Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order $(1,1)$, on an asymptotically Euclidean manifold. We first prove a two term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $Q=(1+|x|^2)(1-\Delta)$ on $\mathbb{R}^d$.


Introduction
Let (X, g) be a d-dimensional asymptotically Euclidean manifold. On X we consider a self-adjoint positive operator P , elliptic in the SG-calculus of order (m, n) with m, n ∈ (0, ∞). By the compact embedding of weighted Sobolev spaces, the resolvent is compact and hence the spectrum of P consists of a sequence of eigenvalues 0 < λ 1 ≤ λ 2 ≤ . . . → +∞.
The goal of this article is to study the Weyl law of P , that is, the asymptotics of its counting function, (1) N(λ) = #{j : λ j < λ}.
Hörmander [15] proved, for a positive elliptic self-adjoint classical pseudodifferential operator of order m > 0 on a compact manifold, the Weyl law It was pointed out that, in general, this is the sharp remainder estimate, since the exponent of λ in the remainder term cannot be improved for the Laplacian on the sphere. It was subsequently shown by Duistermaat and Guillemin [12] that under a geometric assumption there appears an additional term γ ′ λ (d−1)/m and the remainder term becomes o(λ (d−1)/m ).
In the case of SG-operators on manifolds with ends, the leading order of the Weyl asymptotics was found by Maniccia and Panarese [17]. Battisti and Coriasco [1] improved the remainder estimate to O(λ d/ max{m,n}−ǫ ) for some ǫ > 0. For m = n, Coriasco and Maniccia [8] proved the general sharp remainder estimate.
In Theorem 1, we prove the analogue of Hörmander's result for m = n. This provides a more precise remainder term compared to the earlier result given in [1]. If the geodesic flow at infinity generated by the corner component p ψe of the principal symbol of P is sufficiently generic, we have an even more refined estimate, parallel to the Duistermaat-Guillemin theorem, described in Theorem 2. Theorem 1. Let P ∈ Op SG m,m cl (X) be a self-adjoint, positive, elliptic SG-classical pseudodifferential operator on an asymptotically Euclidean manifold X, and N(λ) its associated counting function. Then, the corresponding Weyl asymptotics reads as The coefficients γ j , j = 1, 2, are given by where TR and TR x,ξ are suitable trace operators on the algebra of SG-operators on X. .
Remark 3. The trace operators TR and TR x,ξ appearing in Theorems 1 and 2 were introduced in [1]. The coefficient γ 0 can be calculated as the Laurent coefficient of order −2 at s = d − 1 of ζ(s), the spectral ζ-function associated with P .
Remark 4. To our best knowledge, this is the first result of a logarithmic Weyl law with the remainder being one order lower than the leading term (we refer to [1] for other settings with logarithmic Weyl laws).
Next, we apply our results to the model operator P associated with the symbol p(x, ξ) = x · ξ , z = 1 + |z| 2 , z ∈ R d , that is, P = · √ 1 − ∆. In particular, we observe that the condition on the underlying Hamiltonian flow in Theorem 2 is not satisfied, and compute explicitly the coefficients γ 1 and γ 2 .
Here, the coefficients are where γ = lim n→+∞ n k=1 1 k − log n is the Euler-Mascheroni constant and is the digamma function.
This implies that the Weyl asymptotics of the operator is given by with the same coefficients given in Theorem 5 above.
The paper is organized as follows. In Section 2 we fix most of the notation used throughout the paper and recall the basic elements of the calculus of SG-classical pseudodifferential operators, the associated wave-front set, and the computation of the parametrix of Cauchy problems for SG-hyperbolic operators of order (1, 1). In Section 3 we consider the wave-trace of a SG-classical operator P of order (1, 1). Section 4 is devoted to study the relation between the wave-trace and the spectral ζ-function of P . In Section 5 we prove our main Theorems 1 and 2, while in Section 6 we examine the example given by the model operator P = · D , and prove Theorem 5. We conclude with a short appendix on asymptotically Euclidean manifolds and a few more remarks about aspects of the proofs of the main results.

SG-Calculus on R d
The Fourier transform F : and extends by duality to a bounded linear operator F : The space SG m ψ ,me (R 2d ) becomes a Fréchet space with the seminorms being the best constants in (2). The space of all SG-pseudodifferential operators of order (m ψ , m e ) is denoted by We have the following properties (we refer to, e.g., [2] and [23,Chapter 3] for an overview of the SG-calculus): Op SG m ψ ,me (R d ) is a graded *-algebra; its elements are linear continuous operators from S(R d ) to itself, extendable to linear continuous operators on S ′ (R d ); (2) the differential operators of the form are SG operators of order (m ψ , m e ); (3) If A ∈ Op SG 0,0 (R d ), then A extends to a bounded linear operator (4) there is an associated scale of SG-Sobolev spaces (also known as Sobolev-Kato spaces), defined by The formula involving integrals only holds true for a ∈ S(R 2d ), but the quantization can be extended to any a ∈ S ′ (R 2d ) using the Fourier transform, pull-back by linear transformations, and the Schwartz kernel theorem.
and, for all m ψ , m e , s ψ , s e ∈ R, the operator A ∈ Op SG m ψ ,me (R d ) is a bounded linear operator 2.1. SG-Classical Symbols. We first introduce two classes of SG-symbols which are homogeneous in the large with respect either to the variable or the covariable. For any (1) A symbol a = a(x, ξ) belongs to the class SG m ψ ,me cl(ξ) (R 2d ) if there exist functions a m ψ −i,· (x, ξ), i = 0, 1, . . . , homogeneous of degree m ψ −i with respect to the variable ξ, smooth with respect to the variable x, such that, (2) A symbol a belongs to the class SG m ψ ,me . This means that a(x, ξ) has an asymptotic expansion into homogeneous terms in x.

Definition 6.
A symbol a is called SG-classical, and we write a ∈ SG m ψ ,me , if the following two conditions hold true: (ii) there exist functions a ·,me−k (x, ξ), homogeneous of degree m e − k with respect to the x and smooth in ξ, such that Note that the definition of SG-classical symbol implies a condition of compatibility for the terms of the expansions with respect to x and ξ. In fact, defining σ ψ m ψ −j and σ e and SG m ψ ,me cl(x) , respectively, as it possible to prove that for all j, k ∈ N.

Moreover, the composition of two SG-classical operators is still classical. For
is called the principal symbol of A. This definition keeps the usual multiplicative behaviour, that is, for any A ∈ Op SG where the product is taken component-wise. Proposition 7 below allows to express the ellipticity of SG-classical operators in terms of their principal symbol. Fixing a cut-off function ω ∈ C ∞ c (R d ) as above, we define the principal part of a to be

SG-wavefront sets.
We denote by W the disjoint union which may be viewed as the boundary of the (double) radial compactification of the Therefore, it is natural to define smooth functions on W as follows: By restriction, the principal symbol can be defined as a map σ : SG

Proposition 7. An operator
For A ∈ SG m ψ ,me cl (R d ) we define the following sets (see [8,20]): The SG-wavefront set of a distribution u ∈ S ′ (R d ) is defined as see [2,8,20]. We will decompose the SG-wavefront set of u ∈ S ′ (R d ) into its components in W, namely, The SG-wavefront set is well-behaved with respect to the Fourier transform (see, e.g., [7,Lemma 2.4]):

Complex Powers.
As in the case of closed manifolds, it is possible to define complex powers of SG-pseudodifferential operators. We will only review the crucial properties of complex powers for a positive elliptic self-adjoint operator A ∈ Op SG For the definition and proofs of the following properties, we refer to [1] (cf. also [18,25]).
(iv) If A is a classical SG-operator, then A z is classical and its principal symbol is given by Let s ∈ C with Re(s) > max{d/m e , d/m ψ }. Using the property (v) it is possible to define ζ(s) by where K A z is the Schwartz kernel of A z . We note that the ζ-function may be written as with (λ j ) j∈N the sequence of eigenvalues of A.
Moreover, it can be extended as a meromorphic function with possible poles at the points

Such poles can be of order two if and only if there exist integers j, k such that
2.4. Parametrix of SG-hyperbolic Cauchy problems. Let P ∈ Op SG 1,1 cl (R d ) be a self-adjoint positive elliptic operator. By the construction from [9, Theorem 1.2] (cf. also [3,4,7]), it is possible to calculate a suitable parametrix for the Cauchy problem associated with the wave equation, namely, The solution operator of (6) exists by the spectral theorem and is denoted by where dE is the spectral measure of P . There exists a short time parametrix U (t), which is given by operators defined through the integral kernels By a Duhamel argument, Theorem 16], [11, p. 284]). Since the error term is regularizing, we obtain that for a ∈ C ∞ c ((−ǫ, ǫ), SG 0,0 cl ) with a(0) = 1 (cf. [5,Lemma 4.14]). Let p be the principal part of the full Weyl-quantized symbol of P . The phase function φ satisfies the eikonal equation This implies that we have a Taylor expansion in t of the form For any f ∈ C ∞ (R 2d ), we define the Hamiltonian vector field by and we denote its flow by t → exp(tX f ). For P ∈ Op SG 1,1 cl (R d ), we will collectively denote by X σ(P ) the Hamiltonian vector fields on W • generated by σ • (P ), • ∈ {ψ, e, ψe}, and by t → exp(tX σ(P ) ) the three corresponding flows.
By the group property, U(t + s) = U(t)U(s), we can extend propagation of singularities results for small times to t ∈ R. In [7] the propagation of the SG-wavefront set under the action of SG-classical operators and operator families like U(t) has been studied. In particular, the following theorem was proved there (see also [6]).
Remark 10. In view of (11), Theorem 9 can also be stated in the following way: for , and X f is the Hamiltonian vector field generated by f . In the sequel we will express this fact in the compact form

Wave Trace
We fix a positive elliptic operator P ∈ Op SG 1,1 cl (R 2d ) with ψe-principal symbol p ψe = σ ψe (P ). By the compactness of the embedding of SG-Sobolev spaces, we have that the resolvent (λ−P ) −1 is compact for λ > 0 and hence there exists an orthonormal basis {ψ j } of L 2 consisting of eigenfunctions of P with eigenvalues λ j with the property that 0 < λ 1 ≤ λ 2 ≤ · · · → +∞. Therefore, the spectral measure is given by dE(λ) = ∞ j=1 δ λ j (λ) ·, ψ j ψ j , where δ µ is the delta distribution centered at µ, and we have that The wave trace w(t) is (formally) defined as As usual, w(t) is well-defined as a distribution by means of integration by parts and the fact that P −N is trace-class for N > d (cf. Schrohe [25,Theorem 2.4]).
We will show that the improvement of the Weyl law is only related to the corner component The structure of the singularities of w(t) is more involved. This comes from the fact that the boundary at infinity is not a manifold or equivalently the flow is not homogeneous. In contrast to the case of a closed manifold, the distribution w(t) will not be a conormal distribution near 0, but it turns out that it is a log-polyhomogeneous distribution.
By the previous remark, we have that T B (t) is the Fourier transform of (N ′ B * χ)(λ). We will now calculate the inverse Fourier transform of T B .
Using the Taylor expansion of the phase function, we have that where ψ is smooth in t. Formally, we can write the trace as As in Hörmander [16] we set Note that ellipicity implies thatÃ B (t, λ) < ∞. By the Push-Forward Theorem (cf. Melrose [19] and Grieser and Gruber [14]) it follows from (13) The above implies thatÃ B and A B are log-homogeneous of order d. In particular, we have that We note that the coefficients are determined by derivatives ofÃ B (t, λ) at t = 0 and sinceχ = 1 near t = 0, the specific choice of χ does not change the coefficients.

Relation with the spectral ζ-function
As in the case of pseudodifferential operators on closed manifolds (cf. Duistermaat and Guillemin [12, Corollary 2.2]), the wave trace at t = 0 is related to the spectral ζ-function. This relation extends to the SG setting.
Recall that for a positive self-adjoint elliptic operator P ∈ Op SG 1,1 cl (R d ), the function ζ(s) is defined for Re s > d by ζ(s) = Tr P −s .
In addition, we consider the microlocalized version of ζ(s), defined by for B ∈ Op SG 0,0 cl . Of course, ζ I (s) = ζ(s). By Theorem 8, ζ(s) admits a meromorphic continuation to C with poles of maximal order two at d−k, k ∈ N. This result extends to ζ B (s) and we characterize the Laurent coefficients in terms of the wave trace expansion at t = 0.

Proposition 14.
The function ζ B (s) extends meromorphically to C and has at most poles of order two at the points d − k, k ∈ N. We have the expansion where f is holomorphic near s = d − k and where the w jk , k ∈ N, j = 0, 1, are the coefficients appearing in the asymptotic expansion (12) of N B (λ).
Proof. The meromorphic continuation and the possible location of the poles follow from similar arguments as in [1, Theorem 3.2] (see also the proof of Proposition 16).
Hence, we only have to show that the poles are related to N B (λ).
By partial integration we obtain where ψ(s) = λ d−k−s χ ′ (λ)dλ is holomorphic and ψ(d − k) = 1. Therefore, we have Hence, the integral near s = d − k is given by where f is holomorphic in a neighbourhood of s = d − k. The formulae relating the coefficients A j+1,k and w jk , j = 0, 1, are obtained by comparing the λ-derivative of (12) with (15).
The main advantage in employing the ζ-function is that the coefficients are easier to calculate than for the wave trace.
Proof. This follows from the same arguments as in [1] (cf. the proof of Proposition 16 below), with the modification that the full symbol is a(z) = p(z)#b, where p(z) denotes the full symbol of P z . The principal ψe-symbol of A(z) = P z B is given by For the three-term asymptotics, we compute the third coefficient more explicitly.
Proof. By the analysis performed in [1], it follows that where, for Re s > d, Let us recall the main aspects of the proof of the properties of the four terms ζ j (s), j = 1, . . . , 4, showed in [1].
(1) ζ 1 (s) is holomorphic, since we integrate p(−s), a holomorphic function in s and smooth with respect to (x, ξ), on a bounded set with respect to (x, ξ). (2) Let us first assume Re s > d. Using the expansion of p(−s) with M ≥ 1 terms homogeneous with respect to ξ, switching to polar coordinates in ξ and integrating the radial part, one can write Notice that the last integral is convergent, and provides a holomorphic function in s. Arguing similarly to the case of operators on smooth, compact manifolds, ζ 2 (s) turns out to be holomorphic for Re(s) > d, extendable as a meromorphic function to the whole complex plane with, at most, simple poles at the points s 1 j = d − j, j = 0, 1, 2, . . .
Arguing as in point 2, ζ 3 (s) turns out to be holomorphic for Re s > d, extendable as a meromorphic function to the whole complex plane with, at most, simple poles at the points s 2 k = d − k, k = 0, 1, 2, . . . (4) To treat the last term, both the expansions with respect to x and with respect to ξ are needed. We assume that Re s > d and choose M ≥ 1. We argue as in point 2 to obtain Now, we introduce the expansion with respect to x, switching to polar coordinates and integrating the x-radial variable in the homogeneous terms, for both integrals We end up with

Proof of the Main Theorems
Arguing as in [8], it is enough to prove Theorems 1 and 2 for P ∈ Op SG 1,1 cl (R d ). In such situation, as explained in [1], where the triple (p ψ , p e , p ψe ) is the principal symbol of P .
Theorem 17 (Tauberian theorem). Let N : R → R such that N is monotonically nondecreasing, N(λ) = 0 for λ ≤ 0, and is polynomially bounded as λ → +∞. If Proof of Theorem 1. The first part of Theorem 1 follows directly from the Tauberian theorem and Proposition 12, due to the identity From Proposition 14 it follows that the coefficients w j,k are given by the Laurent coefficients of ζ(s).
We will need a microlocalized version of the Poisson relation.
The proof is a standard argument (cf. Wunsch [26]) and is only sketched here.
Proof of Proposition 18. For t 0 ∈ suppχ and (x, ξ) ∈ Γ, we choose a conic neighborhood U of (x, ξ) such that for all t ∈ (t 0 − ǫ, t 0 + ǫ) with ǫ > 0 sufficiently small. The existence of this neighborhood is guaranteed by the conditions on Γ and suppχ. ChooseB ∈ Op SG 0,0 cl with WF ′ SG (B) ⊂ U. Lemma 11 implies that for any k ∈ N, henceBU(t)B and all its derivatives are trace-class. We obtain the claim by using a partition of unity.
We also define the modified return timẽ where ǫ is given as in (8), and setΠ Γ = inf z∈ΓΠ (z). The main tool to prove Theorem 2 is the next Proposition 19.

Proposition 19. It holds true that
Proof of Theorem 2. The claim follows immediately by Proposition 19, since the assumptions imply that Π(x, ξ) −1 = 0 almost everywhere on W ψe .
We consider a partition of unity on the level of operators such that , A ψe j ∈ Op SG 0,0 cl and R ∈ L(S ′ , S). Furthermore, we assume that WF(A • j ) ⊂ Γ • j . Inserting the partition of unity into the counting function yields Here, ψ k are the eigenfunctions of P with eigenvalue λ k .
By the classical result of Hörmander [15], we have that N ψ j (λ) = (N ψ j * ρ)(λ)+O(λ d−1 ) and by [8] we obtain that N e j (λ) = (N e j * ρ)(λ) + O(λ d−1 ). The operator E λ R is regularising, thus its trace is uniformly bounded. We arrive at It remains to estimate the terms N ψe j (λ) − (N ψe j * ρ)(λ). For this let For 1/T < ǫ, we have by Proposition 12 that This implies that the derivative is given by where w 1,0 is given by Proposition 15. Namely, Together with Proposition 18 this implies that Applying the Tauberian theorem to N ψe j * ρ T yields for λ ≥Π j . Taking the lim sup and summing over all j gives The right hand side is an upper Riemann sum, therefore we obtain the claim by shrinking the partition of unity.
We have to investigate the flow of the principal symbol p ψe in the corner. The Hamiltonian vector field on R 2d is given by First, we show that the angle between x and ξ is invariant under the flow. This follows from Hence, the quantity is preserved by the flow. The Hamiltonian flow Φ ψe (t) : W ψe → W ψe is given by the angular part.
Lemma 20. The differential equation for ω = x/ |x| and θ = ξ/ |ξ| describing the Hamiltonian flow Φ ψe (t) : W ψe → W ψe is given by Proof. We observe that The calculation of ∂ t |x| is straightforward: as claimed. The second equation follows likewise.
Proposition 21. The return time function Π : W ψe → R is given by Proof. The system of differential equations (18) decomposes into d decoupled systems of the form We note that the eigenvalues of the matrix A are given by λ ± = ±i √ 1 − c 2 . Thus, we have that the fundamental solution to the differential equation (18) for (ω, θ) is given by for some unitary matrix S = S(c). The claim follows by choosing the minimal t > 0 with t √ 1 − c 2 ∈ 2πZ and noting that c = ω(0), θ(0) = ω 0 , θ 0 for ω 0 , θ 0 ∈ S d−1 .
Remark 22. Proposition 21 shows that Theorem 2 cannot be applied to P .
Proof of Theorem 5. By the Weyl law, Theorem 1, we have that So it remains to calculate the corresponding Laurent coefficients of ζ(s). With the notation and the results of [1], in view of (16), we have that Under the sterographic projection SP : x → x −1 (1, x) ∈ S d we may identify R d with the interior of S d + = {y = (y 0 , . . . , y d+1 ) : y 0 ≥ 0, |y| = 1}. If we set ρ = |x| −1 , then the Euclidean metric becomes where g S d−1 is the induced metric on the sphere.
For any compact manifold with boundary X with boundary defining function ρ X , we define the space of scattering vector fields sc V(x) := ρ b V(X), where b V(X) is the space tangential vector fields. There is natural vector bundle, sc T X such that the sections of sc T X are exactly the scattering vector fields. The dual bundle is the scattering cotangent bundle, sc T * X. Using the fiberwise stereographic projection, we obtain a manifold with corners sc T * X with boundary defining functions ρ X and ρ Ξ .
The new-formed fiber boundary may be identified with a rescaling of the cosphere bundle, called sc S * X. Since X is a compact manifold with boundary, sc T * X is a compact manifold with corners. The boundary W of sc T * X splits into three components: W e := sc T * ∂X X, W ψ := sc S * X o X, W ψe := sc S * ∂X X. It can be shown (cf. [10]) that the SG-classical symbols SG m ψ ,me cl . All the concepts and notions introduced in the previous parts of this section, for the local model given by R d and its compactification S d + , extend to the setting of a general scattering manifold X. Melrose-Zworski [22] defined for f ∈ ρ −me X ρ −m ψ Ξ C ∞ ( sc T * X) the Hamiltonian vector which generalizes the usual Hamiltonian vector field to the compactified cotangential bundle of asymptotically Euclidean manifolds.
For f ∈ ρ −1 X ρ −1 Ξ C ∞ ( sc T * X), the Hamiltonian vector field is tangential to the boundary and hence its flow exp(t sc X f ) can be restricted to a map exp(t sc X f )| W : W → W that preserves the components W e , W ψ , and W ψe . Note that the flow t → exp(t sc X f )| W depends only on the principal symbol of f .
The propagation of singularities results from [7] now reads as follows: Proposition 25. Let P be an elliptic SG-pseudodifferential operator of order (1, 1) on an asymptotically Euclidean manifolds (X, g). Denote by Φ(t) : W → W the Hamiltonian flow associated with the principal symbol of P . Then WF SG (e −itP u) = Φ(t)(WF SG (u)).
Remark 26. Actually, the results on complex powers, trace operators and spectral asymptotics of SG-classical operators have been proved in detail, in [1] and [8], for operators defined on the subclass of manifolds with (cylindrical) ends. In particular, the results about the Cauchy problems for SG-hyperbolic operators of order (1, 1) yield there a global parametrix U (t), locally represented by operators with kernel given in (7), see [8]. To keep this exposition within a reasonable length, and avoid to deviate from our main focus, the detailed analysis of the extension of such previous results to general scattering manifolds, as well as the proof of some results on the operator SG-wavefront set, tacitly used above, will be illustrated elsewhere.