j,k := {ρ(l, m) | l = j, j + 1, m = -l, . . . , l}, where ρ : {(l, m) | l ≥ |k|, m = -l, . . . , l} → N is the lexicographic ordering. Then the graph G k with vertices {I j,k } ∞ j=|k| and edges {(I j,k , I j ,k ) | I j,k ∩ I j ,k = ∅} is linear. In order to apply Theorem 5 to the restriction of [START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF] to S k , we should consider the projected dynamics onto N j,k := L j,k ⊕ L j+1,k , where L l,k := span{D l k,m | m = -l, . . . , l}. The only spectral gaps in S k are

σ j = |E j+1 k -E j k | = j + 1 I 2 , j ≥ |k|.
Notice that (σ j , l) ∈ Ξ 0 j . We write the electric potentials projected onto N j,k :

E σ j (iB 1 ) = m=-j,...,j a j,k,m δ 3 E (j,k,m),(j+1,k,m+1) + a j,k,-m δ 3 E (j,k,m),(j+1,k,m-1) , E σ j (iB 2 ) = m=-j,...,j a j,k,m δ 3 F (j,k,m),(j+1,k,m+1) -a j,k,-m δ 3 F (j,k,m),(j+1,k,m-1) , E σ j (iB 3 ) = m=-j,...,j -b j,k,m δ 3 F (j,k,m),(j+1,k,m) ,
having used the explicit pairings (39), which can be found in Appendix B, and which describe the transitions excited by the frequency σ j . Note that here the sum does not run over k since we are considering the dynamics restricted to S k . We consider the family of excited modes

F j,k = {E σ j (iB l ), W i (E σ j (iB l )) | l = 1, 2, 3} ⊂ ν 0 j .
We claim that the Lie algebra generated by F j,k , seen as a subset of su((2j + 1) 2 + (2(j + 1) + 1) 2 ), is equal to su((2j + 1) 2 + (2(j + 1) + 1) 2 ). Such an identity has been proved in [START_REF] Boscain | Multi-input Schr ödinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF]Section 3.3], since the projection to N j,k is isomorphic to an analogous projection for the linear molecule. Hence, we conclude that the system ( 21) is an m-tracker in S k .

Non-controllability of the quantum orthogonal accidentally symmetric top

Let us consider separately the case where δ = (δ 1 , δ 2 , 0) t , left out by Theorem 7. The situation in which the dipole lies in the plane orthogonal to the symmetry axis of the molecule (that is, the orthogonal accidentally symmetric-top) is interesting from the point of view of chemistry, since the accidentally symmetric-top molecules present in nature are usually of that kind (see Figure 2.6). In order to study this problem, let us introduce the Wang functions

   S j k,m,γ := 1 √ 2 (D j k,m + (-1) γ D j -k,m ), k = 1, . . . , j, S j 0,m,0 = D j 0,m , k = 0,
for j ∈ N, m = -j, . . . , j, and γ = 0, 1. Due to the k-degeneracy E j k = E j -k in the spectrum of the rotational Hamiltonian H, the functions S j k,m,γ still form an orthogonal basis of eigenfunctions of H. Then we consider the change of basis D j k,m → e -ikθ D j k,m , and we choose θ

∈ [0, 2π) such that e -iθ (δ 2 + iδ 1 ) = i δ 2 1 + δ 2 2 , e iθ (δ 2 -iδ 1 ) = -i δ 2 1 + δ 2 2 . (27)
System (27) describes a rotation in the complex plane of the vector δ 2 ± iδ 1 , by the angle ∓θ.

The composition of this two changes of basis gives us the rotated Wang states S j k,m,γ (θ) := 1 √ 2 (e -ikθ D j k,m + (-1) γ e ikθ D j -k,m ), for k = 0, and S j 0,m,0 = D j 0,m . In the next theorem we express in this new basis a symmetry which prevents the system from being controllable. Proof. We need to prove that the pairings allowed by the controlled vector fields B 1 , B 2 and B 3 conserve the parity of j + γ + k. To do so, let us compute S j k,m,γ (θ), iB 1 S j+1 k+1,m+1,γ (θ) = -c j,k,m e -iθ (δ 2 + iδ 1 ) + c j,k,m e iθ (δ 2 -iδ 1 )

= -2ic j,k,m δ 2 1 + δ 2 2 , (28)
S j k,m,γ (θ), iB 1 S j+1 k+1,m+1,γ (θ) = -c j,k,m e -iθ (δ 2 + iδ 1 ) -c j,k,m e iθ (δ 2 -iδ 1 ) = 0, γ = γ , having used the expression of the Wang functions as linear combinations of Wigner functions, the explicit pairings (31) which can be found in Appendix A, and the choice we made of θ in (27). Then we also have

S j k,m,γ (θ), iB 1 S j k+1,m+1,γ (θ) = 0, S j k,m,γ (θ), iB 1 S j k+1,m+1,γ (θ) = -2ih j,k,m δ 2 1 + δ 2 2 , γ = γ , (29)
having used this time the pairings (38), which can be found in Appendix B. From ( 28) and ( 29) we can see that the allowed transitions only depend on the parity of j + γ and k, in fact we have either transitions between states of the form

j + γ even k even ←→ j + γ odd k odd,
or transitions between states of the form

j + γ even k odd ←→ j + γ odd k even.
The same happens if we replace m + 1 with m -1 and k + 1 with k -1 in (28) and (29). Because of the selection rules [START_REF] Varshalovich | Quantum theory of angular momentum[END_REF], these are the only transitions allowed by the field B 1 . One can easily check, in the same way, that every transition induced by B 2 , B 3 also conserves the parity of j + γ + k.

Appendix A Proof of Proposition 1

In order to write down the matrices in F j , we need to study the resonances between the spectral gaps inside M j . First notice that the degeneracy E j k = E j -k characterizes a symmetric-top molecule and we cannot remove it. We claim that there are no other internal resonances except those due to such a degeneracy. Indeed, we have already noticed in Lemma 2 that a spectral gap of the type λ j k (respectively, σ j ) is different from any spectral gap of the type λ j k , σ j , or η k unless λ j k = λ j k and (k, j) = (k , j ) (respectively, σ j = σ j and j = j ). Moreover,

η k = η k if k = k .
Hence we have the following.

Lemma 3 (Internal resonances). Let I 2 /I 3 / ∈ Q. Then 1. λ j k -resonances: the equation |E j+1 k+1 -E j k | = |E j s+h -E j s |, j ≤ j ≤ j ≤ j + 1, -j ≤ s ≤ j , h ∈ {-1, 0, 1}, implies that j = j, j = j + 1, s = ±k, s + h = ±(k + 1); 2. η k -resonances: the equation |E j k+1 -E j k | = |E j
s+h -E j s |, j ≤ j ≤ j ≤ j + 1, -j ≤ s ≤ j , h ∈ {-1, 0, 1}, implies that j = j = j or j = j = j + 1 and s = ±k, s + h = ±(k + 1); 3. σ j -resonances: the equation

|E j+1 k -E j k | = |E j s+h -E j s |, j ≤ j ≤ j ≤ j + 1, -j ≤ s ≤ j , h ∈ {-1, 0, 1}, implies that j = j, j = j + 1, h = 0, s = ±k.
As a consequence of Lemma 3, part 1, if I 2 /I 3 / ∈ Q, the only transitions driven by the fields iB l , l = 1, 2, 3, excited at frequency λ j k , are the ones corresponding to the following matrix elements (written in the basis of M j given by the Wigner functions) and can be computed using, e.g., [START_REF] Cook | Microwave molecular spectra[END_REF]Table 2.1]:

               D j k,m , iB 1 D j+1 k+1,m±1 = -c j,k,±m (δ 2 + iδ 1 ), D j k,m , iB 1 D j+1 k-1,m±1 = c j,-k,±m (δ 2 -iδ 1 ), D j k,m , iB 2 D j+1 k+1,m±1 = ∓ic j,k,±m (δ 2 + iδ 1 ), D j k,m , iB 2 D j+1 k-1,m±1 = ±ic j,-k,±m (δ 2 -iδ 1 ), D j k,m , iB 3 D j+1 k±1,m = ±id j,±k,m (δ 2 ± iδ 1 ), (30)
where c j,k,m := [(j + k + 1)(j + k + 2)] 1/2 [(j + m + 1)(j + m + 2)] 1/2 4(j + 1)[(2j + 1)(2j + 3)] 1/2 , and

d j,k,m := [(j + k + 1)(j + k + 2)] 1/2 [(j + 1) 2 -m 2 ] 1/2 2(j + 1)[(2j + 1)(2j + 3)] 1/2 .
Now, using a symmetry agrument, we explain how to get rid of one electric dipole component between δ 1 and δ 2 . Indeed, we shall explain in detail the meaning of the rotation (27) which we have already used in the proof of Theorem 9. By the very definition of the Euler angles, one has that the rotation of angle θ around the symmetry axis a 3 is given by γ → γ + θ. This rotation acts on the Wigner functions in the following way

D j k,m (α, β, γ) → D j k,m (α, β, γ + θ) = e ikθ D j k,m (α, β, γ) =: D j k,m (
θ)(α, β, γ), having used the explicit expression of the symmetric states [START_REF] Koch | Quantum control of molecular rotation[END_REF]. Note that these rotated Wigner functions form again an orthogonal basis for L 2 (SO(3)) of eigenfunctions of the rotational Hamiltonian H, so we can also analyze the controllability problem in this new basis. In other words, the inertia ellipsoid of a symmetric-top molecule admits the circle S 1 as symmetry group. This is again a characterizing property of symmetric-top, which is no longer true for asymmetric-top molecules. In this new basis the matrix elements (corresponding to the frequency λ j k ) of the controlled fields are

D j k,m (θ), iB 1 D j+1 k+1,m+1 (θ) = -c j,k,m e -iθ (δ 2 + iδ 1 ), D j k,m (θ), iB 1 D j+1 k-1,m+1 (θ) = c j,-k,m e iθ (δ 2 -iδ 1 ), (31)
and the same happens for all the other transitions described in (30). So, the effect of this change of basis is that we are actually rotating the first two components of the dipole moment, by the angle θ. We can now choose θ ∈ [0, 2π) such that e -iθ (δ 2 + iδ 1 ) = δ 2 1 + δ 2 2 = e iθ (δ 2 -iδ 1 ).

In other words, with this change of basis, we can assume without loss of generality that δ 1 = 0, since we can rotate the vector δ 2 ± iδ 1 and get rid of its imaginary part (note that in (27) and in the proof of Theorem 9 we were rotating the vector δ 2 ± iδ 1 in the other sense, i.e., to get rid of its real part). This will simplify the expression of the controlled fields. Clearly, we could have alternatively chosen to set δ 2 = 0. Note that

W i (E (j,k,m),(j+1,k+1,n) ) = ±F (j,k,m),(j+1,k+1,n) , W i (F (j,k,m),(j+1,k+1,n) ) = ∓E (j,k,m),(j+1,k+1,n) ,
depending on whether I 2 is greater or smaller than I 3 . We shall assume

W i (E (j,k,m),(j+1,k+1,n) ) = -F (j,k,m),(j+1,k+1,n) , W i (F (j,k,m),(j+1,k+1,n) ) = E (j,k,m),(j+1,k+1,n) ,
since the computations in the two cases are completely analogous.

From the identity [e j,k , e n,m ] = δ kn e j,m -δ jm e n,k we get the useful bracket relations

[E j,k , E k,n ] = E j,n , [F j,k , F k,n ] = -E j,n , [E j,k , F k,n ] = F j,n , [E j,k , F j,k ] = 2D j,k , [F j,k , D j,k ] = 2E j,k .
Moreover, two operators coupling no common states commute, that is,

[Y j,k , Z j ,k ] = 0 if {j, k} ∩ {j , k } = ∅, with Y, Z ∈ {E, F, D}.
Finally, we can conveniently represent the matrices corresponding to the controlled vector field (projected onto M j ) in the rotated basis found with the symmetry argument. So, for each k = -j, . . . , j, because of Lemma 3, part 1, and (30), we have

E λ j k (iB 1 ) = m=-j,...,j -c j,k,m δ 2 E (j,k,m),(j+1,k+1,m+1) -c j,k,-m δ 2 E (j,k,m),(j+1,k+1,m-1) + c j,k,m δ 2 E (j,-k,m),(j+1,-k-1,m+1) + c j,k,-m δ 2 E (j,-k,m),(j+1,-k-1,m-1) , (32)
E λ j k (iB 2 ) = m=-j,...,j -c j,k,m δ 2 F (j,k,m),(j+1,k+1,m+1) + c j,k,-m δ 2 F (j,k,m),(j+1,k+1,m-1) + c j,k,m δ 2 F (j,-k,m),(j+1,-k-1,m+1) -c j,k,-m δ 2 F (j,-k,m),(j+1,-k-1,m-1) , (33)
E λ j k (iB 3 ) = m=-j,...,j d j,k,m δ 2 F (j,k,m),(j+1,k+1,m) -d j,k,m δ 2 F (j,-k,m),(j+1,-k-1,m) , (34)
where, with a slight abuse of notation, we write B l instead of Π Mj B l Π Mj . Now we show how the sum over m in (32), ( 33) and (34) can be decomposed, in order to obtain that all the elementary matrices X (j,k,m),(j+1,k+1,m±1) + X (j,-k,m),(j+1,-k-1,m±1) and X (j,k,m),(j+1,k+1,m) -X (j,-k,m),(j+1,-k-1,m) are in L j , for any m, k = -j, . . . , j, where X ∈ {E, F }.

Let us first fix k = 0 and consider

W i (E λ j k (iB 3
E (j,k,m),(j+1,k+1,m) + E (j,k,-m),(j+1,k+1,-m) + -E (j,-k,m),(j+1,-k-1,m) -E (j,-k,-m),(j+1,-k-1,-m) ∈ L j , (35)
for m = 0, . . . , j. In particular, E (j,k,0),(j+1,k+1,0) -E (j,-k,0),(j+1,-k-1,0) is in L j . Hence,

E λ k (iB 1 ) -W i (E λ k (iB 2 )) 2 , E (j,k,0),(j+1,k+1,0) -E (j,-k,0),(j+1,-k-1,0) , (36)
E (j,k,0),(j+1,k+1,0) -E (j,-k,0),(j+1,-k-1,0) = -c j,k,0 δ 2 E (j,k,0),(j+1,k+1,-1) -c j,k,-1 δ 2 E (j,k,1),(j+1,k+1,0) -c j,k,0 δ 2 E (j,-k,0),(j+1,-k-1,-1) -c j,k,-1 δ 2 E (j,-k,1),(j+1,-k-1,0) is also in L j . Define Q 0 = -c j,k,0 δ 2 E (j,k,0),(j+1,k+1,-1) -c j,k,-1 δ 2 E (j,k,1),(j+1,k+1,0) -c j,k,0 δ 2 E (j,-k,0),(j+1,-k-1,-1) -c j,k,-1 δ 2 E (j,-k,1),(j+1,-k-1,0) , Q m = -c j,k,-m δ 2 E (j,k,-m),(j+1,k+1,-m-1) -c j,k,-m-1 δ 2 E (j,k,m+1),(j+1,k+1,m) -c j,k,-m δ 2 E (j,-k,-m),(j+1,-k-1,-m-1) -c j,k,-m-1 δ 2 E (j,-k,m+1),(j+1,-k-1,m) ,
if 0 < m < j, and

Q j = -c j,k,-j δ 2 E (j,k,-j),(j+1,k+1,-j-1) -c j,k,-j δ 2 E (j,-k,-j),(j+1,-k-1,-j-1) .
We have m=s,...,j Q m , E (j,k,s),(j+1,k+1,s) + E (j,k,-s),(j+1,k+1,-s) -E (j,-k,s),(j+1,-k-1,s)

-E (j,-k,-s),(j+1,-k-1,-s) , E (j,k,s),(j+1,k+1,s) + E (j,k,-s),(j+1,k+1,-s)

-E (j,-k,s),(j+1,-k-1,s) -E (j,-k,-s),(j+1,-k-1,-s) = Q s ,
for s = 1, . . . , j. By iteration on s and because of (35), it follows that Q s ∈ L j for every s = 0, . . . , j. Now, since

Q j -c j,k,-j δ 2 = E (j,k,-j),(j+1,k+1,-j-1) + E (j,-k,-j),(j+1,-k-1,-j-1) ∈ L j ,
then ad 2 E (j,k,-j),(j+1,k+1,-j-1) +E (j,-k,-j),(j+1,-k-1,-j-1) (E (j,k,j),(j+1,k+1,j) + E (j,k,-j),(j+1,k+1,-j) -E (j,-k,j),(j+1,-k-1,j) -E (j,-k,-j),(j+1,-k-1,-j) ) = E (j,k,-j),(j+1,k+1,-j) -E (j,-k,-j),(j+1,-k-1,-j) ∈ L j , which, in turns, implies that ad 2 E (j,k,-j),(j+1,k+1,-j) -E (j,-k,-j),(j+1,-k-1,-j) (Q j-1 ) = -c j,k,-j+1 E (j,k,-j+1),(j+1,k+1,-j) -c j,k,-j+1 E (j,-k,-j+1),(j+1,-k-1,-j) ∈ L j .

Iterating the argument, E (j,k,m),(j+1,k+1,m) -E (j,-k,m),(j+1,-k-1,m) ∈ L j , m = -j, . . . , j

and E (j,k,m),(j+1,k+1,m-1) + E (j,-k,m),(j+1,-k-1,m-1) are in L j for m = -j, . . . , j.

By the same argument as above, with in (36) we also have that E (j,k,m),(j+1,k+1,m+1) + E (j,-k,m),(j+1,-k-1,m+1) is in L j for all m = -j, . . . , j. + c j,k,m F (j,-k,m),(j+1,-k-1,m+1) , the arguments above prove that both F (j,k,m),(j+1,k+1,m) -F (j,-k,m),(j+1,-k-1,m) and F (j,k,m),(j+1,k+1,m±1) + F (j,-k,m),(j+1,-k-1,m±1) are in L j for all m = -j, . . . , j.

E λ j k (iB 1 ) -W i (E λ j k (iB 2

If we now replace

B Proof of Proposition 2

Using again [12, Table 2.1] we write the pairings

               D j k,m
, iB 1 D j k+1,m±1 = ∓h j,k,±m (δ 2 + iδ 1 ), D j k,m , iB 1 D j k-1,m±1 = ∓h j,-k,±m (δ 2 -iδ 1 ), D j k,m , iB 2 D j k+1,m±1 = -ih j,k,±m (δ 2 + iδ 1 ), D j k,m , iB 2 D j k-1,m±1 = -ih j,-k,±m (δ 2 -iδ 1 ), D j k,m , iB 3 D j k±1,m = -iq j,±k,m (δ 2 ± iδ 1 ),

where h j,k,m := [j(j + 1) -k(k + 1)] 1/2 [j(j + 1) -m(m + 1)] 1/2 4j(j + 1) , q j,k,m := [j(j + 1) -k(k + 1)] 1/2 m 2j(j + 1) . Note that the k → k transitions are driven by δ 3 . Since we are still assuming that δ 1 = 0, and because of Lemma 3, parts 2 and 3, the expression of the controlled fields excited at the frequencies η k and σ j are E η k (iB 1 ) = l=j,j+1, m=-l,...,l-1 -h l,k,m δ 2 E (l,k,m),(l,k+1,m+1) -h l,k,m δ 2 E (l,-k,m),(l,-k-1,m+1) l=j,j+1, m=-l+1,...,l h l,k,-m δ 2 E (l,k,m),(l,k+1,m-1) + h l,k,-m δ 2 E (l,-k,m),(l,-k-1,m-1) ,

Moreover

E η k (iB 2 ) = l=j,j+1, m=-l,...,l-1 -h l,k,m δ 2 F (l,k,m),(l,k+1,m+1) -h l,k,m δ 2 F (l,-k,m),(l,-k-1,m+1) l=j,j+1, m=-l+1,...,l -h l,k,-m δ 2 F (l,k,m),(l,k+1,m-1) -h l,k,-m δ 2 F (l,-k,m),(l,-k-1,m-1) ,

E η k (iB 3 ) = l=j,j+1, m=-l,...,l -q l,k,m δ 2 F (l,k,m),(l,k+1,m) -q l,k,m δ 2 F (l,-k,m),(l,-k-1,m) ,

and E σ j (iB 1 ) = m,k=-j,...,j a j,k,m δ 3 E (j,k,m),(j+1,k,m+1) + a j,k,-m δ 3 E (j,k,m),(j+1,k,m-1) ,

E σ j (iB 2 ) = m,k=-j,...,j a j,k,m δ 3 F (j,k,m),(j+1,k,m+1) -a j,k,-m δ 3 F (j,k,m),(j+1,k,m-1) , E σ j (iB 3 ) = m,k=-j,...,j -b j,k,m δ 3 F (j,k,m),(j+1,k,m) .

Note that in E η k (-iB 3 ) the term for m = 0 vanishes, since q j,k,0 = 0 for every j, k.

To decouple all the m-degeneracies in the excited modes, we just make double brackets with the elementary matrices that we have obtained above. As an example, using (37) we can decouple the m → m transitions corresponding to the frequency σ j by considering [[W i (E σ j (iB 3 )), E (j,k,m),(j+1,k+1,m) -E (j,-k,m),(j+1,-k-1,m) ], E (j,k,m),(j+1,k+1,m) -E (j,-k,m),(j+1,-k-1,m) ] = E (j,k,m),(j+1,k,m) -E (j,-k,m),(j+1,-k,m) ∈ Lie( P j ).

Making every possible double brackets as above, we obtain, for X ∈ {E, F }, that X (j,k,m),(j+1,k,m±h) + X (j,-k,m),(j+1,-k,m±h) ∈ Lie( P j ), k = 0,

when we start from the matrices in (43), and that X (l,k,m),(l,k+1,m) + X (l,-k,m),(l,-k-1,m) , X (l,k,m),(l,k+1,m±1) + X (l,-k,m),(j,-k-1,m±1)

are in Lie( P j ), l = j, j + 1, m, k = 0, when we start from the matrices in (40), (41), (42). Now we can also generate the missing k = 0 elements of (37) by making double brackets with X (j+1,1,m),(j+1,2,m) + X (j+1,-1,m),(j+1,-2,m) ∈ Lie( P j ). As an example, we have that = F (j,0,m),(j+1,1,m) + F (j,0,m),(j+1,-1,m) ∈ Lie( P j ).

Moreover, also the m = 0 elements in the transitions (42) are in Lie( P j ), as one can check by making a bracket between two transitions obtained in (37) and (44). For example, [E (j,k,0),(j+1,k+1,0) -E (j,-k,0),(j+1,-k-1,0) , E (j+1,k+1,0),(j,k+1,0)

+ E (j+1,-k-1,0),(j,-k-1,0) ] = E (j,k,0),(j,k+1,0) -E (j,-k,0),(j,-k-1,0) ∈ Lie( P j ).

Finally, we apply a three-waves mixing argument (Figure B) in order to decouple the sum over k and -k in every elementary matrices: consider the bracket between the following elements in Lie( P j ) is in Lie( P j ), and hence E (j,k+1,m),(j+1,k,m) and E (j,-k-1,m),(j+1,-k,m) are in Lie( P j ). In this way we can break every k-degeneracy, and finally obtain that Lie( P j ) = su(M j ), which concludes the proof.

3 l=1u 3

 33 l (t)B l (R, δ)ψ(R, t), ψ(•, t) ∈ L 2 (SO(3)), is the rotational Hamiltonian, I 1 , I 2 , I 3 are the moments of inertia

  Figure 1:

Remark 1 .= 0 .

 10 If G is abelian, then ad ∂H ∂p * Thus one recovers the classical Hamiltonian equations.

Theorem 2 .

 2 in the body frame. The corresponding molecule is called a genuine symmetric-top. The third angular momentum P 3 is a conserved quantity for the controlled motion (6) of the genuine symmetry-top molecule.

  (θ) ∈ SO(3) is a rotation of angle θ around the axis e i . The explicit expression of the matrix R(α, β, γ) ∈ SO(3) is R =   cos α cos β cos γ -sin α sin γ -cos α cos β sin γ -sin α cos γ cos α sin β sin α cos β cos γ + cos α sin γ -sin α cos β sin γ + cos α cos γ sin α sin β -sin β cos γ sin β sin γ cos β   .

Remark 4 .

 4 The level k = 0 corresponds to the vanishing of the third angle γ. Then the system becomes a linear molecule which can be described by the spherical harmonics {Y j m | j ∈ N, m = -j, . . . , j} ([23, Chapter 5],[START_REF] Biedenharn | of Encyclopedia of Mathematics and its Applications[END_REF] Section 3.10]). Indeed, Y j m (α, β) = D j 0,m (α, β, γ).

3 Theorem 6 . 3 

 363 recall that the genuine symmetric-top molecule is a symmetric rigid body with electric dipole δ along the symmetry axis: δ = in the principal axis frame on the body. We then introduce the subspaces S k := span{D j k,m | j ∈ N, m = -j, . . . , j}, where span denotes the closure of the linear hull in L 2 (SO(3)). The quantum number k is invariant in the controlled motion of the genuine symmetric-top molecule. That is, if I 1 = I 2 and δ = , the subspaces S k are invariant for any propagator of the Schrödinger equation[START_REF] Vahagn Nersesyan | Global approximate controllability for Schr ödinger equation in higher Sobolev norms and applications[END_REF].

0

  dγe ikγ e -ik γ ) ( sin 2 (β) cos(α)e imα e -im α d j k,m (β)d j k ,m (β)) = 0, using the orthogonality of the functions e ikγ and the explicit form (17) of the matrix R, which yields

Figure 2 :Figure 3 :

 23 Figure 2: Graph of the transitions associated with the frequency λ j k between eigenstates |j, k = |j, k, m := D j k,m (m fixed). Same-shaped arrows correspond to equal spectral gaps.

Figure 4 :

 4 Figure 4: Diagram of the orthogonal accidentally symmetric molecule D 2 S 2 . The electric dipole δ lies in the orthogonal plane to the symmetry axis.

Theorem 9 .

 9 Let I 1 = I 2 and δ = δ 1 = 0 or δ 2 = 0. Then the parity of j + γ + k is conserved.

2 E 2 E

 22 ,...,j d j,k,m δ 2 E (j,k,m),(j+1,k+1,m) -d j,k,m δ 2 E (j,-k,m),(j+1,-k-1,m) , (W i (E λ j k (iB 3 ))) = m=-j,...,j (-1) s 2 2s d 2s+1 j,k,m δ 2s+1 (j,k,m),(j+1,k+1,m) + (-1) s 2 2s (-d j,k,m ) 2s+1 δ 2s+1 (j,-k,m),(j+1,-k-1,m) , for s ∈ N, where ad A (B) = [A, B] and ad n+1 A (B) = [A, ad n A (B)]. Since d j,k,m = d j,k,-m , the invertibility of the Vandermonde matrix gives that

  ,...,j -c j,k,m E (j,k,m),(j+1,k+1,m+1) + c j,k,m E (j,-k,m),(j+1,-k-1,m+1)

(iB 1 )) 2 =

 2 ,...,j -c j,k,-m F (j,k,m),(j+1,k+1,m-1)+ c j,k,-m F (j,-k,m),(j+1,-k-1,m-1) or E λ j k (iB 2 ) -W i (E λ j k m=-j,...,j -c j,k,m F (j,k,m),(j+1,k+1,m+1)

  , iB 1 D j+1 k,m±1 =: a j,k,±m δ 3 , D j k,m , iB 2 D j+1 k,m±1 = ±ia j,k,±m δ 3 , D j k,m , iB 3 D j+1 k,m =: -ib j,k,m δ 3 ,(39)wherea j,k,m := [(j + 1) 2 -k 2 ] 1/2 [(j + m + 1)(j + m + 2)] 1/2 2(j + 1)[(2j + 1)(2j + 3)] 1/2 , b j,k,m := [(j + 1) 2 -k 2 ] 1/2 [(j + 1) 2 -m 2 ] 1/2 (j + 1)[(2j + 1)(2j + 3)] 1/2 .

[[E λ j 0 (iB 3 )

 03 , F (j+1,1,m),(j+1,2,m) + F (j+1,-1,m),(j+1,-2,m) ],F (j+1,1,m),(j+1,2,m) + F (j+1,-1,m),(j+1,-2,m) ]

Figure 5 :

 5 Figure 5: Three-waves mixing around k = 1, -1. The same-shaped arrows correspond to equal spectral gaps, and thus, coupled transitions. The goal of the three-waves mixing is to decouple those arrows.

is an hypergeometric series and n j,k,m is a normalizing factor.

Acknowledgement

The project leading to this publication has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement no. 765267 (QuSCo). Also it was supported by the ANR projects SRGI ANR-15-CE40-0018 and Quaco ANR-17-CE40-0007-01.