Quantum Evolution And Sub-laplacian Operators On Groups Of Heisenberg Type

In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schr{\"o}dinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified structure of the group and describe the semi-classical measures (also called quantum limits) that are associated with this family. This allows us to prove an Egorov's type Theorem describing the quantum evolution of a pseudodifferential semi-classical operator through the semi-group generated by a sub-Laplacian.


Introduction
We consider groups of Heisenberg type, or H-type groups G, which are a special case of simply connected Lie groups stratified of step 2 as described more precisely later. As a step 2 stratified group, its Lie algebra g is equipped with a vector space decomposition such that [v, v] = z = {0} and z is the center of g. Choosing a basis V j of v and identifying g with the Lie algebra of left-invariant vector fields on G, one defines the sublaplacian together with the associated Schrödinger propagator e it∆ G . We are interested in the asymptotic analysis as ε goes to 0 of quantities of the form for φ ∈ C ∞ c (G), T ∈ R, ℵ ∈ R and (ψ ε 0 ) ε>0 a bounded family of L ∞ (R, L 2 (G)) which satisfies (1.2) ∃s, C s > 0, ≤ C s , so that the oscillations of the initial data are exactly of size 1/ε. Taking into account that the operator ∆ G is homogeneous of degree 2 and writing t ε ℵ ∆ G = t ε ℵ+2 ε 2 ∆ G , we choose ℵ > −2. Considering the asymptotics ε → 0 then consists in doing an analysis in large times (times of sizes O ε −ℵ−2 ) simultaneously with the study of the dispersion of the concentration or oscillation effects that are present in the initial data. A consequence of our main results is the next theorem where we denote by M + (Z) the set of finite positive Radon measures on a locally compact Hausdorff set Z (see Definition 2.7 for L ∞ (R, M + (Z))). Theorem 1.1. Let G be a H-type group and (ψ ε 0 ) ε>0 a bounded family of L ∞ (R, L 2 (G)) satisfying (1.2). Any weak limit of the measure |e i t 2ε ℵ ∆ G ψ ε 0 (x)| 2 dx dt is of the form d̺ t (x) ⊗ dt where t → ̺ t is a map in L ∞ (R, M + (G)). Moreover, for almost all t ∈ R, the measure ̺ t writes with t −→ ̺ z * t and t −→ ̺ v * t in L ∞ (R, M + (G)), and has the following properties.
Several aspects are interesting to notice. Firstly, there exists a threshold, ℵ = 0, above which the weak limits of the time-averaged energy density is 0; this means that for sufficiently large scale of times, all the concentrations and oscillations effects have disappeared: the dispersion is complete. A similar picture holds in the Euclidean setting, however the threshold occurs at ℵ = −1 (see [2] and the Appendix in this article). This illustrates the fact that the dispersion is slower in sub-Riemanian geometries than in Euclidean ones, as already noticed in [6,13,5]. Secondly, one observes a decomposition of these weak limits into two parts ̺ t = ̺ v * t + ̺ z * t which turn out to have different transitional regimes: ℵ = −1 for ̺ v * t and ℵ = 0 for ̺ z * t . This splitting is also present in the works [7] about Grushin-Schrödinger equation and [34,10] about sublaplacians on contact manifolds. The part ̺ v * t behaves like in the Euclidean setting and equation (1.3) also presents Euclidean features. However, the other part ̺ z * t looks completely different and is specific to the nilpotent Lie group context, showing that the structure of the limiting objects is more complex than in the Euclidean case.
Similar questions have been addressed for the Laplace operator in different geometries, including compact ones: in the torus and for integrable systems ( [3,2]), in Zoll manifolds (see [28,25] and the review [26]), or on manifolds such as the sphere ( [27]). In contrast with the non-compact case (which is ours here), the compactness of the manifold implies that the complete dispersion of the energy is not possible; furthermore, the weak limits of the energy densities possess structural properties due to the geometry of the manifold, such as invariance by some flows, that may allow for their determination. For example, on compact Riemanian manifolds, such a measure belongs to the set of measures which are invariant under the geodesic flow, and this property is at the root of quantum ergodicity theorem [30,9,33] (see the introductory survey [1] and the articles [35,14] for more recent developments in the topic). The question of quantum ergodicity also arises in subriemanian geometries and have been addressed for contact [34,10] and quasi-contact [29] manifolds. As will be made precise in the next sections, we observe invariance properties by a flow that turns out to coincide with the Reeb flow used in [34,10] when G is the Heisenberg group. Theorem 1.1 is a consequence of the main results of this paper which use the semi-classical approach introduced in [16] for H-type Lie groups and are in the spirit of the article [25] for the treatment of the large time evolution together with the oscillations. They are as follows: (1) The first result is an Egorov's type Theorem on H-type groups (see Theorem 2.5), which describes as ε goes to 0 the asymptotics of quantities of the form for θ ∈ C ∞ c (R), f ∈ L 2 (G) and where the operator Op ε (σ) is the semi-classical operator of a symbol σ as introduced in [16] (see also [15,19,4,32]) All these elements are carefully explained in Section 2.
(2) The second result concerns the structure of the limiting objects when passing to the limit in (1.4). We extend the notion of semi-classical measure introduced in [16] to a timedependent context and analyze the properties of the semi-classical measures associated in 3 that manner with the family (e i t 2ε ℵ ∆ G ψ ε 0 ) ε , depending on the value of ℵ. We give a complete description of these limiting objects in Theorems 2.8 and 2.10 below.
The proof of Theorem 1.1 is based on the fact that, under certain hypothesis on the size of the oscillation, the analysis of the weak limits of the energy density can be deduced from those of its semi-classical measures, which are also called quantum limits in some geometric contexts. This idea was introduced in the 90's in the Euclidean case (see [12,20,24]), and adapted for H-type groups in [16]. The hypothesis on the size of the oscillation of (ψ ε 0 ) ε is a uniform strict ε-oscillation property (see Section 5.2) which guarantees that the oscillations are of sizes ε −1 and is implied by the condition in (1.2). Then, using the semi-classical pseudodifferential operators constructed in [16], we determine the semi-classical measures that are associated with the family e i t 2ε ℵ ∆ G ψ ε 0 and prove Theorem 1.1.
A straightforward generalization of our result would consist in adding a scalar potential ε θ V (x) for a smooth function V defined on G and a parameter θ ∈ R + . Then, one could exhibit regimes depending on the position of θ with respect to ℵ and the vector fields to consider should be modified in a non-trivial manner. One should then consider operations on symbols σ(x, λ) that involve differentials of the potential V (x) and difference operators acting on the operator part of σ(x, λ). A second generalization would be to consider more general stratified and graded groups. This would require to obtain in this more general setting similar results to those of Appendix B which at the moment heavily rely on the special case of H-type groups. However, the authors think this is doable and they have this generalization in mind. They also think that this approach can be adapted to homogeneous spaces.
In the next section, we recall the definition of H-type groups and present our two main results shortly described above, the Egorov theorem 2.5 and the analysis of the semi-classical measures associated with a family of the Schrödinger equation in Theorem 2.10. After some preliminary results on semi-classical symbols in Section 3, we prove both theorems in Section 4. Theorem 1.1 is a consequence of this analysis and is proved in Section 5. An Appendix is devoted to a short description of the Euclidean case and to some technical auxiliary results.

2.1.
H-type groups, notations and definitions. A simply connected Lie group G is said to be stratified of step 2 if its Lie algebra g is equipped with a vector space decomposition and z is the center of g. Via the exponential map exp : g → G which is a diffeomorphism from g to G, one identifies G and g as a set and a manifold. Under this identification, the group law on G (which is generally not commutative) is provided by the Campbell-Baker-Hausdorff formula, and (x, y) → xy is a polynomial map. More precisely, if x = Exp(v x + z x ) and y = Exp(v y + z y ) then . We may identify g with the space of left-invariant vector fields via For any λ ∈ z ⋆ (the dual of the center z) we define a skew-symmetric bilinear form on v by Following [23], we say that G is of H-type (or of Heisenberg type) when, once the inner products on v and on z are fixed, the endomorphism of this skew symmetric form (that we still denote by B(λ)) satisfies This implies in particular that the dimension of v is even. We set dim v = 2d, dim z = p.
The fundamental property (2.2) satisfied by B(λ) considered as an endomorphism on v implies that for all V ∈ v, |B(λ)V | 2 v = |λ| 2 |V | 2 v , and, by linearization, we deduce where Z (λ) is the unique vector of z equal to λ through the identification of z * to z via the inner product, and for all 1 ≤ j 1 , j 2 ≤ d Realisation of the elements in G. Denoting by z = (z 1 , · · · , z p ) the coordinate of Z in a fixed orthonormal basis (Z 1 , · · · , Z p ) of z, and once given λ ∈ z * , we will often use the writing of an element x ∈ G or X ∈ g as where p = (p 1 , · · · , p d ) are the λ-dependent coordinates of P on the vector basis (P 1 , · · · , P d ), by q = (q 1 , · · · , q d ) those of Q on (Q 1 , · · · , Q d ), while the coordinates z = (z 1 , · · · , z p ) of Z are independent of λ. We will also fix an orthonormal basis (V 1 , . . . , V 2d ) of v to write the coordinates of v independently of λ. The inner products on v and z allow us to consider the Lebesgue measure dv dz on g = v ⊕ z. Via the identification of G with g by the exponential map, this induces a Haar measure dx on G. This measure is invariant under left and right translations: Note that the convolution of two functions f and g on G is given by and as in the Euclidean case we define Lebesgue spaces by for q ∈ [1, ∞), with the standard modification when q = ∞.
We define the Schwartz space S(G) as the set of smooth functions on G such that for all α, β in N 2d+p , the function x → x β X α f (x) belongs to L ∞ (G), where X α denotes a product of |α| left invariant vector fields forming a basis of g and x β a product of |β| coordinate functions on G ∼ v × z. The Schwartz space S(G) can be naturally identified with the Schwartz space S(R 2d+p ); in particular, it is dense in Lebesgue spaces.

Dilations.
Since G is stratified, there is a natural family of dilations on g defined for t > 0 as follows: if X belongs to g, we decompose X as X = V + Z with V ∈ v and Z ∈ z and we set This allows us to define the dilation on the Lie group G via the identification by the exponential map: To simplify the notation, we shall still denote by δ t the map exp • δ t • exp −1 . The dilations δ t , t > 0, on g and G form a one-parameter group of automorphisms of the Lie algebra g and of the group G. The Jacobian of the dilation δ t is t Q where is called the homogeneous dimension of G. A differential operator T on G (and more generally any operator T defined on C ∞ c (G) and valued in the distributions of G ∼ R 2d+p ) is said to be homogeneous of degree ν (or ν-homogeneous) when where x has been written as in (2.6). The representations π λ , λ ∈ z * \ {0}, are infinite dimensional. The other unitary irreducible representations of G are given by the characters of the first stratum in the following way: for every ω ∈ v * , we set π 0,ω x = e iω(V ) , x = Exp(V + Z) ∈ G, with V ∈ v and Z ∈ z. 6 The set G of all unitary irreducible representations modulo unitary equivalence is then parametrized by (z * \ {0}) ⊔ v * : We will often identify each representation π λ with its equivalence class; in this case, we may write H λ for the Hilbert space of the representation instead of L 2 (p λ ) ∼ L 2 (R d ); we also set H (0,µ) = C. Note that the trivial representation 1 G corresponds to the class of π (0,ω) with ω = 0, i.e. 1 G := π (0,0) .

2.2.2.
The Fourier transform. In contrast with the Euclidean case, the Fourier transform is defined on G and is valued in the space of bounded operators on L 2 (p λ ). More precisely, the Fourier transform of a function f in L 1 (G) is defined as follows: for any λ ∈ z * , λ = 0, Note that for any λ ∈ z * , λ = 0, we have π λ x * = π λ x −1 and the map π λ x is a group homomorphism from G into the group U (L 2 (p λ )) of unitary operators of L 2 (p λ ), so functions f of L 1 (G) have a Fourier transform (F(f )(λ)) λ which is a bounded family of bounded operators on L 2 (p λ ) with uniform bound: since the unitarity of π λ implies (π λ x ) * L(L 2 (p λ )) = 1.

Plancherel formula.
The Fourier transform can be extended to an isometry from L 2 (G) onto the Hilbert space of measurable families A = {A(λ)} (λ)∈z * \{0} of operators on L 2 (p λ ) which are Hilbert-Schmidt for almost every λ ∈ z * \ {0}, with norm We have the following Fourier-Plancherel formula: where c 0 > 0 is a computable constant. This yields an inversion formula for any f ∈ S(G) and x ∈ G: where Tr denotes the trace of operators of L(L 2 (p λ )). This formula makes sense since for f ∈ S(G), the operators Ff (λ), λ ∈ z * \ {0}, are trace-class and z * \{0} Tr Ff (λ) |λ| d dλ is finite.

2.2.4.
Fourier transform and finite dimension representations. Usually, the Fourier transform of a locally compact group G would be defined on G, the set of unitary irreducible representations of G modulo equivalence, via for a representation π of G, and then considering the unitary equivalence we obtain a measurable field of operators Ff (π), π ∈ G. Here, the Plancherel measure is supported in the subset {class of π λ : λ ∈ z * \ {0}} of G (see (2.8)) since it is c 0 |λ| d dλ. This allows us to identify G and z * \ {0} when considering measurable objects up to null sets for the Plancherel measure. However, our semiclassical analysis will lead us to consider objects which are also supported in the other part of G. For this reason, we also set for ω ∈ v * and f ∈ L 1 (G): 2.2.5. Convolution and Fourier operators. The Fourier transform sends the convolution, whose definition is recalled in (2.7), to composition in the following way: We recall that a convolution operator T with integrable convolution kernel κ ∈ L 1 (G) is defined by T f = f * κ and we have F(T f ) = Fκ Ff by (2.12); hence, T appears as a Fourier multiplier with Fourier symbol Fκ acting on the left of Ff . Consequently, T is invariant under left-translation and bounded on L 2 (G) with operator norm In other words, T is in the space L(L 2 (G)) G of the left-invariant bounded operators on L 2 (G).
2.2.6. The von Neumann algebra of the group. Let us denote by L ∞ ( G) the space of bounded symbols, that is, here, measurable fields of operators σ = {σ(λ) : λ ∈ G} which are bounded in the sense that the essential supremum for the Plancherel measure c 0 |λ| d dλ is naturally equipped with a von Neummann algebra, and is called the von Neumann algebra of the group. As explained above, we already know L ∞ ( G) ⊃ FL 1 (G) by (2.9), but this inclusion is strict. The full Plancherel theorem [11] implies that the von Neumann algebras L ∞ ( G) and the space L(L 2 (G)) G of left-invariant bounded operators on L 2 (G) introduced above are isomorphic via the mapping σ → Op 1 (σ) where Op 1 (σ) is the operator with Fourier operator symbol σ, The isomorphism between L ∞ ( G) and L(L 2 (G)) G allows us to naturally extend the group Fourier transform to distributions κ ∈ S ′ (G) such that the convolution operator f → f * κ is bounded on L 2 (G) by setting that F(κ) is the symbol of the corresponding operator in L(L 2 (G)) G .
The infinitesimal representation of π extends to the universal enveloping Lie algebra of g that we identify with the left invariant differential operators on G. Then for such a differential operator T we have F(T f )(π) = π(T )Ff (π) and we may write π(T ) = F(T ). For instance, if as before X α denotes a product of |α| left invariant vector fields forming a basis of g, then Note that π(X ) α may be considered as a field of unbounded operators on G defined on the smooth vectors of the representations [19].
2.3. The sublaplacian. The sublaplacian on G is defined by One checks easily that ∆ G is a differential operator which is left invariant and homogeneous of degree 2. In this paper, we shall consider its associated Schrödinger equation The operator ∆ G is essentially self-adjoint on C ∞ c (G) (see [19,Section 4.1.3] or [31, Proof of Lemma 12.1]), so the Schrödinger equation has a unique solution for any data ψ 0 ∈ L 2 (G) by Stone's theorem. We keep the same notation for its unique self-adjoint (unbounded) extension to L 2 (G). More precisely, to deal with high-frequencies data, we shall be concerned with the semi-classical Schrödinger equation where ε > 0 is a small parameter taking into account the size of the oscillations of the initial data and τ > 0 is a parameter allowing us to consider large time behaviour, and as the same time the asymptotics ε → 0.
The definition of ∆ G is independent of the chosen orthonormal basis for v -although it depends on the scalar product that we have fixed at the very beginning on v. In particular, choosing the basis fixed in Section 2.1 for any λ ∈ z * \ {0} we have The infinitesimal representation (or Fourier transform) of ∆ G can be computed thanks to the equalities in (2.13), (2.14) and (2.16): at π (0,ω) , ω ∈ v * , it is the number Up to a constant, this is the quantum harmonic oscillator. The spectrum {|λ|(2n + d), n ∈ N} of H(λ) is discrete and the eigenspaces are finite dimensional. To each eigenvalue |λ|(2n + d), we denote by Π In particular, for each n ∈ N, all the eigenspaces V (λ) n , λ ∈ z * \ {0}, are isomorphic, and may be denoted by V n .

The space
is a smooth and compactly supported function from G to S(G). Being compactly supported means that κ x (z) = 0 for x outside a compact of G and any z ∈ G.
Remark 2.1. The algebra A 0 is the space of smoothing symbols with compact support in x. We will recall the definition of the space S −∞ of smoothing symbols introduced in [19] at the beginning of Section 3.4 below. Examples of smoothing symbols include the spectrally defined symbols f (H(λ)) for any f ∈ S(R) [19,Chapter 4].
As the Fourier transform is injective, it yields a one-to-one correspondence between the symbol σ and the function κ: we have σ(x, λ) = Fκ x (λ) and conversely the Fourier inversion formula (2.11) yields The set A 0 is an algebra for the composition of symbols since if In the case of representations of finite dimension, we distinguish between all the finite dimensional representations by replacing λ = 0 with the parameters (0, ω), ω ∈ v * . The operator Fκ x (0, ω) = σ(x, (0, ω)) then reduces to a complex number since H (0,µ) = C.

2.4.2.
Semi-classical pseudodifferential operators. Given ε > 0, the semi-classical parameter, that we use to weigh the oscillations of the functions that we shall consider, we quantify the symbols of A 0 by setting as in [15] (see also [19,4,32]) The kernel of the operator Op ε (σ) is the function , is called the convolution kernel of σ. Note that ε 2 λ can be understood as the action on λ of the dilation induced on G by the dilation δ ε of G (see Remark 3.3 in [15]).
Following [15], the action of the symbols in A 0 on L 2 (G) is bounded: One also has to mention that there exists a symbolic calculus for these operators (see [16]). In this paper, we will mainly use the description of the commutator between the sub-Laplacian and a semi-classical pseudodifferential operator, which comes from the explicit computation and writes: for all σ ∈ A 0 , where H(λ) = F(−∆ G ) has been defined in (2.17) and (2.18).

2.4.3.
The subspace A H of A 0 . The Egorov Theorem that we are going to state in the next section is valid for symbols compactly supported with respect to both the Fourier transform and the spectral decomposition of H(λ): and when its kernel and image contain a finite number of V n , in the sense that we have n ′ = 0, for all but a finite number of integers n, n ′ ∈ N.
One checks readily that A H is a subalgebra of A 0 . It is non-trivial since it contains for instance all the symbols of the form and where b ∈ S(z * ) vanish in a neighbourhood of 0 (see Remark 2.1); for other symbols in A H , see Remark 3.13. Although A H cannot be dense in A 0 for the Fréchet topology of A 0 , we will see in Corollary 3.11 that it satisfies a property of weak density. Besides, symbols σ ∈ A H can be decomposed in commuting and non-commuting symbols according to the following definition.
Lemma 2.4 will be a consequence of Corollary 3.9, see Remark 3.10.
2.5. The Egorov Theorem on H-type groups. For s ∈ R, we define the flow Ψ s on G×(z * \{0}) via In particular, this map may be composed with symbols with support in G × (z * \ {0}) such as the symbols in A H . This and Lemma 2.4 allows us to define the following action on A (d) H : 1) ).
(ii) For the H-diagonal part, we have the following alternative: In Parts (2) and (3), we use the action Φ s defined in (2.22).
It may appear unusual to have an Egorov Theorem holding in the space of distributions in the time variable. However, it is already the case in the Euclidean case when one works with the propagator of a Schrödinger operator with matrix-valued potential − ε 2 2 ∆ Id + V (x) with V matrix-valued (see [22,17,18]). The proof of this Theorem is postponed until Section 4.
2.6. Time averaged semi-classical measures and the quantum limits. We now want to pass to the limit ε → 0 in expressions of the form (1.4) and to identify the limiting objects, together with their properties. For this purpose, we follow [15, Section 5] with slightly different notation and introduce the following vocabulary for operator valued measures: Definition 2.6. Let Z be a complete separable metric space, and let ξ → H ξ a measurable field of complex Hilbert spaces of Z.
. By convention and if not otherwise specified, a representative of the class Γdγ is chosen such that Tr H ξ Γ = 1. In particular, if H ξ is 1-dimensional, Γ = 1 and Γdγ reduces to the measure dγ. One checks readily that M ov (Z, (H ξ ) ξ∈Z ) equipped with the norm · M is a Banach space.
When the field of Hilbert spaces is clear from the setting, we may write , for short. For instance, if ξ → H ξ is given by H ξ = C for all ξ, then M(Z) coincides with the space of finite Radon measures on Z. Another example is when Z is of the form Z = Z 1 × G where Z 1 is a complete separable metric space, and H (z 1 ,λ) = H λ , where the Hilbert space H λ is associated with the representation of λ ∈ G (that is, using the description in (2.8), H λ is equivalent to L 2 (p λ ) if the representation corresponds to λ ∈ z * \ {0} and H (0,ω) = C if λ = 0 and the representation corresponds to (0, ω) with ω ∈ v * ).
We will often consider measurable bounded maps of the time variable, valued in the space of measures that are positive as scalar-valued or operator-valued measures: Definition 2.7. If X denotes the Banach space M(Z) or more generally M ov (Z) as in Definition 2.6, then L ∞ (R, X + ) denotes the space of maps of t ∈ R and valued in X in L ∞ (R, X) with positive values for almost every t ∈ R.
2.6.1. Time-averaged semi-classical measures. With a bounded family (u ε ) ε>0 in L ∞ (R, L 2 (G)), we associate the quantities the limits of which are characterized by a map in L ∞ (R, M + ov (G × G)).
Given the sequence (ε k ) k∈N , the map t → Γ t dγ t is unique up to equivalence. Besides, We call the map t → Γ t dγ t satisfying Theorem 2.8 (for some subsequence ε k ) a time-averaged semi-classical measure of the family (u ε (t)). Note that we have not assumed any estimate of the form (1.2) on the family u ε in order to define its time averaged semi-classical measure; such additional property will however be useful to determine the limits of the time-averaged densities associated with u ε in terms of time-averaged semi-classical measures, as we shall see in Section 5.
(1) Note that this result can be generalised to any graded Lie group: for any bounded family (u ε ) ε>0 in L ∞ (R, L 2 (G)), there exist a sequence (ε k ) k∈N in (0, +∞) with for every θ ∈ C ∞ c (R) and σ ∈ A 0 . (2) In the case of this article where G is H-type, the special structure of G implies that Γ t dγ t consists of two pieces, one localized above λ ∈ z * \ {0} and another one which is scalar above v * , see (2.8).
satisfies the following alternatives: The existence of the semi-classical measure Γ t dγ t follows from Theorem 2.8, while its additional properties come from the fact that ψ ε (t) solves the Schrödinger equation. Point (i) of Theorem 2.8 is a consequence of (i) of Theorem 2.5. It will then appear that we will need to only use symbols which commute with H(λ).
Before closing this section, we discuss why the invariance of a semi-classical measure by vector fields imply that it is 0 (as in (3) of (ii)) or that its support has special properties (as in (3) of (iii)). (3) in (iii). The invariance properties have consequences that have been already studied in the Euclidean case in [8,Lemma 3.6]. We adapt them to the setting of (3) of (ii) in the following way. First let N G : G → [0, +∞) be defined via

Proof of Part (3) in (ii) and Part
by [15,Section 2.3], it is continuous. We also define the usual quasi-norm on G via We can now define the continuous function N : we deduce |Exp(s Z (λ) )x| 4 = |V | 4 + |Z + sZ (λ) | 2 + N G (λ) 4 . As a consequence, if K is any compact subset of G × (z * \ {0}), then there exist constants α 1 , β 1 , α 2 , β 2 , s 0 > 0 such that which is enough for the proof of Lemma 3.6 in [8]. The measure Tr(Γ t )dγ t which is invariant under the flow Ψ s is 0 above K. A similar argument can be performed for (3) of (iii) since the only invariant set by the action of Ξ s is the set G × {0} and if K is a compact subset of G × v * such that K ∩ (G × {0}) = ∅, then there exists α 1 , β 1 , α 2 , β 2 , s 0 > 0 such that for s ≥ 0 , Therefore, the measure ς t (x, ω) is supported on G × {ω = 0}.
3. The C * -algebra A associated with semi-classical symbols In this section we introduce the C * -algebra formalism which can be associated with semi-classical symbols. The properties of this algebra, introduced in Section 3.1 are at the roots of our analysis and allow us to prove Theorem 2.8 in Section 3.2. The proofs of Theorems 2.5 and 2.10 will use several ingredients. First, it requires the analysis of the symbolic properties of the eigenprojectors Π (λ) n performed in Section 3.4. Then, in order to pass to the limits in the relations of the Egorov theorem 2.5, we will need to approximate general symbols in A 0 by symbols belonging to the class A H , which is done in Section 3.5. Finally, it will use symbols that commute with H, the space of which is studied in Section 3.6.
3.1. The C * -algebra A and its states. We introduce the algebra A which is the closure of A 0 for the norm · A given by Clearly, A is a sub-C * -algebra of the tensor product of the commutative C * -algebra C 0 (G) of continuous functions on G vanishing at infinity together with L ∞ ( G).
It turns out that one can identify its spectrum in the following way: Proposition 3.1. The set A is a separable C * -algebra of type 1. It is not unital but admits an approximation of identity. Besides, if π 0 ∈ G and x 0 ∈ G, then the mapping extends to a continuous mapping ρ x 0 ,π 0 : A → L(H π 0 ) which is an irreducible non-zero representation of A. Furthermore, the mapping is a homeomorphism which allows for the identification of A with G × G.
The proof follows the lines of [15,Section 5]. It utilises the fact that, by definition, the C * algebra C * (G) of the group G is the closure of FS(G) for sup λ∈ G · L(H λ ) and that the spectrum of C * (G) is G. This implies readily that A may be identified with the C * -algebra of continuous functions which vanish at infinity on G and are valued on C * (G) and that its spectrum is as described in Proposition 3.1. Furthermore, the algebraic span of the symbols of the form τ (x, λ) = a(x)b(λ) with a(x) in C ∞ c (G) and the Fourier multiplier b(λ) in S −∞ is dense in A. Notice that their boundedness is easier to obtain since Op ε (τ ) simply is the composition of the operator of multiplication by a(x) and of the Fourier multiplier b(ε 2 λ), and one has (3.2) Op ε (τ ) L(L 2 (G)) ≤ sup x∈G, λ∈ G τ L(L 2 (p λ )) .
We can also describe the states of the C * -algebra A. Proof. This proposition is a corollary of Proposition 4.1 in [16]. Its proof follows the lines of [15,Section 5] and is performed in details in the Appendix of [16].
The description of the states of the C * -algebra A yields the following corollary: Then ℓ extends uniquely to a continuous bilinear map ℓ : L 1 (R) × A → C for which we keep the same notation. Furthermore, there exists a unique map t → Γ t dγ t in L ∞ (R, M + ov (G× G)) satisfying Γ t dγ t M = 1 for almost all t ∈ R, and: Proof. The estimate in (3.5) implies that the bilinear map ℓ extends uniquely into a continuous bilinear map on L 1 (R) × A for which we keep the same notation. Furthermore, for each σ ∈ A, we identify the continuous linear map θ → ℓ(θ, σ) on the Banach space L 1 (R) with the function ℓ σ ∈ L ∞ (R) given via Note that ℓ σ L ∞ (R) ≤ σ A and that the map σ → ℓ σ is a linear mapping on A to L ∞ (R). Hence, we can view the map L : t → (σ → ℓ σ (t)) as a measurable bounded map from R to the Banach space A * . Moreover, the assumption in (3.6) implies that hence ℓ σ * σ (t) ≥ 0 for almost every t ∈ R. In other words, σ → ℓ σ (t) is a state for almost every t ∈ R. Hence, the map L ∈ L ∞ (R, A * ) is valued in the set of state of A. Corollary 3.3 with the identification of A * with M ov (G × G) allows us to conclude.
With these concepts in mind, one can now sketch the proof of Theorem 2.8 as its arguments are an adaptation of the ones in [15,16].
3.3. Some comments on time-dependent states. The result in Proposition 3.4 calls for some comments which will not be used in the following paper but are of interest in themselves. Indeed, with a straightforward adaptation of the arguments given for the proof of Proposition 3.1, we obtain an analogue description for the closure of C ∞ (J)⊗A 0 if J is a compact interval, and of C ∞ c (R)⊗A 0 if J = R. Note first that in both cases, the closure is for the norm and we identify them respectively with the C * -algebra C(J, A) of continuous functions on J valued in A when J is a compact interval and with the Banach space C 0 (R, A) of continuous functions on R valued in A and vanishing at infinity when J = R. In order to unify the presentation, we may write C 0 (J, A) for C(J, A) when J is a compact interval of R.
Then, an analysis similar to the one of the proof of Proposition 3.1 gives that the C * -algebra C 0 (J, A) for J compact interval and for J = R is a separable C * -algebra of type 1. It is not unital but admits an approximation of identity. Its spectrum may be identified with J × G × G. Its states may be identified with the elements Γ dγ in M + ov (J × G × G) satisfying J×G× G Tr (Γ(t, x, λ)) dγ(t, x, λ) = 1, Finally, a map ℓ : C ∞ c (R) × A 0 → C as in Proposition 3.4. extends uniquely into a continuous linear map on L 1 (R) × A, and also into a state of C(J, A) up to the normalisation |J| for any compact interval. Using the characterisation above and the uniqueness, we can define a pair (γ, Γ) ∈ M + ov (R × G × G), unique up to equivalence, such that here γ is a measure on R × G × G. This is a weaker result than the one obtained in Proposition 3.4, which states that the measure above is absolutely continuous with respect to dt, and this explains why we proceed in this manner.
Note that we can proceed as in Proposition 3.2 and Corollary 3.3 to obtain a description of the dual of C 0 (J, A) that we will use later on: Following [19] (Section 5.2 for any graded nilpotent Lie group and Section 6.5 for the Heisenberg group), the class S m of symbols of order m in G consists of fields of operators σ(x, λ) such that for each α, β ∈ N n and γ ∈ R we have Using the dilation induced on G by the one of G, one defines for m ∈ R, m-homogeneous fields of operators σ(x, λ) by asking that σ(x, ε·λ) = ε m σ(x, λ) for all x ∈ G, almost all λ ∈ G and dε-almost all ε > 0 (in the preceding formula, ε · λ = ε 2 λ for λ ∈ z * and ε · (0, ω) = (0, εω) for ω ∈ v * ). In parallel to what is done in the Euclidean setting, one then defines regular m-homogeneous symbols as the setṠ m of m-homogeneous fields of operators σ(x, λ) which satisfy for any α, β ∈ N n , γ ∈ R: In both the inhomogeneous and homogeneous case, it was proved that an equivalent characterisation is (3.7) and (3.8) respectively, for all α, β but only γ = 0. Proposition 3.6. Let n ∈ N. The spectral projectors Π (λ) n associated with H(λ), λ ∈ z * \ {0}, form a field Π n of operators which is a homogeneous symbol inṠ 0 .
Remark 3.7. A remark on the notations: we shall use Π n when denoting the field of operators acting on L 2 (p λ ) and write Π (λ) n when some λ ∈ z * is fixed.
The proof of Proposition 3.6 relies on the spectral expression: where C n is any circle of the complex plane with centre 2n + d and radius ρ ∈ (0, 2), and the following lemma: Lemma 3.8. The field of operators |λ|I L 2 (p λ ) , λ ∈ z * \ {0}, yields a homogeneous regular symbol inṠ 2 .
Proof of Lemma 3.8. The first thing to notice is that |λ|H(λ) −1 is a bounded operator (as a selfadjoint operator with bounded eigenvalues). Then, we look at ∆ q |λ|I L 2 (p λ ) . The kernel corresponding to the symbol |λ|I L 2 (p λ ) is the distribution δ v=0 ⊗F −1 z |λ|, so the corresponding convolution operator on G acts only on the central component.
For q = z j z k , ∆ q |λ|I L 2 (p λ ) is up to a constant the group Fourier transform of the distribution δ v=0 ⊗ F −1 z ∂ λ j ∂ λ k |λ|. We see that for q = z k , [q] = 2 and Therefore sup Recursively, we obtain for any monomial q, we have the estimates required by (3.8): where [q] denotes the degree of homogeneity of q. This shows Lemma 3.8.
Proof of Proposition 3.6. In order to show that the field of operators consisting of the Π (λ) n is inṠ 0 , it suffices to show that (H(λ) − |λ|z) −1 is inṠ 2 with uniform semi-norms estimates of the form (3.8) with respect to z ∈ C n because of (3.9), Lemma 3.8 and The rest of the statement will then follow by [15] (see Section 4.2 therein).

3.5.
Approximation of H-diagonals and anti-H-diagonals symbols. We now use the symbolic properties of Section 3.4 for the projections Π (λ) n to decompose symbols as described in Lemma 2.4.
However, σ (n,n ′ ) is usually not smoothing but may be weakly approximated by the smoothing symbols ψ(uH)Π n σΠ n ′ ψ(uH) with u → 0 with ψ as in Proposition 3.6. Proposition 3.6 allows us to modify the previous construction to symbols which also depend on x ∈ G in order to obtain the following weak approximation of the Π n -projections of smoothing symbols: Corollary 3.9. We fix a smooth function ψ : R → [0, 1] satisfying ψ ≡ 0 on (−∞, 1/2) and ψ ≡ 1 on (1, +∞). Let σ ∈ S −∞ .
The same ideas also gives the approximations of symbols in A 0 by symbols in A H : Corollary 3.11. Let σ ∈ A 0 . Then there exists a sequence (σ n ) n∈N of symbols in A H converging to σ in the following sense: • for each λ ∈ z * \ {0}, we have the convergence σ n (x, λ) −→ σ(x, λ) in SOT of L 2 (p λ ) as n → +∞ uniformly in x ∈ G, and • we have the convergence σ n (x, ·) → σ(x, ·) in SOT of L ∞ ( G) as n → +∞ uniformly in x ∈ G.
Corollary 3.11 follows from Proposition 3.6 and Proposition 4.6 of [15] together with a smooth cut-off in λ ∈ z * . The latter property is granted by the following lemma: (1) For any g ∈ S(z * ), the symbol given by ) be a sequence of functions which are bounded by 1 and converges to the constant function 1 pointwise. Then (g n σ(x, ·)) n∈N converges to σ(x, ·) in SOT of Proof. Let κ σ be the kernels associated with the symbols which may be identified with a map in C ∞ c (G; S(G)). Part (1) follows from the kernel κ g associated with the symbol g(λ) being the central distribution δ v=0 ⊗ F −1 z g and the kernel associated with g(λ)σ being κ g * κ σ . Part (2) follows from the Lebesgue dominated convergence theorem and the Plancherel formula.
Remark 3.13. If σ = {σ(x, π), (x, π) ∈ G× G} ∈ S −∞ , then constructing as above ψ(uH)Π n σΠ n ψ(uH) yields a smoothing symbol commuting with H(λ). The closure of their span form a much larger class than the class of spectral multipliers of H(λ). Indeed, spectral multipliers are constant on the vector sets V n which is not the case of ψ(uH)Π n σΠ n ψ(uH). The reader can refer to [19,Chapter 4] for considerations on symbols that are functions of H(λ).

3.6.
The sub-C * -algebra B of H-commuting symbols. In order to prove Theorem 2.10, we shall use symbols in A which commute with H(λ). The set B of such symbols satisfies the following properties: Lemma 3.14. Let B be the sub-C * -algebra of A consisting of symbols σ ∈ A which commute with H(λ), i.e. σ(x, λ)H(λ) = H(λ)σ(x, λ) for almost every (x, λ) ∈ G × G.
(1) The space B contains the symbols of the form a(x)σ(λ) with a ∈ C ∞ c (G) and σ smoothing and commuting with H(λ), and the algebraic span of these symbols are dense in B.
(2) The states of B are in one-to-one correspondence as in Proposition 3.2 with the measures Γdγ ∈ M + ov (G × G) such that Γ = n∈N Π n ΓΠ n . Proof of Lemma 3.14. Part (1) is readily checked. Let us prove Part (2). Let ℓ be a state of B. For any N ∈ N and u ∈ (0, 1), Part (5) of Corollary 3.9 implies that ℓ N,u (σ) := N n=0 ℓ(σ (n,n,u) ), defines a continuous linear functional ℓ N,u on A which is a state or 0. We denote by Γ N,u dγ N,u ∈ M + ov (G × G) the measure corresponding to ℓ N,u by Proposition 3.2. We observe that for any N 1 ≤ N 2 and 2u 1 < u 2 < u 3 /2 n,u 3 ) ).
The uniqueness in Proposition 3.2 implies that there exists Γdγ ∈ M + ov (G× G) of mass 1, commuting with H(λ) and such that Γ N,u dγ N,u = N n=0 ψ(uH)Π n ΓΠ n ′ ψ(uH)dγ. This implies Part (2). It readily follows from Lemma 3.14 that the pure states of the C * -algebra B are given either by |v v| δ x 0 ⊗ δ π λ 0 for some v ∈ V n when λ 0 ∈ z * \ {0} and x 0 ∈ G, or by δ x 0 ⊗ δ π (0,ω 0 ) for some ω 0 ∈ v * and x 0 ∈ G. One can thus describe easily the dual of B by identification with the subset

Proof of Theorems 2.5 and 2.10
The core of our results are Theorems 2.5 and 2.10 from which Theorem 1.1 will derive in Section 5.3, so we focus on these two first statements. Their proofs rely on a careful analysis of the commutator [Op ε (σ), .

4.1.3.
A more precise computation. In order to prove the rest of Theorems 2.5 and 2.10, we write down more precisely the equalities we have obtained above using (2.20) It is actually convenient to use the notation ℓ ε (θ, σ) introduced in (2.23). By Theorem 2.8, we know that up to extraction of a subsequence ε k , ℓ ε (θ, σ) has a limit that we denote by ℓ ∞ (θ, σ). With these notations, equation (4.2) writes

4.2.
Proof of (ii) Theorems 2.10 and 2.5. In this paragraph, we prove Parts (ii) in Theorems 2.10 and 2.5. We consider symbols supported away from λ = 0 and commuting with H(λ).
We are going to use some properties that are summarised in the following technical lemma, the proof of which is in Appendix B.

4.2.2.
A more precise computation. Before focusing on the cases τ ≥ 2, let us make more explicit the computation in Section 4.2.1 (which is valid for any τ ) by setting for any σ as in Lemma 4.1: Rewriting (4.4), we obtain Theorem 2.5 (i) and Lemma 4.1 (3) together with (4.5) give for any σ ∈ A H , as the terms in ∆ G cancel each other.
We set for any σ as in Lemma 4.1: Theorem 2.8 and (4.6) imply while passing to the limit by Theorem 2.8, we have for any τ > 0: (2) and (3) of Theorem 2.5 (ii). We now consider σ ∈ A (d)

Proof of Parts
H . We may assume that σ = Π n σΠ n for some fixed n ∈ N.
In the case τ = 2, the computations in Section 4.2.2 give (using first equation (4.6) and then (4.7)) d ds An integration over s shows Part (1).

Proof of Part
The weak density of A H in A 0 (see Corollary 3.9) implies that Part (2) of Theorem 2.10 (ii).

4.2.5.
Proof of Part (3) Theorem 2.10 (ii). Here τ > 2. For any σ ∈ A H satisfying σ = Π n σΠ n for some fixed n ∈ N, passing to the limit in (4.10) give (in view of (4.8)) The weak density of A H in A 0 (see Corollary 3.9) implies that Γ n,t (x, λ)dγ t (x, λ) is invariant under Z (λ) away from λ = 0. As this is true for every n ∈ N, the measure 1 λ =0 Tr(Γ t )dγ t = 1 λ =0 n∈N Tr(Γ n,t )dγ t is invariant under the flow Ψ s defined in (2.21). This concludes the proof of Theorem 2.10 (ii) in view of the discussion at the end of Section 2.6.2.
. This map as well as its restriction to the sub-C * -algebra B of H-commuting symbols (see Section 3.6) are surjective.
We observe that the map Θ : σ → σ| G×{λ=0} maps A 0 on C ∞ c (G; S(v * )) in the following way where we write σ ∈ A 0 as σ(x, λ) = κ x (λ) with the map x → κ x in C ∞ c (G; S(G)). In fact, Θ(A 0 ) = C ∞ c (S(v * )) and we can easily construct a right inverse via where χ ∈ C ∞ c (z * ) with χ(0) = 1. From this, we easily check that Θ extends to a C * -algebra morphism from A onto C 0 (G × v * ). However, we now give below another argument for the surjectivity of Θ which has the advantage that it also holds for its restriction to B.
Proof of Lemma 4.2. Using the notation just above, Θ : σ → σ| G×{λ=0} maps A 0 on C ∞ c (G; S(v * )) and extends to a C * -algebra morphism from A to C 0 (G × v * ). The set Θ(A) is a sub-C * -algebra of C 0 (G × v * ), their spectrum are included accordingly with equality if and only if they are equal. Any state of C 0 (G × v * ) is given by a measure in M + (G × v * ) which may be viewed as an operator valued measure in M + ov (G × G) vanishing on G × {λ = 0}. This shows that the spectrum of the commutative algebra Θ(A) is G × v * , so Θ(A) = C 0 (G × v * ). The same argument holds for B.
(2) If τ = 1, using (2.14), we obtain from which we deduce in the sense of distributions by Lemma 4.2 (3) If τ > 1, using again (2.14) and Lemma 4.2 yields ℓ ∞ (θ, V · π λ (V )σ) = 0 and this implies that the measure ς t (x, ω) is invariant under the flow Ξ s , s ∈ R, defined by And this invariance translates in terms of localisation of the support of ς t in view of the discussion of the end of Section 2.6.2, whence Part (iii) (3) of Theorem 2.10 . In the situation when τ ∈ (0, 1], Part (iii) (1) and (2) of Theorem 2.10 comes from the resolution of the transport equations satisfied by ς t by using the continuity of t → ς t that is proved in the next section. This concludes the proof of Theorem 2.10.
The proof of Proposition 4.3 relies on (4.2) and Lemma 3.14. (1) The weak continuity of the map t → Γ t dγ t granted in Proposition 4.3 allows us to solve the transport equations of (ii) Point (1), and so for (i) and (ii) Point (2).
(2) The proof of Proposition 4.3 implies that, under the assumptions of Proposition 4.3, if ε k is the sub-sequence realising a semi-classical measure Γ t dγ t , then we have for all t ∈ R meaning that one can pass to the limit t-by-t and not only when averaged in time as in the original statement of Theorem 2.10.
Proof of Proposition 4.3. For any σ ∈ B, (4.2) gives in the sense that we may assume the distribution t → ℓ ε,t (σ) to be continuous and even C 1 on R and that d dt ℓ ε,t (σ) is uniformly bounded with respect to t in a bounded interval of R and ε ∈ (0, 1). Consider a sequence (ε j ) j∈N in (0, 1) converging to 0 as j → ∞. By the Arzéla-Ascoli theorem and (4.12), we can extract a subsequence (ε j k ) k∈N such that, as k → ∞, ε j k → 0 and (ℓ ε j k ,· (σ)) k∈N converges to a continuous function t → ℓ t (σ) locally uniformly on R for all σ ∈ B (this requires to consider a dense subset of B and a diagonal extraction procedure). We proceed as in the proof of Theorem 2.8 (see also [15,16]) using the C * -algebra B with its properties in Lemma 3.14 instead of A: either L := lim sup k→0 ψ ε j k 0 L 2 (G) = 0 and ℓ t = 0 or L −1 ℓ t is a state of B for each t ∈ R. Let Γ t dγ t ∈ M ov (G × G) with Γ t = n∈N Π n Γ t Π n corresponding to ℓ t . Up to a further extraction of a converging subsequence (for which we keep the same notation), we may assume that Γ t dγ t coincides with the semi-classical measure in Part (i) and Part (ii) (1) of Theorem 2.10, whence the result.

Uniform ε-oscillation and marginals of semi-classical measures
We prove here Theorem 1.1 in Section 5.3, using the notion of ε-oscillation explained in Section 5.2, which allows to relate the weak limits of energy density and marginals of semi-classical measures, that we first study in Section 5.1.
Part (3) follows from the commutativity of σ with H and Lemma B.2.