Poisson and Szeg\"{o} kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szeg\"{o} kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order terms, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range $O\left(\lambda^{\epsilon-1/2}\right)$ in all the variables involved, where $\lambda\rightarrow+\infty$ is the asymptotic parameter, rather than rescale according to Heisenberg type.


Introduction
It was shown by Bruhat and Whitney ( [BW]) that a real-analytic compact and connected manifold M has an essentially unique complexification (M, J), that is, a complex manifold in which M sits as a totally real submanifold. If furthermore κ is a real-analytic Riemannian metric on M, it was proved independently by Guillemin and Stenzel ([GS-1991], [GS-1992]) and by Lempert and Szöke ([Lem], [LS], [Sz]) thatM can be endowed with a canonically determined Kähler structure (M , J, Ω), with the following two properties. First, the symplectic manifold (M , Ω) is symplectomorphic to a tubular neighbourhood of the zero section in the cotangent bundle T ∨ M of M, endowed with its canonical symplectic structure (and the symplectomorphism, of course, is the identity on M). Second, the square norm of κ pulls back onM to a Kähler potential for Ω, while the norm pulls back onM \ M to a solution of the complex Monge-Ampère equation. Viewed the other way, this construction endows the cotangent bundle of M, near the zero section, with a canonical compatible complex structure, with the property that the Riemannian and Monge-Ampère foliations coincide. This intrinsic complex struture was called adapted by Lempert and Szöke. The sphere bundles of sufficiently small radii τ > 0 in T ∨ M correspond to boundaries X τ of strictly pseudoconvex domains in M τ ⊆M, so-called Grauert tubes, corresponding to disc bundles in T ∨ M; furthermore, the homogeneous geodesic flow on the sphere bundles is closely related to the flow of the Reeb vector field on X τ .
A basic problem in this context, partly motivated by certain analogies with the setting of positive line bundles on complex projective manifolds, is to study the asymptotic concentration behaviour of the eigenfunctions of the generator of the homogeneous geodesic flow (transported to X τ ). This involves the local study of certain smoothed spectral spectral projectors, which are the Toeplitz counterpart of commonly studied objects in spectral theory (see e.g. [DG], [GrSj]). However, in sharp contrast to the line bundle setting, the latter flow is generally not CR-holomorphic. This is a source of difficulty in the adaptation to the Grauert tube setting of the Szegö kernel techniques that have proved successful in the line bundles setting ([Z-1998], [SZ], [BSZ]).
A related issue, which is instead genuinely intrinsic to the Grauert tube setting, concerns analytic continuation of the eigenfunctions of the nonnegative Laplacian ∆ of (M, κ). Let (ϕ j ) ∞ j=1 be a complete orthonormal system of L 2 (M) composed of eigenfuntions of ∆. Being real-analytic, each ϕ j extends to a holomorphic functionφ j to some sufficiently small Grauert tube M τ ⊆M ; here in principle τ depends on j. It was a deep discovery of Boutet de Monvel ( [BdM-1978], [BdM-1979]) that for sufficiently small τ > 0 every ϕ j extends to M τ (see also [GS-1992], [Leb], [S-2014], [S-2015], [Z-2012], [Z-2017] for discussions and different proofs). As explained in [Z-2007], [Z-2010], [Z-2012], the relation between the eigenfunctions ϕ j and their analytic continuationsφ j is governed by the so-called Poisson-wave operator, obtained by analytically continuing the Schwartz kernel of the wave operator on M (in both time and space). The study of the asymptotic distribution of the analytic continued eigenfunctions, pioneered by Zeldtich, involves certain 'complexified spectral projectors', bearing resemblance to the smoothed spectral projectors above.
In two striking recent papers ( [CR1], [CR2]), Chang and Rabinowitz have made significant progress in pushing forward the analogy between the line bundle and the Grauert tube settings. Their analysis rests on two pillars. One is the description, due to Zelditch, of certain unitary groups of Toeplitz operators as 'dynamical Toeplitz operators' ([Z-1997], [Z-2010], [Z-2012]); another is a clever use of the Heisenberg (or normal) local coordinates adapted to a hypersurface in a complex manifold introduced by Folland and Stein in [FS1] and [FS2].
The goal of the present paper is partly to present a gentle introduction to the promising and efficient approach of Chang and Rabinowitz, and partly to provide some complementary results that may be useful in future applications. We shall restrict the present discussion to the near-diagonal situation. In particular, we shall give near-diagonal scaling asymptotics for the smoothing kernels hinted at above, in a range O λ δ−1/2 in all variables involved, rather than according to Heisenberg type. More precisely, in an appropriate set of local coordinates centered at x ∈ X τ , we shall consider rescaled dispacements of the form form x + (θ/ √ λ, v/ √ λ) with (θ, v) ≤ C k δ (here ǫ ∈ (0, 1/6), say). Furthermore, we shall provide an explicit numerical determination of the leading order factor in the asymptotic expansion. Also, we shall give a bound in the degree of the polynomials in the rescaled variables (θ, v) that appear in the lower order terms of the expansion; this is useful in ensuring that, when (θ, v) are allowed to expand at a controlled pace as above, one actually obtains an asymptotic expansion.
In order to give a more precise description of the content of this paper, it is in order to premise a more detailed account of the geometric setting involved.
Let M be a d-dimensional compact connected real-analytic (in the following, C ̟ ) manifold. As mentioned, there is a complex manifold (M, J), the Bruhat-Whitney complexification of M [BW], in which M embeds as a totally real submanifold. If  : M ֒→M is the inclusion, thenM is uniquely determined locally along (M), up to unique biholomorphism. Therefore, since (M, J) and (M , −J) are both complexifications of M, there is an antiholomorphic involution σ :M →M with fixed locus (M) ( [BW], [GS-1991], [LGS]). We shall identify M with (M) in the following.
One callsM τ the open Grauert tube of radius τ , and √ ρ the tube function of (M, κ). We shall denote by H(X τ ) ⊂ L 2 (X τ ) the Hardy space of X τ , and by Π τ : L 2 (X τ ) → H(X τ ) (1) the orhogonal projector, that is, the Szegö projector of the CR manifold X τ 1 . One also has the following alternative perspective. Rather than starting from the complexification (M , J), and then considering a Kähler structure Ω on (M , J) canonically induced by κ, one can start instead from the canonical symplectic structure Ω can on the cotangent bundle T ∨ M, locally given by dq∧dp, where q denotes local coordinates on M and p the induced fiberwise linear coordinates on the cotangent spaces; then one introduces a compatible complex structure J ad on a suitable neighborhood of the zero section in (T ∨ M, Ω can ), again canonically induced by κ. In fact, J ad is uniquely determined by the condition that the maps C → T M parametrizing the leaves of the Riemann foliation are holomorphic, when suitably restricted. Lempert and Szöke call J ad adapted ([GS-1991], [Lem], [LS], [Sz]).
More explicitly, if (m, v) ∈ T M and v = 0, let γ : R → M denote the unique geodesic with initial condition (m, v), andγ : I → T M its velocity vector. For every b > 0, let N b : T M → T M denote dilation by b. Consider the smooth map ψ γ : a + ı b ∈ C → N b (γ(a)) ∈ T M. (2) The submanifolds ψ γ (C \ R) are the leaves of the Riemann foliation of T M \ M 0 (here M 0 is the zero section); J ad is characterized by the property that the maps ψ γ are holomorphic from a suitable strip S ⊆ C to T ǫ M (see [PW-1991] and [LS], §3-4, [Sz]). After [LS] and [LGS], the two approaches are related by the imaginary time exponential map of κ (see also the discussions in [Z-2007], [Z-2010], [Z-2012] ). Let us identify T M and T ∨ M by κ, and view Ω can as a symplectic structure on T M. Given m ∈ M, let exp m : U m → M be the exponential map at m for κ; here U m ⊆ T m M is some neighborhood of the origin. Let As M is compact, we may assume that for some ǫ > 0 one hasŨ such that E ǫ (m, 0) = m ∀ m ∈ M; E ǫ has the following properties: 1. E ǫ intertwines the square norm function · 2 : T M → R and ρ: hence it maps T τ M toM τ , for all τ ∈ (0, ǫ); 2. for sufficiently small ǫ > 0, E ǫ is a C ̟ symplectomorphism between (T ǫ M, Ω can ) and (M ǫ , Ω), that is, and similarly replacing ǫ with any τ ∈ (0, ǫ); 3. consequently, E ǫ intertwines the Hamiltonian flow of · on (T ǫ M \ (0), Ω can ), which is the homogeneous geodesic flow, with the Hamiltonian flow of √ ρ on (M \ M, Ω); 4. the relation E ǫ * (Ω) = Ω can also implies that E ǫ * (J) is a compatible complex structure on (T ǫ M, Ω can ), and in fact J ad = E ǫ * (J); 5. E ǫ intertwines the Riemann foliation of (M, κ) with the Monge-Ampère foliation of √ ρ ( [LS], [GLS]).
If M ′ ⊆M is open and f ∈ C ∞ (M ′ ), υ f ∈ X(M ′ ) will denote the Hamiltonian vector field of f with respect to Ω. By the above, the homogeneous geodesic flow on T ∨ M \ M 0 is intertwined by E ǫ with the flow of υ √ ρ oñ M \ M. The following holds (see §3).
1. α is invariant under the flow of υ √ ρ ; equivalently, λ can is preserved by the homogeneous geodesic flow; 2. if  τ : X τ ֒→M is the inclusion, then α τ :=  τ * (α) is a contact form; 3. the cone is symplectic (for the standard symplectic structure of T ∨ X τ ); 4. being tangent to X τ , υ √ ρ induces by restriction a smooth vector field υ τ √ ρ ∈ X(X τ ), and by the above the flow of υ τ √ ρ preserves the volume form on X τ ; 5. consequently, the differential operator is formally self-adjoint on L 2 (X τ ); These facts have the following consequence. Consider the composition in the terminology of [G] and [BdM-G], D τ √ ρ is a self-adjoint first-order Toeplitz operator on X τ . Its principal symbol (as a Toeplitz operator) is, by definition, the restriction of σ(D τ √ ρ ) to Σ τ , and is therefore strictly positive. Hence D τ √ ρ is an elliptic Toeplitz operator. By the theory in §2 of [BdM-G] (especially Proposition 2.4), the spectrum of D τ √ ρ is discrete, bounded from below and has only +∞ as accumulation point (see §3.4 below).
In order to obtain spectral or eigenfunction asymptotics, it is common to consider smoothed versions of the spectral kernels associated to each eigenvalue (see e.g. [DG] and [GrSj]). In the present setting, the C ∞ function on X τ ×X τ is the Schwartz kernel of the orthogonal projector Π τ j : [CR1] and [CR2], let us consider the 'smoothed spectral projector' 2 We shall use the same symbol for operators and their distributional kernels whose distributional kernel is the C ∞ function of (λ, x, y) heuristically, (15) is a slight smoothing of a spectral projector kernel relative to a spectral band travelling to the right as λ → +∞. The near-diagonal asymptotics of (15) encode information about the local concentration behaviour of the ρ j,k 's, and globally on the global asymptotic distribution of the λ j 's. For a discussion in the specific context of Grauert tubes, see [Z-2012] and [Z-2017]; for applications to the Toeplitz quantization of Kähler manifolds, with an emphasis on scaling asymptotics, see [P-2009], [P-2010], [P-2012], [P-2017], [P-2018]). For recent work on general CR manifolds, see [HHMS].
In [CR1] an [CR2], the authors provide near-diagonal and near-graph scaling asymtptotics for (15), strikingly similar to the Szegö kernel asymptotics holding in the line bundle setting ([Z-1998], [SZ], [BSZ]) and to those holding for Toeplitz spectral projectors in [P-2010]. In the present work, we shall survey the approach of [CR1], focusing on the near-diagonal case, and provide some complements to their results.
Prior to precise statements, some additional notation is in order. To begin with, we introduce an invariant, denoted ψ 2 , which is related to a Hermitian vector space and ubiquitously appears in various guises as the exponent controlling equivariant Szegö kernel scaling asymptotics [SZ], [BSZ].
Definition 1. Let V be a finite-dimensional compòex vector space, and let h : V × V → C be a Hermitian product. Then g := ℜ(h) and ω := −ℑ(h) are, respectively, an Euclidean scalar product and a symplectic form on V (viewed as a real vector space), compatible with the complex structure. If · h is the norm function on V associated to h (or g), let us define Clearly, ψ h 2 (u, v) is positively homogeneous of degree 2 in the pair (u, v), and of degree 1 in h (or ω).
For instance, if V = C k and h st : C k × C k → C is the standard Hermitian product, then h st = g st − ı ω st , where g st and ω st denote, respectively, the standard Euclidean product and the standard symplectic structure on R 2k ∼ = C k . Then ψ 2 := ψ hst : C k × C k → C is given by If k = dim C (V ), and B is an orthonormal basis of (V, h), let M B : V → C k be the associated unitary isomorphism. Then ψ h 2 = ψ 2 • (M B × M B ). With this in mind, when no confusion seems likely, we shall occasionally leave dependence on h implicit and write ψ 2 for ψ h 2 . We shall set ω := 1 2 Ω; thus (M , ω, J) is a Kähler manifold, with associated Riemannian metricκ := 1 2κ . If x ∈M, with tangent space T xM , we correspondingly have a function As mentioned, the near-diagonal asymptotic expansion for Π τ χ,λ at x ∈ X τ in [CR1] rests on the choice of so-called Heisenberg local coordinates (called normal in [FS1], [FS2]). The concept of Heisenberg local coordinates is twofold: one first introduces Heisenberg local coordinates onM centered at x, and adapted to X τ ; then Heisenberg local coordinates on X τ centered at x. The latter will be induced by the former by restriction and projection. More precisely, Heisenberg local coordinates onM will be a system of holomorphic local coordinates centered at x, in which the defining equation φ τ := ρ − τ 2 for X τ takes a certain canonical form (Definition 29). Let U ⊂M be an open neighbourhood of x ∈ X τ on which Heisenberg local coordinates (z 0 , z 1 , . . . , z d−1 ) : U → C d (centered at x and adapted to X τ ) have been chosen. Set θ := ℜ(z 0 ) : U → R and U τ := X τ ∩ U; then ϕ τ : (θ, z 1 , . . . , z d−1 )| U τ : U τ → R × C d−1 will be a system of Heisenberg local coordinates on X τ centered at x. We shall generally redefine the z i 's and omit symbols of restriction, and write z ′ = (z 1 , . . . , z d−1 ). Alternatively, we shall use real notation and write Furthermore, we shall often adopt the additive short-hand x + (θ, u) := ϕ τ −1 (θ, u). Actually, it will be convenient to work with a slightly more restrictive class of local coordinates, that will be called normal Heisenberg local coordinates adapted to X τ at x (see §3.3.3).
In the line bundle setting, X τ is a fixed-radius circle bundle in the dual of the polarizing line bundle. Translation in θ may be then be assumed to correspond to fiberwise rotation, hence to the flow of the Reeb vector field, which is CR-holomorphic.
In the Grauert tube setting, instead, neither is the Reeb flow generally CR-holomorphic nor may it be assumed to correspond to translation in θ. Namely, let R τ ∈ X(X τ ) be the Reeb vector field of (X τ , α τ ). While we do have R τ (x) = ∂/∂θ| x (see (60) below), there is no reason to expect that R τ = ∂/∂θ on U τ (see e.g. Theorem 18.5 in [FS1]). Hence the curves θ → x + (θ, 0) deflect from the trajectories of R τ , which are a rescaling of the geodesic flow. This is a sharp difference with the line bundle situation, and contributes to making the derivation of the asymptotics technically more involved.
We need one last piece of notation before formulating the near-diagonal scaling asymptotics of the smoothed spectral projectors Π τ χ,λ . Let Γ τ t : X τ → X τ be, with abuse of language warranted by the previous identifications, the homogeneous geodesic flow at time t (to be precise, this is the flow of υ √ ρ ).
Definition 2. If χ ∈ C ∞ c (R) and x ∈ X τ , let us set for some suitably small ǫ > 0. Then the following holds.
for certain smooth functions B j ∈ C ∞ (X τ ).
We recover the near-diagonal asymptotic expansion of Chang and Rabinowitz (Theorem 1.1 of [CR1]) rescaling according to Heisenberg type, holding θ j and v j fixed.
Being of class C ω , each ϕ j,k admits a holomorphic extensionφ j,k to some open neighborhood of M inM , which a priori depends on (j, k). Boutet de Monvel discovered that a much stronger result is true: there exists τ 0 > 0 such that every ϕ j,k extends holomorphically toM τ 0 . Therefore, for τ ∈ (0, τ 0 ] the restrictionφ τ j,k :=φ j,k | X τ is a CR function. This collective extension property is closely related to the analytic extension of the Schwartz kernel of the Poisson operator for τ > 0 (see [BdM-1978], [GS-1992], [GLS], [Z-2010], [Z-2012], [CR1] and [CR2] for discussion and motivation). Assuming that the ϕ j,k 's are real, the distributional kernel of U(ı τ ) admits the spectral representation which is globally real-analytic on M × M for any τ > 0 [Z-2012]. If τ > 0 is sufficiently small, analytic extension in m, followed by restriction to X τ , yields the kernel As an operator, P τ is a Fourier integral operator with complex phase of positive type and order −(d−1)/4. It is in fact a Fourier-Hermite operator in the sense of [BdM-G], adapted to a homogeneous symplectic equivalence χ τ : The complexified Poisson kernel P τ governs the analytic continuation of eigenfunctions: for any j,φ τ j,k = e τ µ j P τ (ϕ j,k ).
The composition is a Fourier integral operator with complex phase of positive type and of degree −(d − 1)/2 on X τ ; it is in fact a Fourier-Hermite operator adapted to the identity of Σ τ . The definition of (20) depends on the choice of a Riemannian density on X τ ; given this, we may identify its distributional kernel with the generalized function where U τ j ∈ C ∞ (X τ × X τ ) is given by As (P τ (ϕ j,k )) j,k is not an orthonormal system, neither U C (2 ı τ ) nor U τ j are orthogonal projectors. Nonetheless, U C (2 ı τ ) plays a role in the asymptotic study of analytic extensions reminiscent of Π τ ( §6 of [Z-2010]).
Suppose as above that χ ∈ C ∞ 0 (−ǫ, ǫ) , for a suitably small ǫ > 0. The asymptotic concentration of the complexified eigenfunctions pertaining to a spectral band drifting to infinity is probed by the following smoothed version of (21): Heuristically, P τ χ,λ ∈ C ∞ (X τ × X τ ) is a complex (tempered) analogue of the smoothed spectral projector kernel (15). The diagonal restriction of P τ χ,λ is the non-negative function The complexified Poisson operator is a special instance of the complexified Poisson wave operator (see (158) below), which was proved by Zelditch to be describable in terms of dynamical Toeplitz operators (see e.g. [Z-2012], especially §8-9). Building on this, and on their use of the normal local coordinates of Folland and Stein, Chang and Rabinowitz proved in [CR1] a near-diagonal asymptotic expansion for P τ χ,λ very similar to the one holding for Π τ χ,λ . The corresponding version that we shall provide here runs parallel to Theorem 3.
1. Suppose C, δ > 0 are constants. Then (with the notation of Definition 2) 2. Suppose x ∈ X τ , and let us choose a system of normal Heisenberg local coordinates on X τ centered at x. If C > 0 and δ ∈ (0, 1/6), uniformly for a certain constant γ τ 0,0 (x) (to specified below) and where, as a function of (θ 1 , v 1 , θ 2 , v 2 ), F τ j is a polynomial of degree ≤ 3 j and parity j.
Corollary 8. Under the assumptions of Theorem 6, for certain smooth functions Q τ j ∈ C ∞ (X τ ). We shall leave it to the reader to formulate the corresponding analogue of Corollary 5. As an application of the on-diagonal asymptotic expansions for Π τ χ,λ and P τ χ,λ , we provide local Weyl laws by a standard argument, similar to the one in [P-2009] for the line bundle setting (inspired by the discussion in [GrSj]). The stated equality γ τ 0,0 (x) = τ (d−1)/2 will follow by comparison with the local Weyl law in Proposition 3.8 of [Z-2014]. For x ∈ X τ , let us note the identities where notation is as in (13) and (21). Let us further define, for x ∈ X τ and λ ∈ R, where H is the Heaviside function.
Proposition 9. Uniformly in x ∈ X τ , we have for λ → +∞ and Comparing with Proposition 3.8 of [Z-2014] we finally conclude

An example
The geometric setting is clarified by the following example (see [PW-1991], [GS-1991], and §2.1 of [Z-2007] for this and other model examples).

Consider the compact torus
here, by abuse of notation, elements of R d are identified with their classes in M, and the sum is meant in T . The complexified exponential map E u : In particular, writing z = u + ı v, we see that the standard symplectic structure Ω := (ı/2) dz ∧ dz pulls back to Hence, the pull-back of the standard complex structure onM by E is indeed compatible with Ω can .
which is holomorphic. Hence J ad = E * (J) is the adapted complex structure of T M, in the terminology of [LS].
Let us henceforth identify T M withM by E, and consider the functions Thus ρ is a Kähler potential for Ω can . A direct computation shows that the differential form ı ∂ ∂ √ ρ, which is with constant rank 2d − 2. Furthermore, its 2-dimensional kernel at u + ı v is generated by the tangent vectors (v, 0) and (0, v); by (28), this is the tangent space of the Riemann foliation, which therefore coincides with the Monge-Ampère foliation.
Since dρ = 2 j v j dv j and Ω can = j du j ∧ dv j , letting υ ρ denote the Hamiltonian vector field of ρ with respect to Ω we have Let us set Then α(R) ≡ 1, and R is itself a Hamiltonian vector field so that X τ is the boundary of a strictly pseudoconvex domain inM . Let  τ : X τ ֒→M be the inclusion, and set α τ :=  τ * (α). Then R is tangent to X τ and restricts to a vector field R τ on X τ , which satisfies In other words, R τ is the Reeb vector field of the contact manifold (X τ , α τ ). Similarly, υ √ ρ is also tangent to X τ , and in view of (29) it restricts along which (with the identification provided by E) corresponds to the homogeneous geodesic flow. Finally, let us consider the cones

Notation
For the reader's convenience, we collect here some of the notation and conventions adopted in the paper.
so that the Fourier inversion formula has the form Densities and functions. Our manifolds will be endowed with naturally given volume forms, and we shall identify densities, half-densities and functions.
Schwartz kernels. Given a manifold N, we shall use the same letter for a continuous linear operator F : Induced vector fields. Given a smooth action of a Lie group G on a manifold R, for any ξ ∈ g (the Lie algebra of G) we shall denote by ξ R ∈ X(R) (the Lie algebra of smooth vector fields on R) the vector field induced by ξ.
Riemannian distance functionκ τ is the Riemannian metric on X τ given by the restriction ofκ, and dist X τ : X τ × X τ → R denotes the corresponding distance function.
Reeb vector fields. R is the vector field of (M, κ) (Definition 16); it is tangent to every X τ , and restricts along X τ to the Reeb vector field R τ of (X τ , α τ ).
Volume form. vol R X τ is the Riemannian volume form on X τ , in terms of which the Hilbert structure on L 2 (X τ ) is defined.
Geodesic flow. Γ τ t : X τ → X τ denotes the homogeneous geodesic flow along X τ at time t.

The geometric setting 3.2.1 The homogeneous geodesic flow
As remarked, E ǫ intertwines the homogenous geodesic flow on T ǫ M \M 0 (that is, the Hamiltonian flow with respect to Ω can of the norm function induced by κ on T M) with the Hamiltonian flow of √ ρ onM \ M with respect to Ω.
Neither flow is generally holomorphic (of course, the former is if and only if so is the latter).
Recall that we identify T M and T ∨ M by means of κ.
Lemma 11. The canonical 1-form λ can on T ∨ M is invariant under the homogenous geodesic flow. Similarly, α in (8) is invariant under the flow of υ √ ρ .
Proof. The two statements are equivalent by (7). To verify the former, we may work in a local coordinate chart (q, p) for the cotangent bundle associated to a system of local coordinates q for M. Let K = (k ij ), where κ ij = κ ij (q), denote the inverse metric tensor, and set ̺ := · 2 ; then Since locally Ω can = dq ∧ dp, the Hamiltonian vector field of √ ρ is Since locally λ can = p dq, Hence the Lie derivative of λ can along V √ ̺ is For a more general statement, see Lemma 21.

The Reeb vector field of (M, k)
As shown in [GS-1991], the condition that √ ρ satisfies the complex Monge-Ampère equation may be reformulated in terms of the norm of the gradient of ρ. Let us briefly recall the argument in [GS-1991]. Let Ξ ∈ X(M) be defined by the identity ι(Ξ) Ω = α.
Corollary 13. With Ξ as in (31), the square norm of Ξ with respect toκ is Let us set Corollary 15. R is uniquely determined in X(M \ M) by the conditions Definition 16. We shall call R in (40) the Reeb vector field of (M, κ).
The motivation for this definition is the following. Suppose τ ∈ (0, ǫ). Since R is tangent to X τ , it restricts to a vector field R τ ∈ X(X τ ). Furthermore, the Lie derivative has vanishing pull-back to X τ ; therefore L R τ (α τ ) = 0. In other words, R τ ∈ X(X τ ) is the (genuine) Reeb vector field of (X τ , α τ ). We also have Corollary 13 implies: hence √ ρ is a distance function onM \ M forκ. While the flow of υ √ ρ is intertwined by E ǫ with the homogeneous geodesic flow, the trajectories of the gradient vector field grad √ ρ = J(υ √ ρ ) are unit speed geodesics forκ, perpendicular to the hypersurfaces X τ and minimizing the distance between them (see §3 of [LS]). By (42), the flows of R τ and υ τ √ ρ on X τ are related by a rescaling by the factor −1/τ in the time variable.

The volume form on X τ
The Riemannian volume form on the Kähler manifold (M ǫ , Ω, J) is it pulls back under E ǫ to the symplectic volume form vol can := 1 d! Ω ∧d can . There are various natural alternatives in the literature for a volume form on X τ ; let us dwell to specify the choice in this paper.
Given that that grad √ ρ is a unit normal vector field to X τ by (43), the Riemannian volume form is An alternative choice is the contact volume form Let us clarify the relation between vol R X τ and vol C X τ .
The two equalities are clearly equivalent. As to the latter, Our choice for a volume form on X τ will be vol R X τ . Let us consider its homogeneity properties. For λ > 0 let δ λ : be the vector field correlated with Ξ by E ǫ . Since locally Ω = dq ∧ dp and α = −p dq, we have which is homogeneous of degree zero with respect to δ λ . Therefore the same holds of Ξ with respect to δ ′ λ . Since √ ρ is homogenous of degree 1,

Induced vector fields and Hamiltonians
The Kähler structure makes TM ǫ into a complex Hermitian vector bundle. Being everywhere non-vanishing, υ √ ρ spans onM ǫ \ M a 1-dimensional complex subundle V of (TM , J): Hence there is onM ǫ \ M a decomposition of (TM, J) as the orthogonal direct sum of complex vector sub-bundles Let T ⊂ V be the real vector subbundle generated onM ǫ \ M by υ √ ρ ; thus The C ∞ sections of T ⊕ H are the smooth vector fields that are tangent to X τ , for every τ ∈ (0, ǫ). The decomposition (47) restricts to a corresponding orthogonal direct sum decomposition for the tangent bundle T X τ : Lemma 20. We have Hence T x ⊆ ker(α x ). One argues similarly for H.
For a vector bundle E onM ǫ \ M, let Γ(E) be the space of its smooth sections.

Heisenberg local coordinates
As emphasized in the Introduction, Chang and Rabinowitz in [CR1] and [CR2] considerably simplified the application of the ideas and techniques from the line bundle setting in [Z-1998], [BSZ] and [SZ] to the Grauert tube context, and their approach is partly based on Folland and Stein's construction of Heisenberg local coordinates for a strictly pseudoconvex hypersurface in a complex manifold [FS1], [FS2].

Heisenberg-type order
The notion of Heisenberg local coordinates on a complex manifold adapted to a strictly pseudoconvex hypersurface rests on the concept of Heisenberg-type order of vanishing of a smooth function at a given point with respect to a local holomorphic chart ( [FS1], §14 and §18).
We shall occasionally abridge the notation O k ϕ (M ) to O k ϕ . By the inductive definition, i.e. any f ∈ O k ϕ is a (finite) linear combination of products f 1 · · · f k with f j ∈ O 1 ϕ .
for any k, l ≥ 1.
Lemma 24. Suppose f ∈ m x (M ) Then the following conditions are equivalent: Proof. Let 0 < d k < D k be constants, depending only on k ≥ 1, such that for any pair a, b ≥ 0 one has Then f is a sum of products of the form f 1 · · · f k with each f j ∈ O 1 ϕ . We may thus assume that f itself is of this form. Hence there is a constant C > 0 (depending on f ) such that |f j (y)| ≤ C |z ′ (y)| + |z 0 (y)| 1/2 (j = 1, . . . , k); Hence 2. holds. Conversely, assume that 2. holds. Then f ∈ O 1 ϕ by definition for k = 1. Suppose that k ≥ 2. Then In some neighbourhood U of x inM , let us define g : U → C by Hence g ∈ O 1 ϕ , and f (y) = g(y) · |z ′ (y)| + |z 0 (y)| 1/2 k−1 .
Let us focus on C k ϕ .
Definition 25. For a, b ∈ Z d ≥0 , consider the monomial function of z := z 0 · · · z d−1 ∈ C d given by The weighted degree of P a,b is Proposition 26. Suppose f ∈ C ∞ (M) x . Then the following conditions are equivalent.
1. f ∈ C k ϕ ; 2. every monomial contributing to the Taylor expansion of f • ϕ −1 at 0 has weighted degree ≥ k.
A monomial P a,b (z) pulls back to

Heisenberg type order and holomorphic extensions
A notational clarification is in order. Let be given a d-dimensional complex manifold Z, with complex structure J. We shall denote by Z the conjugate complex manifold (that is, Z = Z as differentiable manifolds, but with complex structure −J). In particular, let J 0 be the standard complex structure on C d ; an open subset A ⊆ C d is a complex manifold with the induced complex structure, which shall also be denoted J 0 . Then A is the same open subset, endowed with the complex structure −J 0 .
On the other hand, let c : C d → C d denote complex conjugation. We shall set A c := c(A), with the complex structure J 0 . Then c yields by restriction an anti-holomorphic diffeomorphism c : A → A c , or equivalently a biholomorphism c : A → A c .
Consider a holomorphic local chart (U, ϕ, A) of Z; thus U ⊆ Z is an open subset with the complex structure J and ϕ : U → A is a biholomorphism for J and J 0 . We obtain two 'holomorphic charts' for Z, both defined on U: (U , ϕ, A), and (U , c • ϕ, A c ) (to be precise, both are biholomorphisms, but only the latter is a genuine holomorphic chart). Suppose hence it locally admits a power series expansion in the conjugate variables z j 's.
is holomorphic on A c ; hence it locally admits a power series expansion in the standard variables z j 's.
The diagonal ∆ ⊂M ×M is a totally real submanifold, real-analytically diffeomorphic toM . Let (U, ϕ, A) be holomorphic local chart for M centered at x. Then (U × U , ϕ × ϕ, A × A) is a 'holomorphic local chart' forM ×M centered at (x, x). Let z j and u j denote, respectively, the standard complex linear coordinates on the two factors of C d ×C d , respectively. If F :M ×M → C is holomorphic, then F • (ϕ × ϕ) −1 can be expanded in a power series in the z j 's and the u j 's.
Any real-analytic function f onM may be viewed as a real-analytic function on ∆; as such, it has a holomorphic extensionf to an open neighborhood of ∆ inM ×M. Suppose that f ∈ C k ϕ for some k ≥ 1. By Proposition 26, the only contributions to the power series expansion of f • ϕ −1 ∈ C k come from monomials (52) such that 2 (a 0 + b 0 ) + d−1 j=1 (a j + b j ) ≥ k. On the other hand, the holomorphic extension of (52) to C d × C d has the form When we match Heisenberg type ordering with holomorphic extension of real analytic functions, we are thus led to introduce the following two rings.
Definition 27. Let k ≥ 1 be an integer.
1. O k will denote the ring of germs of holomorphic functions F on C d ×C d at (0, 0) with the following property: if a monomial (54) gives a nontrivial contribution to the power series expansion of F , then 2 (a 0 + b 0 ) + d−1 j=1 (a j + b j ) ≥ k; 2. O k ϕ×ϕ will denote the ring of germs of holomorphic functions F oñ Corollary 28. Let (U, ϕ, A) be a holomorphic local chart for M centered at x. Then the following holds.

Heisenberg local coordinates adapted to a hypersurface
In §18 of [FS1], a special system of local holomorphic coordinates is constructed on a complex manifold near a point lying on a strictly pseudoconvex hypersurface; this construction was profitably put to use in [CR1] and [CR2] to study the asymtptotics of Szegö and Poisson kernels on Grauert tubes. In such a system of coordinates, the local geometry of the hypersurface is well approximated by the local geometry of the Heisenberg group; for this reason, these systems of coordinates are naturally referred to as Heisenberg local coordinates (see e.g. [SZ], [CR1], [CR2]), although they were originally called normal coordinates in [FS1]. In our setting, up to a simple rescaling this amounts to the existence of coordinates as in following definition.
Definition 29. If τ ∈ (0, ǫ) and x ∈ X τ ⊆M ǫ , a system of Heisenberg local coordinates onM adapted to X τ at x is a holomorphic local chart (U, ϕ, A) forM centered at x, with the following properties: 3. the defining function φ τ := ρ − τ 2 for X τ takes the form where f ∈ C 3 (clearly f is real valued and real-analytic).
Caveat 30. The norm · in (55) is for now simply the Euclidean norm in the given coordinate system, but it will be shown below to have an metric intrinsic meaning (see (63)).
As mentioned in the Introduction, it will be convenient to make a slightly more specific choice of coordinates (without altering the previous properties).
Since f ∈ C 3 and is real-analytic, for suitable coefficients c ∈ R and a l , b j ∈ C we have where R 3 is the third order remainder, i.e. a convergent power series near 0 in (z, z) involving only monomials of total ordinary total degree ≥ 3. If we make the change of variables and replace z by w in (55), we reduce to the case where all a j = 0 (possibily with a new R 3 ). With this adjustment, we shall refer to (U, ϕ, A) as a system of normal Heisenberg local coordinates adapted to X τ at x.
With abuse of notation, if γ is a locally defined differential form onM , we shall occasionally also denote by γ its local coordinate representation ϕ −1 * (γ).
Proposition 33. Let (U, ϕ, A) be a normal Heisenberg local chart forM, centered at x and adapted to X τ . Then the local coordinate expressions of φ τ , α and Ω are as follows: where R j denotes an expression of the appropriate type (function, differential 1-or 2-form, metric tensor respectively) vanishing to j-th order at the origin.
In Heisenberg local coordinates forM at x ∈ X τ , (w, u) ∈ C × C d−1 corresponds to the real tangent vector Corollary 34. With V as in (62), the square norm of V with respect toκ is As in the Introduction, let us set ω := 1 2 Ω; thus the Riemannian metric on the Kähler manifold (M ǫ , ω, J) isκ := 1 2κ . With V as in (62) then 3.3.4 Heisenberg local coordinates on X τ .
The local expression for φ τ in Proposition 33 yields an estimate on ℑ(z 0 ) on X τ .
Definition 37. We shall call (U τ , ϕ τ , A τ ) the Heisenberg local chart for X τ at x induced by (U, ϕ, A), and say that (U τ , ϕ τ , A τ ) is a normal Heisenberg local chart for X τ if so is (U, ϕ, A) forM . We shall often use additive notation for ϕ τ , in the following ways. First, if x ′ ∈ U τ and ϕ τ (x ′ ) is as in (64), we shall write x ′ = x + θ(x ′ ), z ′ (x ′ ) . When viewing z ′ ∈ C d−1 as an element of R 2d−2 , we shall use bold notation and write x ′ = x + θ(x ′ ), v(x ′ ) . Furthermore, let us identify T x X τ with R × C d−1 , by letting (a, u) ∈ R × C d−1 correspond to the tangent vector We shall then also write x + W := ϕ τ −1 (a, u) = x + (a, u).
Remark 38. By (60), if the tangent vectors on the right hand side of (65) are meant in terms of the coordinates onM , they are actually all tangent to X τ at x. Hence (65) may as well be interpreted in terms of the (restricted) local coordinates on X τ . Thus the additive short-hand x + W has different meanings according to whether we think of W as tangent to X τ and refer to ϕ τ , or toM and refer to ϕ. The context should clarify the potential ambiguity.
We can extend the notion of Heisenberg-type order of vanishing to functions on X τ with respect to (U τ , ϕ τ , A τ ), by the following variant of Definition 22 (see §18 of [FS1]).
Definition 39. Let (U τ , ϕ τ , A τ ), ϕ τ = (θ, z ′ ), be a system of Heisenberg local coordinates on X τ cetnered at x. Let J x (X τ ) be the ring of germs of (non necessarily smooth, real or complex) functions on X τ at x; let m x (X τ )✂ J x (X τ ) be the ideal of those germs that vanish at x. Let C ∞ (X τ ) x ⊆ J x (X τ ) be the subring of germs of smooth functions. Suppose f ∈ m x (X τ ). Then . Let (U τ , ϕ τ , A τ ) be induced by the system of Heisenberg local coordinates (U, ϕ, A) adapted to X τ at x. The definition of O k ϕ τ entails the following.
Lemma 40. Let  τ : X τ ֒→M be the inclusion. Then Proof. The statement follows readily from the definition in case k = 1. For . The claim follows from this and (51) since  τ * is multiplicative morphism.
Let us express vol R X τ (x) in terms of ϕ τ (recall (45) and Corollary 18). By The latter factor is the standard volume form on C d−1 ∼ = R 2d−2 in the linear coordinates z ′ .
Thus ℑ f (z 0 , z ′ ) vanishes to third order at x (that is, at the origin). Since f is holomorphic, f itself vanishes to third order at x. We conclude the following.

The geodesic flow in Heisenberg coordinates
Since the vector field R of Definition 16 is tangent to the compact hypersurfaces X τ , it is complete onM \ M. Given x ∈ X τ , let us choose a system of Heisenberg normal coordinates (U, ϕ, A) adapted to X τ at x. Let Λ x : R → X τ be the integral curve of R passing through x at t = 0. For t sufficiently small, Let us write z 0 (t) = θ(t) + ı η(t), where θ(t) = ℜ z 0 (t) and η(t) = ℑ z 0 (t) ; in view of (60), we have Hence θ(t) − t, η(t), and z ′ (t) vanish to second order at the origin. Thus where f and F are smooth and vanish to second order at 0 ∈ R.
Since Λ x is an integral curve of R, Expressing this condition by means of Proposition 33 yields Thus θ(t) = t + R 3 (t). We conclude: Lemma 42. If Λ x : R → X τ is the integral curve of R through x, then in normal Heisenberg local coordinates for X τ at x we have In additive notation as in Definition 37, Λ x (t) = x + t + R 3 (t), R 2 (t) .
Since any smooth function vanishing to first order at x is in C 1 ϕ τ (X τ ), we reach the following conclusion (a slight refinement of Lemma 3.6 of [CR1]).
Corollary 43. Suppose y = x + (θ, u) ∈ U τ and let Λ y : R → X τ be the integral curve of R with initial condition y. Then for t small we have where f (t, ·, ·) and (every component of ) f(t, ·, ·) are in C 1 ϕ τ (X τ ) (that is, they are O 1 ϕ τ ).
The previous statement may be converted into one concerning the homogeneous geodesic flow. The latter is intertwined by E ǫ with the flow of υ √ ρ ; on the other hand by (42) we have υ τ √ ρ = −τ R τ on X τ . Thus γ(·) is an integral curve of R τ if and only if γ(−τ ·) is an integral curve of υ τ √ ρ . Let us denote by Γ τ t : X τ → X τ the restricted geodesic flow at time t.
Hence by Proposition 33 We conclude the following.
Assume that L W (α τ ) = 0. Then the flow of W preserves vol C X τ by (45); hence it also preserves vol R X τ by Corollary 18. Therefore the flow of W induces in a standard manner a one-parameter group of unitary automorphisms of L 2 (X τ ). It follows that, as a differential operator on X τ , W is skew-symmetric; thus the Toeplitz operator By [BdM-G], Toeplitz operators on X τ have well-defined principal symbols, which are smooth functions on the closed symplectic cone Let us compute the principal symbol σ(W) of W at (x, r α τ x ). We consider a system of normal Heisenberg local coordinates onM adapted to X τ at x (Definition 29), and the corresponding Heisenberg local chart for X τ at x (Definition 37); we denote the latter by ϕ τ = (θ, z ′ ). By Proposition 33, On the other hand, by (60) and (72), Thus, If λ > 0, therefore, W is a positive self-adjoint Toeplitz operator, so that its spectrum is discrete, bounded from below and accumulates at +∞ (see [BdM-G]). This applies in particular if W = υ √ ρ by (42), with λ = √ ρ.

The Szegö kernel and its phase
Recall that L 2 (X τ ) denotes the Hilbert space of square summable functions on X τ with respect to vol R X τ in §3.2.3, H(X τ ) ⊆ L 2 (X τ ) is the corresponding Hardy space, and Π τ : L 2 (X τ ) → H(X τ ), the Szegö projector, is the orthogonal projector. By [BdM-S], Π τ is a Fourier integral operator with complex phase; its wave front WF(Π τ ) = Σ τ ♯ is the anti-diagonal of Σ τ in (10): More precisely, up to a smoothing term the distributional kernel of Π τ (a.k.a. the Szegö kenel of X τ ) is microlocally of the form where the amplitude s τ and the phase ψ τ are as follows (see [BdM-S]).
1. s τ is a semiclassical symbol admitting an asymptotic expansion 2. ψ τ satisfies ℑ(ψ τ ) ≥ 0 and is essentially determined along the diagonal of X τ by the Taylor expansion of the defining function φ τ ; in the present real-analytic setting we may assume that whereφ τ denotes the holomorphic extension of φ τ toM ×M (see the discussion preceding Definition 27).
Let us express ψ τ in the neighbourhood of (x, x) in X τ ×X τ using normal Heisenberg local coordinates ϕ τ = (θ, z ′ ) on X τ centered at x, defined on an open subset U τ ⊆ X τ . Let ψ ωx 2 be as in (16). Proposition 47. Suppose that where the latter term denotes a power series in the indicated variables, involving only terms of total degree ≥ 3.
Proof. Let notation be as in Proposition 47. Then d (x,x) The statement then follows from Proposition 33.
Again, the following statement follows from the general theory of [BdM-S], but it can also be verified by direct inspection of (80). Let dist X τ : X τ ×X τ → R be the Riemannian distance function.
Corollary 49. There are a neighborhood X ′ ⊆ X τ × X τ of the diagonal and a constant C τ > 0 such that

The leading order term of the amplitude
We aim to determine the evaluation s τ 0 (x, x) of the leading order term in (77); we shall follow the argument in §4 of [BdM-S], and apply Proposition 47.
Remark 51. Recall that Π τ is the Szegö kernel for the Hermitian structure on L 2 (X τ ) associated to vol R X τ . Integration with respect to vol R X τ in a variable y will be denoted by the short-hand dV X τ (y). A different choice of volume form would clearly lead to a different result.
Proof. By the idempotency of Π τ , for all (x, x) ∈ X τ × X τ we have The singular support of Π τ is S.S.(Π τ ) = diag(X τ ) (the diagonal in X τ × X τ ). Hence we need only consider the situation for (x ′ , x ′′ ) close to diagonal.
Suppose then x ∈ X τ and that x ′ , x ′′ both belong to a small neighbourhood of x. For the same reason, only a smoothing term (not contributing to the asymptotic expansion of the amplitude) is lost, if integration in (81) is restricted to a suitable neighborhood of x. We may thus multiply the integrand by some cut-off function in y, identically equal to 1 near x, and assume that x ′ , x ′′ , y belong to some δ-neighbourhood of x; for notational simplicity, the latter cut-off will be left implicit. In view of (76), up to a smoothing contribution we have where, setting v = u σ, Since x ′ , x ′′ , y all belong to a δ-neighborhood of x, therefore, iterated 'integration by parts' in y shows that only a negligible contribution in u → +∞ is lost, if integration in σ is restricted to a suitable neighborhood of 1. Hence the asymptotics in u → +∞ are unaltered, if the integrand is multiplied by γ(σ), with γ ∈ C ∞ 0 (R + ) identically equal to 1 on (ǫ ′ , 1/ǫ ′ ), for some ǫ ′ > 0. Thus integration in σ may also be assumed to be compactly supported. Again, the latter cut-off will be left implicit.
As in [BdM-S], in order to evaluate (83) asymptotically we first look for stationary points of Υ τ when x ′ = x ′′ = x. To this end, let us fix Heisenberg local coordinates on X τ at x, and set y = x + (θ, v). By Proposition 47, Here (θ, v) ∼ 0, and the only real critical point near the origin is (1, 0, 0). The Hessian matrix at the critical point is Hence by Theorem 2.3 of [MS] we may apply the complex version of the stationary phase Lemma, with Recalling (67), to leading order in u we obtain for (83) where ψ 1 (x ′ , x ′′ ) is the critical value of a holomorphic extension of Υ τ . More precisely, letX τ denote a complexification of the real-analytic manifold X τ . The real-analytic function Υ τ (x ′ , x ′′ ; ·, ·) : X τ × R → C, depending on the parameter (x ′ , x ′′ ) ∈ X τ × X τ , extends uniquely to a holomorphic functioñ Υ τ (x ′ , x ′′ ; ·, ·) to an open neighbourhood of X τ × R inX τ × C. We have seen that Υ τ (x, x; ·, ·) admits a unique and non-degenerate crtitical point near (x, 1), namely (x, 1). By the theory in [MS],Υ τ (x ′ , x ′′ ; ·, ·) has a unique critical point This is (a special case of) Proposition 4.8 of [BdM-S]. Nonetheless, we provide the proof below for the reader's convenience and because the argument appears somewhat more concrete in the current real-analytic setting.
Let us premise a few remarks. Let us fix normal Heisenberg coordinates onM adapted to X τ at x, and with abuse of notation identify functions on M with their coordinate representation. As above, let us identifyM (as a differentiable manifold) with the totally real submanifold ∆M := diag(M ) ⊂ M ×M . In local coordinates, we shall write Z for x + Z, where Z = (z 0 , z ′ ). Thus Z is mapped to (Z, Z) ∈ ∆M . In addition, φ τ may be written (locally near x) as a convergent power series φ τ (Z, Z); its holomorphic extention tõ M ×M is then locally given byφ τ Z, W . By (78), By the embedding X τ ֒→M ∼ = ∆M ⊂M ×M , we can locally realize the real-analytic hyprsurface X τ as the manifold of C d × C d X τ ′ := (Z, Z) :φ τ Z, Z = 0 .
The complexification X τ is then locally describable as the holomorphic hy- (to be precise, here (Z, W ) belongs to a neighbourhood of the diagonal).
We can now conclude the proof of Theorem 50. By (82), Proposition 52), and idempotency, Π τ is a Fourier integral operator with phase ψ τ and a symbol of order d − 1 whose leading order term must coincide with the one in (76). Since y c (x, x) = x, equating the leading order coefficients in (77) and (82), we obtain The claim follows.

Dynamical Toeplitz operators
As mentioned in the Introduction, the homogenous geodesic flow is generally not holomorphic for J ad . Equivalently, the (1, 0)-component of υ √ ρ needn't be holomorphic onM \ M. Therefore, when viewed a differential operator, υ τ √ ρ does generally not preserve the Hardy space H(X τ ). A natural replacement is the self-adjoint, first order Toeplitz operator D τ √ ρ : The latter generates the 1-parameter group of unitary Toeplitz operators In view of (11), Hence, heuristically U τ √ ρ (t) is a Toeplitz quantization of the geodesic flow at time −t. In the notation of the Introduction, the distributional kernel U τ √ ρ (t; ·, ·) ∈ D ′ (X τ × X τ ) of (96) admits the spectral description Arguing, say, as in §12 of [GrSj], and using (15) and (30), one obtains It was shown by Zelditch that, up to smoothing Toeplitz operators, U τ √ ρ (t) is a 'dynamical Toeplitz operator' associated to the geodesic flow at time −t, composed with a suitable pseudodifferential operator (see e.g. §5.3 of [Z-2010]). To express this precisely, recall that Γ τ t : X τ → X τ denotes the geodesic flow along X τ (Corollary 44); for t ∈ R let us set Π τ t := Γ τ t * • Π τ . Thus Π τ t has Schwartz kernel Π τ t (x, y) := Π τ (Γ τ t (x), y). Remark 53. When defining the pull-back under a diffeomorphism, one ought to distinguish whether Π τ is referred to as a (generalized) function, density, or half-density. There is no ambiguity in the present case, since these are being identified by means of vol R X τ , which is invariant under Π τ t .
Theorem 54. (Zelditch) There exist a zeroth order polyhomogeneous complete classical symbol of the form and a zeroth order pseudodifferential operator where ≃ means 'equal modulo smoothing Toeplitz operators'. Furthermore, the principal symbol σ τ t,0 (x) is the inverse of the L 2 -pairing of two normalized Gaussian functions, related to complex structures J x and J tx , respectively, where J t is the push-forward of J under the flow of υ τ √ ρ at time t; in particular, σ τ 0,0 (x) = 1.
Up to smoothing terms, the Schwartz kernel of Π τ −t has the form where ψ τ and s τ are as in (76). Using classical results on the composition of pseudodifferential and Fourier integral operators ( [Sh], [T]), we reach the following conclusion.
Lemma 55. Up to smoothing terms, the Schwartz kernel of the composition P τ t := P τ t • Π τ −t has the form where 4 Near-diagonal asymptotics for Π τ χ,λ Before delving into the proof of Theorem 3, let us premise some notation. The choice of a normal Heisenberg local chart ϕ τ for X τ at x determines an isomorphism T x X τ ∼ = R × R 2d−2 ; a general υ ∈ T x X τ will be written accordingly as a pair υ = (θ, u). The subspaces R × {0} and {0} × R 2d−2 correspond, respectively, to T τ (x) and H τ (x) (see (48)). By (63), the isomorphism C d−1 ∼ = {0} × R 2d−2 → H τ (x) is unitary, when H τ (x) is endowed with the Hermitian structure associated to ω x = 1 2 Ω x . With the notation of Corollary 45, let (a x , A x ) ∈ R × R 2d−2 be defined by where F τ 2 vanishes to second order at the origin. We may then reformulate the conclusion of Corollary 44 writing We shall verify a posteriori that (a x , A x ) = (0, 0). Definition 56. With (a x , A x ) as in (104), let us set We further define Proof of Theorem 3. By (98) and (100), we can rewrite (15) in the following form: where ∼ stands for 'has the same asymptotics for λ → +∞ as'.
The wave front set of Π τ is given by (75). Therefore, given that the geodesic flow preserves α (Lemma 11), the wave front of Hence the singular supports of Π τ and P τ t •Π τ −t are, respectively, the diagonal and the graph of Γ τ −t in X τ × X τ . Let c > 0 be such that Then U ′ := {U ′ 1 , U ′ 2 } is an open cover of X τ ; let γ ′ 1 + γ ′ 2 = 1 be a partion of unity subordinate to U ′ . Thus (110), with γ ′ j in place of γ j . For y ∈ U ′ 2 , the function y → Π τ (x 1 , y) is C ∞ , and an adaptation of the previous argument implies that Π τ χ,λ (x 1 , where integration in y is now over a small neighborhood of x 1 . We may assume that x 1 , x 2 and every y in the support of the integrand in Π τ χ,λ (x 1 , x 2 ) ′ 1 belong to a Heisenberg coordinate neighborhood centered at some x ∈ X τ . Without altering the asymptotics, Π τ may be represented as a Fourier integral operator (76), and apply Lemma 55 and (103). Then (107) may be rewritten . With the rescaling u λ u and v λ v, (113) becomes We shall let y = x + (θ, u) in Heisenberg local coordinates on X τ at x; then dV X τ (y) by V(θ, u) dθ du, and by (67) Integration in (θ, u, t) is compactly supported near the origin. By (115), Proposition 33, Corollaries 44 and 48 It follows by a standard 'partial integration' argument in the compactly supported variables (θ, t) that the contribution of the locus where (u, v) ≫ 0 contributes negligibly to the asymtptotics. Similarly, if ǫ ≪ 1, integration by parts in t shows that the contribution of the locus where v ≪ 1/τ is also negligible. Finally, integration by parts in θ yields a similar conclusion for u.
We thus obtain the following reduction.
Lemma 59. There exists D ≫ 0 such that the following holds. Let h ∈ C ∞ 0 (1/(2D), 2 D) be such that h ≡ 1 on (1/D, D). Then only a negligible contribution to the asymptotics of (114) is lost, if the integrand is multiplied by h(u) · h(v). Hence, integration in (u, v) may be assumed to be compactly supported in (1/(2D), 2 D) 2 .
Proof of Statement 1) of Theorem 3. There exist constants 0 < a ≤ A such that is an open cover of X τ . If y ∈ U λ 1 , for any t ∈ (−ǫ, ǫ) we have By Corollary 49, in the same range Integrating by parts in v, we obtain that the contribution of U λ 1 to the asymptotics of (114) On the other hand, if y ∈ U λ 2 then ∂Ψ τ ∂u = |ψ τ (y, Integrating by parts in u, we obtain that the contribution of U λ 2 is also O (λ −∞ ).
We focus on statement 2. Let us set Under the assumptions, Let h τ be a Riemannian metric on X τ that in a sufficiently small neighborhood of x is given in Heisenberg local coordinates by Thus h τ x =κ τ x by Corollary 34. Let dist X τ : X τ ×X τ → R be the Riemannian distance of h τ . By the latter remark, For r > 0, let B x (r) ⊆ X τ be the open ball centered at x for h τ . Let r be small enough that B x (r) ⊂ U τ , and consider the open cover of X τ B := B x (r), B x (r/2) c .
is a partition of unity subordinate to the rescaled open cover The asymptotics of Π τ χ,λ (x 1λ , x 2λ ) are given by (114) with (x 1 , x 2 ) replaced by (x 1λ , x 2λ ).
We now make explicit the dependence of Ψ τ where Q k is a polynomial of degree ≤ 3 k.
On the other hand, the asymptotic expansion (77) yields where S τ k,l is homogenous of degree l. Similarly, the asymptotic expansion in Lemma 55 yields where again R τ k,l is homogeneous of degree l, and we have used that r τ 0, 0 (x, x) = s τ 0 (x, x).
Multiplying (139), (140), (141), and the Taylor expansion of χ and V at the origin, we obtain an asymptotic expansioñ (u,t,v,θ,u) where B k (x; ·) is a polynomial of degree ≤ 3 k in (θ j , v j , u, t, θ), while β is compactly supported and identically equal to 1 in a suitable neighborhood of the origin. The latter is indeed an asymptotic expansion for δ ∈ (0, 1/6). Furthermore, fractional powers of λ arise from Taylor expansion in (θ j , v j , u), while the asymptotic expansion for the amplitude in the Szegö kernel parametrix is by descending integer powers. Hence P j will be even for j even (corresponding to integer powers of λ), and odd for j odd (corresponding to fractional powers).
Inserting (142) in (136), we obtain an asymptotic expansion for the integrand which, in view of (134) or the previous remark on the domain of integration, can be integrated term by term. Each term is an oscillatory integral with phase (132) in the parameters (t, v, θ, u), and depending parametrically on the other parameters. Now we remark that the asymptotics of (135) are unaltered, if integration in (t, θ) is restricted to a suitable compact set. In fact, since ∂ u Υ τ = θ 1 − θ, ∂ v Υ τ = θ + τ t − θ 2 , integration by parts in (u, v) implies the following.
We shall leave the latter cut-off implicit in the following, and simply assume henceforth that integration in (θ, t) is over a compact neighborhood of θ 1 , (θ 2 − θ 1 )/τ .
The proof of the following is omitted.
The third order remainder at P s is zero.
Putting all these asymptotic expansions together, we obtain an asymptotic expansion for I λ (u) of the form I λ (u) ∼ e ı √ λ θ 1 −θ 2 τ · λ 2d−3 (2 π 2 τ 2 ) d−1 · χ(0) e Sc(θ j ,v j ,u) ·β 1 λ −δ u, 1 + k≥1 λ −k/2 F k (x; θ j , v j , u) where F k (x; ·) is a polynomial of degree ≤ 3 k and parity k, and β 1 is an appropriate cut-off function identically equal to one near the origin. The asymptotic expansion (147) may be integrated term by term. In view of the rapidly decreasing exponential e Sc(θ j ,v j ,u) , we obtain We compute the leading order term using (146). With the change of variable Let us set v ′ j := 1 √ τ v j , a ′ := 1 √ τ a x (θ 1 , θ 2 ). With the further change of variable w = 1 √ 2 r, we obtain where L and L ′ are multi-indexes, l + |L| + |L ′ | ≤ 3 j, and l + |L| + |L ′ | has the same parity as j. In turn, in view of (149), the Gaussian integral in (155) may be written as a linear combination of summands of the form where each D (a) θ j ,v j a first order differential operator with constant coefficients and no zeroth order term in (θ j , v j ), with a + |B| + |C| = l + |L| + |L ′ |. It then follows from (106) and (153) that the coefficient of λ d−1− k 2 has the form where again P k is a polynomial of degree ≤ 3 k and parity k.
5 Proof of Theorem 6

Preliminaries for Theorem 6
In addition to the general setting of Theorem 3, we need the description, also due to Zelditch, of the wave group in the complex domain as a dynamical Toeplitz operator ([Z-2010], Proposition 7.1; [Z-2012], especially §8-9; [Z-2014], §4; see also the discussions in [CR1] and [CR2]). This is the analogue of the description of U τ √ ρ (t) as a dynamical Toeplitz operator recalled in §3.6.
Alternatively, U C (t + 2 ı τ ) is a Fourier integral operator with complex phase of positive type on X τ , of degree −(d − 1)/2; in the terminology of [BdM-G], it is in fact a Fourier-Hermite operator adapted to the symplectomorphism Σ τ → Σ τ induced by the homogeneous geodesic flow at time t.
First, t appears in place of −t in the amplitude and phase. This difference is compensated by a change of variables, which replaces χ(t) with χ(−t); given the rescaling t → t/ √ λ in (125), the leading order term of the expansion is not affected.
Second, the leading order term of the amplitude has been multiplied by a factor γ τ 0,0 (x)·(u τ ) − d−1 2 . In view of the rescaling u → λ u in (114), this change entails an additional factor γ τ 0,0 (x) · λ − d−1 2 in front of the resulting asymptotic expansion. Furthermore, by Lemma 63 at the critical point u τ = 1.

The pointwise Weyl law
Proof of Proposition 9. We shall prove (26); the same argument, with obvious adaptations, also proves (27).
where H is the Heaviside function. Let us consider the following positive measures L and T τ x on R. First, L is the Lebesgue measure. Second, Let us endow R × R with the product measure L × T . By the Fubini Theorem (see e.g. Ch. 8 of [R]), the claim will follow by comparing both sides of (163). The former integral in (163) is on the last line, we have made use of (15) and Theorem 1.27 of [R]. In view of Corollary 4, (164) implies that as λ → +∞ On the other hand, the latter integral in (163) Lemma 69. For λ → +∞, we have Proof. It follows from Corollary 4 that for λ ≫ 0 Hence there exist C 0 , C 1 > 0 such that Thus for suitable C ′ , C ′′ > 0 for any λ, t ∈ R for appropriate constants A, B > 0.