Yohann Le Floch

Álvaro Pelayo

San Vu

Yohann Le

V Ũ Ngo

Inverse spectral theory for semiclassical Jaynes-Cummings systems

Introduction

A natural question in semiclassical analysis is whether the knowledge of the joint spectrum of a quantum integrable system allows to determine the classical dynamics of the underlying integrable system. Pursuing this question in such generality has been made possible thanks to the development of semiclassical analysis with microlocal techniques (see for instance the recent books by Dimassi-Sjöstrand [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF], Guillemin-Sternberg [START_REF] Guillemin | Semi-classical analysis[END_REF], and Zworski [START_REF] Zworski | Semiclassical analysis[END_REF] and the references therein) which nowadays permits a constant interaction between symplectic geometry and spectral theory. In particular, these techniques led to the resolution of the inverse spectral question in a number of cases; for instance: (i) compact toric integrable systems, in the context of Berezin-Toeplitz quantization [START_REF] Charles | c: Isospectrality for quantum toric integrable systems[END_REF]; (ii) semiglobal inverse problem near the so called "focus-focus" singularities of 2D integrable systems, in the context of -pseudodifferential quantization [START_REF] Pelayo | c: Semiclassical inverse spectral theory for singularities of focus-focus type[END_REF]; (iii) inverse theory for the Laplacian on surfaces of revolution [START_REF] Zelditch | The inverse spectral problem for surfaces of revolution[END_REF]; (iv) 1-dimensional pseudodifferential operators with Morse symbol [START_REF] Ngo | Symplectic inverse spectral theory for pseudodifferential operators[END_REF]. The flexibility of microlocal analysis makes us hope that more general integrable systems will be treated in the future. An interesting step is to understand what happens for semitoric systems on 4-dimensional phase spaces [START_REF] Pelayo | Constructing integrable systems of semitoric type[END_REF], which form an important extension of toric systems. Definition 1.1. A C ∞ classical integrable system F := (J, H) : M → R 2 on a connnected symplectic 4-dimensional manifold (M, ω) is semitoric if:

(H.i) J is the momentum map of an effective Hamiltonian circle action.

(H.ii) The singularities of F are non-degenerate with no hyperbolic component. (H.iii) J is a proper map (i.e., the preimages of compact sets are compact).

A quantum semitoric integrable system (P, Q) is given by two semiclassical commuting self-adjoint operators whose principal symbols form a classical semitoric integrable system. The notion of semiclassical operators that we use is defined in Section 2.4; it includes standard semiclassical pseudodifferential operators, and Berezin-Toeplitz operators.

When (M, ω) is four-dimensional, toric systems form a particular class of semitoric systems for which F is the momentum map of a Hamiltonian T 2 -action. The symplectic classification of toric systems was done by Delzant [START_REF] Delzant | Hamiltoniens périodiques et images convexes de l'application moment[END_REF], and the quantum spectral theory in the case of Berezin-Toeplitz quantization was carried out in [START_REF] Charles | c: Isospectrality for quantum toric integrable systems[END_REF]. A simple corollary of this spectral theory is that the image F (M ), which is the so-called Delzant polytope, can be recovered from the joint spectrum; in view of the Delzant theorem, this implies that the joint spectrum completely determines the triple (M, ω, F ) up to toric isomorphism.

The main difference with the toric case is that focus-focus singularities can appear in a semitoric system, making the system more difficult to describe. For instance, if there is at least one focus-focus singularity, the image of the moment map is no longer a convex polygon. Moreover, new symplectic invariants appear; according to [START_REF] Pelayo | c: Semitoric integrable systems on symplectic 4-manifolds[END_REF][START_REF] Pelayo | Constructing integrable systems of semitoric type[END_REF], a semitoric system is determined up to isomorphisms 1 by five symplectic invariants:

(1) the number of focus-focus singular values of the system;

(2) a Taylor series i,j∈N a ij X i Y j for each focus-focus singularity ( [START_REF] Ngo | On semi-global invariants for focus-focus singularities[END_REF][START_REF] Pelayo | c: Semitoric integrable systems on symplectic 4-manifolds[END_REF]); (3) a height invariant h > 0 measuring the volume of certain reduced spaces at each focus-focus singularity; (4) a polygonal invariant (in fact, a family of polygons) obtained by unwinding the singular affine structure of the system; (5) an index associated with each pair of focus-focus singularities, called the twisting index.

Therefore, if one is able to recover these five invariants from the semiclassical joint spectrum of a quantum integrable system quantizing (J, H), then in effect one can recover the triple (M, ω, F ) up to the appropriate notion of isomorphism. From [START_REF] Pelayo | c: Semiclassical inverse spectral theory for singularities of focus-focus type[END_REF], (2) can be recovered. The goal of this paper is to show that this is the case for invariants (1) to (4). Let us be more precise and state our main result. We will say that a semitoric integrable system F = (J, H) is simple if it satisfies the following: if m is a focus-focus critical point for F , then m is the unique critical point of the level set J -1 (J(m)).

Our main theorem is the following. 1 The notion of isomorphism for semitoric systems is recalled in Definition 2.1.

Theorem A. Let (P, Q) be a quantum simple semitoric system on M for which the Bohr-Sommerfeld rules hold. Then from the knowledge of the semiclassical joint spectrum JointSpec(P, Q) + O( 2), one can recover the four following invariants of the associated classical semitoric system:

(1) the number m f of focus-focus values, (2) the Taylor series associated with each focus-focus value, (3) the height invariant associated with each focus-focus value, (4) the polygonal invariant of the system.

Of course, Theorem A is not entirely satisfactory if one has in mind the problem of completely recovering the classical system from the joint spectrum of its quantum counterpart. However, there is one case where we can say more: for the simplest examples of semitoric integrable systems, which we call systems of Jaynes-Cummings type. The characteristic of such a system is to display only one focus-focus singularity. One of the simplest yet most important models in classical and quantum mechanics was proposed by Jaynes and Cummings [START_REF] Jaynes | Comparison of quantum and semiclassical radiation theories with application to the beam maser[END_REF][START_REF] Cummings | Stimulated emission of radiation in a single mode[END_REF] in 1963, and it is now known as the Jaynes--Cummings model. 2 The Jaynes-Cummings model is obtained by coupling a spin with a harmonic oscillator. In this way one obtains a physical system with phase space S 2 × R 2 and Hamiltonian functions J := u 2 +v 2 2 + z, H = ux+vy 2 , where (x, y, z) denotes the point in the 2-sphere S 2 ⊂ R 3 and (u, v) denote points in R 2 (J is the momentum map for the combined rotational S 1actions about the origin in R 2 and about the vertical axes on S 2 ). Recently the second and third authors described in full the semiclassical spectral theory of this system [START_REF] Pelayo | c: Hamiltonian dynamics and spectral theory for spinoscillators[END_REF]. Definition 1.2. A classical integrable system F := (J, H) : M → R 2 on a symplectic 4-manifold (M, ω) is of Jaynes-Cummings type if:

(a) F is a semitoric system; (b) F has one, and only one, singularity of focus-focus type. A quantum integrable system (P, Q) of Jaynes-Cummings type is given by two semiclassical commuting self-adjoint operators whose principal symbols form a classical integrable system of Jaynes-Cummings type.

The Jaynes-Cummings model is a particular example of a system of Jaynes-Cummings type. Jaynes-Cummings type systems form a large class of integrable Hamiltonian systems because the structure of the singularity in part (b) is extremely rich, and it is classified by a Taylor series i,j∈N a ij X i Y j , according to [START_REF] Ngo | On semi-global invariants for focus-focus singularities[END_REF] (two such singularities are symplectically equivalent if and only if each and everyone of the coefficients in the Taylor series coincide for both singularities). Moreover, by [START_REF] Ngo | On semi-global invariants for focus-focus singularities[END_REF][START_REF] Pelayo | Constructing integrable systems of semitoric type[END_REF] every such Taylor 2 The Jaynes-Cummings model was initially introduced to describe the interaction between an atom prepared in a mixed state with a quantum particle in an optical cavity. It was found to apply to many physical situations (quantum chemistry, quantum optics, quantum information theory, etc.) because it represents the easiest way to have a finite dimensional state (like a spin) interact with an oscillator.

series can be realized as the Taylor series invariant of an integrable system (in fact, of many inequivalent such systems). Accordingly, the moduli space of Jaynes-Cummings type systems is, from the point of view of Hamiltonian dynamics, extremely rich. As a corollary of Theorem A, we solve the inverse spectral problem for quantum integrable systems of Jaynes-Cummings type.

Theorem B. Let (P, Q) be a quantum integrable system of Jaynes-Cummings type on M for which the Bohr-Sommerfeld rules hold. Then from the knowledge of JointSpec(P, Q)+O( 2), one can recover the principal symbol σ(P, Q) up to isomorphisms of semitoric integrable systems.

This theorem gives the first global inverse spectral result that the authors are aware of for integrable Hamiltonian systems with focus-focus singularities (and hence no global action-angle variables). In the context of Hamiltonian toral actions (eg. toric integrable systems), all singularities are of elliptic type, which is strongly related to the dynamical and spectral rigidity of such systems [START_REF] Charles | c: Isospectrality for quantum toric integrable systems[END_REF]. We believe that allowing focus-focus singularities, which have a much larger moduli space, is an important step forward in the study of the inverse spectral problem for general integrable systems.

The problem treated in this paper belongs to a class of semiclassical inverse spectral questions which has attracted much attention in recent years, e.g. [21,[START_REF] Gurarie | Semiclassical eigenvalues and shape problems on surfaces of revolution[END_REF][START_REF] Hall | Diophantine tori and non-selfadjoint inverse spectral problems[END_REF][START_REF] De Verdière | A semi-classical inverse problem I: Taylor expansions[END_REF][START_REF] Pelayo | Semiclassical quantization and spectral limits of pseudodifferential and Berezin-Toeplitz operators[END_REF][START_REF] Phan | Spectral monodromy of non-self-adjoint operators[END_REF][START_REF] Ngo | Symplectic inverse spectral theory for pseudodifferential operators[END_REF], which goes back to pioneer works of Bérard [START_REF] Bérard | Quelques remarques sur les surfaces de révolution dans R3[END_REF], Brüning-Heintze [START_REF] Brüning | Ernst Spektrale Starrheit gewisser Drehflächen[END_REF], Colin de Verdière [START_REF] De Verdière | Spectre conjoint d'opérateurs pseudo-différentiels qui commutent I[END_REF][START_REF] De Verdière | Spectre conjoint d'opérateurs pseudo-différentiels qui commutent II[END_REF], Duistermaat-Guillemin [START_REF] Duistermaat | The spectrum of positive elliptic operators and periodic bicharacteristics[END_REF], and Guillemin-Sternberg [START_REF] Guillemin | Homogeneous quantization and multiplicities of group representations[END_REF], in the 1970s/1980s, and are closely related to inverse problems that are not directly semiclassical but do use similar microlocal techniques for some integrable systems, as in [START_REF] Zelditch | The inverse spectral problem for surfaces of revolution[END_REF] (see also [START_REF] Zelditch | The inverse spectral problem[END_REF] and references therein).

We conclude this section by a natural question. The following corollary of Theorem A directly follows from the symplectic classification [START_REF] Pelayo | c: Semitoric integrable systems on symplectic 4-manifolds[END_REF] of semitoric systems: Corollary 1.3. Let (P, Q) and (P ′ , Q ′ ) be quantum simple semitoric systems on M and M ′ , respectively, for which the Bohr-Sommerfeld rules hold. If

JointSpec(P, Q) = JointSpec(P ′ , Q ′ ) + O( 2 ), (1)
and if the twisting index invariants of σ(P, Q) and σ(P ′ , Q ′ ) are equal, then σ(P, Q) and σ(P ′ , Q ′ ) are isomorphic as semitoric integrable systems.

In view of this result, one question remains: can one obtain the twisting index invariant of a semitoric system from the data of the joint spectrum of the corresponding quantum system? A positive answer to this question would lead to the definition of a new quantum invariant which would be quite robust (since the twisting index between two focus-focus singularities is just an integer).

Preliminaries

Let (M, ω) be a smooth, connected 4-dimensional symplectic manifold.

2.1. Integrable systems. An integrable system (J, H) on (M, ω) consists of two Poisson commuting functions J, H ∈ C ∞ (M ; R) i.e. :

{J, H} := ω(X J , X H ) = 0, whose differentials are almost everywhere linearly independent 1-forms. Here X J , X H are the Hamiltonian vector fields induced by J, H, respectively, via the symplectic form ω: ω(X J , •) = -dJ, ω(X H , •) = -dH. Moreover, the function F = (J, H) will be assumed to be proper throughout this paper.

For instance, let M 0 = T * T 2 be the cotangent bundle of the torus T 2 , equipped with canonical coordinates (x 1 , x 2 , ξ 1 , ξ 2 ), where x ∈ T 2 and ξ ∈ T *

x T 2 . The linear system

(J 0 , H 0 ) := (ξ 1 , ξ 2 ) is integrable. An isomorphism of integrable systems (J, H) on (M, ω) and (J ′ , H ′ ) on (M ′ , ω ′ ) is a diffeomorphism ϕ : M → M ′ such that ϕ * ω ′ = ω and ϕ * (J ′ , H ′ ) = (f 1 (J, H), f 2 (J, H))
for some local diffeomorphism (f 1 , f 2 ) of R 2 . This same definition of isomorphism extends to any open subsets U ⊂ M , U ′ ⊂ M ′ (and this is the form in which we will use it later). Such an isomorphism will be called semiglobal if U, U ′ are respectively saturated by level sets

{J = const 1 , H = const 2 } and {J ′ = const ′ 1 , H ′ = const ′ 2 }. If F = (J, H) is an integrable system on (M, ω), consider a point c ∈ R 2
that is a regular value of F , and such that the fiber Λ c = F -1 (c) is compact and connected. Then, by the action-angle theorem [START_REF] Duistermaat | On global action-angle variables[END_REF], a saturated neighborhood of the fiber is isomorphic in the previous sense to the above linear model on M 0 = T * T 2 . Therefore, all such regular fibers (called Liouville tori ) are isomorphic in a neighborhood.

However, the situation changes drastically when the condition that c be regular is violated. For instance, it has been proved in [START_REF] Ngo | On semi-global invariants for focus-focus singularities[END_REF] that, when c is a so-called focus-focus critical value, an infinite number of equations has to be satisfied in order for two systems to be semiglobally isomorphic near the critical fiber.

2.2. Semitoric systems. Semitoric systems (Definition 1.1) form a particular class of integrable systems admitting an S 1 symmetry. It is therefore natural to introduce a suitable notion of isomorphism for such systems, which mixes the general notion defined in the previous section with the more rigid one coming from Hamiltonian S 1 -manifolds.

Definition 2.1. The semitoric systems (M 1 , ω 1 , F 1 := (J 1 , H 1 )) and (M 2 , ω 2 , F 2 := (J 2 , H 2 )) are isomorphic if there exists a symplectomorphism ϕ :

M 1 → M 2 such that ϕ * (J 2 , H 2 ) = (J 1 , h(J 1 , H 1 )) for a smooth h such that ∂h ∂H 1 > 0. 2.
3. The period lattice. Let F = (J, H) be an integrable system on a 4-dimensional symplectic manifold. For any regular value c of F , the set of points (t, u) ∈ R 2 such that the vector field tX J + uX H has a 2π-periodic flow on Λ c is a sublattice of R 2 called the period lattice [START_REF] Duistermaat | On global action-angle variables[END_REF]. When c varies in the set of regular values of F , the collection of the period lattices is a Lagrangian subbundle of T * R 2 , called the period bundle.

Coming back to our case where F = (J, H) is a semitoric system, there is a natural way to construct a basis of this lattice. Firstly, since J generates a S 1 -action, (1, 0) belongs to the period lattice. Secondly, define two real numbers τ 1 (c), τ 2 (c) as follows: choose a point m ∈ Λ c , and define τ 2 (c) > 0 as the time of first return for the Hamiltonian flow associated with H to the trajectory of the Hamiltonian flow of J passing through m. Let τ 1 (c) ∈ [0, 2π) be the time that it takes to come back to m following the flow of X J . Because of the commutativity of the Hamiltonian flows of J and H, the values of τ 1 (c), τ 2 (c) do not depend on the choice of the starting point m ∈ Λ c . The vector field τ 1 (c)X J + τ 2 (c)X H defines a 1-periodic flow; hence, if we define

(2) ζ 1 (c) = τ 1 (c) 2π , ζ 2 (c) = τ 2 (c) 2π , then (ζ 1 (c), ζ 2 (c)
) and (0, 1) form a basis of the period lattice.

2.4. Semiclassical operators. Let I ⊂ (0, 1] be any set which accumulates at 0. If H is a complex Hilbert space, we denote by L(H) the set of linear (possibly unbounded) self-adjoint operators on H with a dense domain.

A space Ψ of semiclassical operators is a subspace of ∈I L(H ), containing the identity, and equipped with a weakly positive principal symbol map, which is an R-linear map

σ : Ψ → C ∞ (M ; R),
with the following properties:

(1)

σ(I) = 1; (normalization) (2) if P, Q are in Ψ and if the composition P • Q is well defined and is in Ψ, then σ(P • Q) = σ(P )σ(Q); (product formula) (3
) if σ(P ) 0, then there exists a function → ǫ( ) tending to zero as → 0, such that P -ǫ( ), for all ∈ I. (weak positivity) If P = (P ) ∈I ∈ Ψ, the image σ(P ) is called the principal symbol of P . There are two major examples of such semiclassical operators. One is given by semiclassical pseudodifferential operators, as described for instance in [START_REF] Dimassi | Spectral asymptotics in the semi-classical limit[END_REF] or [START_REF] Zworski | Semiclassical analysis[END_REF]. The second one is less well known, but developing very fast: semiclassical (or Berezin) -Toeplitz operators, as described in [START_REF] Borthwick | Semiclassical spectral estimates for Toeplitz operators[END_REF][START_REF] Charles | Berezin-Toeplitz operators, a semi-classical approach[END_REF][START_REF] Charles | Symbolic calculus for Toeplitz operators with half-forms[END_REF][START_REF] Charles | Toeplitz operators and Hamiltonian Torus Actions[END_REF][START_REF] Charles | Semi-classical properties of geometric quantization with metaplectic correction[END_REF][START_REF] Ma | Toeplitz operators on symplectic manifolds[END_REF][START_REF] Schlichenmaier | Berezin-Toeplitz quantization for compact Kähler manifolds[END_REF] following the pioneer work [START_REF] Boutet De Monvel | The spectral theory of Toeplitz operators[END_REF].

2.5. Semiclassical spectrum. Recall that when A and B are unbounded self-adjoint operators, they are said to commute when their projector-valued spectral measures commute.

If P = (P ) ∈I and Q = (Q ) ∈I are semiclassical operators on (H ) ∈I , in the sense of Section 2.4, we say that they commute if for each ∈ I the operators P and Q commute. If P and Q commute, we may define for fixed , the joint spectrum of (P , Q ) to be the support of the joint spectral measure. It is denoted by JointSpec(P , Q ). If H is finite dimensional (or, more generally, when the joint spectrum is discrete), then

JointSpec(P , Q ) = (λ 1 , λ 2 ) ∈ R 2 | ∃v = 0, P v = λ 1 v, Q v = λ 2 v .
The joint spectrum of P, Q is the collection of all joint spectra of (P , Q ), ∈ I. It is denoted by JointSpec(P, Q). For convenience of the notation, we will also view the joint spectrum of P, Q as a set depending on .

2.6. Joint spectrum and image of the joint principal symbol. Proposition 2.2. If F := (J, H) : M → R 2 is an integrable system on a 4-dimensional connected symplectic manifold and P, Q are commuting semiclassical operators with principal symbols J, H : M → R, then

E / ∈ F (M ) ⇒ ∃ε > 0 ∃ 0 ∈ I ∀ 0 ∈ I, JointSpec(P , Q ) ∩ B(E, ε) = ∅.
This proposition is well-known for pseudodifferential and Toeplitz operators; it is interesting to notice that, in fact, it directly follows from the axioms we chose for semiclassical operators in Section 2.4.

Proof. If E = (E 1 , E 2 ) does not belong to F (M ), then the function f = (J -E 1 ) 2 + (H -E 2 ) 2
never vanishes. Thus, by the normalization, the product rule and the weak positivity of the principal symbol (items 1, 2 and 3 in section 2.4), we have

(3) (P -E 1 ) 2 + (Q -E 2 ) 2 C,
for some constant C > 0, when is small enough. If fact, since F (M ) is closed (because F is proper), the same holds uniformly when E varies is a small ball. Let Π Q (dλ) and Π P (dµ) be the spectral measures of P and Q respectively (now is fixed). We have

(P -E 1 ) 2 + (Q -E 2 ) 2 = (λ -E 1 ) 2 Π P (dλ) + (µ -E 2 ) 2 Π Q (dµ).
Suppose that (E 1 , E 2 ) belongs to the joint spectrum of (P, Q). Then for each n 0 one can find a vector u n of norm 1 such that

u n ∈ Ran(Π P ([E 1 -1 n , E 1 + 1 n ])) ∩ Ran(Π Q ([E 2 -1 n , E 2 + 1 n ])). Then (P -E 1 ) 2 u n , u n = [E 1 - 1 n ,E 1 + 1 n ] (λ -E 1 ) 2 Π P (dλ)u n , u n 1 n 2 | Π P (dλ)u n , u n | 1 n 2 .
Similarly, (Q

-E 2 ) 2 u n , u n 1 n 2 .
Letting n → ∞, we contradict [START_REF] Brüning | Ernst Spektrale Starrheit gewisser Drehflächen[END_REF]. Thus E ∈ JointSpec(P,Q), which proves the proposition. 2.7. Bohr-Sommerfeld rules. Recall that the Hausdorff distance between two bounded subsets A and

B of R 2 is d H (A, B) := inf{ǫ > 0 | A ⊆ B ǫ and B ⊆ A ǫ },
where for any subset X of R 2 , the set X ǫ is

X ǫ := x∈X {m ∈ R 2 | x -m ǫ}.
If (A ) ∈I and (B ) ∈I are sequences of uniformly bounded subsets of R 2 , we say that

A = B + O( N ) if there exists a constant C > 0 such that d H (A , B ) C N
A ∩ D = B ∩ D + O( 2 ).
Definition 2.3. Let F := (J, H) : M → R 2 be an integrable system on a 4-dimensional connected symplectic manifold. Let P and Q be commuting semiclassical operators with principal symbols J, H : M → R. We say that JointSpec(P, Q) satisfies the Bohr-Sommerfeld rules if for every regular value c of F there exists a small ball B(c, ǫ c ) centered at c, such that, ( 4)

JointSpec(P, Q) = g (2π Z 2 ∩ D) + O( 2 ) on B(c, ǫ c ), with g = g 0 + g 1 ,
where g 0 , g 1 are smooth maps defined on a bounded open set D ⊂ R 2 , g 0 is a diffeomorphism into its image, c ∈ g 0 (D) and the components of g -1 0 = (A 1 , A 2 ) form a basis of action variables. this situation, if is small enough, then g is a diffeomorphism into its image, and its inverse admits an asymptotic expansion in non-negative powers of for the C ∞ topology; we call (g ) -1 an affine chart for JointSpec(P, Q).

Bohr-Sommerfeld rules are known to hold for integrable systems of pseudodifferential operators (thus M is a cotangent bundle) [START_REF] Charbonnel | Comportement semi-classique du spectre conjoint d'opérateurs pseudo-différentiels qui commutent[END_REF][START_REF] Ngo | Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type[END_REF], or for integrable systems of Toeplitz operators on prequantizable compact symplectic manifolds [START_REF] Charles | Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators[END_REF]. It would be interesting to formalize the minimal semiclassical category where Bohr-Sommerfeld rules are valid.

Note that action variables are not unique. Thus, if (g ) -1 is an affine chart for JointSpec(P, Q) and B ∈ GL(2, Z) then B • (g ) -1 is again an affine chart. In view of the discussion in Section 2.3, this remark implies the following proposition.

Proposition 2.4. If F is a semitoric system, then in Definition 2.3, we can assume that A 1 (c 1 , c 2 ) = c 1 . Therefore, there exists an integer k such that the actions A 1 , A 2 satisfy:

(5) dA 1 = dc 1 , dA 2 = (ζ 1 + k)dc 1 + ζ 2 dc 2 ,
where ζ 1 , ζ 2 are defined in (2).

Main Result

We state in this section a more precise version of our main result, Theorem A. Let M ST be the set of semitoric systems (i.e. triples (M, ω, F ) satisfying Definition 1.1) modulo isomorphisms (as defined in Definition 2.1).

For each of the four invariants (1), ( 2), (3), or (4) mentioned in Section 1, we may define a map I j , j = 1, 2, 3, 4, from M ST with value in the appropriate space corresponding to the invariant (we refer to [START_REF] Pelayo | c: Semitoric integrable systems on symplectic 4-manifolds[END_REF] for these spaces; here we simply denote them by B j , j = 1, 2, 3, 4, as their precise definition is not important for our purpose).

Let Q ST be the set of all quantum simple semitoric systems for which the Bohr-Sommerfeld rules hold, equipped with the natural arrow σ : Q ST → M ST induced by the principal symbol map. We introduce now the joint spectrum map

JS : Q ST -→ P(R 2 ) I (P, Q) -→ JointSpec(P, Q),
where we recall that I is the set where the semiclassical parameter varies. Let us denote by P 2 the set of equivalence classes of -dependent subsets of R 2 with respect to the equality modulo O( 2 ) on every ball, and JS : Q ST → P 2 the quotient map of JS. Let Σ ⊂ P 2 be the range of JS, i.e. the subset of all joint spectra of semitoric systems, modulo O( 2 ). Then Theorem A can be rephrased as follows:

Theorem 3.1. For each j = 1, 2, 3, 4, there exists a map Îj : Σ → B j such that the following diagram is commutative:

Q ST JS / / σ Σ Îj M ST I j / / B j Corollary 3.2.
If two quantum simple semitoric systems for which the Bohr-Sommerfeld rules hold have the same joint spectrum modulo O( 2), then the underlying classical systems have the same set of invariants (1), ( 2), (3), (4). In particular, if two quantum Jaynes-Cummings type systems for which the Bohr-Sommerfeld rules hold have the same joint spectrum modulo O( 2 ), then the underlying classical systems are isomorphic.

Proof of Theorem 3.1

Let P, Q be a quantum simple semitoric system with joint principal symbol F = (J, H). Remember that we want to prove that the knowledge of the joint spectrum of P, Q modulo O( 2 ) allows to recover invariants (1) to (4). For the sake of clarity, we divide the proof into five steps.

Step 1. We recover the image F (M ) thanks to Proposition 2.2. Indeed, choose a point E = (E 1 , E 2 ) in R 2 ; assume that the following condition holds:

(C) for every ε > 0 and for every 0 ∈ I, there exists

0 in I such that JointSpec(P , Q ) ∩ B(E, ε) = ∅.
Then Proposition 2.2 implies that E belongs to F (M ). Conversely, assume E ∈ B r , where B r is the set of regular values of F . Because of the Bohr-Sommerfeld rules, there exists a small ball around E in R 2 in which the joint spectrum is a deformation of the lattice Z 2 . Hence when is small enough, this ball always contains some element of the joint spectrum (the number of joint eigenvalues grows like -2 ), which says that Condition (C) holds. Let S be the set of E ∈ R 2 for which (C) holds. We have

B r ⊂ S ⊂ F (M ).
But we know from [START_REF] Ngo | Moment polytopes for symplectic manifolds with monodromy[END_REF]Proposition 2.9] that the closure of B r equals F (M ). Therefore, S = F (M ), which proves that the image F (M ) can be recovered from the joint spectrum.

Note that this step would also work with a weaker hypothesis than the Bohr-Sommerfeld rules. For instance, having a C ∞ 0 functional calculus for the semiclassical operators, or being able to construct microlocal quasimodes (which is common in pseudodifferential or Toeplitz analysis) would be sufficient for recovering F (M ).

Step 2. In this step, we show how to recover the periods of the classical system at regular values from the knowledge of the joint spectrum. In order to do so, we adapt an argument from [START_REF] Ngo | Symplectic inverse spectral theory for pseudodifferential operators[END_REF] for the resolution of a similar inverse problem in dimension 2. Although in our case we are working in dimension 4, which makes the study more difficult, the situation is also simpler by some aspects, because we know from [START_REF] Ngo | Moment polytopes for symplectic manifolds with monodromy[END_REF]Theorem 3.4] that the regular fibers are connected.

Let c 0 be a regular value of F , and let B be a ball centered at c 0 in which the joint spectrum is described by the Bohr-Sommerfeld rules (4). Let D and g be as in the statement of the latter. We can assume that g is a diffeomorphism from g -1 (B) into B. We recall that

JointSpec(P , Q ) ∩ B = g (2π Z 2 ∩ D) ∩ B + O( 2 ),
where B = B + O( 2 ). Now, let χ be a non-negative smooth function with compact support K ⊂ B, equal to 1 on a compact subset of B. We consider the spectral measure

D(λ, ) = c∈JointSpec(P,Q)∩B χ(c)δ c (λ)
where δ c is the Dirac distribution at c. Let F stand for the semiclassical Fourier transform, so that

F (f )(ξ) = 1 (2π ) 2 R 2 exp -i -1 x, ξ f (x)dx,
for smooth, compactly supported functions f , and introduce

Z(t, ) = (2π ) 2 F (D(•, ))(t) = c∈JointSpec(P,Q)∩B χ(c) exp -i -1 c, t .
Thanks to the Bohr-Sommerfeld conditions, we may estimate this quantity as

Z(t, ) = s∈2π Z 2 ∩D ϕ t (g (s), ) + O( ) with ϕ t (s, ) = χ(g (s)) exp -i -1 g (s), t .
Because χ(g (s)) = 0 if s / ∈ D, this yields

Z(t, ) = α∈Z 2 ϕ t (g (2π α), ) + O( ).
By the Poisson summation formula, we thus obtain

Z(t, ) = β∈Z 2 Z β (t, ) + O( ) with Z β (t, ) = 1 (2π ) 2 R 2 exp -i -1 ( β, s + g (s), t ) χ(g (s))ds.
Since g is a diffeomorphism from g -1 (B) into B, we can use the change of variables c = g (s), s = f (c), which yields:

Z β (t, ) = 1 (2π ) 2 R 2 exp -i -1 ( β, f (c) + c, t ) χ(c)| det J f (c)|dc, which means that Z β (t, ) = F (ψ β )(t) where ψ β (c) = exp -i -1 β, f (c) χ(c)| det J f (c)| is a WKB function with phase θ β (c) = -β, f 0 (c) = -β 1 A 1 (c) -β 2 A 2 (c).
Since by equation ( 5)

∇θ β (c) = - β 1 + β 2 (ζ 1 (c) + k) β 2 ζ 2 (c) ,
the associated Lagrangian submanifold is the set

(6) (c, t) ∈ R 4 | (t 1 , t 2 ) = -(β 1 + β 2 (ζ 1 (c) + k), β 2 ζ 2 (c)) .
One can easily check that this submanifold is indeed Lagragian: the 1-form

ν = (β 1 + β 2 (ζ 1 (c) + k)) dc 1 + β 2 ζ 2 (c)dc 2 is closed, as ν = d(β 1 A 1 + β 2 A 2 ).
Since the Jacobian | det J f (c)| does not vanish in the support K of χ, this implies that the semiclassical wavefront set of

Z β (•, ) is WF (Z β (•, )) = (c, t) ∈ R 4 | (t 1 , t 2 ) = -(β 1 + β 2 (ζ 1 (c) + k), β 2 ζ 2 (c)), c ∈ K = L β (K)
To obtain a similar result on Z(•, ), we still need to sum over β ∈ Z 2 . Let

t 0 = (t 0 1 , t 0 2 ) ∈ (R * ) 2 , ε > 0, and let ρ ∈ C ∞ 0 (B(t 0 , ε)).
M = ε + t 0 min K |ζ 2 | max 1, min K |ζ 2 | + |k| + max K |ζ 1 | .
Proof. For such a solution, we have t ε + t 0 , thus

(β 1 + β 2 (ζ 1 (c) + k)) 2 + β 2 2 ζ 2 (c) 2 (ε + t 0 ) 2 , which implies that (7) |β 2 | ε + t 0 min K |ζ 2 | .
Since we also have

|β 1 + β 2 (ζ 1 (c) + k)| ε + t 0 ,
we deduce from the previous inequality that ( 8)

|β 1 | (ε + t 0 ) 1 + |k| + max K |ζ 1 | min K |ζ 2 | ,
which proves the result.

Using the proof of the non-stationary phase lemma, we can write for such a β and any N 1 (2π ) 2 Z β (t, ) equals

i max(|β 1 |, |β 2 |) N R 2 exp -i -1 ( β, f (c) + c, t ) L N (a(c, ))dc,
where a(•, ) is compactly supported and admits an asymptotic expansion in non-negative powers of in the C ∞ topology, and L is the differential operator defined as

Lu = ∇ max(|β 1 |, |β 2 |) u |V | 2 V with V (c) = - t 1 + β 1 + β 2 (ζ 1 (c) + k) t 2 + β 2 ζ 2 (c) . Introduce the function b = (max(|β 1 |, |β 2 |)/|V | 2 )V ; one has |b(c)| = t 1 + β 1 + β 2 (ζ 1 (c) + k) max(|β 1 |, |β 2 |) 2 + t 2 + β 2 ζ 2 (c) max(|β 1 |, |β 2 |) 2 -1/2 .
Then b is uniformly bounded on K for β such that max(|β 1 |, |β 2 |) > M and, for every ℓ ∈ N 2 , there exists a constant C ℓ such that

|∂ c ℓ b| = |∂ c 1 ℓ 1 ∂ c 2 ℓ 2 b| C ℓ on K. Consequently, there exists a constant CN > 0 such that |ρ(t)Z β (t, )| CN max(|β 1 |, |β 2 |) N when max(|β 1 |, |β 2 |) > M .
Therefore, for N 4, we have

β∈Z 2 max(|β 1 |,|β 2 |)>M |ρ(t)Z β (t, )| ĈN N
for some constant ĈN > 0. This shows that only a finite number of terms contribute to ρ(t)Z(t, ) up to O( ∞ ), hence

WF (ρZ(•, )) ⊂ (c 1 , c 2 , -β 1 -β 2 (ζ 1 (c) + k), -β 2 ζ 2 (c)) ∈ R 4 | (c 1 , c 2 ) ∈ K, max(|β 1 |, |β 2 |) > M } and finally WF (Z(•, )) = (c 1 , c 2 , -β 1 -β 2 (ζ 1 (c) + k), -β 2 ζ 2 (c)) ∈ R 4 | (c 1 , c 2 ) ∈ K, β ∈ Z 2 = L(K),
which is exactly the restriction of the period bundle over K (see Section 2.3).

The last part of this step is to explain how one can extract the functions (τ 1 , τ 2 ) from the data of L(K) = β∈Z 2 L β (K), which is the disjoint union of smooth surfaces in R 4 . Endow R 4 with the coordinates (x 1 , x 2 , x 3 , x 4 ), and introduce the plane

Π = {x ∈ R 4 | x 1 = c 0 1 , x 2 = c 0 2 }, for a fixed c 0 ∈ K. Then the set E = L(K) ∩ Π = {(c 0 1 , c 0 2 , -β 1 -β 2 (ζ 1 (c 0 ) + k), -β 2 ζ 2 (c 0 )) | β ∈ Z 2 }
is discrete, and the set {x 4 | x ∈ E} ∩ R * + is bounded from below. Let F be the set of points in E with minimal coordinate x 4 ; then

F = {(c 0 1 , c 0 2 , ζ 1 (c 0 ) + k -β 1 , ζ 2 (c 0 )) | β 1 ∈ Z}.
Again, the set {x 3 | x ∈ F} ∩ R * + is bounded from below, and the point of this set with minimal coordinate

x 3 is (c 0 1 , c 0 2 , ζ 1 (c 0 ), ζ 2 (c 0 )). The connected component of this point in L(K) is the graph of the function c ∈ K → (ζ 1 (c), ζ 2 (c)) = 1 2π (τ 1 (c), τ 2 (c)).
Step 3. Let us now explain how to recover the position of the focus-focus values from the joint spectrum. Thanks to step 1, we know F (M ). By [START_REF] Ngo | Moment polytopes for symplectic manifolds with monodromy[END_REF]Theorem 3.4], we know that the boundary of F (M ) consists of the singularities of elliptic-elliptic and transversally elliptic type, and that the only singular values in the interior of F (M ) are the images of the focusfocus singularities. Let A be any point lying on the boundary ∂F (M ). Let C 1 , . . . , C m f be the images of the focus-focus points in F (M ), labelled in such a way that

J(m 1 ) < J(m 2 ) < . . . < J(m m f ),
where for i in {1, . . . , m f }, m i is the only focus-focus-point in F -1 (C i ).

Consider the distance d = min 1 i m f A -C i and let j ∈ {1, . . . , m f } be such that d = A -C j ; since C j lies in the interior of F (M ), we have that d > 0. Let B r be the set of regular values of F ; for every ε in (0, d], the intersection

X ε = B(A, ε) ∩ F (M )
of the ball of radius ε centered at A with the interior of F (M ) is contained in B r . Thus, from step 2, we can compute the function τ 2|X ε from the joint spectrum. It follows from [37, proposition 3.1] that τ 2 has a logarithmic behavior near C j . Hence, if τ 2|X ε can be extended to a continuous function on B(A, ε) ∩ F (M ), then necessarily ε < d. This allows to find d; the point C j belongs to the circle C of radius d centered at A. Furthermore, the only points in C ∩ F (M ) where τ 2 admits a logarithmic singularity are some of the C i (including C j ), that we recover this way.

We obtain the positions of the other focus-focus values by applying this method recursively. For instance, we recover another point C k by considering circles of growing radius centered at C j , and so on (let us recall that m f is finite).

Step 4. Since we now know precisely the position of the focus-focus values, [START_REF] Pelayo | c: Semiclassical inverse spectral theory for singularities of focus-focus type[END_REF]Theorem 3.3] implies that the Taylor series invariant associated with each focus-focus singularity can be recovered from the joint spectrum.

Step 5. In this step, we prove that from the data of the joint spectrum, one can deduce the polygonal invariant introduced in [START_REF] Ngo | Moment polytopes for symplectic manifolds with monodromy[END_REF].

Recall that a map U ⊂ R n → V ⊂ R n is integral affine on U if it is of the form x ∈ U → Ax + b, where A ∈ GL(n, Z) and b ∈ R n . An integral affine structure on a smooth n-dimensional manifold is the data of an atlas (U i , ϕ i ) such that for all i, the transition function ϕ i • ϕ -1 j is integral affine. As a consequence of the action-angle theorem, the integrable system (J, H) induces an integral affine structure on the set B r of regular values of F . The charts are action variables, that is maps ϕ : U → R 2 where U is a small open subset of B r and ϕ • F generates a T 2 -action.

Let ε i ∈ {-1, 1} and let ℓ ε i i be the vertical segment starting at the focusfocus value C i , going upwards (respectively downwards) if ε i = 1 (respectively ε i = -1), and ending at the boundary of F (M ). Set ℓ ε = i ℓ ε i i . Theorem 4.2 ([38, Theorem 3.8]). For ε ∈ {-1, 1} m f , there exists a homeomorphism Φ from B = F (M ) to ∆ = Φ(B) ⊂ R 2 such that:

(1) Φ |B\ℓ ε is a diffeomorphism into its image, (2) Φ |Br\ℓ ε is affine: it sends the integral affine structure of B r to the standard integral affine structure of R 2 , (3) Φ preserves J: Φ(x, y) = (x, Φ 2 (x, y)), (4) Φ |Br\ℓ ε extends to a smooth multi-valued map from B r to R 2 and for any i ∈ {1, . . . , m f } and any c ∈ li , then

lim (x,y)→c x<x i dΦ(x, y) = 1 0 1 1 lim (x,y)→c x>x i dΦ(x, y), (5)
∆ is a rational convex polygon.

The polygon ∆ is the sought invariant; in fact, the real invariant is a family of such polygons, more precisely the set of all such ∆ for all possible choices of ε ∈ {-1, 1} m f and all their images by linear maps leaving the vertical direction invariant. We refer the reader to [START_REF] Pelayo | c: Semitoric integrable systems on symplectic 4-manifolds[END_REF]Section 4.3] for more precise statements. Proof. Once a starting point c 0 ∈ B r is chosen (which, by convention, is taken to be on the left of the first focus-focus critical value, when these values are ordered by non-decreasing abscissae), the affine map Φ |Br\ℓ ε is uniquely determined by the affine structure. Indeed, the set B r \ ℓ ε is simply connected and Φ is the developing map of the induced affine structure. The crucial observation is that it follows from the construction in [START_REF] Ngo | Moment polytopes for symplectic manifolds with monodromy[END_REF] that the map Φ |B\ℓ ε is the natural extension of Φ |Br\ℓ ε to the boundary of B r \ℓ ε away from the half-lines ℓ ε , and this boundary consists of elliptic (or transversally elliptic) singularities. Precisely, the extension is obtained as follows. Near a 1-dimensional family of transversally elliptic singularities, we use the normal form due to Miranda and Zung [START_REF] Miranda | Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems[END_REF]: if c e is a transversally elliptic value, there exist a symplectomorphism ϕ from a neighborhood of F -1 (c e ) in M to a neighborhood of {I = x = ξ = 0} in T * S 1 × R 2 with coordinates ((θ, I), (x, ξ)) and standard symplectic form dI ∧ dθ + dξ ∧ dx, which sends the set {F = constant} to the set {I = constant, x 2 + ξ 2 = constant}, and a smooth function g such that (F • ϕ -1 )(θ, I, x, ξ) = g(I, x 2 + ξ 2 ) where ϕ -1 , g are defined. Let (A 1 , A 2 ) be an affine chart for B r inside this neighborhood where the normal form holds. Since (I, (x 2 + ξ 2 )/2) is also an affine chart, there exists a matrix A = α β γ δ ∈ GL(2, Z) such that (9) A 1 (c) = αI + β(x 2 + ξ 2 )/2; A 2 (c) = γI + δ(x 2 + ξ 2 )/2

where m = ϕ -1 (θ, I, x, ξ) ∈ M is a regular point for F and c = F (m). Since α, β, γ, δ are constant, Formula (9) naturally gives the required extension of (A 1 , A 2 ) (and hence Φ) to the boundary near c e . Near an elliptic-elliptic point, we can apply the same reasoning, using Eliasson's normal form [START_REF] Eliasson | Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case[END_REF].

In view of the proposition, step 5 will be treated as soon as we show that the integral affine structure on B r can be recovered from the joint spectrum JointSpec(P, Q) up to O( 2 ), which can be done as follows. From the previous steps, we can recover F (M ) and the position of the focus-focus values C i = (x i , y i ), 1 i m f . Therefore, we know the set of regular values B r , which is the interior of F (M ) minus the focus-focus critical values.

In a small ball B 0 ⊂ B r , we can construct action variables (A 1 , A 2 ). Indeed, from step 2 we can recover the functions τ 1 , τ 2 on B 0 . Fixing a point s ∈ B 0 , we can pick for every point c ∈ B 0 a smooth path γ c : [0, 1] → B 0 such that γ c (0) = s, γ c (1) = c and compute

A (0) 1 (c) = c 1 ; A (0) 2 (c) = 1 0 ζ 1 (γ c (t)) ζ 2 (γ c (t))
, γ ′ c (t) dt, where we recall that τ i = 2πζ i for i = 1, 2. In this way, we have constructed the integral affine structure of B r from the joint spectrum. In remains to apply Proposition 4.3 to construct Φ, and hence ∆ by Theorem 4.2.

Step 6. It only remains to prove that we can recover the height invariant associated with each focus-focus singularity from the joint spectrum. In order to do so, let i ∈ {1, . . . , m f } and consider a sequence (Y n ) n∈N of points of B such that every Y n has the same abscissa as C i and ordinate smaller than the one of C i , and such that Y n -→ n→+∞ C i . We may assume that Y 0 lies on the boundary of B. Let Φ be a homeomorphism from B to ∆ as in the previous step. Then the point

P = lim n→+∞ Φ(Y n )
is well-defined and P is the image of the focus-focus value in the polygon ∆. The height invariant that we seek is the difference between the ordinate of P and the ordinate of Φ(Y 0 ).

Remark 4.4. Another way of obtaining the height invariant associated with C i would have been to count the joint eigenvalues lying on a vertical line below C i and use a Weyl law to relate this number to the volume of the set J -1 (C i ) ∩ {H < H(m i )}. Although it may seem more natural than our method, it is also more technical, and that is why we have chosen not to treat the problem this way.

Figure 1 .

 1 Figure 1. Semiclassical joint spectrum of the Jaynes-Cummings model.

  for all ∈ I. If A or B are not uniformly bounded, we shall say that A = B + O( N ) on a ball D if there exists a sequence of sets D , all diffeomorphic to D, such that D = D + O( 2 ) and

Lemma 4 . 1 .

 41 If there exists a solution (c, t) of (6) with t in the support of ρ, then β is such that max(|β 1 |, |β 2 |) M where M is defined as

Proposition 4 . 3 .

 43 Given any ε ∈ {-1, 1} m f , the corresponding polygon ∆ = ∆ ε is determined by the integral affine structure of B r .

Acknowledgements. Part of this paper was written at the Institute for Advanced Study (Princeton, NJ) during the visit of the last two authors in July 2014, and they are very grateful to Helmut Hofer for the hospitality. AP was partially supported by NSF DMS-1055897, the STAMP Program at the ICMAT research institute (Madrid), and ICMAT Severo Ochoa grant Sev-2011-0087. VNS is partially supported by the Institut Universitaire de France, the Lebesgue Center (ANR Labex LEBESGUE), and the ANR NOSEVOL grant.