From open quantum systems to open quantum maps

In this paper we show that for a class of open quantum systems satisfying a natural dynamical assumption (see §2.2) the study of the resolvent, and hence of scattering, and of resonances, can be reduced, in the semiclassical limit, to the study of open quantum maps, that is of finite dimensional quantizations of canonical relations obtained by truncation of symplectomorphisms derived from the classical Hamiltonian flow (Poincaré return maps). We first explain the result in a simplified setting. For that consider the Schrödinger operator (1.1) and let Φ t be the corresponding classical flow on T * R n ∋ (x, ξ): Φ t (x, ξ) def = (x(t), ξ(t)) , x ′ (t) = 2ξ(t) , ξ ′ (t) = -dV (x(t)) , x(0) = x , ξ(0) = ξ . Figure 1. An example of a potential, V ∈ C ∞ c (R 2 ), to which the results apply: the Hamiltonian flow is hyperbolic on the trapped set in a range of energies -see [38,Appendix c]. In this example each energy surface p -1 (E) is three dimensional, so the Poincaré section is two dimensional as shown in Fig. 2. Equivalently, this flow is generated by the Hamilton vector field The energy shift by -1 allows us to focus on the quantum and classical dynamics near the energy E = 0, which will make our notations easier 1 . We assume that the Hamiltonian flow has no fixed point at this energy: dp↾ p -1 (0) = 0. The trapped set at any energy E is defined as (1.4) K E def = {(x, ξ) ∈ T * R n : p(x, ξ) = E , Φ t (x, ξ) remains bounded for all t ∈ R} . The information about spectral and scattering properties of P = P (h) in (1.1) can be obtained by analyzing the resolvent of P , R(z) = (Pz) -1 , Im z > 0 , and its meromorphic continuation -see for instance [33] and references given there. More recently semiclassical properties of the resolvent have been used to obtain local smoothing and Strichartz estimates, leading to applications to nonlinear evolution equations -see [14] for a recent result and for pointers to the literature. In the physics literature the Schwartz kernel of R(z) is referred to as Green's function of the potential V . The operator P has absolutely continuous spectrum on the interval [-1, ∞); nevertheless, its resolvent R(z) continues meromorphically from Im z > 0 to the disk D(0, 1), in the sense that χR(z)χ, χ ∈ C ∞ c (R n ), is a meromorphic family of operators, with poles independent of the choice of χ ≡ 0 (see for instance [41,Section 3] and [39,Section 5]). The multiplicity of the pole z ∈ D(0, 1) is given by m R (z) where the integral runs over a sufficiently small circle around z. We now assume that at energy E = 0, the flow Φ t is hyperbolic on the trapped set K 0 and that this set is topologically one dimensional. Hyperbolicity means [24,Def. 17.4.1] that at any point ρ = (x, ξ) ∈ K 0 the tangent space to the energy surface splits into the neutral (RH p (ρ)), stable (E - ρ ), and unstable (E + ρ ) directions: (1.5) 1 There is no loss of generality in this choice: the dynamics of the Hamiltonian ξ 2 + Ṽ (x) at some energy E > 0 is equivalent with that of ξ 2 + Ṽ /E -1 at energy 0, up to a time reparametrization by a factor √ E. The same rescaling holds at the quantum level. A schematic view of a Poincaré section Σ = ⊔ j Σ j for K E inside p -1 (E). The flow near K E can be described by an ensemble of symplectomorphisms between different components Σ j -see §2.2 for abstract assumptions and a discussion why they are satisfied when the flow is hyperbolic on K E and K E has topological dimension one. The latter condition simply means that the intersections of K E with Σ j 's are totally disconnected. this decomposition is preserved through the flow, and is characterized by the following properties: (1.6) ∃ C > 0, ∃ λ > 0, d exp tH p (ρ)v ≤ C e -λ|t| v , ∀ v ∈ E ∓ ρ , ±t > 0 . When K 0 is topologically one dimensional we can find a Poincaré section which reduces the flow near K 0 to a combination of symplectic transformations, called the Poincaré map F : see Fig. 2 for a schematic illustration and §2.2 for a precise mathematical formulation. The structural stability of hyperbolic flows [24,Thm. 18.2.3] implies that the above properties will also hold for any energy E in a sufficientlys short interval [-δ, δ] around E = 0, in particular the flow near K E can be described through a Poincaré map F E . Under these assumptions, we are interested in semiclassically locating the resonances of the operator P (h) in a neighbourhood of this energy interval: where δ, M 0 are independent of h ∈ (0, 1]. Here the h log(1/h)-size neighbourhood is natural in view of results on resonance free regions in case of no trapping -see [26]. To characterize the resonances in R(δ, M 0 , h) we introduce a family of "quantum propagators" quantizing the Poincaré maps F E . Theorem 1. Suppose that Φ t is hyperbolic on K 0 and that K 0 is topologically one dimensional. More generally, suppose that P (h) and Φ t satisfy the assumptions of §2.1- §2.2. Then, for any δ > 0 small enough and any M 0 > 0, there exists h 0 > 0 such that there exists a family of matrices, {M(z, h), z ∈ R(δ, M 0 , h), h ∈ (0, h 0 ]} , holomorphic in the variable z, and satisfying such that for any h ∈ (0, h 0 ], the zeros of give the resonances of P (h) in R(δ, M 0 , h), with correct multiplicities. The matrices M(z, h) are open quantum maps associated with the Poincaré maps F Re z described above: for any L > 0, there exist a family of h-Fourier integral operators, {M(z, h)}, quantizing the Poincaré maps F Re z (see §2.3.2 and §3.3), and projections Π h (see The statement about the multiplicities in the theorem says that (1.8) A more precise version of Theorem 1, involving complex scaling and microlocally deformed spaces (see §3.4 and §3.5 respectively), will be given in Theorem 2 in §5.4. In particular Theorem 2 gives us a full control over both the cutoff resolvent of P , χR(z)χ, and the full resolvent (P θ,Rz) -1 of the complex scaled operator P θ,R , in terms of the family of matrices M(z, h); for this reason, the latter is often called an effective Hamiltonian for P . The mathematical applications of Theorem 1 and its refined version below include simpler proofs of fractal Weyl laws [43] and of the existence of resonance free strips [31]. The advantage lies in eliminating flows and reducing the dynamical analysis to that of maps. That provides an implicit second microlocalization without any technical complication (see [43, §5]). The key is a detailed understanding of the operators M(z, h) stated in the theorem. Relation to semiclassical trace formulae. The notation ζ(z, h) in the above theorem hints at the resemblance between this determinant and a semiclassical zeta function. Various such functions have been introduced in the physics literature, to provide approximate ways of computing eigenvalues and resonances of quantum chaotic systems -see [47,20,10]. These semiclassical zeta functions are defined through formal manipulations starting from the Gutzwiller trace formula -see [42] for a mathematical treatment and references. They are given by sums, or Euler products, over periodic orbits where each term, or factor is an asymptotic series in powers of h. Most studies have concentrated on the zeta function defined by the principal term, without h-corrections, which strongly resembles the Selberg zeta function defined for surfaces of constant negative curvature. However, unlike the case of the Selberg zeta function, there is no known rigorous connection between the zeroes of the semiclassical zeta function and the exact eigenvalues or resonances of the quantum system, even in the semiclassical limit. Nevertheless, numerical studies have indicated that the semiclassical zeta function admits a strip of holomorphy beyond the axis of absolute convergence, and that its zeroes there are close to actual resonances [10,48]. The traces of M(z, h) k , k ∈ N admit semiclassical expressions as sums over periodic points, which leads to a formal representation of as a product over periodic points. That gives it the same form as the semiclassical zeta functions in the physics literature. In this sense, the function ζ(z, h) is a resummation of these formal expressions. As will become clear from its construction below, the operator M(z, h) is not unique: it depends on many choices which affect the remainder term O(h L ) in (1.7). However, the zeroes of ζ(z, h) in R(δ, M 0 , h) are the exact resonances of the quantum Hamiltonian. Comments on quantum maps in the physics literature. Similar methods of analysis have been introduced in the theoretical physics literature devoted to quantum chaos. The classical case involves a reduction to the boundary for obstacle problems: when the obstacle consists of several strictly convex bodies, none of which intersects a convex hull of any other two bodies, the flow on the trapped set is hyperbolic. The reduction can then be made to boundaries of the convex bodies, resulting with operators quantization Poincaré maps -see Gaspard and Rice [16], and for a mathematical treatment Gérard [18], in the case of two convex bodies, and [28, §5.1], for the general case. Fig. 3 illustrates the trapped set in the case of three discs. The semiclassical analogue of the two convex obstacle, a system with one closed hyperbolic orbit, was treated by Gérard and the second author in [19]. The approach of that paper was also based on the quantization of the Poincaré map near this orbit. A reduction of a more complicated quantum system to a quantized Poincaré map was proposed in the physics literature. Bogomolny [4] studied a Schrödinger operator P (h) with discrete spectrum, and constructed a family of energy dependent quantum transfer operators T (E, h), which are integral operators acting on a hypersurface in the configuration The section is the union of the three coball bundles of circle arcs (in red) parametrized by s (the length parameter on the circle, horizontal axis), and cos ϕ (vertical axis), where ϕ is the angle between the velocity after impact and the tangent to the circle. Green, blue,red strips correspond to different regions of forward escape; they are bounded by components of the stable manifold. The trapped set, T , shown in yellow, is the intersection of the latter with the unstable manifold. space. These transfer operators are asymptotically unitary as h → 0. The eigenvalues of P (h) are then obtained, in the semiclassical limit, as the roots of the equation det(1 -T (E)) = 0. Smilansky and co-workers derived a similar equation in the case of closed Euclidean 2-dimensional billiards [13], replacing T (E) by a (unitary) scattering matrix S(E) associated with the dual scattering problem. Prosen [35] generalized Bogomolny's approach to a nonsemiclassical setting. Bogomolny's method was also extended to study quantum scattering situations [17,32]. Open quantum maps have first been defined in the quantum chaos literature as toy models for open quantized chaotic systems (see [29, §2.2], [30, §4.3] and references given there). They generalized the unitary quantum maps used to mimic bound chaotic systems [11]. Some examples of open quantum maps on the 2-dimensional torus or the cylinder, have been used as models in various physical settings: Chirikov's quantum standard map (or quantum kicked rotator) was first defined in the context of plasma physics, but then used as well to study ionization of atoms or molecules [9], as well as transport properties in mesoscopic quantum dots [46]. Other maps, like the open baker's map, were introduced as clean model systems, for which the classical dynamics is well understood [36,30]. The popularity of quantum maps mostly stems from the much simplified numerical study they offer, both at the quantum and classical levels, compared with the case of Hamiltonian flows or the corresponding Schrödinger operators. For instance, the distribution of resonances and resonant modes has proven to be much easier to study numerically for open quantum maps, than for realistic flows [7,37,29,25,27]. Precise mathematical definitions of quantum maps on the torus phase space are given in [29, §4.3-4.5]. Organization of the paper. In the remainder of this section we give assumptions on the operator P and on the corresponding classical dynamical system, in particular we introduce a Poincaré section Σ and map associated with the classical flow. We refer to results of Bowen and Walters [8] to show that these assumptions are satisfied if the trapped set supports a hyperbolic flow, and is topologically one dimensional, which is the case considered in Theorem 1. In §3 we recall various tools needed in our proof: pseudodifferential calculus, the concept of semiclassical microlocalization, local h-Fourier integral operators associated to canonical tranformation (these appear in Theorem 1), complex scaling (used to define resonances as eigenvalues of nonselfadjoint Fredholm operators), microlocally deformed spaces, and Grushin problems used to define the effective Hamiltonians. In §4 we follow a modified strategy of [42] and construct a microlocal Grushin problem associated with the Poincaré map on Σ. No knowledge of that paper is a prerequisite but the self-contained discussion of the problem for the explicit case of S 1 given in [42, §2] can illuminate the complicated procedure presented here. In [42, §2] one finds the proof of the classical Poisson formula using a Grushin problem approach used here. Because of the hyperbolic nature of the flow the microlocal Grushin problem cannot directly be made into a globally well-posed problem -see the remark at the end of §4. This serious difficulty is overcome in §5 by adding microlocal weights adapted to the flow. This and suitably chosen finite dimensional projections lead to a well posed Grushin problem, with an effective Hamiltonian essentially given by a quantization of the Poincaré map: this fact is summarized in Theorem 2, from which Theorem 1 is a simple corollary. Here we carefully state the needed assumptions on quantum and classical levels. 2.1. Assumptions on the quantum Hamiltonian P (h). Our results apply to operators P (h) satisfying general assumptions given in [31, §3.2] and [43, (1.5), (1.6)]. In particular, they apply to certain elliptic differential operators on manifolds X of the form where R > 0 is large and X R is a compact subset of X. The reader interested in this higher generality should consult those papers. Here we will recall these assumptions only in the (physical) case of differential operators on X = R n . We assume that (2.1) where and a α (x, h) = a α (x) is independent of h for |α| = 2. Furthermore, for some C 0 > 0 the functions a α (x, h) have holomorphic extensions to they are bounded uniformly with respect to h, and a α (x, h) = a 0 α (x) + O(h) on that set. Let P (x, ξ) denote the (full) Weyl symbol of the operator P , so that P = P w (x; hD; h), and assume 2), uniformly with respect to (ξ, h) ∈ K×]0, 1] for any compact set K ⋐ R n (here, and below, ⋐ means that the set on the left is a pre-compact subset of the set on the right). We also assume that P is classically elliptic: (2.4) and that P is self-adjoint on L 2 (R n ) with domain H 2 (R n ). The Schrödinger operator (1.1) corresponds to the choices |α|=2 a α ξ α = |ξ| 2 , a α ≡ 0 for |α| = 1, and a 0 (x) = V (x) -1. The assumption (2.3) show that we can also consider a slowly decaying potential, as long as it admits a holomorphic extension in (2.2). The dynamical assumptions we need roughly mean that the flow Φ t on the energy shell p -1 (0) ⊂ T * X can be encoded by a Poincaré section, the boundary of which does not intersect the trapped set K 0 . The abstract assumptions below are satisfied when the flow is hyperbolic on the trapped set which is assumed to be topologically one dimensional -see Proposition 2.1. To state the assumption precisely, we notice that is the semi-classical principal symbol of the operator P (x, hD; h). We assume that the characteristic set of p (that is, the energy surface p -1 (0)) is a simple hypersurface: (2.6) dp = 0 on p -1 (0). Like in the introduction, we denote by the flow generated by the Hamilton vector field H p (see (1.2)). Our assumptions on p(x, ξ) ensure that, for E close to 0, we still have no fixed point in p -1 (E), and the trapped set K E (defined in (1.4)) is a compact subset of p -1 (E). We now assume that there exists a "nice" Poincaré section for the flow near K 0 , namely finitely many compact contractible smooth hypersurfaces Σ k ⊂ p -1 (0), k = 1, 2, . . . , N with smooth boundaries, such that (2.7) H p is transversal to Σ k uniformly up to the boundary, (2.9) We call Poincaré section the disjoint union The functions ρ → ρ ± (ρ), ρ → t ± (ρ) are uniquely defined (ρ ± (ρ) will be called respectively the successor and predecessor of ρ). They remain well-defined for ρ in some neighbourhood of K 0 in p -1 (0)) and, in such a neighbourhood, depend smoothly on ρ away from Σ. In order to simplify the presentation we also assume the successor of a point ρ ∈ Σ k belongs to a different component: The section can always be enlarged to guarantee that this condition is satisfied. For instance, for K 0 consisting of one closed orbit we only need one transversal component to have (2.7)-(2.8); to fulfill (2.10) a second component has to be added. We recall that hypersurfaces in p -1 (0) that are transversal to H p are symplectic. In fact, a local application of Darboux's theorem (see for instance [23, §21.1]) shows that we can make a symplectic change of variables in which p = ξ n and ) can be chosen as coordinates on Σ. Since ω↾ p -1 (0) = n-1 j=1 dξ j ∧ dx j , that means that ω↾ Σ is nondegenerate. The local normal form p = ξ n will be used further in the paper (in its quantized form). The final assumption guarantees the absence of topological or symplectic peculiarities: There exists a set Σ k ⋐ T * R n-1 with smooth boundary, and a symplectic diffeomorphism κ k : Σ k → Σ k which is smooth up the boundary together with its inverse. We assume that κ k extends to a neighbourhood of Σ k in T * R n . (2.11) In other words, there exist symplectic coordinate charts on Σ k , taking values in Σ k . The following result, due to Bowen and Walters [8], shows that our assumptions are realized in the case of 1-dimensional hyperbolic trapped sets. Proposition 2.1. Suppose that the assumptions of §2.1 hold, and that the flow Φ t ↾ K 0 is hyperbolic in the standard sense of Eqs. (1.5,1.6). Then the existence of Σ satisfying (2.7)-(2.11) is equivalent with K 0 being topologically one dimensional. Remark. Bowen and Walters [8] show more, namely the fact that the sets {Σ k } can be chosen of small diameter, and constructed such that Σ ∩ K 0 forms a Markov partition for the Poincaré map. Small diameters ensures that (2.11) holds, while, as mentioned before, (2.10) can always be realized by adding some more components. Proposition 2.1 shows that the assumptions of Theorem 1 imply the dynamical assumptions made in this section. The proof of [38,Appendix c] shows that the following example of "three-bumps potential", x k = (cos(2πk/3), sin(2πk/3)) , satisfies our assumptions as long as R > 1 is large enough (see Fig. 1). The Poincaré map. Here we will analyze the Poincaré map associated with the Poincaré section discussed in §2.2, and its semiclassical quantization. More precisely, let us identify Σ k 's with Σ k using κ k given in (2.11), so that the Poincaré section The map f : T -→ T , ρ -→ f (ρ) def = ρ + (ρ) (see the notation of (2.9)) is the Poincaré map for Φ t ↾ K 0 . It is a Lipschitz bijection. The decomposition T = k T k allows us to define the arrival and departure subsets of T : For each k we call J + (k) ⊂ {1, . . . , N} the set of indices i such that D ik is not empty (that is, for which T i is a successor of T k ). Conversely, the set J -(i) refers to the predecessors of T i . Using this notation, the map f obviously decomposes into a family of Lipschitz bijections f ik : D ik → A ik . Similarly to the maps ρ ± , each f ik can be extended to a neighbourhood of D ik , to form a family of local smooth symplectomorphisms where Since our assumption on K 0 is equivalent with the fact that the reduced trapped set T is totally disconnected, we may assume that the sets {D ik } i∈J + (k) (resp. the sets {A ik } k∈J -(i) ) are mutually disjoint. We will call Notice that, for any index i, the sets D i , A i both contain the set T i , so they are not disjoint. We will also define the tubes T ik ⊂ T * X containing the trajectories between D ik and A ik : (2.12) See Fig. 4 for a sketch of these definitions, and Fig. 5 for an artistic view of T ik The maps F ik will be grouped into the symplectic bijection F between k D k and k A k . We will also call F the Poincaré map, which can be viewed as a symplectic relation on Σ. We will sometimes identify the map F ik with its action on subsets of T * R n-1 . Using the continuity of the flow Φ t , we will show in §4.1.1 that the above structure can be continuously extended to a small energy interval z ∈ [-δ, δ]. The Poincaré map for the flow in p -1 (z) will be denoted by F z = (F ik,z ) 1≤i,k≤N (see §4.1.1 for details). In the case of K 0 supporting a hyperbolic flow, a structural stability of K z holds in a stronger sense: the flows Φ t ↾ Kz and Φ t ↾ K 0 are actually orbit-conjugate (that is, conjugate up to time reparametrization) by a homeomorphism close to the identity. [24, Thm. 18.2.3]. 2.3.2. Quantization of the Poincaré map. In this section we make more explicit the operator M(z, h) used in Theorem 1. The semiclassical tools we are using will be recalled in §3. Let us first focus on a single component and which is associated with the symplectomorphism F ik . (h-FIOs are defined in §3.3, and WF ′ h is defined in (3.9) below). Being associated to the symplectic map F ik means the following thing: for any a ∈ C ∞ c ( A ik ), the quantum operator Op w h (a) transforms as follows when conjugated by M ik (h) where the symbol α ik ∈ S δ (T * R n-1 ) is independent of a, α ik = 1 on some neighbourhood of T k in Σ k , and b ∈ S δ (T * R n-1 ), for every δ > 0. Here Op w h denotes the semiclassical Weyl quantization on R 2(n-1) (see eq.(3.1)), and S δ (T * R n-1 ) is the symbol class defined in §3.1. The necessity to have δ > 0 in (2.14) comes from the slightly exotic nature of our Fourier integral operator, due to the presence of some mild exponential weights -see §3.5 below. The property (2.14), which is a form of Egorov's theorem, characterizes M ik (h) as a semiclassical Fourier integral operator associated with F ik (see [42,Lemma 2] and [15, §10.2] for that characterization). We can then group together the M ik (h) into a single operator-valued matrix (setting We call this M(h) a quantization of the Poincaré map F . The operators M(z, h) in Theorem 1 will also holomorphically depend on z ∈ R(δ, M 0 , h), such that for each z ∈ R(δ, M 0 , h) ∩ R the family (M(z, h)) h∈(0,1] is an h-Fourier integral operator of the above sense. Comment on notation. Most of the estimates in this paper include error terms of the type O(h ∞ ), which is natural in all microlocal statements. To simplify the notation we adopt the following convention (except in places where it could lead to confusion): with norms appropriate to context. Since most estimates involve functions u microlocalized to compact sets, in the sense that, u - ), the norms are almost exclusively L 2 norms, possibly with microlocal weights described in §3. 5. means an open neighbourhood of the set A inside the set B. Starting with §4, we denote the Weyl quantization of a symbol a by the same letter a = a w (x, hD). This makes the notation less cumbersome and should be clear from the context. Finally, we warn the reader that from §4 onwards the original operator P is replaced by the complex scaled operator P θ,R , whose construction is recalled in §3.4. Because of the formula (3.13), that does not affect the results formulated in this section. In this section we present background material and references needed for the proofs of the theorems. 3.1. Semiclassical pseudodifferential calculus. We start by defining a rather general class of symbols (that is, h-dependent functions) on the phase space where . Most of the time we will use the class with δ = 0 in which case we drop the subscript. When m = k = 0, we simply write S(T * R d ) or S for the class of symbols. In the paper d = n (the dimension of the physical space) or d = n-1 (half the dimension of the Poincaré section), and occasionally (as in (2.13)) d = 2n -2, depending on the context. The quantization map, in its different notational guises, is defined as follows , ξ e i x-y,ξ /h u(y)dydξ , (3.1) and we refer to [12,Chapter 7] for a detailed discussion of semiclassical quantization (see also [40,Appendix]), and to [15,Appendix D.2] for the semiclassical calculus for the symbol classes given above. ) is exactly represented by a unique (full) symbol a(x, ξ; h). It is useful to consider only certain equivalence classes of this full symbol, thus defining a principal symbol map -see [15,Chapter 8]: The combination . The main property of this principal symbol map is to "restore commutativity": Certain symbols in S m,0 (T * R d ) admit an asymptotic expansion in powers of h, such symbols (or the corresponding operator) are called classical, and make up the subclass S m,0 cl (T * R d ) (the corresponding operator class is denoted by Ψ m,0 cl (R d )). For any operator A ∈ Ψ m,0 cl (R d ), its principal symbol σ h (A) admits as representative the h-independent function a 0 (x, ξ), first term in (3.2). The latter is also usually called the principal symbol of a. In §3.5 we will introduce a slightly different notion of leading symbol, adapted to a subclass of symbols in S(T * R) larger than S cl (T * R d ). The semiclassical Sobolev spaces, H s h (R d ) are defined using the semiclassical Fourier transform, F h : Unless otherwise stated all norms in this paper, • , are L 2 norms. We recall that the operators in Ψ(R d ) are bounded on L 2 uniformly in h, and that they can be characterized using commutators by Beals's Lemma (see [12,Chapter 8] and [43,Lemma 3.5] for the S δ case): For a given symbol a ∈ S(T * R d ) we follow [42] and say that the essential support is contained in a given compact set The essential support is then the intersection of all such K's. Here S denotes the Schwartz space. the semiclassical wavefront set of A. (In this paper we are concerned with a purely semiclassical theory and will only need to deal with compact subsets of T * R d . Hence, we won't need to define noncompact essential supports). Microlocalization. We will also consider spaces of L 2 functions (strictly speaking, of h-dependent families of functions) which are microlocally concentrated in an open set The semiclassical wave front set of u ∈ H(V ) is defined as: The condition (3.7) can be equivalently replaced with a w u = O S (h ∞ ), since we may always take a ∈ S (T * R d ). This set obviously satisfies WF h (u) ⋐ V . Notice that the condition does not characterize the individual functions u(h), but the full sequence as h → 0. We will say that an h-dependent family of operators T = (T (h)) h∈(0,1] : Such a family of operators is microlocally defined on V if one only specifies (or considers) its action on states u ∈ H(V ), modulo O S ′ →S (h ∞ ). For instance, T is said to be asymptotically uniformly bounded on H(V ) if Two tempered operators T, T ′ are said to be microlocally equivalent on V , iff for any If there exists an open subset W ⋐ T * R k and L ∈ R such that T maps any u ∈ H(V ) into a state T u ∈ h -L H(W ), then we will write and we say that T is defined microlocally in W × V . For such operators, we may define only the part of the (twisted) wavefront set which is inside W × V : (3.9) , is an extension of the microlocally defined T (h), see [15,Chapter 10]. Local h-Fourier integral operators. We first present a a class of globally defined h-Fourier integral operators following [42] and [15,Chapter 10]. This global definition will then be used to define Fourier integral operators microlocally. Let (A(t)) t∈[-1,1] be a smooth family of selfadjoint pseudodifferential operators, , where the dependence on t is smooth, and WF h (A(t)) ⊂ Ω ⋐ T * R d , in the sense of (3.5). We then define a family of operators The family (U(t)) t∈[-1,1] is an example of a family of unitary h-Fourier integral operators, associated to the family of canonical transformations κ(t) generated by the (timedependent) Hamilton vector fields H a 0 (t) . Here the real valued function a 0 (t) is the principal symbol of A(t) (see (3.2)), and the canonical transformations κ(t) are defined through If U = U(1), say, and the graph of κ( 1) is denoted by C, we conform to the usual notation and write Here the twisted graph In words, U is a unitary h-Fourier integral operator associated to the canonical graph C (or the symplectomorphism κ(1) defined by this graph). Locally all unitary h-Fourier integral operators associated to canonical graphs are of the form U(1), since each local canonical transformation with a fixed point can be deformed to the identity, see [42,Lemma 3.2]. For any χ ∈ S(T * R d ), the operator U(1) χ w , with χ ∈ S(T * R d ) is still a (nonunitary) h-Fourier integral operator associated with C. The class formed by these operators, which are said to "quantize" the symplectomorphism κ = κ(1), depends only on κ, and not on the deformation path from the identity to κ. This can be seen from the Egorov characterization of Fourier integral operators -see [42,Lemma 2] or [15, §10.2]. Let us assume that a symplectomorphism κ is defined only near the origin, which is a fixed point. It is always possible to locally deform κ to the identity, that is construct a family of symplectomorphisms κ(t) on T * R d , such that κ(1) coincides with κ in some neighbourhood V of the origin [42,Lemma 3.2]. If we apply the above construction to get the unitary operator U(1), and use a cutoff χ ∈ S(T * R d ), supp χ ⋐ V , then the operator U(1)χ w is an h-Fourier integral operator associated with the local symplectomorphism κ↾ V . Furthermore, if there exists a neighbourhood For an open set V ⋐ R d and κ a symplectomorphism defined in a neighbourhood V of V , we say that a tempered operator T satisfying is equivalent to a unitary h-Fourier integral operator associated with κ↾ V ρ , as defined by the above procedure. The microlocally defined operators can also be obtained by oscillatory integral constructions -see for instance [31, §4.1] for a brief self-contained presentation. An example which will be used in §4.1 is given by the standard conjugation result, see [42,Proposition 3.5] or [15,Chapter 10] for self-contained proofs. Suppose that P ∈ Ψ m,0 cl (R d ) is a semi-classical real principal type operator, namely its principal symbol p = σ h (P ) is real, independent of h, and the Hamilton flow it generates has no fixed point at energy zero: p = 0 =⇒ dp = 0. Then for any ρ 0 ∈ p -1 (0), there exists a canonical transformation, κ, mapping V = neigh((0, 0), and a unitary microlocal h-Fourier integral operator U : H(V ) → H(κ(V )) associated to κ, such that While ξ n is the (classical) normal form for the Hamiltonian p in V , the operator hD xn is the quantum normal form for P , microlocally in V . The definition of h-Fourier integral operators can be generalized to graphs C associated with certain relations between phase spaces of possibly different dimensions. Namely, if a relation , then one can associate with this relation (microlocally in some neighbourhood) a family of h-Fourier integral operators T : The important property of these operators is that their composition is still a Fourier integral operator associated with the composed relations. 3.4. Complex scaling. We briefly recall the complex scaling method of Aguilar-Combes [1] -see [41], [39], and references given there. In most of this section, this scaling is independent of h, and allows to obtain the resonances (in a certain sector) for all operators P (h), h ∈ (0, 1], where P (h) satisfies the assumptions of §2.1. For any 0 ≤ θ ≤ θ 0 and R > 0, we define Γ θ,R ⊂ C n to be a totally real deformation of R n , with the following properties: If R is large enough, the coefficients of P continue analytically outside of B(0, R), and we can define a dilated operator: where P is the holomorphic continuation of the operator P , and ũ is an almost analytic extension of u ∈ C ∞ c (Γ θ,R ) from the totally real submanifold Γ θ,R to neigh(Γ θ,R , C n ). The operator P θ,Rz is a Fredholm operator for 2θ > arg(z + 1) > -2θ. That means that the resolvent, (P θ,Rz) -1 , is meromorphic in that region, the spectrum of P θ,R in that region is independent of θ and R, and consists of the quantum resonances of P . To simplify notations we identify Γ θ,R with R n using the map, and using this identification, consider P θ,R as an operator on R n , defined by (S -1 θ,R ) * P θ,R S * θ,R (here S * means the pullback through S) We note that this identificaton satisfies The identification of the eigenvalues of P θ,R with the poles of the meromorphic continuation of (Pz) -1 : , and in fact, the existence of such a continuation, follows from the following formula (implicit in [39], and discussed in [45] This is initially valid for Im z > 0 so that the right hand side is well defined, and then by analytic continuation in the region where the left hand side is meromorphic. The reason for the Fredholm property of (P θ,Rz) in D(0, sin(2θ)) comes from the properties of the principal symbol of P θ,R -see Fig. 6. Here for convenience, and for applications to our setting, we consider P θ,R as an operator on L 2 (R n ) using the identification S θ,R above. Its principal symbol is given by where the complex arguments are allowed due to the analyticity of p(x, ξ) outside of a compact set -see §2.1. We have the following properties This implies, for R large enough, For our future aims, it will prove convenient to actually let the angle θ explicitly depend on h: as long as θ > ch log(1/h), the estimates above guarantee the Fredholm property of (P θ,Rz) for z ∈ D(0, θ/C), by providing approximate inverses near infinity. We will indeed take θ of the order of h log(1/h), see (3.31). 3.5. Microlocally deformed spaces. Microlocal deformations using exponential weights have played an important role in the theory of resonances since [21]. Here we take an intermediate point of view [26,43] by combining compactly supported weights with the complex scaling described above. We should stress however that the full power of [21] would allow more general behaviours of p(x, ξ) at infinity, for instance potentials growing in some directions at infinity. Let us consider an h-independent real valued function ), and rescale it in an h-dependent way: For A ∈ Ψ m,0 (R d ), we consider the conjugated operator where The semiclassical calculus of pseudodifferential operators [12, Chapter 7],[15, Chapter 4, Appendix D.2] and (3.17) show that Since G w 0 L 2 →L 2 ≤ C 0 , functional calculus of bounded self-adjoint operators shows that exp(±tG w (x, hD)) ≤ h -tC 0 M , so we obtain the bound, with δ > 0 arbitrary small. Applying this bound,we may write (3.18) as In turn, this expansion, combined with Beals's characterization of pseudodifferential operators (3.4), implies that the exponentiated weight is a pseudodifferential operator: Using the weight function G, we can now define our weighted spaces. Let H k h (R d ) be the semiclassical Sobolev spaces defined in (3.3). We put ), but the Hilbert norms are different. In the case of L 2 , that is of k = 0, we simply put H 0 G = H G . The mapping properties of P = p w (x, hD) on H G (R d ) are equivalent with those of P G def = e -G w /h P e G w /h on L 2 (R d ), which are governed by the properties of the (full) symbol p G of P G : formula (3.19) shows that At this moment it is convenient to introduce a notion of leading symbol, which is adapted to the study of conjugated operators such as P G . For a given Q ∈ S(T * R d ), we say that for any γ ∈ (0, 1). This property is obviously an equivalence relation inside S(T * R d ), which is weaker than the equivalence relation defining the principal symbol map on Ψ h (see §3.1). In particular, terms of the size h log(1/h) are "invisible" to the leading symbol. For example, the leading symbols of p G and p are the same. If we can find q independent of h, then it is unique. For future use we record the following: is self-adjoint (with respect to the Hilbert norm on H G ). Then this operator admits a real leading symbol. Conversely, if q ∈ S(T * R d ) is real, then there exists Proof. This follows from noting that has the same leading symbol as Q w (x, hD), and that self-adjointness of The weighted spaces can also be microlocalized in the sense of §3.2: for V ⋐ T * R d , we define the space In other words, H G (V ) = e G w (x,hD)/h H(V ). This definition depends only on the values of the weight G in the open set V . For future reference we state the following Lemma 3.2. Suppose T : H(V ) → H(κ(V )) is an h-Fourier integral operator associated to a symplectomorphism κ (in the sense of §3.3), and is asymptotically uniformly bounded (in the sense of (3.8)). is also asymptotically uniformly bounded with respect to the deformed norms. Proof. Since the statement is microlocal we can assume that V is small enough, so that T ≡ T 0 A in V , where T 0 is unitary on L 2 (R d ) and A ∈ Ψ h . As in the proof of Lemma 3.1 the boundedness of (3.25) is equivalent to considering the boundedness of where A κ * G def = e -(κ * G) w (x,hD)/h Ae (κ * G) w (x,hD)/h . Because of (3.19), we have uniform boundedness of A κ * G on L 2 . Unitarity of T 0 means that it is sufficient to show the uniform boundedness of . † Alternatively, we can compare exp(M log(1/h)G w κ ) with (exp(M log(1/h)G κ )) w and use product formulae for pseudodifferential operators -see [43,Appendix] or [15,Section 8.2]. This proves uniform bounded of globally defined operators T 0 A, and the asymptotic uniformly boundedness in the sense of (3.8) of T on spaces of microlocally localized functions. 3.6. Escape function away from the trapped set. In this section we recall the construction of the specific weight function G which, up to some further small modifications, will be used to prove Theorems 1 and 2. Let K E ⊂ p -1 (E) be the trapped set on the E-energy surface, see (1.4), and define The construction of the weight function is based on the following result of [19,Appendix]: for any open neighbourhoods U, V of K, U ⊂ V , there exists These properties mean that G 1 is an escape function: it increases along the flow, and strictly increases along the flow on p -1 ([-δ, δ]) away from K (as specified by the neighbourhood V ). Furthermore, H p G is bounded in a neighbourhood of p -1 (0). Since such a function G 1 is necessarily of unbounded support, we need to modify it to be able to use H G -norms defined in §3.5 (otherwise methods of [21] could be used and that alternative would allow more general behaviours at infinity, for instance a wide class of polynomial potentials). For that we follow [43, § §4.1,4.2,7.3] and [31, §6.1]: G 1 is modified to a compactly supported G 2 in a way allowing complex scaling estimates (3.16) to compensate for the wrong sign of H p G 2 . Specifically, [31, Lemma 6.1] states that for any large R > 0 and δ 0 ∈ (0, 1/2) we can construct G 2 with the following properties: with M > 0 a fixed constant. Then, in the notations of §3.5, we will be interested in the complex-scaled operator ) , for a scaling angle depending on h: Inserting the above estimates in (3.22), we get provided that we choose [31, §6.1] Assuming that the constant M 0 appearing in the statement of Theorem 1 satisfies for δ > 0 and h > 0 small enough, the rectangle R(δ, M 0 , h) is contained in the uncovered region in Fig. 6, hence the scaling by the angle (3.29) gives us access to the resonance spectrum in the rectangle R(δ, M 0 , h). In §5.3 we will need to further adjust M 0 with respect to M 1 . In this section we recall some linear algebra facts related to the Schur complement formula, which are at the origin of the Grushin method we will use to analyze the operator P θ,R . For any invertible square matrix decomposed into 4 blocks, we have = q 11 q 12 q 21 q 22 =⇒ p -1 11 = q 11q 12 q -1 22 q 21 , provided that q -1 22 exists (which implies that q 22 , and hence p 11 , are square matrices). We have the analogous formula for q -1 22 : q -1 22 = p 22p 21 p -1 11 p 12 . One way to see these simple facts is to apply gaussian elimination to The formula for the inverse of p 11 leads to the construction of effective Hamiltonians for operators (quantum Hamiltonians) P : H 1 → H 2 . We first search for auxiliary spaces H ± and operators R ± for which the matrix of operators is invertible for z running in some domain of C. Such a matrix is called a Grushin problem, and when invertible the problem is said to be well posed. When successful this procedure reduces the spectral problem for P to a nonlinear spectral problem of lower dimension. Indeed, if dim H -= dim H + < ∞, we write and the invertibility of (Pz) : H 1 → H 2 is equivalent to the invertibility of the finite dimensional matrix E -+ (z). The zeros of det E -+ (z) coincide with the eigenvalues of P (even when P is not self-adjoint) because of the following formula: (3.33) tr valid when the integral on the left hand side is of trace class -see [44,Proposition 4.1] or verify it using the factorization (3.32). Here z denotes an integral over a small circle centered at z. The above formula shows that dim ker(Pz) = dim ker E -+ (z). The matrix E -+ (z) is often called an effective Hamiltonian for the original Hamiltonian P -see [44] for a review of this formalism and many examples. In the physics literature, this reduction is usually called the Feshbach method. We illustrate the use of Grushin problems with a simple lemma which will be useful later in §5.3. Lemma 3.3. Suppose that where H j and H ± are Banach spaces. If P -1 : H 2 → H 1 exists then P is a Fredholm operator ⇐⇒ R + P -1 R -: H -→ H + is a Fredholm operator , Proof. We apply the factorization (3.32) with p 11 = P , p 12 = R -, p 21 = R + , p 22 = 0. Since the first factor is invertible we only need to check the the Fredhold property and the index of the second factor: 1 and the lemma is immediate. In this section we recall and extend the analysis of [42] to treat a Poincaré section Σ ⊂ p -1 (0) for a flow satisfying the assumptions in §2.2. In [42] a Poincaré section associated to a single closed orbit was considered. The results presented here are purely microlocal in the sense of §3.2, first near a given component Σ k of the section, then near the trapped set K 0 . In this section P is the original differential operator, but it could be replaced by its complex scaled version P θ,R , since the complex deformation described in §3.4 takes place far away from K 0 . Also, when no confusion is likely to occur, we will often denote the Weyl quantization χ w of a symbol χ ∈ S(T * R d ) by the same letter: χ = χ w . Microlocal study near Σ k . First we focus on a single component Σ k of the Poincaré section, for some arbitrary k ∈ {1, ..., N}. Most of the time we will then drop the subscript k. Our aim is to construct a microlocal Grushin problem for the operator , and δ will be chosen small enough so that the flow on Φ t ↾ K Re z is a small perturbation of Φ t ↾ K 0 . We call Ω k = κ k ( Ω k ) the neighbourhood of Σ k in T * X in the range of κ k . The "width along the flow" ǫ > 0 is taken small enough, so that the sets {Ω k , k = 1, . . . , N} are mutually disjoint, and it takes at least a time 20ǫ for a point to travel between any Ω k and its successors. The symplectic maps κ k allow us to extend the Poincaré section Σ to the neighbouring energy layers p are still transversal to the flow in p -1 (z). Using this extension we may continuously deform the departure sets , and by consequence the tubes T jk into tubes T jk (z) ⊂ p -1 (z) through a direct generalization of (2.12). The tube T jk (z) intersects Σ j (z) on the arrival set A jk (z) ⊂ Σ j (z); notice that for z = 0, the latter is in general different from κ j,z ( A jk ) (equivalently A jk (z) = κ -1 j,z (A jk (z)) is generally different from A jk (0)). These tubes induce a Poincaré map F jk,z bijectively relating D jk (z) with A jk (z). The following Lemma, announced at the end of §2.3.1, shows that for |z| small enough the interesting dynamics still takes place inside these tubes: the trapped set is stable with respect to variations of the energy. Lemma 4.1. Provided δ > 0 is small enough, for any z ∈ [-δ, δ] the trapped set K z ⋐ ⊔ jk T jk (z). As a consequence, in this energy range the Poincaré map associated with Σ(z) fully describes the dynamics on K z . Proof. From our assumption in §2.1, there exists a ball B(0, R) (the "interaction region") such that, for any E ∈ [-1/2, 1/2], the trapped set K E must be contained inside B(0,R) X, z ≈ 0, will "escape fast" in the past or in the future, because the Hamilton vector field is close to the one corresponding to free motion, 2 j ξ j ∂ x j . Hence we only need to study the behaviour of points in p -1 (z) ∩ T * B(0,R) R n . Let us define the escape time from the interaction region T * B(0,R) X: For any E ∈ [-1/2, 1/2], the trapped set K E can be defined as the set of points in p -1 (E) for which t esc (ρ) = ∞. Let us consider the neighbourhood of K 0 formed by the interior of the union of tubes, (⊔T ik ) • . By compactness, the escape time is bounded from above outside this neighbourhood, that is in p -1 (0) ∩ T * B(0,R) X \ (⊔T ik ) • , by some finite t 1 > 0. By continuity of the flow Φ t , for δ > 0 small enough, the escape time in the deformed neighbourhood p -1 (z) ∩ T * B(0,R) X \ (⊔T ik (z)) • will still be bounded from above by 2t 1 : this proves that K z ⋐ ⊔T ik (z). A direct consequence is that the reduced trapped sets T j (z) For any set S(z) depending on the energy in the interval z ∈ [-δ, δ], we use the notation We will extend the notation to complex values of the parameter z ∈ R(δ, M 0 , h), identifying S(z) with S(Re z). For z ∈ R(δ, M 0 , h), we consider the microlocal Poisson operator ) , which obviously satisfies the equation (hD xnz) K(z) v + = 0. For v + microlocally concentrated in a compact set, the wavefront set of K(z) v + is not localized in the flow direction. On the other hand, the Fourier integral operator U is welldefined and unitary only from Ω to Ω. Therefore, we use a smooth cutoff function χ Ω , χ Ω = 1 in Ω, χ Ω = 0 outside Ω ′ a small open neighbourhood of Ω (say, such that |x n | ≤ 2ǫ inside Ω ′ ), and define the Poisson operator ), to a microlocal solution of the equation (Pz)u = 0 in Ω, with u ∈ H(Ω ′ ). As we will see below, the converse holds: each microlocal solution in Ω is parametrized by a function v + ∈ H( Σ). In a sense, the solution u = K(z)v + is an extension along the flow of the transverse data v + . More precisely, K(z) is a microlocally defined Fourier integral operator associated with the graph Equivalently, this relation associates to each point (x ′ , ξ ′ ) ∈ Σ a short trajectory segment through the point κ(x ′ , 0; ξ ′ , Re z) ∈ Σ(Re z). We use the notation C -since this relation is associated with the operator R -defined in (4.13) below. Back to the normal form hD xn , let us consider a smoothed out step function, We notice that the commutator (i/h)[hD xn , χ 0 ] = χ ′ 0 (x n ) is localized in the region of the step and integrates to 1: this implies the normalization property , where •, • is the usual Hermitian inner product on L 2 (R n ). Notice that the right hand side is independent of the precise choice of χ 0 . We now bring this expression to the neighbourhood of Σ through the Fourier integral operator χ w Ω U. This implies that the Poisson operator K(z) satisfies: Here the symbol χ is such that χ w ≡ U χ w 0 U * inside Ω, so χ is equal to 0 before Φ -ǫ (Σ) and equal to 1 after Φ ǫ (Σ) (in the following we will often use this time-like terminology referring to the flow Φ t ). In (4.7), we are only concerned with [P, χ w ] microlocally near Ω, since the operator χ w Ω U is microlocalized in Ω ′ × Ω ′ . Hence, at this stage we can ignore the properties of the symbol χ outside Ω ′ . The expression (4.7) can be written Fixing a function χ with properties described after (4.7) and writing χ = χ f (where f is for forward), we define the operator (4.9) (from here on we denote χ = χ w in similar expressions). This operator "projects" any u ∈ H(Ω) to a certain transversal function v + ∈ H( Σ). But it is important to notice that R + (z) is also well-defined on states u microlocalized in a small neighbourhood of the full trapped set K: the operator χ w Ω [(i/h)P, χ f ] cuts off the components of u outside Ω. Hence, we may write The equation (4.8) shows that this projection is compatible with the above extension of the transversal function: This shows that transversal functions v + ∈ H( Σ) and microlocal solutions to (Pz)u = 0 are bijectively related. Since inside Ω), we have the bounds Just as K(z) * , R + (z) is a microlocally defined Fourier integral operator associated with the relation namely the inverse of C -given in (4.5). In words, this relation consists of taking any ρ ∈ Ω ∩ p -1 (Re z) and projecting it along the flow on the section Σ(z). We now select a second cutoff function χ b with properties similar with χ f , and satisfying also the nesting property (4.12) With this new cutoff, we define the operator Starting from a transversal data u -∈ H( Σ), this operator creates a microlocal solution in Ω and truncates by applying a pseudodifferential operator with symbol H p χ b . Like K(z), it is a microlocally defined Fourier integral operator associated with the graph C -. its norm is bounded by 4.1.3. Solving a Grushin problem. We are now equipped to define our microlocal Grushin problem in Ω. Given v ∈ H(Ω), v + ∈ H( Σ), we want to solve the system with u ∈ L 2 (X) a forward solution, and u -∈ H( Σ). Let us show how to solve this problem. First let u be the forward solution of (i/h)(Pz) u = v, microlocally in Ω. That solution can be obtained using the Fourier integral operator U in (4.3) and the easy solution for hD xn . We can also proceed using the propagator to define a forward parametrix: T 0 e -it(P -z)/h dt. The time T is such that Φ T (Ω) ∩ Ω = ∅ (from the above assumption on the separation between the Ω k we may take T = 5ǫ). By using the model operator hD xn , one checks that the parametrix E(z) transports the wavefront set of v as follows: In general, u does not satisfy R + (z) u = v + , so we need to correct it. For this aim, we solve the system (4.17 Indeed, the property (Pz) K(z) ≡ 0 ensures that (i/h)(Pz) u = -R -(z)u -. We then obtain the identities The second identity uses the nesting assumption (H p χ f )χ b = H p χ f , and the last one results from (4.8). This shows that the Ansatz (4.18) solves the system (4.17). Finally, (u = u+ u, u -) solves (4.14) microlocally in Ω× Σ, for v ∈ H(Ω) and v + ∈ H( Σ) respectively. Furthermore, these solutions satisfy the norm estimate The form of the microlocal construction in this section is an important preparation for the construction of our Grushin problem in the next section. In itself, it only states that, for v microlocalized near Σ, (i/h)(Pz)u = v can be solved microlocally near Σ in the forward direction. Microlocal solution near K. We will now extend the construction of the Grushin problem near each Σ k , described in §4.1, to obtain a microlocal Grushin problem near the full trapped set K. This will be achieved by relating the construction near Σ k to the one near the successor sections Σ j . We now need to restore all indices k ∈ {1, . . . , N} in our notations. 4.2.1. Setting up the Grushin problem. We recall that ) is the space of functions microlocally concentrated in Σ k (see (3.6)). For u ∈ L 2 (X) microlocally concentrated in neigh( K, T * X), we define realizing a smoothed-out step from 0 to 1 along the flow near Σ k . Similarly, we define Each R k -(z) was defined in (4.13) in terms of a cutoff function χ k b ∈ C ∞ c (T * X) which also changes from 0 to 1 along the flow near Σ k , and does so before χ k f . Below we will impose more restrictions on the cutoffs χ k b . With these choices, we now consider the microlocal Grushin problem The aim of this section is to construct a solution (u, u -) microlocally concentrated in a small neighbourhood of is concentrated in a sufficiently small neighbourhood of the same set. To achieve this aim we need to put more constraints on the cutoffs χ k b . We assume that each χ k b ∈ C ∞ c (T * X) is supported near the direct outflow of T k . To give a precise condition, let us slightly modify the energy-thick tubes T jk (see (2.12), (4.2)) by removing or adding some parts near their ends: With this definition, the short tubes T -- jk do not intersect the neighbourhoods Ω k , Ω j , while the long tubes T ++ jk intersect both (see Fig. 7). We then assume that (4.23) and supp χ k b is contained in a small neighbourhood of that set. Furthermore, we want the cutoffs {χ k b } k=1,...,N to form a microlocal partition of unity near K 0 : there exists a neighbourhood V 0 of K containing all long tubes: and such that (4.25) These conditions on χ k b can be fulfilled thanks to the assumption (2.10) on the section Σ. A schematic representation of these sets and cutoffs is shown in Fig. 7 + , initially concentrated inside the neighbourhood Ω ′ k , can be extended along the flow to a larger set Ω + k , which intersects the successors Σ j (z) of Σ k (z) and contains the union of tubes j∈J + (k) T ++ jk (we remind that j = k according to assumption (2.10)). This can be done by extending the symplectomorphism κ k , the associated unitary Fourier integral operator U k , and replace the cutoff function χ Ω k by a function χ Ω + k supported in the set Ω + k ; we can then define the extended Poisson operator as: In that case, we take as our Ansatz (4.26) . Schematic representation of (part of) the neighbourhoods V 1 ⊂ V 0 of K 0 (resp. green shade and green dashed contour), some sections Σ k (thick black) and arrival sets A kj ⊂ Σ k (red). We also show the tubes T ±± Due to the assumption (4.23), the cutoff χ k b effectively truncates the solution only near the sections Σ k (z) and Σ j (z), j ∈ J + (k), but not on the "sides" of supp χ k b . Hence, the expression , and other components supported near the arrival sets A jk (z) ⊂ Ω j , due to the "step down" of χ k b near A jk (z). The assumption (4.25) ensures that (4.30) [(i/h)P, χ k b ] ≡ -[(i/h)P, χ j b ] microlocally near A jk (z) , so the expression in (4.29) reads Now, for each j ∈ J + (k) we notice that K + k (z) v k + is a solution of (Pz)u = 0 near A jk (z), so this solution can also be parametrized by some transversal data "living" on the section Σ j (z) (see the discussion before (4.5)). This data obviously depends linearly on v k + , which defines the monodromy operator M jk (z): The operators M jk (z) are microlocally defined from D k ⊂ Σ k to A jk (z) ⊂ Σ j , they are zero on H( D ℓk ) for ℓ = j. The identity (4.8) provides an explicit formula: . Before further describing these operators, let us complete the solution of our Grushin problem. Combining (4.31) with (4.32), we obtain This shows that the problem (4.22) in the case v = 0 and a single v . We now consider the Grushin problem with v = 0, v + = (v 1 + , . . . , v N + ) with each v k + microlocalized in D k . By linearity, this problem is solved by From the above discussion, u is microlocalized in the neighbourhood V 0 of K, while u j -is microlocalized in D j ∪ A j (z). Let us now come back to the monodromy operators. The expression (4.33) shows that M jk (z) is a microlocal Fourier integral operator. Since we have extended the solution K k (z) v k + beyond Ω k , the relation associated with the restriction of K + k (z) on H( D jk ) is a modification of (4.5), of the form ρ)); ρ), ρ ∈ D jk , -ǫ ≤ t ≤ t max + ǫ} , such that the trajectories cross Σ j . On the other hand, the relation C + associated with R j + (z) is identical with (4.11). By the composition rules, the relation associated with M jk (z) is } . This is exactly the graph of the Poincaré map F jk,z : D jk (z) → A jk (z), seen through the coordinates charts κ k,z , κ j,z . When z is real, the identity (4.8) implies that M jk (z) : H( D jk ) → H( A jk (z)) is microlocally unitary. Also, the definition (4.33) shows that this operator depends holomorphically of z in the rectangle R(δ, M 0 , h). To lowest order, the z-dependence takes the form ) is an extension of the return time associated with the map F jk,z on D jk . For z ∈ R(δ, M 0 , h), this operator satisfies the asymptotic bound (each tube T -+ jk intersects Ω k only near D k , see figure 7). Let us first assume that v is microlocally concentrated inside a short tube T -- jk . We use the forward parametrix E(z) of (i/h)(Pz) given in (4.15) with the time Also, the choice of the time T ensures that E(z)v is a microlocal solution of (Pz)u = 0 near A jk (z), so . Thus, we can solve (4.37) by taking The propagation of wavefront sets given in (4.16) shows that u j -∈ H( A jk (z)), and that WF h (u) ⊂ T +- jk does not intersect the "step up" region of the forward cutoffs χ ℓ f , so that and apply the same construction. The only notable difference is the fact that R k + (z)u may be a nontrivial state concentrated in ∪ |t|≤ǫ D k . In both cases, we see that u + u - h -M 0 (tmax+2ǫ) v . By linearity, the above procedure allows to solve (4.37) for any v microlocalized inside the neighbourhood V 0 . This solution produces a term R + u, which can be solved away using the procedure of §4.2.2. Notice that R + u h -M 0 (tmax+ǫ) v . We summarize the construction of our microlocal Grushin problem in the following Proposition 4.2. For δ > 0 small enough, there exist neighbourhoods of K = K δ in T * X, V + and V -, and neighbourhoods of κ -1 j ( T j ) in Σ j , V j + , and Here R ± (z) are given by (4.20) and (4.21). Furthermore, the solutions satisfy the norm estimates , where t max is the maximal return time defined in (4.27). One possible choice for the above sets is Proof. Take v ∈ H(V 1 ), and call ( u, u -) the solution for the inhomogeneous problem (4.37). Then the propagation estimate (4.16) implies that u is concentrated inside the larger neighbourhood V 0 ⊂ ∪ j,k T ++ jk (see (4.24)), while u j -∈ H( A j (z)). We have R k + (z) u ∈ H( D k ) so, provided the data satisfies v k + ∈ D k , the computations of §4.2.2 show how to solve the homogeneous problem with data (v + -R + (z) u). That solves the full problem. The expressions (4.35) show that the solutions to the homogeneous problem ( u, u k -) are microlocalized, respectively, in V 0 and in Remark. The proof of the proposition shows that the neighbourhoods V k + and V k -are different. For given data (v, v + ), the solutions (u, u -) will not in general be concentrated in the same small set as the initial data. This, of course, reflects the fact that a neighbourhood V of K 0 is not invariant under the forward flow, but escapes along the unstable direction. In order to transform the microlocal Grushin problem described in this proposition into a well-posed problem, we need to take care of this escape phenomenon. This will be done using escape functions in order to deform the norms on the spaces L 2 (X) (as described in §3.5), but also on the auxiliary spaces L 2 (R n-1 ). The difficulty described in the remark at the end of §4 will be resolved by modifying the norms on the space L 2 (X) × L 2 (R n-1 ) N , through the use of exponential weight functions as described in §3.5. These weight functions will be based on the construction described in §3. 6. In most of this section we will consider the scaled operator P θ,R globally, so we cannot replace it by P any longer. To alleviate notation, we will write this operator (5.1) where C 0 is the constant appearing in (2.2), and M 1 > 0 is a constant (it will be required to satisfy (3.31) once we fix the weight G, and is larger than M 0 appearing in Theorem 1). We will first discuss the local construction near each Σ k and then, as in the previous section, adapt it to construct a global Grushin problem. Our first task is still microlocal: we explain how a deformation of the norm on L 2 (X) by a suitable weight function G can be used to deform the norms on the N auxiliary spaces L 2 (R n-1 ), microlocally near Σ k . 5.1. Exponential weights near Σ k . As in §4.1, in this subsection we work microlocally in the neighbourhood Ω k of one component Σ k (Ω k is the neighbourhood described in §4.1); we drop the index k in our notations. Notice that the complex scaling has no effect in this region, so P ≡ P θ,R . We will impose a constraint on the weight function G near Σ, and construct a weight functions g on Σ. The construction of the local solution performed in §4.1 will then be studied in these deformed spaces. Take a function g 0 ∈ C ∞ c (R n-1 ), and use it to define Then, using the Fourier integral operator U given in (4.3), one can construct a weight function G 0 ∈ S(T * X) such that This weight satisfies G 0,0 = G 0 • κ -1 in Ω, and the invariance property (5.3) [P (h), G w 0 (x, hD)] ≡ 0 microlocally in Ω . As in §3.5, we rescale these weight functions by (5.4) Still using the model hD xn , one can easily check the intertwining property e -G w (x,hDx;h)/h K(z) ≡ K(z) e -g w (x ′ ,hD x ′ ;h)/h : H( Σ) → H(Ω ′ ) . (5.5) Using the weights G and g we define the microlocal Hilbert spaces H G (Ω ′ ) and H g ( Σ) by the method of §3.5. We need to check that the construction of a microlocal solution performed in §4.1.2 remains under control with respect to these new norms. Lemma 5.1. The operators satisfy the analogue of (4.7). Namely, taking a cutoff χ jumping from 0 to 1 near Σ as in §4.1.2, then any v + ∈ H g ( Σ) will satisfy Proof. From the cutoff χ we define the deformed symbol χ G through χ w G (x, hD) def = e -G w (x,hD)/h χ w (x, hD) e G w (x,hD)/h . The symbol calculus of §3.5 shows that χ G also jumps from 0 to 1 near Σ, so that (returning to the convention of using χ for χ w ) Hg . In the second line we used (5.5), the third line results from P ≡ P G , due to (5.3), and the last one from (4.7) applied to χ G . Equation (5.5) shows that, for z ∈ R(δ, M 0 , h), the operators K(z) and R ± (z) defined respectively in (4.9) and (4.13), satisfy the same norm estimates with respect to the new norms as for the L 2 norms: The arguments presented in §4.1 carry over to the weighted spaces, and the microlocal solution to the problem (4.14) constructed in §4.1.3 satisfies the norm estimates (5.9) Given a function G 0,0 (x, ξ) satisfying H p G 0,0 = 0 in Ω, one can iteratively construct a full symbol G 0 of the form (5.2), such that (5.3) holds. Now, the lower order terms in G 0 may change the norms only by factors 1 + O(Mh log(1/h)) , so the same norm estimates hold if we replace G 0 by its principal symbol G 0,0 in the definition of the new norms. As a result, we get the following Proposition 5.2. Take g 0 ∈ C ∞ c (T * R n-1 ), G 0 (x ′ , x n , ξ ′ , ξ n ) = g 0 (x ′ , ξ ′ ), G 0 ∈ C ∞ c (X) satisfying G 0 = G 0 • κ in Ω, and Then, the estimates (5.7-5.9) hold in the spaces H G (Ω), H g ( Σ). Globally defined operators and finite rank weighted spaces. In this section we transform our microlocal Grushin problem into a globally defined one. This will require transforming all the microlocally defined operators (R ± (z), M jk (z)) into globally defined operators acting on L 2 (X) or L 2 (R n-1 ). Because our analysis took place near the trapped set K 0 , we will need to restrict our auxiliary operators to some subspaces of L 2 (R n-1 ) obtained as images of some finite rank projectors. These subspaces are composed of functions microlocalized near K 0 . To show that the resulting Grushin problem is well-posed (invertible), the above construction must be performed using appropriately deformed norms on the spaces L 2 (X), L 2 (R n-1 ), obtained by using globally defined weight functions G, g j . Our first task is thus to complete the constructions of these global weights, building on §3.6 and §5.1. Global weight functions. We will now construct global weight functions G ∈ C ∞ c (X), g j ∈ C ∞ c (T * R n-1 ) (one for each section Σ j ). For this, we will use the construction of an escape function away from K 0 presented in §3.6, and modify it near the Poincaré section so that it takes the form required in Proposition 5.2, and allows us to define auxiliary escape functions g j . These weight functions will allow us to to define finite rank realizations of the microlocally defined operators R ± (z) and M(z). Our escape function G 0 ∈ S(T * X) is obtained through a slight modification of the weight G 2 (x, ξ) described in (3.28). The modification only takes place near the trapped set K, and in particular near the sections Σ j . The following lemma is easy to verify. Lemma 5.3. Let {Ω j , } j=1,...,K be the neighbourhoods of Σ j described in §4.1.1, Ω ′ j and Ω ′′ j be small neighbourhoods of Ω j , Ω j ⋐ Ω ′ j ⋐ Ω ′′ j , and let V be a small neighbourhood of K δ (see (3.26)). Then there exists G 0 ∈ C ∞ c (T * X) such that H p G 0 = 0 on Ω ′ j , H p G 0 ≥ 0 on T * B(0,3R) X, H p G 0 ≥ -δ 0 on T * X. (5.10) Besides, using the coordinate charts κ j : Ω ′ j → Ω ′ j (see §4.1.1), we can construct G 0 such that G 0 • κ j ↾ Ω ′ j is independent of the energy variable ξ n ∈ [-δ, δ]. The last assumption (local independence on ξ n ) is not strictly necessary, but it simplifies our construction below, making the auxiliary functions g j independent of z -see Proposition 5.2. For the set V we assume that V ⋐ V 1 , where V 1 is the set defined in (4.38). As a consequence, there exists a set V ′ 1 , with V ⋐ V ′ 1 ⋐ V 1 with the following property. Consider the the parametrix E(z) (4.15) with the time T = t max + 5ǫ. Then there exists t 1 > 0 such that, for any ρ ∈ p -1 ([-δ, δ]) \ V ′ 1 , the trajectory segment {Φ t (ρ), 0 ≤ t ≤ T } spends a time t ≥ t 1 outside of W . The main consequence of this property is a strict increase of the weight along the flow outside V ′ 1 : (5.11) ∀ρ ∈ T * B(0,2R) X ∩ p -1 ([-δ, δ]) \ V ′ 1 , G 0 (Φ T (ρ)) -G 0 (ρ) ≥ t 1 . (Here we use the fact that T is small enough, so that a particle of energy z ≈ 0 starting inside T * B(0,2R) at t = 0 will remain inside T * B(0,3R) up to t = T .) The set V will be further characterized in the next subsection. . Schematic representation (inside some energy layer p -1 (z)) of the neighbourhood V and the sets S k , S j . The departure/arrival sets D kj , A kj are similar to the ones appearing in figure 4. The sets S k , S j are represented through their images in Σ k , Σ j through κ k,z , κ j,z . We showe 3 trajectories staying inside V all the time, and one ending outside S k . Q j = Q j (x ′ , ξ ′ ; h) ∈ S(T * R n-1 ) , with leading symbol q j independent of h (the leading symbol is meant in the sense of (3.23)). We choose that leading symbol to be real and have the following properties: q j (ρ) < 0 , ρ ∈ S j , q j (ρ) > 0 , ρ ∈ T * R n-1 \ S j , lim inf ρ→∞ q j (ρ) > 0. (5.15) Lemma 3.1 shows that one can choose Q j so that Q w j (x ′ , hD x ′ ) : H g j (R n-1 ) -→ H g j (R n-1 ) is self-adjoint. Under the assumptions (5.15), we know that Q j has discrete spectrum in a fixed neighbourhood of R -when h > 0 is small enough. Let (5.16) H j def = Π j H g j (R n-1 ) , where Π j def = 1l R -Q w j (x ′ , hD x ′ ) , that is, Π j is the spectral projection corresponding to the negative spectrum of Q w j . In particular, (5.17) Π j Hg j →Hg j = 1, dim(H j ) ∼ c j h 1-n , c j > 0 . 5.4. Invertibility of the Grushin problem. We can transform this approximate solution into an exact one. The system (5.38) can be expressed as an approximate inverse of P(z): (5.40) For h small enough the operator I + R(h) can be inverted by a Neumann series, so we obtain an exact right inverse of P(z), E(z) = E(z) (I + R(z)) -1 . Since P(z) is of index zero, E(z) is also a left inverse, which proves the well-posedness of our Grushin problem (5.22). Theorem 2. We consider h > 0 small enough, and z ∈ R(δ, M 0 , h). For every (v, v + ) ∈ H G × H, there exists a unique (u, u -) ∈ H 2 G × H such that where R ± (z) are defined by (5.18) and (5.20). The estimates (5.39) hold, so if we write then the following operator norms (between the appropriate Hilbert spaces) are bounded by: (5.42) E , E + , E -, E -+ = O(h -M 0 (2tmax+2ǫ) ). Moreover, we have a precise expression for the effective Hamiltonian: