On the Spectral Asymptotics of Operators on Manifolds with Ends

We deal with the asymptotic behaviour for $\lambda\to+\infty$ of the counting function $N_P(\lambda)$ of certain positive selfadjoint operators $P$ with double order $(m,\mu)$, $m,\mu>0$, $m\not=\mu$, defined on a manifold with ends $M$. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier Integral Operators associated with weighted symbols globally defined on $\mathbb{R}^n$. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for $N_P(\lambda)$ and show how their behaviour depends on the ratio $\frac{m}{\mu}$ and the dimension of $M$.


Introduction
The aim of this paper is to study the asymptotic behaviour, for λ → +∞, of the counting function N P (λ) = λ j ≤λ 1 where λ 1 ≤ λ 2 ≤ . . . is the sequence of the eigenvalues, repeated according to their multiplicities, of a positive order, selfadjoint, classical, elliptic SG-pseudodifferential operator P on a manifold with ends. Explicitly, SG-pseudodifferential operators P = p(x, D) = Op p on R n can be defined via the usual left-quantization Pu(x) = 1 (2π) n e ix·ξ p(x, ξ)û(ξ)dξ, u ∈ S(R n ), starting from symbols p(x, ξ) ∈ C ∞ (R n × R n ) with the property that, for arbitrary multiindices α, β, there exist constants C αβ ≥ 0 such that the estimates (1.1) |D α ξ D β x p(x, ξ)| ≤ C αβ ξ m−|α| x µ−|β| hold for fixed m, µ ∈ R and all (x, ξ) ∈ R n × R n , where y = 1 + |y| 2 , y ∈ R n . Symbols of this type belong to the class denoted by S m,µ (R n ), and the corresponding operators constitute the class L m,µ (R n ) = Op (S m,µ (R n )). In the sequel we will sometimes write S m,µ and L m,µ , respectively, fixing once and for all the dimension of the (non-compact) base manifold to n. These classes of operators, introduced on R n by H.O. Cordes [6] and C. Parenti [30], see also R. Melrose [26], M.A. Shubin [33], form a graded algebra, i.e., L r,ρ • L m,µ ⊆ L r+m,ρ+µ . The remainder elements are operators with symbols in S −∞,−∞ (R n ) = (m,µ)∈R 2 S m,µ (R n ) = S(R 2n ), that is, those having kernel in S(R 2n ), continuously mapping S ′ (R n ) to S(R n ). An operator P = Op p ∈ L m,µ and its symbol p ∈ S m,µ are called SG-elliptic if there exists R ≥ 0 such that p(x, ξ) is invertible for |x| + |ξ| ≥ R and p(x, ξ) −1 = O( ξ −m x −µ ).
In such case we will usually write P ∈ EL m,µ . Operators in L m,µ act continuously from S(R n ) to itself, and extend as continuous operators from S ′ (R n ) to itself and from H s,σ (R n ) to H s−m,σ−µ (R n ), where H s,σ (R n ), s, σ ∈ R, denotes the weighted Sobolev space H s,σ (R n ) = {u ∈ S ′ (R n ) : u s,σ = Op π s,σ u L 2 < ∞}, π s,σ (x, ξ) = ξ s x σ .
Continuous inclusions H s,σ (R n ) ֒→ H r,ρ (R n ) hold when s ≥ r and σ ≥ τ, compact when both inequalities are strict, and An elliptic SG-operator P ∈ L m,µ admits a parametrix E ∈ L −m,−µ such that for suitable K 1 , K 2 ∈ L −∞,−∞ = Op (S −∞,−∞ ), and it turns out to be a Fredholm operator. In 1988, E. Schrohe [31] introduced a class of non-compact manifolds, the so-called SG-manifolds, on which it is possible to transfer from R n the whole SGcalculus. In short, these are manifolds which admit a finite atlas whose changes of coordinates behave like symbols of order (0, 1) (see [31] for details and additional technical hypotheses). The manifolds with cylindrical ends are a special case of SG-manifolds, on which also the concept of SG-classical operator makes sense: moreover, the principal symbol of a SG-classical operator P on a manifold with cylindrical ends M, in this case a triple σ(P) = (σ ψ (P), σ e (P), σ ψe (P)) = (p ψ , p e , p ψe ), has an invariant meaning on M, see Y. Egorov and B.-W. Schulze [13], L. Maniccia and P. Panarese [24], R. Melrose [26] and Section 2 below. We indicate the subspaces of classical symbols and operators adding the subscript cl to the notation introduced above.
The literature concerning the study of the eigenvalue asymptotics of elliptic operators is vast, and covers a number of different situations and operator classes, see, e.g., the monograph by V.J. Ivrii [22]. Then, we only mention a few of the many existing papers and books on this deeply investigated subject, which are related to the case we consider here, either by the type of symbols and underlying spaces, or by the techniques which are used: we refer the reader to the corresponding reference lists for more complete informations. On compact manifolds, well known results were proved by L. Hörmander [19] and V. Guillemin [15], see also the book by H. Kumano-go [23]. On the other hand, for operators globally defined on R n , see P. Boggiatto, E. Buzano, L. Rodino [2], B. Helffer [16], L. Hörmander [20], A. Mohammed [27], F. Nicola [28], M. A. Shubin [33]. Many other situations have been considered, see the cited book by V.J. Ivrii. On manifolds with ends, T. Christiansen and M. Zworski [5] studied the Laplace-Beltrami operator associated with a scattering metric, while L. Maniccia and P. Panarese [24] applied the heat kernel method to study operators similar to those considered here.
Here we deal with the case of manifolds with ends for P ∈ EL m,µ cl (M), positive and selfadjoint, such that m, µ > 0, m µ, focusing on the (invariant) meaning of the constants appearing in the corresponding Weyl formulae and on achieving a better estimate of the remainder term. Note that the situation we consider here is different from that of the Laplace-Beltrami operator investigated in [5], where continuous spectrum is present as well: in fact, in view of Theorem 3.2, spec(P) consists only of a sequence of real isolated eigenvalues {λ j } with finite multiplicity.
As recalled above, a first result concerning the asymptotic behaviour of N P (λ) for operators including those considered in this paper was proved in [24], giving, for λ → +∞, for m > µ.
Note that the constants C 1 , C 2 , C 1 0 above depend only on the principal symbol of P, which implies that they have an invariant meaning on the manifold M, see Sections 2 and 3 below. On the other hand, in view of the technique used there, the remainder terms appeared in the form o(λ n min{m,µ} ) and o(λ n m log λ) for m µ and m = µ, respectively. An improvement in this direction for operators on R n had been achieved by F. Nicola [28], who, in the case m = µ proved that while, for m µ, showed that the remainder term has the form O(λ n min{m,µ} −ε ) for a suitable ε > 0. A further improvement of these results in the case m = µ has recently appeared in U. Battisti and S. Coriasco [1], where it has been shown that, for a suitable ε > 0, Even the constant C 2 0 has an invariant meaning on M, and both C 1 0 and C 2 0 are explicitly computed in terms of trace operators defined on L m,m cl (M). In this paper the remainder estimates in the case m µ are further improved.
More precisely, we first consider the power Q = P 1 max{m,µ} of P (see L. Maniccia, E. Schrohe, J. Seiler [25] for the properties of powers of SG-classical operators). Then, by studying the asymptotic behaviour in λ of the trace of the operator ψ λ (−Q), ψ λ (t) = ψ(t)e −itλ , ψ ∈ C ∞ 0 (R), defined via a Spectral Theorem and approximated in terms of Fourier Integral Operators, we prove the following Then, the following Weyl formulae hold for λ → +∞: The order of the remainder is then determined by the ratio of m and µ and the dimension of M, since In particular, when max{m,µ} min{m,µ} ≥ 2, the remainder is always O(λ n max{m,µ} ). Examples include operators of Schrödinger type on M, that is P = −∆ g + V, ∆ g the Laplace-Beltrami operator in M associated with a suitable metric g, V a smooth potential that, in the local coordinates x ∈ U N ⊆ R n on the cylindrical end growths as x µ , with an appropriate µ > 0 related to g. Such examples will be discussed in detail, together with the sharpness of the results in Theorem 1.1, in the forthcoming paper [4], see also [3].
The key point in the proof of Theorem 1.1 is the study of the asymptotic behaviour for λ → +∞ of integrals of the form where a and ϕ satisfy certain growth conditions in x and ξ (see Section 3 for more details). The integrals I(λ) represent in fact the local expressions of the trace of ψ λ (−Q), obtained through the so-called "geometric optic method", specialised to the SG situation, see e.g. S. Coriasco [7,8], S. Coriasco and L. Rodino [11]. To treat the integrals I(λ) we proceed similarly to A. Grigis and J. Sjöstrand [14], B. Helffer and D. Robert [17], see also H. Tamura [34]. The paper is organised as follows. Section 2 is devoted to recall the definition of SG-classical operators on a manifold with ends M. In Section 3 we show that the asymptotic behaviour of N P (λ), λ → +∞, for a positive self-adjoint operator P ∈ L m,µ cl (M), m, µ > 0, is related to the asymptotic behaviour of oscillatory integrals of the form I(λ). In Section 4 we conclude the proof of Theorem 1.1, investigating the behaviour of I(λ) for λ → +∞. Finally, some technical details are collected in the Appendix.

SG-classical operators on manifolds with ends
From now on, we will be concerned with the subclass of SG-operators given by those elements P ∈ L m,µ (R n ), (m, µ) ∈ R 2 , which are SG-classical, that is, P = Op p with p ∈ S m,µ cl (R n ) ⊂ S m,µ (R n ). We begin recalling the basic definitions and results (see, e.g., [13,25] for additional details and proofs).
. . , positively homogeneous functions of order µ − k with respect to the variable x, smooth with respect to the variable ξ, such that, for a 0-excision function ω,

Definition 2.2.
A symbol p(x, ξ) is SG-classical, and we write p ∈ S m,µ Moreover, the composition of two SG-classical operators is still classical. For P = Op p ∈ L m,µ cl the triple σ(P) = (σ ψ (P), σ e (P), σ ψe (P)) = (p m,· , p ·,µ , p m,µ ) = (p ψ , p e , p ψe ) is called the principal symbol of P. The three components are also called the ψ-, e-and ψe-principal symbol, respectively. This definition keeps the usual multiplicative behaviour, that is, for any R ∈ L r,ρ cl , S ∈ L s,σ cl , (r, ρ), (s, σ) ∈ R 2 , σ(RS) = σ(S) σ(T), with componentwise product in the right-hand side. We also set As a consequence, denoting by {λ j } the sequence of eigenvalues of P, ordered such that j ≤ k ⇒ λ j ≤ λ k , with each eigenvalue repeated accordingly to its multiplicity, the counting function N P (λ) = λ j ≤λ 1 is well-defined for a SG-classical elliptic selfadjoint operator P, see, e.g., [1,3,4,28]. We now introduce the class of noncompact manifolds with which we will deal: the symbol ∐ C means that we are gluing M and C , through the identification of represents an equivalence class in the set of functions for all ρ ≥ max{δ f , δ g } and γ ∈ S n−1 .
We use the following notation: x |x| ; • f π = f • π : U δ f → C is a parametrisation of the end. Let us notice that, setting F = g −1 π • f π , the equivalence condition (2.1) implies We also denote the restriction of f π mapping U δ f ontoĊ = C \ X byḟ π .
is a finite atlas for M and (Ω N , ψ N ) = (Ċ ,ḟ −1 π ), then M, with the atlas A , is a SG-manifold (see [33]): an atlas A of such kind is called admissible. From now on, we restrict the choice of atlases on M to the class of admissible ones. We introduce the following spaces, endowed with their natural topologies: S ′ (M) denotes the dual space of S (M). Note that, since U δ f is conical, the definition of homogeneous and classical symbol on U δ f makes sense. Moreover, the elements of the asymptotic expansions of the classical symbols can be extended by homogeneity to smooth functions on R n \ {0}, which will be denoted by the same symbols. It is a fact that, given an admissible there exists a partition of unity {θ i } and a set of smooth functions {χ i } which are compatible with the SG-structure of M, that is: Moreover, θ N and χ N can be chosen so that θ N •ḟ π and χ N •ḟ π are homogeneous of degree 0 on U δ . We denote by u * the composition of u : It is now possible to give the definition of SG-pseudodifferential operator on M: Definition 2.7. Let M be a manifold with a cylindrical end. A linear operator P : 3) K P , the Schwartz kernel of P, is such that The most important local symbol of P is p N . Our definition of SG-classical operator on M differs slightly from the one in [24]: The notions of ellipticity can be extended to operators on M as well: Definition 2.9. Let P ∈ L m,µ cl (M) and let us fix an exit map f π . We can define local objects Then, with any P ∈ L m,µ cl (M), it is associated an invariantly defined principal symbol in three components σ(P) = (p ψ , p e , p ψe ). Finally, through local symbols given by

Spectral asymptotics for SG-classical elliptic self-adjoint operators on manifolds with ends
In this section we illustrate the procedure to prove Theorem 1.1, similarly to [14], [16], [34]. The result will follow from the Trace formula (3.6), (3.7), the asymptotic behaviour (3.8) and the Tauberian Theorem 3.6. The remaining technical points, in particular the proof of the asymptotic behaviour of the integrals appearing in From now on, when we write P ∈ EL m,µ cl (M) we always mean its unique closed extension, defined in Proposition 3.1. As standard, we denote by ̺(P) the resolvent set of P, i.e., the set of all λ ∈ C such that λI − P maps H m,µ (M) bijectively onto L 2 (M). The spectrum of P is then spec(P) = C \ ̺(P). The next Theorem was proved in [24]. Given a positive selfadjoint operator P ∈ EL m,µ cl (M), m, µ > 0, µ m, we can assume, without loss of generality 1 , 1 ≤ λ 1 ≤ λ 2 . . . Define the counting function N P (λ), λ ∈ R, as 1 Considering, if necessary, P + c in place of P, with c ∈ R a suitably large constant.
Clearly, N P is non-decreasing, continuous from the right and supported in [0, +∞).
If we set Q = P 1 l , l = max{m, µ} (see [25] for the definition of the powers of P), Q turns out to be a SG-classical elliptic selfadjoint operator with σ(Q) = (p We denote by {η j } the sequence of eigenvalues of Q, which satisfy η j = λ 1 l j : we can then, as above, consider N Q (η). It is a fact that N Q (η) = O(η n l ), see [24]. From now on we focus on the case µ > m > 0: the case m > µ > 0 can be treated in a completely similar way, exchanging the role of x and ξ. So we can start from a closed positive selfadjoint operator Q ∈ EL m,1 and the series converges in the L 2 (M) norm (cfr., e.g., [14]). Clearly, for all t ∈ R, U(t) is a unitary operator such that Let us fix ψ ∈ S(R). We can then define the operator ψ(−Q) either by using the formula or by means of the vector-valued integral so the definition makes sense and gives an operator in L(L 2 (M)) with norm bounded by ψ L 1 (R) .
The following Lemma, whose proof can be found in the Appendix, is an analog on M of Proposition 1.10.11 in [16]: Clearly, we then have By the analysis in [7,8], [10], [11] (see also [9]), the above Cauchy Problem 2 See the Appendix for some details concerning the extension to the manifold M of the results on R n proved in [7,8], [10], [11].
with phase and amplitude functions such that , and similar, in Theorem 3.4 and in the sequel, also mean that the seminorms of the involved elements in the corresponding spaces (induced, in the mentioned cases, by (1.1)), are uniformly bounded with respect to t ∈ (−T, T).
where the local coordinates in the right hand side depend on k and, to simplify the notation, we have omitted the corresponding coordinate maps. By the choices of ψ, θ k and χ k we obtain . 3 Trivially, for j = 1, . . . , N − 1, q k and a k can be considered SG-classical, since, in those cases, they actually have order −∞ with respect to x, by the fact that q k (x, ξ) vanishes for x outside a compact set.
Then In view of Theorem 3.6 and Remark 3.7, to complete the proof of Theorem 1.1 we need to show that (3.8) holds. To this aim, as explained above, this Section will be devoted to studying the asymptotic behaviour for |λ| → +∞ of (4.1) Since q −1 (x, ξ) ∈ O( x −1 ξ −m ) for |x| + |ξ| ≥ R > 0, it is not restrictive to assume that this estimate holds 4 on the whole phase space, so that, for a certain constant A > 1, For two functions f, g, defined on a common subset X of R d 1 and depending on parameters y ∈ Y ⊆ R d 2 , we will write f ≺ g or f (x, y) ≺ g(x, y) to mean that there exists a suitable constant c > 0 such that | f (x, y)| ≤ c|g(x, y)| for all (x, y) ∈ X × Y. The notation f ∼ g or f (x, y) ∼ g(x, y) means that both f ≺ g and g ≺ f hold.
Remark 4.1. The ellipticity of q yields, for λ < 0, From now on any asimptotic estimate is to be meant for λ → +∞.
We will make use of a partition of unity on the phase space: the supports of its elements will depend on suitably large positive constants k 1 , k 2 > 1. We also assume, as it is possible, λ ≥ λ 0 , again with an appropriate λ 0 >> 1. As we will see below, the values of k 1 , k 2 and λ 0 depend only on q and its associated seminorms.

Proposition 4.2. Let H 1 be any function in C
Proof. Write , and the assertion follows integrating by parts with respect to t in the first integral of (4.4).

Remark 4.3.
We actually choose k 1 > 4AC > 2AC, since this will be needed in the proof of Proposition 4.7 below, see also subsection A.3 in the Appendix.
To estimate I 1 (λ), we will apply the Stationary Phase Theorem. We begin by rewriting the integral I 1 (λ), using the fact that ϕ is solution of the eikonal equation associated with q and that q is a classical SG-symbol. Note that then In view of the Taylor expansion of ϕ at t = 0, recalling the property q(x, ξ) = ω(x)q e (x, ξ) + S m,0 (x, ξ), ω a fixed 0-excision function, we have, for some 0 < δ 1 < 1, where the subscript e denotes the x-homogeneous (exit) principal parts of the involved symbols, which are all SG-classical and real-valued, see [10].
Let us now consider I 2 (λ). We follow a procedure close to that used in the proof of Theorem 7.7.6 of [21]. However, since here we lack the compactness of the support of the amplitude with respect to x, we need explicit estimates to show that all the involved integrals are convergent, so we give below the argument in full detail.

Remark 4.5. Incidentally, we observe that a rough estimate of
An even less precise result would be the bound λ n m , using the convergence of the integral with respect to x in the whole R n , given by − n m + n < 0.
The next Lemma is immediate, and we omit the proof:  Explicitely,

The main result of this Section is
We will prove Proposition 4.7 through various intermediate steps. First of all, arguing as in the proof of (4.5), exchanging the role of x and ξ, we note that, for all x ∈ R n , ς ∈ S n−1 , where we have used Lemma 4.6. By the symbolic calculus, remembering that λζ ≥ k m 2 > 1 on supp U 2 , we can rewrite the expressions above as We now prove that, modulo an O(|λ| −∞ ) term, we can consider an amplitude such that, on its support, the ration ζ/ζ 0 is very close to 1. To this aim, take H 3 ∈ C ∞ 0 (R) such that 0 ≤ H 3 (υ) ≤ 1, H 3 (υ) = 1 for |υ| ≤ 3 2 ε and H 3 (υ) = 0 for |υ| ≥ 2ε, with an arbitrarily fixed, small enough ε ∈ 0, 1 2 , and set Proof. Since 0 < m < 1, in view of (1.1), (4.6), and (4.8), we can choose k 2 > 1 so large that, for an arbitrarily fixed ε ∈ 0, 1 2 , for any λ ≥ λ 0 , ζ ∈ (0, +∞) satisfying uniformly with respect to (X, Y) ∈ S X ×S Y ⊇ supp U 2 (.; λ). Then, F 2 is non-stationary 1 m ), the assertion follows by repeated integrations by parts with respect to t, using the operator and recalling Remark 4.5.

Proposition 4.9.
With the choices of ε, T > 0, k 1 , k 2 , λ 0 > 1 above, we can assume, modulo an O(λ n−1 ) term, that the integral with respect to x in J 2 (λ) is extended to the set {x ∈ R n : x ≤ κλ}, with Proof. Indeed 7 , if κ <κ = 2k 1 k 2 −m , we can split J 2 (λ) into the sum Observing that, on supp U 2 , switching back to the original variables, the first integral in (4.11) can be treated as I 1 (λ), and gives, in view of Proposition 4.4, an O(λ n−1 ) term, as stated.
Since, by the choice of k 2 , |∂ ζ G(ζ; Y; λ)| ≤ k 0 < 1, uniformly with respect to Y ∈ S n−1 × {x ∈ R n : x ≤ κλ}, λ ≥ λ 0 , G has a unique fixed point ζ * 0 = ζ * 0 (Y; λ), smoothly depending on the parameters, see the Appendix for more details. Since (4.13) we can assume that λζ ≥ k m 2 and the choices of the other parameters imply, on supp V 2 , So we have proved that, on supp V 2 , (4.14) By (1.1), (4.12), and ζ * 0 = G(ζ * 0 ; Y; λ), (X, Y) ∈ S X × S Y ⊇ supp V 2 (.; λ), we also find uniformly with respect to λ ≥ λ 0 . The proof is complete. The next Lemma says that the presence in the amplitude of factors which vanish at X = X * 0 implies the gain of negative powers of λ: Lemma 4.12. Assume α ∈ Z 2 + , |α| > 0, W is smooth, W k (X, Y; λ) ≺ x k , k ∈ Z + , and has a SG-behaviour as the factors appearing in the expression of V 2 . Then where W has the same SG-behaviour, support and x-order 8 of V 2 . 8 Including the powers of ζ.
Proof. By arguments similar to those used in the proof of Proposition 4.8, on supp W Assume that the first condition in (4.15) holds. Under the hypotheses, if α 1 > 0, we can first insert e iλF 2 (X,Y;λ) = L α 1 ζ e iλF 2 (X,Y;λ) in the left hand side of (4.16), where , and integrate by parts α 1 times. Similarly, if α 2 > 0, we , and integrate by parts α 2 times. The assertion then follows, remembering that ζ-derivatives of W produce either an additional ζ −1 factor or a lowering of the exponent of ζ − ζ * 0 , and that ζ, ζ * 0 ∼ x −1 on supp W. The proof in the case that the second condition in (4.15) holds is the same, using first L ζ and then L t .
Proof of Proposition 4.7. Define, τ ∈ (0, λ −1 0 ], and consider the Taylor expansion of J τ (s) of order 2N − 1, N > 1, so that Remark 4.5 and Lemma 4.12 imply that |J 1]: indeed, it is easy to see, by direct computation, that G can be bounded by linear combinations of expressions of the form 3 , with W k , k ∈ Z + , having the required properties. Then, the bound of G 2N will always contain a term of the type t 3N W 3N (X, Y; λ)(ζ − ζ * 0 (Y; λ)) 3N , which corresponds to the (minimun) value |α| = 3N in (4.15).
Each term J (k) τ (0), k = 0, . . . , 2N − 1, has the quadratic phase function Q, which of course also satisfies ∂ ζ Q(X, Y; τ −1 ) ≻ x |t|, ∂ t Q(X, Y; τ −1 ) ≻ x |ζ − ζ * 0 (Y; τ −1 )|. Then, denoting by Γ the Taylor expansion of G at X * 0 of order 3N, we observe that G k − Γ k can be bounded by polynomial expressions in X − X * 0 of the kind appearing in the right hand side of (4.15), with |α| = N + k (cfr. the proof of Theorem 7.7.5 in [21]). Setting We now apply the Stationary Phase Method to T k τ and prove that which is a consequence of (4.18) with M evaluated with τ −1 in place of λ. Recalling (4.14), it follows that the inverse matrix M −1 satisfies, on supp V 2 , in view of the ellipticity of the involved symbols. Then, the operators L j,k,Y,τ , j, k ∈ Z + , do not increase the x-order of the resulting function with respect to that of their arguments, (iλΓ) k V 2 , which is the same of V 2 , uniformly with respect to τ. The proof of (4.18) then follows by Theorem 7.6.1, the proof of Lemma 7.7.3 and formula (7.6.7) in [21], see also [17,18]. Indeed, by the mentioned results, for any j ∈ Z + , τ ∈ (0, λ −1 0 ]. We can then integrate J τ (1) and its asymptotic expansions with respect to Y ∈ S n−1 × {x ∈ R n : x ≤ κλ} and find Let us now focus on the leading coefficient, given by 9 To confirm this, first note that ζ * 0 (Y; λ) → ζ 0 (Y), λ → +∞, for any (Y; λ) belonging to the support of the integrand, see the Appendix. Moreover, the integrand is uniformly bounded by the summable function x − n m , and its support is included in the set S. Then, recalling (4.14) and setting H = |ζ 2 0 det(M)| − 1 2 , The first integral can be estimated as follows. Since by the properties of ζ * 0 (see the Appendix) we find 9 Remember that ψ(0) = 1 and a(0, x, ξ) = 1, for all x, ξ ∈ R n . For ζ = ζ * 0 (Y; λ), the factors H 1 , H 2 , and H 3 are identically equal to 1, see the Appendix. since S −1,0 (x, (λζ * 0 ) 1 m ) << 1. By (4.14), we similarly have   − 1 and n, as claimed. The proof for µ < m is the same, by exchanging step by step the role of x and ξ. 10 We observe that the multiplicative constants appearing in the terms O(λ n−1 ) in the expansions of I 1 (λ) and I 2 (λ) can be estimated by products of expressions in A, C, k 1 , ε, times the same power κ n− n−1 m .
For the sake of completeness, here we illustrate some details of the proof of Theorem 1.1, which we skipped in the previous Sections. They concern, in particular, formula (3.7), which expresses the relation between j ψ(λ − η j ) and the oscillatory integrals examined in Section 4. We mainly focus on the aspects which are specific for the manifolds with ends. We also show more precisely how the constants k 1 , k 2 , λ are involved in the solution of equation (4.12) via the Fixed Point Theorem, completing the proof of Proposition 4.10.
A.1. Solution of Cauchy problems and SG Fourier Integral Operators. Using the so-called "geometric optics method", specialised to che pseudodifferential calculus we use (see [7,8,9,10,11] and [29]), the Cauchy Problem and First of all, we recall that the partition of unity {θ k } and the family of functions {χ k } of Definition 2.7 can be chosen so that (θ k ) * and (χ k ) * are SG-symbols of order (0, 0) on U k , extendable to symbols of the same class defined on R n (see [31]).
(1) The complete symbol of Q depends, in general, on the choice of the admissible atlas, of {θ k } and of {χ k }. Anyway, if {q k } is another complete symbol of Q, κ(x)(q k (x, ξ) −q k (x, ξ)) ∈ S(ϕ k (Ω k ∩ Ω k )) for an admissible cut-off function κ supported in ϕ k (Ω k ∩ Ω k ).
(2) The solution of (3.3) in the SG-classical case and the properties of ϕ k and a k in (3.5) were investigated in [10] (see also [29], Section 4). In particular, it turns out that ϕ k ∈ C ∞ ((−T k , T k ), S 1,1 cl ), T k > 0. According to [8], page 101, for every SG phase functions ϕ of the type involved in the definition of V(t) we also have, for all x ∈ R n : with a constant C > 0 not depending on t, x, ξ. The function Φ t,ξ (x) := ∇ ξ ϕ(t, x, ξ) turns out to be a (SG-)diffeomorphism, smoothly depending on the parameters t and ξ (see [7]).
Before proving Theorem 3.4, we state a technical Lemma, whose proof is immediate and henceforth omitted.
We remark that, since a manifold with ends is, in particular, a SG-manifold, the charts (Ω k , ψ k ) and the functions {θ k }, {χ k }, can be chosen such that 11 • for a fixed δ > 0, each coordinate open set U k = ψ k (Ω k ), k = 1, . . . , N, contains an open subset W k such that • the supports of θ k and χ k , k = 1, . . . , N, satisfies hypotheses as the supports of θ and χ in Lemma A.15 (see, e.g., Section 3 of [31] for the construction of functions with the required properties).
Proof of Theorem 3.4. We will write R ≡ S when R − S ∈ L −∞,−∞ (M) and χ k ⊳ χ k when the functions χ k , χ k are smooth, non-negative, supported in Ω k , satisfy χ k χ k = χ k and (χ k ) * , ( χ k ) * are SG-symbols of order (0, 0) on U k = ψ k (Ω k ). Obviously, R ∈ L −∞,−∞ (M) implies R V ∈ C ∞ ((−T, T), L −∞,−∞ (M)). To simplify notation, in the computations below we will not distinguish between the functions χ k , θ k , etc., and their local representations. V(t) obviously satisfies (A.2). To prove (A.1), choose functions ζ k , υ k supported in Ω k such that θ k ⊳ ζ k ⊳ χ k ⊳ υ k . Then Q ≡ N k=1 θ k Q k χ k and, for all k = 1, . . . , N, Qχ k ≡ υ k Q k χ k (see [6], Section 4.4; cfr. also [23]), so that That the first term in the sum (A.3) is smoothing comes from the SG symbolic calculus in R n and the observations above, since sym ([Q k , χ k ] ζ k ) ∼ 0. The same property holds for each k in the second term, provided t ∈ I T k , T k > 0 small enough. In fact, by Theorems 7 and 8 of [7], (1 − ζ k ) V k (t) θ k is a SG FIO with the same phase function ϕ k and amplitude w k such that with suitable SG-symbols b jα defined in terms of ϕ k and a k . By Remark A.14 and Lemma A.15, w k ∼ 0 for |t| small enough. The proof that V(t) satisfies (A.1) is completed once we set T = min{T 1 , . . . , T N }. The last part of the Theorem can be proved as in [14], Proposition 12.3, since, setting W(t) := U(−t) V(t), it is easy to see D t W(t) ≡ 0, so that W(0) = I ⇒ W(t) ≡ I ⇒ V(t) ≡ U(t), with smooth dependence on t, as claimed. 11 Actually, this is relevant only for k = N. Then, by e k ∈ S(M) and the fact that (θ r ) * = θ r • ψ −1 r is supported and at most of polynomial growth in U r , it turns out that we can extend (θ r e k ) * and (θ s e k ) * to elements of S(R n ). By an argument similar to the proof of Proposition 1.10.11 in [16] (or by direct estimates of the involved seminorms, as in [14]), (k rs J ) * → (k rs ) * in S(R n × R n ) when J → +∞, with (k rs ) * kernel of (θ r ψ(−Q) θ s ) * . This proves that ψ(−Q) = N r,s=1 θ r ψ(−Q) θ s is an operator with kernel K ψ (x, y) = N r,s=1 k rs (x, y) ∈ S(M × M). The proof of Theorem 3.6 is essentially the one in [14], while the proof of Lemma A.16 comes from [16]: we include both of them here for convenience of the reader. Now, observe that

A.2. Trace formula and asymptotics for
where H(τ) is the Heaviside function. Bringing the series under the integral sign, we can write (A.5) We can then conclude that R(λ) = O(λ n m −1 ), λ ≥ 1, since ψ ∈ S, and this, together with (A.4) and (A.5), completes the proof.