Quantum ergodicity and quantum limits for sub-Riemannian Laplacians
aa r X i v : . [ m a t h . SP ] J un Quantum ergodicity and quantum limits for sub-RiemannianLaplacians
Yves Colin de Verdi`ere ∗ Luc Hillairet † Emmanuel Tr´elat ‡ June 26, 2018
Abstract
This paper is a proceedings version of [6], in which we state a Quantum Ergodicity (QE)theorem on a 3D contact manifold, and in which we establish some properties of the QuantumLimits (QL).We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an orientedcontact distribution. There exists a privileged choice of the contact form, with an associatedReeb vector field and a canonical volume form that coincides with the Popp measure. We statea QE theorem for the eigenfunctions of any associated sR Laplacian, under the assumptionthat the Reeb flow is ergodic. The limit measure is given by the normalized canonical contactmeasure. To our knowledge, this is the first extension of the usual Schnirelman theorem toa hypoelliptic operator. We provide as well a decomposition result of QL’s, which is validwithout any ergodicity assumption. We explain the main steps of the proof, and we discusspossible extensions to other sR geometries.
The property of Quantum Ergodicity (QE) is defined as follows. Let M be a metric space, endowedwith a measure µ defined on a suitable compactification of M , and let T be a self-adjoint operatoron L ( M, µ ), bounded below and having a compact resolvent (and hence a discrete spectrum).Let ( φ n ) n ∈ N ∗ be a (complex-valued) Hilbert basis of L ( M, µ ), consisting of eigenfunctions of T ,associated with the ordered sequence of eigenvalues λ · · · λ n · · · . We say that QE holdsfor T is there exist a probability measure ν on M and a density-one sequence ( n j ) j ∈ N ∗ of positiveintegers such that the sequence of probability measures | φ n j | dµ converges weakly to ν . Themeasure ν may be different from some scalar multiple of µ , and may even be singular with respectto µ .Microlocal versions of QE hold true in general and are stated in terms of pseudo-differentialoperators, yielding a result of the kind h Op( a ) φ n j , φ n j i → Z Σ a d ˜ ν, ∗ Universit´e de Grenoble, Institut Fourier, Unit´e mixte de recherche CNRS-UJF 5582, BP 74, 38402-Saint Martind’H`eres Cedex, France ( [email protected] ). † Universit´e d’Orl´eans, F´ed´eration Denis Poisson, Laboratoire MAPMO, route de Chartres, 45067 Orl´eans Cedex2, France ( [email protected] ). ‡ Sorbonne Universit´es, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, InstitutUniversitaire de France, F-75005, Paris, France ( [email protected] ). a of order 0, where ˜ ν is a lift to T ⋆ M of the measure ν and Σ is a subsetof T ⋆ M (that have to be identified).Such a QE property provides an insight on the way eigenfunctions concentrate when consideringhighfrequencies.In the existing literature, QE results exist in different contexts, but always for elliptic operators.The first historical QE result is due to Shnirelman in 1974 (see [20] for a sketch of proof), statingthat, on a compact Riemannian manifold ( M, g ), if the geodesic flow is ergodic, then we have QEfor any orthonormal basis of eigenfunctions of the Laplace-Beltrami operator, with dν equal to thenormalized Riemannian volume (and d ˜ ν equal to the normalized Liouville measure). The completeproof has been obtained later in [5, 21]. Then, the Shnirelman theorem has been extended to thecase of manifolds with boundary in [9, 23], to the case of discontinuous metrics in [13] and to thesemiclassical regime in [10].We provide here a Shnirelman theorem in a hypoelliptic context. Theorem 1.
Let M be compact connected 3D manifold (without boundary), endowed with anarbitrary volume form dµ and with a Riemannian contact structure. We consider the associatedsub-Riemannian Laplacian △ sR . If the Reeb flow is ergodic for the Popp measure, then we haveQE, and the limit measure is the probability Popp measure. Several definitions are due to understand the content of this theorem. In Section 2, we are thusgoing to recall: • what is a sub-Riemannian Laplacian; • what are a contact structure, the Reeb flow, and the Popp measure.We will then provide a more general (microlocal) version of Theorem 1. We also provide a secondresult, valid without any ergodicity assumption, stating that every quantum limit that is supportedon the characteristic manifold of the sR Laplacian is invariant under the Reeb flow, and that mostQL satisfy the previous assumption on their support.We will explain the main steps of the proof in Section 3, and we will conclude in Section 4 bygiving some perspectives and open problems. We first recall that a sub-Riemannian (sR) structure is a triple (
M, D, g ), where M is a smoothmanifold, D is a smooth subbundle of T M (called horizontal distribution), and g is a fiberedRiemannian metric on D . Example 1.
We speak of a contact structure when M is of dimension 3 and D is a contactdistribution, that is, we can write D = ker α locally around any point, for some one-form α suchthat α ∧ dα = 0 (locally). The local Darboux model in R at the origin is given by α = dz + x dy ,and then, locally, we can write D = Span( X, Y ), with X and Y the vector fields defined by X = ∂ x and Y = ∂ y − x∂ z . Note that D is of codimension one, and the Lie bracket [ X, Y ] = − ∂ z generatesthe missing direction. 2 ub-Riemannian Laplacian. In order to define a sub-Riemannian Laplacian △ sR , let us choosea smooth volume form dµ on M , the associated measure being denoted by µ . Let L ( M, µ ) be theset of complex-valued functions u such that | u | is µ -integrable over M . Then, in whole generality, −△ sR is the nonnegative self-adjoint operator on L ( M, µ ) defined as the Friedrichs extension ofthe Dirichlet integral Q ( φ ) = Z M k dφ k g ∗ dµ, where the norm of dφ is calculated with respect to the co-metric g ⋆ on T ⋆ M associated with g .The sR Laplacian △ sR depends on the choice of g and of dµ .Let us give two equilavent definitions of △ sR : • We have △ sR φ = div µ ( ∇ sR φ )for every smooth function φ on M , where div µ is the divergence operator associated withthe volume form dµ , defined by L X dµ = div µ ( X ) dµ for any vector field X on M , and wherethe horizontal gradient ∇ sR φ of a smooth function φ is the unique section of D such that g q ( ∇ sR φ ( q ) , v ) = dφ ( q ) .v , for every v ∈ D q . • If ( X , . . . , X m ) is a local g -orthonormal frame of D , then ∇ sR φ = P mi =1 ( X i φ ) X i , and △ sR = − m X i =1 X ⋆i X i = m X i =1 (cid:0) X i + div µ ( X i ) X i (cid:1) Hypoellipticity.
A well known theorem due to H¨ormander (see [11]) states that, under theassumption Lie( D ) = T M , the operator △ sR is hypoelliptic (and even subelliptic), has a compactresolvent and thus a discrete spectrum λ λ · · · λ n → + ∞ . Remark 1.
Denoting by h X ( q, p ) = h p, X ( q ) i the Hamiltonian associated with a vector field X (in canonical coordinates), the principal symbol of −△ sR is σ P ( −△ sR ) = m X i =1 h X i = g ⋆ , (it coincides with the co-metric g ⋆ ), and its sub-principal symbol is zero.Actually, in our main result, △ sR may be any self-adjoint second-order differential operatorwhose principal symbol is g ⋆ , whose sub-principal symbol vanishes. Remark 2.
The Hamiltonian function g ⋆ induces a Hamiltonian vector field X g which generatesthe so-called normal geodesics of the sR structure (see [17]). We assume that the manifold M is compact and of dimension 3, and that D is an oriented contactdistribution, that is, we can write D = ker α for some one-form α globally defined such that α ∧ dα = 0 (see [6] for the extension to the non-orientable case).In order to define the canonical contact measure and the Reeb vector field, we need to normalizethe contact form defining D . There exists a unique contact form α g such that dα g ( X, Y ) = 1 forany positive g -orthonormal local frame ( X, Y ) of D . Equivalently, ( dα g ) | D coincides with thevolume form induced by g on D . 3 opp measure. We define the density dP = | α g ∧ dα g | on M . In general, dP differs from dµ . The corresponding measure P is called the Popp measure in the existing literature (see [17],where it is defined in the general equiregular case). Of course, here, this measure is the canonicalcontact measure associated with the normalized contact form α g . We also define the probabilityPopp measure ν = PP ( M ) . Reeb flow.
The
Reeb vector field Z of the contact form α g is defined as the unique vector fieldsuch that ι Z α g = 1 and ι Z dα g = 0. Equivalently, Z is the unique vector field such that[ X, Z ] ∈ D, [ Y, Z ] ∈ D, [ X, Y ] = − Z mod D, for any positive orthonormal local frame ( X, Y ) of D . The Reeb flow R t is defined as the flowgenerated on M by the vector field Z . Remark 3.
The measure ν is invariant under the Reeb flow R t . This invariance property isimportant in view of identifying a QE result. Remark 4.
Denoting by h Z the Hamiltonian associated with Z , geodesics with highfrequenciesin h Z oscillate around the trajectories of ± Z . From the point of view of semiclassical analysis, thispart of the dynamics is expected to be the dominant one. We consider the cotangent space ( T ⋆ M, ω ), endowed with its canonical symplectic form. LetΣ ⊂ T ⋆ M be the characteristic manifold of −△ sR . We have Σ = ( g ⋆ ) − (0) = D ⊥ (annihilator of D ), and Σ coincides with the cone of contact forms defining D . In particular, we haveΣ = { ( q, sα g ( q )) ∈ T ⋆ M | q ∈ M, s ∈ R } . In the 3D contact case, we have as well Σ = { h X = h Y = 0 } if D = Span( X, Y ), with (
X, Y ) alocal g -orthonormal frame of D . Shnirelman theorem in the 3D contact case.
We are now in a position to provide a moreprecise statement of Theorem 1.
Theorem 2.
Let M be a smooth connected compact three-dimensional manifold, equipped with anarbitrary smooth volume form dµ . Let D ⊂ T M be a smooth oriented contact subbundle, and let g be a smooth Riemannian metric on D . Let △ sR be the sR Laplacian associated with the contactsub-Riemannian structure ( M, D, g ) and with the volume form dµ .We assume that the Reeb flow is ergodic . Then, for any real-valued orthonormal Hilbert basis ( φ n ) n ∈ N ∗ of L ( M, µ ) consisting of eigenfunctions of △ sR associated with the eigenvalues ( λ n ) n ∈ N ∗ ,there exists a density-one sequence ( n j ) j ∈ N ∗ of positive integers such that lim j → + ∞ (cid:10) Aφ n j , φ n j (cid:11) = 12 Z M (cid:0) a ( q, α g ( q )) + a ( q, − α g ( q )) (cid:1) dν, for every classical pseudo-differential operator A of order with principal symbol a .In particular, we have lim j → + ∞ Z M f | φ n j | dµ = Z M f dν, for every continuous function f on M . The Reeb flow is ergodic on (
M, ν ) if any measurable invariant subset of M is of measure 0 or 1. h , i stands for the (Hermitian) inner product in L ( M, µ ).We prove in [6] that the above result still holds with a complex-valued eigenbasis, provided theprincipal symbol a satisfies a ( q, α g ( q )) = a ( q, − α g ( q )), for every q ∈ M . We also prove that theresult is valid as well if D is not orientable, without any additional assumption (note that, in thatcase, the contact form and thus the Reeb vector field Z are not defined globally).Note that the subsequence of density one depends on the choice of the eigenbasis ( φ n ) n ∈ N ∗ . Westress that our result is valid for any choice of a smooth volume form dµ on M . Remark 5.
The classical Shnirelman theorem is established in the Riemannian setting under theassumption that the Riemannian geodesic flow is ergodic on ( S ⋆ M, λ L ), where the limit measure isthe Liouville measure λ L on the unit cotangent bundle S ⋆ M of M . In contrast, here the Liouvillemeasure on the unit bundle g ⋆ = 1 has infinite total mass (where g ⋆ is the co-metric on T ⋆ M associated with g ), and hence the QE property cannot be formulated in terms of the geodesic flow.Another interesting difference is that, in the Riemannian setting, QE says that most eigenfunc-tions equidistribute in the phase space, whereas here, in the 3D contact case, they concentrate onΣ = D ⊥ , the contact cone that is the characteristic manifold of △ sR .Riemannian case SR caseErgodicity assumption geodesic flow on ( S ∗ M, Liouville) Reeb flow on ( M, Popp)Quantum limit Liouville measure on S ∗ M Popp measure on M Microlocal concentration on S ∗ M on S Σ = Σ ∩ {| h Z | = 1 } Remark 6 (Examples of ergodic Reeb flows in dimension 3) . Let us give two general constructionsproviding examples of ergodic Reeb flows on 3D contact manifolds: • Geodesic flows: Let X be a two-dimensional compact Riemannian surface, endowed with aRiemannian metric h , and let M = S ⋆ X be the unit cotangent bundle of X . The closedthree-dimensional manifold M is then naturally endowed with the contact form α defined asthe restriction to M of the Liouville 1-form λ = p dq . Let Z be the associated Reeb vectorfield. Identifying the tangent and cotangent bundles of X with the metric h , the set M isviewed as the unit tangent bundle of X . Then Z is identified with the vector field on the unittangent bundle of X generating the geodesic flow on S ⋆ X . Therefore, with this identification,the Reeb flow is the geodesic flow on M .This geodesic flow is ergodic for instance if the curvature of X is negative. • Hamiltonian flows: Let (
W, ω ) be a symplectic manifold of dimension 4, and let M be asubmanifold of W of dimension 3, such that there exists a vector field v on a neighborhoodof M in W , satisfying L v ω = ω (Liouville vector field), and transverse to M . Then the one-form α = ι v ω is a global contact form on M , and we have dα = ω . Note that, if ω = dλ isexact, then the vector field v defined by ι v ω = λ is Liouville (in local symplectic coordinates( q, p ) on W , we have v = p ∂ p ). If the manifold M is moreover a level set of an Hamiltonianfunction h on W , then the Reeb flow on M (associated with α ) is a reparametrization of theHamiltonian flow restricted to M .If D = ker α is moreover endowed with a Riemannian metric g , then α g = hα for somesmooth function h (never vanishing). Let us then choose the metric g such that h = 1. Then,the Reeb flow is ergodic on ( M, ν ) if and only if the Hamiltonian flow is ergodic on (
W, ω ).5 uantum limits in the 3D contact case. Let ( ψ j ) j ∈ N ∗ be an arbitrary orthonormal family of L ( M, µ ). We set µ j ( a ) = h Op( a ) ψ j , ψ j i , for every j ∈ N ∗ , and for every classical symbol a of order0. The measure µ j is asymptotically positive, and any closure point (weak limit) of ( µ j ) j ∈ N ∗ is aprobability measure on the sphere bundle S ⋆ M , called a quantum limit (QL), or a semi-classicalmeasure , associated with the family ( ψ j ) j ∈ N ∗ .Theorem 1 says that, under the ergodicity assumption of the Reeb flow, the probability Poppmeasure ν , which is invariant under the Reeb flow, is the “main” quantum limit associated withany eigenbasis.The following result provides an insight on quantum limits of eigenfunctions in the 3D contactcase in greater generality, without any ergodicity assumption. In order to state it, we identify S ⋆ M = ( T ⋆ M \ { } ) / (0 , + ∞ ) with the union of the unit cotangent bundle U ⋆ M = { g ⋆ = 1 } andof the sphere bundle S Σ = (Σ \ { } ) / (0 , + ∞ ) which is a two-fold covering of M . Each fiber isobtained by compactifying a cylinder with two points at infinity. Moreover, the Reeb flow can belifted to S Σ. Theorem 3.
Let ( φ n ) n ∈ N ∗ be an orthonormal Hilbert basis of L ( M, µ ) , consisting of eigenfunc-tions of △ sR associated with the eigenvalues ( λ n ) n ∈ N ∗ .1. Let β be a QL associated with the family ( φ n ) n ∈ N ∗ . Using the above identification S ⋆ M = U ⋆ M ∪ S Σ , the measure β can be identified to the sum β = β + β ∞ of two mutually singularmeasures such that: • β is supported on U ⋆ M and is invariant under the sR geodesic flow associated with thesR metric g , • β ∞ is supported on S Σ and is invariant under the lift of the Reeb flow.2. There exists a density-one sequence ( n j ) j ∈ N ∗ of positive integers such that, if β is a QLassociated with the orthonormal family ( φ n j ) j ∈ N ∗ , then the support of β is contained in S Σ ,i.e., β = 0 in the previous decomposition. The proof of that result follows from arguments used to prove Theorem 1 (see [6] for details).
The simplest example is given by an invariant metric on a compact quotient of the Heisenberggroup. The spectral decomposition of the Heisenberg Laplacian is then explicit and can serve as amodel to derive our main result.Let G be the three-dimensional Heisenberg group defined as G = R with the product rule( x, y, z ) ⋆ ( x ′ , y ′ , z ′ ) = ( x + x ′ , y + y ′ , z + z ′ − xy ′ ) . The contact form α H = dz + x dy and the vector fields X H = ∂ x and Y H = ∂ y − x∂ z are left-invariant on G . Defining the discrete co-compact subgroup Γ of G by Γ = { ( x, y, z ) ∈ G | x, y ∈√ π Z , z ∈ π Z } , we then define the three-dimensional compact manifold manifold M H = Γ \ G ,and we consider the horizontal distribution D H = ker α H , endowed with the metric g H such that( X H , Y H ) is a g H -orthonormal frame of D H . With this choice, we have α H g = α H .The Reeb vector field is given by Z H = − [ X H , Y H ] = ∂ z . The Lebesgue volume dµ = dx dy dz coincides with the Popp volume dP , and we consider the corresponding sub-Riemannian Laplacian △ H = X H + Y H (here, the vector fields have divergence zero).We refer to this sub-Riemannian case as the Heisenberg flat case ( M H , △ H ). The cotangentspace T ⋆ M H is endowed with its canonical symplectic form.6t is proved in [4] that the spectrum of −△ H is given by { λ ℓ,m = (2 ℓ + 1) | m | | m ∈ Z \ { } , ℓ ∈ N } ∪ { µ j,k = 2 π ( j + k ) | j, k ∈ Z } , where λ ℓ,m is of multiplicity | m | . From this, we infer two things.The first (already known, see, e.g., [15]) is that the spectral counting function has the asymp-totics N ( λ ) ∼ ∞ X l =0 λ (2 l + 1) = π λ = P ( M H )32 λ , as λ → + ∞ . It is interesting to compare that result with the corresponding result for a RiemannianLaplacian on a three-dimensional closed Riemannian manifold, for which N ( λ ) ∼ Cλ / .The second thing is the following. Microlocally, in the cone C c = { p z > c ( p x + p y ) } forsome c > p = ( p x , p y , p z ) in the cotangent space), the sub-RiemannianLaplacian can be written as a commuting product −△ H = R H Ω H = Ω H R H , with • R H = p Z ⋆H Z H (pseudo-differential operator of order 1), • Ω H = U H + V H (harmonic oscillator), where U H and V H are the pseudo-differential operatorsof order 1 / U H = 1 i R − H X H = Op W h X H p | h Z H | ! , V H = 1 i R − H Y H = Op W h Y H p | h Z H | ! , where the notation Op W stands for the Weyl quantization. Note that [ U H , V H ] = ± id(according to the sign of h Z H ) and exp(2 iπ Ω H ) = id.In terms of symbols, since X H and Y H have divergence zero, we have factorized σ ( −△ H ) = h X H + h Y H (full symbol) as | h Z H | (cid:18) h XH √ | h ZH | (cid:19) + (cid:18) h YH √ | h ZH | (cid:19) ! . We fix an arbitrary (orthonormal) eigenbasis ( φ n ) n ∈ N ∗ of △ sR . We recall that the spectral countingfunction is defined by N ( λ ) = { n | λ n λ } . Definition 1.
For every bounded linear operator A on L ( M, µ ) , we define the Ces`aro mean E ( A ) ∈ C by E ( A ) = lim λ → + ∞ N ( λ ) X λ n λ h Aφ n , φ n i , whenever this limit exists. We define the variance V ( A ) ∈ R + by V ( A ) = lim sup λ → + ∞ N ( λ ) X λ n λ |h Aφ n , φ n i| .
7t is easy to see that V ( A ) E ( A ⋆ A ), and that, if A is a compact operator on L ( M, µ ), then E ( A ) = 0 and V ( A ) = 0. Definition 2.
The local Weyl measure w △ is the probability measure on M defined by Z M f dw △ = lim λ → + ∞ N ( λ ) X λ n λ Z M f | φ n | dµ, for every continuous function f : M → R , whenever the limit exists. In other words, w △ = weak lim λ → + ∞ N ( λ ) X λ n λ | φ n | µ. The microlocal Weyl measure W △ is the probability measure on S ⋆ M = S ( T ⋆ M ) , the co-spherebundle, defined as follows: we identify positively homogeneous functions of degree on T ⋆ M withfunctions on the bundle S ⋆ M . Then, for every symbol a : S ⋆ M → R of order zero, we have Z S ⋆ M a dW △ = lim λ → + ∞ N ( λ ) X λ n λ h Op + ( a ) φ n , φ n i , where Op + is a positive quantization, whenever the limit exists. In other words, W △ is the weaklimit of the probability measures on S ⋆ M defined by a N ( λ ) X λ n λ h Op + ( a ) φ n , φ n i . The microlocal Weyl measure, if it exists, does not depend on the choice of the quantizationand of the choice of the orthonormal eigenbasis (because it is a trace). Moreover, W △ is even withrespect to the canonical involution of S ⋆ M . General path towards QE.
In order to establish QE, we follow the general path of proofdescribed in [22]. Defining the spectral counting function by N ( λ ) = { n | λ n λ } , the first stepconsists in establishing a microlocal Weyl law : E ( A ) = Z ST ⋆ M a dW △ , (1)for every classical pseudo-differential operator A of order 0 with a principal symbol a , where ST ⋆ M is the unit sphere bundle of T ⋆ M . This Ces´aro convergence property can usually be establishedunder weak assumptions, without any ergodicity property.The second step consists in proving the variance estimate V ( A − E ( A )id) = 0 . (2)The variance estimate usually follows by combining the microlocal Weyl law (1) with ergodicityproperties of some associated classical dynamics and with an Egorov theorem.Then QE follows from the two properties above. Indeed, for a fixed pseudo-differential operator A of order 0, it follows from (2) and from a well known lemma due to Koopman and Von Neumann(see, e.g., [18, Chapter 2.6, Lemma 6.2]) that there exists a density-one sequence ( n j ) j ∈ N ∗ of positiveintegers such that (cid:10) Aφ n j , φ n j (cid:11) → E ( A ) as j → + ∞ . Using the fact that the space of symbols oforder 0 admits a countable dense subset, QE is then established with a diagonal argument. This lemma states that, given a bounded sequence ( u n ) n ∈ N of nonnegative real numbers, the Ces´aro mean n P n − k =0 u k converges to 0 if and only if there exists a subset S ⊂ N of density one such that ( u k ) k ∈ S converges to0. We recall that S is of density one if n { k ∈ S | k n − } converges to 1 as n tends to + ∞ . .2 The microlocal Weyl law In order to prove Theorem 2, we follow the general path above, and we start by providing amicrolocal Weyl law, identifying the microlocal Weyl measure in the 3D contact case.
Theorem 4.
Let A be a classical pseudo-differential operator of order with principal symbol a .We have X λ n λ h Aφ n , φ n i = P ( M )64 λ (1 + o(1)) Z M (cid:0) a ( q, α g ( q )) + a ( q, − α g ( q )) (cid:1) dν, as λ → + ∞ . In particular, it follows that N ( λ ) ∼ P ( M )32 λ , and that E ( A ) = 12 Z M (cid:0) a ( q, α g ( q )) + a ( q, − α g ( q )) (cid:1) dν = Z S Σ a dW △ , where S Σ = (Σ \ { } ) / (0 , + ∞ ) is the co-sphere bundle of Σ . Our proof of Theorem 4, done in [6], consists in first establishing a local Weyl law, by computingheat traces with the sR heat kernel and using the Karamata tauberian theorem, and then in inferingfrom that local law, the microlocal one. The latter step is performed as follows. We first provethat, if the microlocal Weyl measure W △ exists, then Supp( W △ ) ⊂ S Σ. This follows from the factthat, outside of the characteristic manifold Σ, the operator △ sR is elliptic, and therefore classicalarguments of [8, 12], using Fourier Integral Operators and wave propagation, yield that X λ n λ |h Aφ n , φ n i| = O (cid:0) λ / (cid:1) , as λ → + ∞ , for every pseudo-differential operator of order 0 whose principal symbol vanishes ina neighborhood of Σ. Since we already know that N ( λ ) ∼ Cλ , the concentration on Σ follows.It can be noted that these arguments are as well valid for general sub-Riemannian structures, notonly in the 3D contact case.Then, we prove that, since the horizontal distribution D is of codimension 1 in T M , and sincewe have already identified the local Weyl measure w △ , then the microlocal Weyl measure W △ exists and is equal to half of the pullback of w △ by the double covering S Σ → M which is therestriction of the canonical projection of T ⋆ M onto M . This latter argument is also valid in amore general context, as soon as D is of codimension 1. Remark 7.
A remark which is useful in order to understand why our result does not depend on thechoice of the measure is the following. Let us consider two sR Laplacians △ µ and △ µ associatedwith two different volume forms (but with the same metric g ). We assume that µ = h µ with h a positive smooth function on M . We define the isometric bijection J : L ( M, µ ) → L ( M, µ )by Jφ = hφ . Then J △ µ J − = △ µ + h △ µ ( h − ) id, and therefore △ µ is unitarily equivalent to △ µ + W , where W is a bounded operator. Having in mind the model situation given by the Heisenberg flat case ( M H , △ H ), in the general3D contact case we are able to establish a Birkhoff normal form, in the spirit of a result by Melrosein [14, Section 2], which implies in particular that, microlocally near the characteristic cone, all3D contact sub-Riemannian Laplacians (associated with different metrics and/or measures) areequivalent. 9e define the positive conic submanifolds Σ ± = { ( q, sα g ( q )) ∈ T ⋆ M | ± s > } of T ⋆ M . In theHeisenberg flat case, the characteristic cones are defined accordingly.Given k ∈ N ∪ { + ∞} and given a smooth function f on T ⋆ M , the notation f = O Σ ( k ) meansthat f vanishes along Σ at order k (at least). The word flat is used when k = + ∞ . Classical normal form.Theorem 5.
Let q ∈ M be arbitrary. There exist a conic neighborhood C q of Σ + q in ( T ⋆ M, ω ) and ahomogeneous symplectomorphism χ from C q to ( T ⋆ M H , ω H ) , satisfying χ ( q ) = 0 and χ (Σ + ∩ C q ) ⊂ Σ + H , such that σ P ( −△ H ) ◦ χ = σ P ( −△ sR ) + O Σ ( ∞ )Of course, a similar result can be given for Σ − .In order to establish this normal form, we first endow R with a symplectic form ˜ ω , withan appropriate conic structure, and with an Hamiltonian function H , such that, for any givencontact structure and any q ∈ M , there exists a homogeneous diffeomorphism χ from a conicneighborhood C q of Σ + q to R such that χ ∗ ˜ ω = ω + O Σ (1) and H ◦ χ = σ P ( −△ sR ). Thanks tothe Darboux-Weinstein lemma, we modify χ into a homogeneous diffeomorphism χ such that χ ∗ ˜ ω = ω and H ◦ χ = σ P ( −△ sR ) + O Σ (3). Finally, we improve the latter remainder to a flatremainder O Σ ( ∞ ), by solving an infinite number of cohomological equations in the symplecticconic manifold ( R , ˜ ω ). Quantum normal form.
By quantizing the above Birkhoff normal form, we obtain the followingresult.
Theorem 6.
For every q ∈ M , there exists a (conic) microlocal neighborhood ˜ U of Σ q in T ⋆ M suchthat, considering all the following pseudo-differential operators as acting on functions microlocallysupported in ˜ U , we have −△ sR = R Ω + V + O Σ ( ∞ ) where • V ∈ Ψ is a self-adjoint pseudo-differential operator of order , • R and Ω are self-adjoint pseudo-differential operators of order , of respective principal sym-bols satisfying σ P ( R ) = | h Z | + O Σ (2) , σ P (Ω) = ( h X + h Y ) / | h Z | + O Σ (3) , • [ R, Ω] = 0 mod Ψ −∞ , • exp(2 iπ Ω) = id mod Ψ −∞ . The latter remainders are in the sense of pseudo-differential operators of order −∞ . In the flatHeisenberg case, there are no remainder terms, and we recover the operators R H and Ω H . Thepseudo-differential operators R and Ω can be seen as appropriate perturbations of R H and Ω H ,designed such that the last two items of Theorem 6 are satisfied.Note that the operators R and Ω depend on the microlocal neighborhood ˜ U under consideration.This neighborhood can then be understood as a chart in the manifold T ⋆ M , in which the quantumnormal form is valid. 10 .4 Sketch of proof of the variance estimate in the Heisenberg flat case As explained at the end of Section 3.1, in order to establish QE, it suffices to prove that V ( A ) = 0for every pseudo-differential operator A of order 0 satisfying E ( A ) = 0.We provide the proof of that fact in the Heisenberg flat case, that is, when −△ sR = R H Ω H =Ω H R H . For simplicity of notation, we drop the index H in what follows. Remark 8.
In the Heisenberg flat case, the microlocal factorization of −△ sR is global. This factavoids many technicalities. In [6] where the complete proof is done, we have to use microlocal chartswhere the quantum normal form of Theorem 6 is valid. We have also to take care of remainderterms, since in the general case we do not have an exact factorization, and handling the additionalterms raises additional difficulties.Let A be a pseudo-differential operator of order 0 satisfying E ( A ) = 0. Let a = σ P ( A ) be itsprincipal symbol. Preparation of A . We claim that we can modify A without changing a | Σ , so as to assume that[ A, Ω] = 0.Indeed, it suffices to average with respect to Ω, as follows. Setting A s = exp( is Ω) A exp( − is Ω),for s ∈ R , we have, by the Egorov theorem, σ P ( A s ) = a ◦ exp( t~σ P (Ω)). Since σ P (Ω) = ( h X + h Y ) / | h Z | = 0 along Σ, it follows that σ P ( A s ) | Σ = a . Now, setting ¯ A = π R π A s ds , we have σ P ( ¯ A ) | Σ = a . Using that exp(2 iπ Ω) = id and that dds A s = i [Ω , A s ], we infer that [Ω , ¯ A ] = 0. Thenwe replace A with ¯ A . Averaging with respect to R . Let us now set A t = exp( − itR ) A exp( itR ), for t ∈ R . By theEgorov theorem, we have a t = σ P ( A t ) = a ◦ exp( t~h Z ), where ~h Z is the Hamiltonian vector fieldgenerated by the Hamiltonian function h Z . For T >
0, we define ¯ A T = T R T A t dt . To prove that V ( A ) = 0, it suffices to prove that V ( A − ¯ A T ) = 0 , (3)and that lim T → + ∞ V ( ¯ A T ) = 0 . (4)The next two lemmas are devoted to prove (3) and (4). Ergodicity will be used to prove (4). Lemma 1.
We have V ( A − A t ) = 0 , for every t ∈ R . As a consequence, we have V ( A − ¯ A T ) = 0 ,for every T > .Proof. Since ddt A t = i [ A t , R ], it suffices (by Cauchy-Schwarz) to prove that V ([ A t , R ]) = 0. Usingthe Jacobi identity, and the fact that [ R, Ω] = 0, we compute ddt [ A t , Ω] = i [[ A t , R ] , Ω] = i [ R, [Ω , A t ]] + i [ A t , [ R, Ω]] = − i ad( R ) . [ A t , Ω] , and since [ A, Ω] = 0 by construction, we infer that [ A t , Ω] = e − it ad( R ) [ A, Ω] = 0.Now, for every integer n , we have h [ A t , R ] φ n , φ n i = h A t Rφ n , φ n i − h RA t φ n , φ n i = − λ n h A t Rφ n , R Ω φ n i + 1 λ n h RA t R Ω φ n , φ n i because φ n = − λ n △ sR φ n = 1 λ n h [ RA t R, Ω] φ n , φ n i = 0 11ecause [ R, Ω] = 0 and [ A t , Ω] = 0. Hence V ( A − A t ) = 0 for every t .Using the Fubini theorem and the Jensen inequality, we easily infer that V ( A − ¯ A T ) = 0. Lemma 2.
We have lim T → + ∞ V ( ¯ A T ) = 0 .Proof. We have V ( ¯ A T ) E ( ¯ A T ¯ A ∗ T ), with σ P ( ¯ A T ¯ A ∗ T ) = | a T | = | T R T a t dt | . By the microlocalWeyl law (Theorem 4), we have E ( ¯ A T ¯ A ∗ T ) = 12 X ± Z M | a T ( q, ± α g ( q )) | dν, but, using the Reeb flow R t , f ± T ( q ) = a T ( q, ± α g ( q )) = 1 T Z T a t ( q, ± α g ( q )) dt = 1 T Z T a ( R t ( q ) , ± α g ( R t ( q ))) dt By ergodicity of the Reeb flow, and using the von Neumann ergodic theorem, we have f ± T L −→ T → + ∞ Z M a ( q, ± α g ( q )) dν = 2 E ( A ) = 0 , and the result follows, under the additional assumption that a is even with respect to Σ. If theeigenfunctions are real-valued then this assumption can actually be dropped (see [6]). Reeb flow and spectral theory.
In addition to QE, there are several other relationships be-tween the Reeb flow and the spectral theory of sR Laplacians in the 3D contact case. For example,in the case where M = S , the Popp volume is a spectral invariant, which coincides with theinverse of the Arnold asymptotic linking number of the Reeb orbits. Moreover, using a globalnormal form, one can figure out a close relationship between the spectrum of an elliptic Toeplitzoperator that is the quantization of the Reeb flow and the spectrum of △ sR . These argumentslead to conjecture that the periods of the periodic orbits of the Reeb flow are spectral invariantsof △ sR . Measures in sR geometry.
A general problem consists of showing the existence of the (micro)-local Weyl measures (defined in Definition 2), and of identifying it.In contrast to Riemannian geometry, in sR geometry there are several possible choices of in-trinsic volume forms, for instance: the standard Hausdorff volume (which is defined with arbitrarycoverings), the spherical Hausdorff volume (which is defined with ball coverings only, with ballsof constant radius), the Popp volume. The latter one, which is always a smooth volume, wasintroduced and defined in the equiregular case in [17]. Note that for left-invariant (Lie group) sub-Riemannian manifolds, the Popp measure and the standard and spherical Hausdorff measures areleft-invariant, and thus are proportional to the left-Haar measure. In other words they coincide upto a multiplying scalar. In particular in the 3D contact case, the Popp measure and the standardand spherical Hausdorff measures are proportional, but the constant is not equal to 1, and is notknown.All these measures do not coincide in general (if there is no group structure). More precisely,it is proved in [1] that the Popp measure and the spherical Hausdorff measure are proportionalfor regular sR manifolds of dimensions 3 and 4, but in dimension larger than or equal to 5, the12pherical Hausdorff measure is not smooth in general and therefore differs from the Popp measure(which is always smooth by definition for equiregular distributions).The notion of (microlocal, or local) Weyl measure is very natural in the spectral context, butseems to be new and has never been studied in sR geometry. It is therefore of interest to compareit with other notions of measures and to investigate the following kind of question: does the Weylmeasure coincide with an appropriate Hausdorff measure? What is its Radon-Nikodym derivativewith respect to the Popp measure?
Equiregular sR structures.
In Theorem 4, we have obtained a microlocal Weyl law in the3D contact case. It is likely that the microlocal Weyl formula can be generalized to equiregularsub-Riemannian structures.Let (
M, D, g ) be a sub-Riemannian structure, where M is a compact manifold of dimension d . Let △ sR be a sub-Riemannian Laplacian on M . We assume that Lie( D ) = T M (H¨ormander’sassumption, implying hypoellipticity). We say that the horizontal distribution D is equiregular , ifthe sequence of subbundles ( D i ) i ∈ N ∗ defined by D = { } , D = D , D i +1 = D i + [ D i , D ], is suchthat the dimension of D iq dot not depend on q ∈ M , for every integer i . Let r be the smallestpositive integer such that D r = T M . We set Σ j = D ⊥ j , for j = 0 , . . . , r .In the equiregular case, it is known (see [15]) that N ( λ ) ∼ Cst λ Q/ , with Q = P ri =1 i dim( D i /D i − )(Hausdorff dimension). Besides, as already said in Section 3.2, we already know that, if themicrolocal Weyl measure exists, then supp( W △ ) ⊂ S Σ = S Σ . Actually, we conjecture thatsupp( W △ ) = S Σ r − , and that this result on the support is even valid in general, not only in the equiregular case. Contact case in dimension . Theorem 2 has been established in the 3D contact case. Itis natural to investigate the general contact case, however, already in the contact 5D case, thesituation is much more complicated, and we cannot expect to have a Birkhoff normal form atthe infinite order as in Theorem 5. Indeed, in dimension 5, the characteristic manifold Σ is ofcodimension 4, and we have two harmonic oscillators, which may have resonances according to thechoice of the coefficients of the sR metric. We expect however to be able, in that case, to derive aBirkhoff normal form at finite orders, and thanks to a quantized version of that normal form, tobe able to derive QE under generic assumptions. Indeed, resonances should only occur on sets ofmeasure zero in the generic case.
The role of singular curves.
We briefly recall the definition of a singular curve. Let (
M, D, g )be a sR manifold. A curve q ( · ) : [0 , → M is said to be horizontal if it is absolutely continuousand if ˙ q ( t ) ∈ D q ( t ) for almost every t ∈ [0 , q ∈ M be arbitrary. Let Ω( q ) be the set of allhorizontal curves q ( · ) such that q (0) = q . It is clear that Ω( q ) is a Banach manifold, modeled on L (0 , R m ) (with m = rank( D )). Now, let q ∈ M be another arbitrary point. The set Ω( q , q )of all q ( · ) ∈ Ω( q ) such that q (1) = q may not be a manifold. Indeed, defining the end-pointmapping end q : Ω( q ) → M by end q ( q ( · )) = q (1), we have Ω( q , q ) = (end q ) − ( q ), and themapping end q need not be a submersion. By definition, a singular curve q ( · ) is an horizontal curvethat is a critical point of end q (0) , or, in other words, such that the differential d end q (0) is not offull rank.There are many clues leading one to think that singular curves may have a strong impacton the asymptotic spectral properties of the sR Laplacians. For instance, we expect that theyhave an impact on the Schwartz kernel in the representation through Feynman integrals. This13epresentation has the formexp( ith △ ) f ( x ) = Z q ( · ) ∈ Ω( x ) e ih R t L ( q ( s ) , ˙ q ( s )) ds f ( q ( t )) dγ ( q ( t )) , and if we fix the terminal points of the paths, then we “disintegrate” the measure to obtain theSchwartz kernel. We expect that singular curves cause a singularity in the kernel, which differsfrom the usual one. In order to understand such issues, we provide hereafter several examples. Martinet case.
We consider a smooth compact connected manifold M of dimension 3, and ageneric horizontal distribution D of rank 2, defined by D = ker α , with α a nontrivial one-form on M such that α ∧ dα vanishes transversally on a surface S of dimension 2. This surface is calledthe Martinet surface . Let g be a Riemannian metric on D .The local model in R near 0 is given by α = dz − x dy . Then D is locally spanned by thetwo vector fields X = ∂ x and Y = ∂ y + x ∂ z . Note that S = { x = 0 } and that [ X, Y ] = 2 x∂ z vanishes along S . A Lie bracket of length two, [ X, [ X, Y ]] = 2 ∂ z , is required to generate the missingdirection, along S . Outside of S , the distribution D is of contact type.Note that the characteristic manifold Σ is not symplectic. This is equivalent to the existenceof nontrivial singular curves. The Martinet case is indeed a well known sR model in which thereare nontrivial singular curves, that are minimizing, and locally foliate the Martinet surface S (see[17]).We assume that L = D ∩ T S is a line bundle over S (otherwise, in the general case, L may havegeneric singularities): hence S is either a torus or a Klein bottle. Let dP be the Popp volume on M \ S . Near S , we have dP ∼ dν ⊗ | dφφ | , where ν is a smooth measure on S and φ = 0 is a localequation of S .Given any smooth measure µ on M , we consider the corresponding sR Laplacian, and someeigenbasis of it. We are able to obtain the following local Weyl law (see [7]). Proposition 1.
For every continuous function f on M , we have X λ n λ Z M f | φ n | dµ ∼ R S f dν λ log λ, as λ → + ∞ . In particular, N ( λ ) ∼ ν ( S )32 λ log λ . Corollary 1.
There exists a density-one subsequence ( n j ) j ∈ N ∗ such that lim j → + ∞ Z M f | φ n j | dµ = 0 , for every continuous function f on M , vanishing near S . Note that Proposition 1 cannot be obtained from the asymptotic estimates of the heat kernelalong the diagonal, because those estimates are not uniform: along the diagonal, the asymptoticsis in t − / on S , and in C ( q ) t − outside of S , with C ( q ) bounded but not integrable near S . Theproof of Proposition 1 consists of first computing the asymptotic behavior of Tr( f e − t △ ) as t → + in the flat Martinet case, by rescaling. Then, using the locality of the heat kernel expansion alongthe diagonal, the result is obtained in the locally flat case. Using Dirichlet-Dirichlet bracketing,the result is then established in small cubes with a flat metric, and is finally extended using localquasi-isometries and minimax considerations.It might be expected that the singular flow plays an important role in the asymptotic spectralproperties of the sR Laplacian. In particular, it is tempting to think that a good assumption is14he ergodicity of the abnormal geodesics (lifts of the singular curves in the cotangent space) inthe characteristic distribution ker ω | Σ , at least, under the assumption that there indeed exists acharacteristic vector field leaving ν invariant. Relationship between singular curves and magnetic fields.
This paragraph is inspired bythe paper [16]. Let us consider a manifold N of dimension 2, endowed with a flat Riemannianmetric and with its canonical Riemannian volume. We consider the Laplacian △ = − ( ∂ x − a∂ z ) − ( ∂ y − b∂ z ) , with z ∈ R / π Z and A = a dx + b dy the magnetic potential on N . Setting X = ∂ x − a∂ z and Y = ∂ y − b∂ z , we have [ X, Y ] =
B ∂ z where dA = B dx ∧ dy is the magnetic field. We assume that B vanishes on a closed curve Γ with a non-zero differential. Then α = a dx + b dy + dz definesa Martinet distribution on M = N × R / π Z , and the Martinet surface is S = Γ × R / π Z . Wehave, then, α g = B ( a dx + b dy + dz ) and ν = k dB k | ds dz | , where s is the arc-length along Γ. Theprevious expressions are still valid if the metric is not flat. The characteristic direction is givenby A ( ˙ γ ( s )) ds + dz = 0 if Γ is parametrized by the arc-length. It is always possible to choose agauge A near Γ such that A ( ˙ γ ) = A is constant. Using this gauge, the measure ν is invariantunder the vector field k dB k ( ∂ s − A ∂ z ). The corresponding dynamics is ergodic if 2 πA / length(Γ)is irrational.Another case of interest is the quasi-contact case in dimension 4. It is related to magnetic fieldsin dimension 3. In this case, there exist some nontrivial singular curves that correspond to linesof a magnetic field. It seems that the singularities can only occur at isolated points. We mightthen expect to have QE if there is only one such point, and otherwise a tunnel effect might occurbetween two different points. Almost-Riemannian geometry.
For the sake of simplicity, we restrict ourselves to almost-Riemannian (aR) structures on surfaces. Let M be a smooth compact connected manifold ofdimension 2. An aR structure on M is locally given by two vector fields X and Y that generate,outside of a singular curve S , the tangent space of the surface M . We assume that, on S , thevector fields X , Y and [ X, Y ] generate
T M . The simplest example is the so-called
Grushin case and is given locally by X = ∂ x and Y = x∂ y . Outside of S , the vector fields X and Y define theRiemannian metric g = ν X + ν Y , where ( ν X , ν Y ) is dual to ( X, Y ). The volume µ aR associatedwith g is singular along S . In the Grushin case, we have g = dx + ( dy/x ) and dµ aR = | dx dy | / | x | .Let dµ be any volume form on M . We consider the aR Laplacian △ aR = − X ⋆ X − Y ⋆ Y , wherethe adjoints are taken with respect to µ . It follows from Remark 7 that the aR Laplacians associatedwith two different volumes are unitary equivalent up to a potential of the order of d ( · , S ) − . If µ = µ aR then the resulting operator is singular on S , however it is essentially selfadjoint on M \ S (see [3]).In this Grushin case, we prove in [7] a result that is similar to the Martinet case, namely, thatfor every continuous function f on M , we have X λ n λ Z M f | φ n | dµ ∼ R S f dν π λ log λ, as λ → + ∞ (with, as well, Corollary 1 as a consequence). Remark 9.
In any of the above singular cases (Martinet, almost-Riemannian), one may think ofapplying the desingularization procedure of [19]. For instance, the Martinet case can be seen as theprojection of the so-called Engel case, which refers to an equiregular horizontal distribution of rank15 in dimension 4; the Grushin case can be seen as the projection of the 3D contact case. However,in view of obtaining a microlocal Weyl law, it is not clear whether or not this desingularizationcan be used in a relevant way, because when projecting one has to use integrals of the Schwartzkernel outside of the diagonal, that are not known.
Remark 10.
In order to study sR structures in which the rank of the horizontal distribution D may not be constant, a more general definition of a sR structure has been introduced (althoughnot exactly with those terms) in [2].A relative tangent bundle on a manifold M is a couple ( H, ξ ), where π : H → M is a smoothfibration and ξ : H → T M is a smooth bundle morphism over M . The horizontal distribution isdefined by D = ξ ( H ) ⊂ T M . A vector field X on M is said to be horizontal if there exists a section u of H such that X = ξ ( u ). An horizontal curve q ( · ) : [0 , → M is an absolutely continuous curvefor which there exists a section u of H such that ˙ q ( t ) = ξ q ( t ) ( u ( t )), for almost every t ∈ [0 , M, H, ξ, g ), where M is a manifold, ( H, ξ ) is a relative tangent bundle on M , and g is a Riemannianmetric on H . This definition may be used in order to provide a unifying context for usual sR andaR geometries. Microlocal properties of SR wave equations.
It is interesting to investigate properties ofwave equations associated with a sR Laplacian. One of the first questions that are arising is thequestion of microlocalization. More precisely, let us assume that the wave front of Cauchy data( u , u ) is a subset of a cone K with compact base contained in an open cone C . Does thereexist ε ( K, C ) > C for | t | ε ? This question is a priori far from trivial because the sR cut-locusand the sR conjugate locus of a point always contain this point in their closure. Another natural(simpler) question is to establish finite propagation of waves in the sR context. Quantum limits.
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