Journées Équations aux dérivées partielles XXX, ?? mai–?? juin ???? GDR 1151 (CNRS)

On the distribution of resonances for some asymptotically hyperbolic manifolds

R. G. Froese and P. D. Hislop

Abstract We establish a sharp upper bound for the resonance counting function for a class of asymptotically hyperbolic manifolds in arbitrary dimension, including convex, cocompact hyperbolic manifolds in two dimensions. The proof is based on the construction of a suitable paramatrix for the absolute S-matrix that is unitary for real values of the energy. This paramatrix is the S-matrix for a model Laplacian corresponding to a separable metric near infinity. The proof of the upper bound on the resonance counting function requires estimates on the growth of the relative scattering phase, and singular values of a family of integral operators.

1. Introduction

We sketch an outline of the proof of a sharp upper bound on the counting function for resonances (or scattering poles) for the Laplace operator on asymptotically hyperbolic manifolds of dimension greater than or equal to two. This class of manifolds includes infinite volume, convex, co-compact hyperbolic manifolds in dimension two (cf. [8, 14]). Our main technical results are two: A bound on the singular values of a regularization of the scattering operator, and an estimate on the growth of a relative scattering phase. The key observation is that a parametrix for the absolute S-matrix can be constructed which is unitary on the critical line. This construction uses the fact that the manifold admits a product structure and a boundary-defining function in a neighborhood of infinity. Our main theorem is the following.

Theorem 1.1 Let X be an n-dimensional asymptotically hyperbolic manifold satisfying Hypothesis 1 given below. Let ν(r) denote the number of scattering poles contained in the semicircle of radius r > 0 centered at the origin. Then, for each n ≥ 2, there exists a constant 0 < C_{n} < ∞, such that we have the upper bound

ν(r) ≤ C_{n}(r^{n} + 1).

Research partially supported by NSERC and NSF grant 9707049.

It is know from explicitly computable examples of hyperbolic manifolds that this upper bound is sharp. There have been many results on the distribution of scattering poles for hyperbolic manifolds. Guillopé and Zworski [9] proved an upper bound of the form ν(r) ≤ C(r^{1+n} + 1), for complete manifolds of constant negative curvature at infinity in dimension n ≥ 3. Using the Selberg zeta function associated with the convex, co-compact hyperbolic manifold, Patterson and Perry [18] proved the sharp upper bound ν(r) ≤ C_{n}(r^{n} + 1), for arbitrary even n ≥ 2. Using some results of Bunke and Olbrich [1], Patterson and Perry were able to extend their results for odd n odd. Quite recently, Perry [22] proved a Poisson formula for convex, cocompact hyperbolic manifolds used this to prove lower bounds of the same type on the resonance counting function. The methods of the Selberg zeta function appear to be very rigid. Our work is an attempt to use the S-matrix directly in order to prove these bounds. This allows us to include perturbations of the metric having rapid decay, as will be described below. Unfortunately, our present work cannot be applied directly to convex, co-compact hyperbolic manifolds of dimension n ≥ 3. Our methods do prove, however, the following result. For any two constants ϵ_{1} > 0 and ϵ_{2} > 0, let ν_{ϵ_{1},ϵ_{2}}(r) be the number of scattering resonances in the sector π + ϵ_{1} < argz < −ϵ_{2} in the lower-half complex plane. Then, for any asymptotically hyperbolic manifold, there exists a constant C_{n,ϵ_{1},ϵ_{2}} > 0, so that ν_{ϵ_{1},ϵ_{2}}(r) ≤ C_{n,ϵ_{1},ϵ_{2}}(r^{n} + 1). We do not presently have enough control on the parametrix used in the proof of this estimate in order to extend the results to the full half-disk.

The case of dimension n = 2 is special. Guillopé and Zworski [10, 11] proved upper and lower bounds on the resonance counting function ν(r), and very precise asymptotics on the relative scattering phase. This analysis is facilitated by the fact that in two dimensions, the metric near an end at infinity is an exact product metric. In [11], the Guillopé and Zworski prove that there exists finite, nonzero constants c and C such that cr^{2} ≤ ν(r) ≤ Cr^{2}. Refining this analysis, Zworski [26] proved that the density of scattering poles in a subconic neighborhood near the real axis is controlled by the Hausdorff dimension of the limit set of the group of hyperbolic isometries Γ.

Upper bounds and asymptotics for scattering resonances have been the topics of many recent papers. We refer to the review article of Zworski [27] for an account of much of this work.

2. Preliminaries and Outline of the Proof

It is well-known that the absolute S-matrix S(k) for an asymptotically hyperbolic manifold X is a pseudodifferential operator of complex order −2ik, k ∈ C [13]. The meromorphic continuation of the S-matrix in the hyperbolic case has been proven in several works, including [1, 15, 16, 19, 6]. An interesting feature of the absolute S-matrix for asymptotically hyperbolic manifolds is that it is not of the form S(k) = 1 + T (k), for some operator T (k) in the trace class, as in the case of Schrödinger operators with potentials that decay as |x|^{−(n+ϵ)} near infinity. We can, however, construct a paramatrix S_{0}(k) so that S_{0}(k)^{−1}S(k) = 1 + T (k), with T (k) trace class. Furthermore, the operator S_{0}(k) extends meromorphically to the entire complex plane with computable poles. Such an operator is constructed in [6] (and, implicitly, in [1]) for a general class of geometrically finite, infinite volume hyperbolic manifolds, including cusps. The difficulty with that paramatrix is that S_{0}(k) is not unitary on the ℑk = 0 axis. This parametrix, however, can be used to obtain the estimates on the resonance counting function for a sector mentioned in the introduction. In this note, we sketch the construction of a paramatrix for S(k) that is unitary on the ℑk = 0 axis, and satisfies the same conditions mentioned above. Once this paramatrix is constructed, the proof of the upper bound on the resonance counting function will follow from Lemma 2.1, as explained below. Because the Smatrix satisfies the functional relation S(k)S(−k) = 0, for k ∈ I C, the poles of S(k) for ℑk < 0 correspond to the zeros of S(k) for ℑk > 0. It will be more convenient to count the number of zeros of S(k) for ℑk > 0.

With reference to the relation S_{0}(k)^{−1}S(k) = 1 + T (k), let T (k) be an operatorvalued function analytic on the half-space ℑk > 0, continuous onto the real axis, and in the trace class for ℑk ≥ 0. We further assume that (1 + T (k)) is unitary for ℑk = 0, and that T (0) = 0. Let F (k) ≡ det (1 + T (k)). This function is analytic for ℑk > 0 and continuous onto ℑk = 0. For any r > 0, we denote by ν(r) the number of zeros z_{i} of F (k) satisfying ℑz_{i} > 0 and |z_{i}| < r, including multiplicity. We define an averaged quantity N(r) by

N(r) = ∫_{0}^{r} \frac{ν(t)}{t} dt.  (1)

Since |F (k)| = 1, for ℑk = 0, we define the scattering phase s(k), for k ∈ IR, by s(k) ≡ (−i/2π) log F (k). We will see that this is an analogue of the scattering phase in the usual Schrödinger operator setting.

Lemma 2.1 Let F (k) = det (1 + T (k)) be as defined above. Then, we have

N(r) ≤ \frac{1}{2π} ∫_{0}^{r} t^{−1}|s(t) − s(−t)|dt + \frac{1}{2π} ∫_{0}^{π} log |F (re^{iθ})| dθ.  (2)

The proof of this simple lemma is given in [3]. Let us note that we can recover an upper bound on ν(r) from one on N(r) by using the positivity and monotonicity of ν(r). We have

rν(r) ≤ ∫_{r}^{2r} ν(t)dt ≤ 2r ∫_{r}^{2r} \frac{ν(t)}{t}dt,  (3)

from which it follows that ν(r) ≤ 2(N(2r) − N(r)). Consequently, it suffices to obtain an upper bound on N(r). This requires an upper bound on the scattering phase s(t) of the form |t|^{n}, and an estimate on the growth of |F (re^{iθ})|, as r → ∞, of the form e^{C(r^{n}+1)}.

In the next section, we review the structure of the S-matrix associated to X, and show how to construct the S-matrix S_{0}(k) using the geometry of the manifold X. We will denote by P the Laplace-Beltrami operator on X with an asymptotically hyperbolic metric g. The spectral theory of P is well-understood [6]. The approximate S-matrix S_{0}(k) is associated with an approximate operator P_{0} defined on the end of X obtained by removing a compact set U. The scattering phase s(t) is the usual scattering phase associated with the pair (P_{0}, P). In section 4, we will estimate the relative scattering phase defined by s(k) = (−i/2π) log {det S_{0}(k)^{−1}S(k)}, for k ∈ IR. This estimate is obtained by following the method of Robert [23, 24] (see also Christiansen [2]). The method of Robert requires the construction of a pseudodifferential operator A_{0} which satisfies [P_{0}, A_{0}] = P_{0}, near infinity, up to a smoothing operator. The operator A_{0} has a symbol a that is a global escape function for the symbol of the approximate operator P_{0}. The idea behind the construction of this symbol is given in section 5. In section 6, we estimate the singular values of certain integral operators. These estimates are the key to estimating F (re^{iθ}). The estimate on the growth of the scattering phase and on the singular values are the two main steps in the proof of Theorem 1.1. The details and proofs will be given in [4].

3. The Geometry of Asymptotically Hyperbolic Manifolds

We introduce the class of manifolds for which our methods work. The key assumption is that the metric can be well-approximated by a product metric near infinity. This model metric will provide a model operator P_{0} for which we can apply the usual methods of scattering theory to the pair (P_{0}, P).

3.1. Description of the Metric

Let X be a precompact Riemannian manifold with metric g. We assume that the boundary ∂X of X is a nonempty, smooth, compact manifold, and such that the closure \overline{X} is compact. A boundary-defining function for \overline{X} is a real-valued function x defined on a neighborhood X\U, for a compact set U ⊂ X, of the boundary ∂X of X. A boundary-defining function satisfies ∂X = \{p ∈ \overline{X} | x(p) = 0\}, and dx|∂X ≠ 0.

A manifold X of the above type is called asymptotically hyperbolic if for some compact U ⊂ X, the neighborhood of ∂X given by X\U, admits a product decomposition X ∼ ∂X × (0, ϵ), for some ϵ > 0, and that there exists a boundary-defining function x : X\U → IR, satisfying ∥dx∥ |∂X = 1, so that metric g on X\U has the form

\frac{dx^{2} + h(x, dx, y, dy)}{x^{2}}.  (4)

Furthermore, the function h has the property that h|∂X is a smooth metric on ∂X. It was observed in the article of Mazzeo and Melrose [16] that for metrics of this form, the sectional curvature computed along a smooth geodesic near ∂X, and running to the boundary point p ∈ ∂X, approaches −1. We will write (x, y) for coordinates on X\U near ∂X, with y denoting local coordinates on the compact manifold ∂X.

It is well-known, and proved explicitly in [13], that coordinates can chosen so that h(x, y, dx, dy) is independent of dx. We will assume that this has been done. The form of the metric g near ∂X suggests a model for the end of X near ∂X given by X\U. There, we define a model metric g_{0} as follows. Let h_{0}(y, dy) ≡ h(0, y, dy) be the metric on ∂X associated with h. We define the metric g_{0} on X\U by

g_{0} = \frac{dx^{2} + h_{0}(y, dy)}{x^{2}}.  (5)

The model space for the manifold (X, g) is the manifold (X\U, g_{0}). Note that the model metric is separable, and hence we will be able to do computations involving g_{0} explicitly. We need to make one crucial assumption on the behavior of h as x → 0.

Hypothesis 1. In any local coordinate chart for ∂X, the difference

h(x, y, dy) − h_{0}(y, dy) = O(x^{n−1+ϵ}),  (6)

for some ϵ > 0, and uniformly with respect to y ∈ ∂X.

This hypothesis is satisfied, for example, by a class of perturbations of convex, cocompact hyperbolic manifolds in two dimensions.

3.2. The Laplace-Beltrami and Model Operators

The Laplace-Beltrami operator P on L^{2}(X, dV_{g}), where dV_{g} is the volume form associated with the metric g given in (4), is constructed from the asymptotically hyperbolic metric g on X (we can actually allow more general P). In the coordinates (x, y) near ∂X, the operator P has the form

P = −x^{n}(det h)^{−1/2} \frac{∂}{∂x}x^{2−n}(det h)^{1/2} \frac{∂}{∂x} + x^{2}Δ_{h}(x),  (7)

where we write Δ_{h}(x) for the nonnegative second-order elliptic operator in ∂X, depending parametrically on x, and given by

Δ_{h}(x) ≡ −(det h)^{−1/2} \frac{∂}{∂y_{i}} (det h)^{1/2}h^{ij} \frac{∂}{∂y_{j}} .  (8)

Here, the matrix h^{ij} is the inverse of (n − 1) × (n − 1) matrix h appearing in (12). The operator P is self-adjoint on L^{2}(X, dV_{g}). We note that this Hilbert space admits the decomposition, relative to a boundary-defining function x and product decomposition of X\U, as

L^{2}(X, dV_{g}) = L^{2}(U, dV_{g}) ⊕ L^{2}(∂X × [0, ϵ], x^{−n}\sqrt{det(h)} dx d^{n−1}y).  (9)

We next consider the model manifold (X\U, g_{0}). In the coordinates (x, y), the (formal) model operator P_{0} has the form

P_{0} = −x^{n} \frac{∂}{∂x}x^{2−n} \frac{∂}{∂x} + x^{2}Δ_{h_{0}},  (10)

where we denote by Δ_{h_{0}} = Δ_{h}(0) the nonnegative Laplace-Beltrami operator on ∂X with the induced metric h_{0}(y, dy) as in (5). In order to obtain a self-adjoint operator P_{0} on the Hilbert space L^{2}(∂X × [0, ϵ], dV_{g_{0}}), where dV_{g_{0}} is the volume form for the metric g_{0} in (13), we impose Dirichlet boundary conditions on the surface x = ϵ.

It will be convenient to map X\U = ∂X × [0, ϵ] onto ∂X × [− log ϵ, ∞) through a change to geodesic coordinates by the transformation r ≡ − log x. Near X\U, the metric g becomes

g = dr^{2} + e^{2r}h(e^{−r}, y, dy) and dV_{g} = e^{r(n−1)}\sqrt{det(h)(x, y)} drd^{n−1}y. (11)

In these coordinates, the model metric g_{0} becomes

g_{0} = dr^{2} + e^{2r}h_{0}(y, dy) and dV_{g_{0}} = e^{r(n−1)}\sqrt{det(h_{0})(y)} drd^{n−1}y.  (12)

Let us rescale the r variable so that ϵ = e^{−1}. This is effected by a change of the boundary-defining function to u = (ϵe)^{−1}x. One can easily check that the metrics g and g_{0} have the same forms as in (4) and (5) with a new function h̃. The metric induced on ∂X by taking h̃ at u = 0 is conformally equivalent to the metric h_{0}.

It is convenient for calculations involving only the model operator P_{0} to make one further unitary change of Hilbert space to remove the factor e^{(n−1)r} from the volume forms appearing in (11) and (12). We change to the Hilbert space L^{2}(∂X × [1, ∞), dr dV_{h_{0}}(y)). The volume form dV_{h_{0}} = \sqrt{det(h_{0})} d^{n−1}y is the one on the compact manifold ∂X associated with the metric h_{0}. After performing the standard unitary transformations, the model Laplacian P_{0} on L^{2}(∂X × [1, ∞), dr dV_{h_{0}}) has the form

P_{0} = −∂_{r}^{2} + e^{−2r}Δ_{h_{0}} + c_{n}^{2},  (13)

where c_{n} = (n − 1)/2, and Δ_{h_{0}} is the operator

Δ_{h_{0}} = −(det h_{0})^{−1/2} \frac{∂}{∂y_{i}} (det h_{0})^{1/2}h_{0}^{ij} \frac{∂}{∂y_{j}} ,  (14)

and Dirichlet boundary conditions are imposed at r = 1. This form is convenient for calculations because the operator P_{0} is separable.

For comparison with the operator P, however, we must make a unitary change of Hilbert space so that the Hilbert space associated with the end X\U can be naturally identified with a subspace of L^{2}(X, dV_{g}) as in (9). For this, it suffices to define the unitary map V as V : L^{2}(X\U, drdV_{h_{0}}) → L^{2}(X\U, drdV_{h}), by V f = { det h_{0}/ det h}^{1/4}f. We define the operator P̃_{0}, unitarily equivalent to P_{0}, by P̃_{0} = V P_{0}V^{−1}. We can now compare the operators P and P̃_{0} acting on functions in L^{2}(X\U, dV_{g}) = L^{2}(∂X × [1, ∞), \sqrt{det(h)} dr d^{n−1}y). Let χ be a smooth function of compact support in X so that (1 − χ) is supported in the end X\U. It follows from Hypothesis 1 and a simple calculation that the coefficients of (P − P̃_{0})(1 − χ) are O(e^{−(n−1+ϵ)r}). We will denote the difference P −P̃_{0} by R_{1}. The estimates mentioned above imply that for any f ∈ S(IR), the operators R_{1}(1 − χ)f(P), and R_{1}f(P_{0})(1 − χ), are trace class on L^{2}(X, dV_{g}).

3.3. The S-Matrix

The S-matrix S(k), for k ∈ IR, is defined through the asymptotics of the solution of a Dirichlet problem at infinity. Let f ∈ C_{0}^{∞}(∂X). Except for possibly a countable set of exceptional points k, there exists a unique solution u of the problem

(Δ − k^{2} − (\frac{n − 1}{2})^{2})u = 0, in X,  (15)

having the asymptotic form

u(x, y) ∼ x^{\frac{n−1}{2} −ik}f_{+}(x, y) + x^{\frac{n−1}{2} +ik}f_{−}(x, y),  (16)

as x → 0, and f_{−}(0, y) = f. The S-matrix is defined as the map

S(k)f = f_{+}(0, y).  (17)

In a recent work of Joshi and Sà Barreto [13], the S-matrix is computed in terms of a boundary-defining function x and metric as in (4). Let Δ_{h_{0}} be the nonnegative Laplace-Beltrami operator on ∂X with the metric h_{0}. They prove that if |h(x, ydy) − h_{0}(y, dy)| = x^{r}f(y), then

S(k) = 2^{2ik} \frac{Γ(ik)}{Γ(−ik)}Δ_{h_{0}}^{−ik} mod OP S^{2ℑk−r}(∂X).  (18)

As for the model manifold with the product metric (5), it is easy to compute (cf. [8, 13]) the S-matrix exactly. It is given by

S_{0}(k) = 2^{2ik} \frac{Γ(ik)}{Γ(−ik)}Δ_{h_{0}}^{−ik} + R_{∞}(k).  (19)

The remainder operator R_{∞} ∈ OP S^{−∞}(∂X) is exactly computable in terms of a ratio of modified Bessel functions. The model Hilbert space L^{2}(∂X×[1, ∞), dr\sqrt{det(h_{0})} d^{n−1}y) admits a direct sum decomposition relative to the spectral resolution of the Laplace-Beltrami operator Δ_{h_{0}}. Let {λ_{j}^{2} ≥ 0} denote the eigenvalues of Δ_{h_{0}}. Let c_{0}(k) denote the common factor c_{0}(k) = 2^{2ik}Γ(ik)Γ(−ik)^{−1}. The operator R_{∞}(k), restricted to the eigenspace of Δ_{h_{0}} corresponding to eigenvalue λ_{j}^{2}, is the operator of multiplication by

R_{∞}(k)_{j} = −c_{0}(k)λ_{j}^{−2ik} (\frac{2 sin(ikπ)}{π}) (\frac{K_{ik}(λ_{j})}{I_{−ik}(λ_{j})}) .  (20)

Notice that S_{0}(k) can also be written as

S_{0}(k) = c_{0}(k)Δ_{h_{0}}^{−ik} (\frac{I_{ik}(\sqrt{Δ_{h_{0}}})}{I_{−ik}(\sqrt{Δ_{h_{0}})}}) .  (21)

¿From this expression, it follows that

S_{0}(k)^{−1} = c_{0}(k)^{−1}Δ_{h_{0}}^{ik} + R_{∞}(−k).  (22)

It will be convenient to introduce another operator R̃_{∞}(k) defined by

R̃_{∞}(k) = c_{0}(k)^{−1}Δ_{h_{0}}^{ik}R_{∞}(k)
= \frac{2 sin(ikπ)}{π} (\frac{K_{ik}(\sqrt{Δ_{h_{0}}})}{I_{ik}(\sqrt{Δ_{h_{0}}})}) .  (23)

In terms of this operator, the inverse of S_{0}(k) can be written as

S_{0}(k)^{−1} = (1 + R̃_{∞}(k))c_{0}(k)^{−1}Δ_{h_{0}}^{ik}.  (24)

Let R_{n}(k) ∈ OP S^{2ℑk−(n−1+ϵ)}(∂X) denote the remainder in the formula (18) for S(k). It follows from (23)–(24) that S_{0}(k)^{−1} is a paramatrix for S(k) and that

S_{0}(k)^{−1}S(k) − 1 = c_{0}(k)^{−1}Δ_{h_{0}}^{ik}R_{n}(k) + R̃_{∞}(k)c_{0}(k)^{−1}Δ_{h_{0}}^{ik}R_{n}(k) + R̃_{∞}(k)
= T_{1}(k) + T_{2}(k) + T_{3}(k).  (25)

This formula, initially valid for ℑk = 0, can be extended to ℑk ≥ 0. The operator T_{3}(k) is a meromorphic, trace-class valued operator for ℑk ≥ 0, with at most a finite number of poles in ℑk > 0, and none on the real k-axis. Following the work of [6], one can compute an explicit formula for R_{n}(k) in terms of certain limits of generalized eigenfunctions for P. ¿From this formula, one proves that T_{1}(k) and T_{2}(k) are trace-class, operator-valued functions on ℑk > 0, and continuous on ℑk = 0. We discuss these terms further in section 6. Finally, the product S_{0}(k)^{−1}S(k) is unitary for ℑk = 0. Consequently, with reference to the discussion at the end of section 2, we study the relative scattering phase defined through log(det(1 + T_{1}(k) + T_{2}(k) + T_{3}(k))), for ℑk = 0. As for the behavior in ℑk > 0, we compute the growth of the determinant through a singular-value estimate. These estimates will yield the upper bound for ν(k), as shown in section 2.

4. Bounds on the Scattering Phase

We prove an upper bound on the scattering phase for a geometrically defined pair of operators (P_{0}, P) introduced in section 3. These operators on the manifold X are associated with the metric g having the form (4), and the model metric g_{0} as in (5), respectively. We define an operator I as the restriction operator I : L^{2}(X, dV_{g}) → L^{2}(X\U, dV_{g}), and I^{∗} is the corresponding extension operator given by the identity map. Given Hypothesis 1, it is well-known in that we have a trace formula,

T r(f(P) − I^{∗}f(P_{0})I) = − ∫ f′(λ)ξ(λ)dλ.  (26)

The function ξ(λ) is the Krein spectral shift function.

Theorem 4.1. Under Hypothesis 1 on the pair (g_{0}, g), the Krein spectral shift function ξ(λ) is the relative scattering phase s(k) for the pair (P_{0}, P), and satisfies the estimate,

|ξ(λ)| ≤ C_{n}(|λ|^{n} + 1),  (27)

for some finite constant C_{n} > 0, independent of λ, and depending on the dimension n.

The identification of the Krein spectral shift function for the pair of Laplace-Beltrami operators (P_{0}, P), described in section 3, with the relative scattering phase for this pair, follows from an application of the Maass-Selberg relation. Here, we sketch the idea behind the proof of the upper bound on the function ξ(λ).

In order to obtain the upper bound in Theorem 4.1, we modify the method of Robert [23] (see also Christiansen [2]) in order to apply it to operators P of the type encountered here. There are two difficulties in treating the pair (P_{0}, P). First, as discussed in section 3, for any smooth function χ with U ⊂ supp χ, we have (P − P_{0})(1 − χ) = R_{1}(1 − χ), and the coefficients of the second-order operator R_{1} decay according to Hypothesis 1. Secondly, we do not have an exact commutator relation like [P_{0}, A_{0}] = P_{0}, as in [23]. Rather, we construct a skew-adjoint conjugate operator A_{0} satisfying

[P_{0}, A_{0}] = P_{0} + R_{N},  (28)

where R_{N} ∈ OP S^{−N}, for arbitarily large N. We will use P_{0} to denote either the operator P_{0}, as in (19), acting on L^{2}(X\U, x^{−n}dx dV_{h_{0}}), with Dirichlet boundary conditions on ∂U, or the operator P̃_{0}, unitarily equivalent to P_{0}, acting on L^{2}(X\U, x^{−n}dx dV_{h}), with Dirichlet boundary conditions on ∂U.

Our goal is to estimate T r{E_{P} (I_{λ}) − I^{∗}E_{P_{0}}(I_{λ})I}, for large λ, where E_{P} (I_{λ}) = χ_{λ}(P), and χ_{λ} is the characteristic function of an interval I_{λ} = [λ_{0}, λ], for some fixed λ_{0} > 0. By standard arguments [23], we can pass from a smooth functon f of compact support, and the operators f(P) and f(P_{0}), to the spectral projectors for P, and P_{0}, respectively, and the interval I_{λ}. Let V be a precompact set with U ⊂ V. Let χ_{U} be a smooth cut-off function with compact support contained in V and χ_{U} | U = 1. For a function f ∈ S(IR^{+}) supported away from 0, we write

T r{f(P) − I^{∗}f(P_{0})I} = T r{χ_{U}(f(P) − I^{∗}f(P_{0})I)}
+T r{(1 − χ_{U})(f(P) − I^{∗}f(P_{0})I)}.  (29)

We will deal with each piece of the trace separately. We call the term containing χ_{U} the interior trace term, and the term with (1 − χ_{U}) the exterior trace term.

It is a standard result that the compactness of the support of χ_{U}, and the ellipticity of P and P_{0} away from ∂X, insure that the operators χ_{U}f(P) and I^{∗}χ_{U}f(P_{0})I are both in the trace class. Well-known asymptotics for the counting function for elliptic operators on compact manifolds allows us to conclude that

|T r{χ_{U}E_{P} (I_{λ}) − I^{∗}χ_{U}E_{P_{0}}(I_{λ})I}| ≤ C|λ|^{n}.  (30)

The exterior trace term is written as

T r{(1 − χ_{U})(f(P) − I^{∗}f(P_{0})I)}
= T r{(1 − χ_{U})(P P^{−1}f(P) − I^{∗}P_{0}P_{0}^{−1}f(P_{0})I}
= T r{P (1 − χ_{U})g(P) − I^{∗}P_{0}(1 − χ_{U})g(P_{0})I}
+T r{[P, χ_{U}]g(P) − I^{∗}[P_{0}, χ_{U}]g(P_{0})I},  (31)

where g(x) = x^{−1}f(x). It follows from the fact that the commutators [χ_{U}, P] and [χ_{U}, P_{0}] have compact support, that each term of the second factor on the right in (31) is trace class. By an argument similar to the one used for the interior term, each term is bounded above by |λ|^{n}. We now concentrate on the leading term,

T r{P (1 − χ_{U})g(P) − I^{∗}P_{0}(1 − χ_{U})g(P_{0})I}.  (32)

Following Robert [23], this term can be written as

T r{P (1 − χ_{U})g(P) − I^{∗}P_{0}(1 − χ_{U})g(P_{0})I}
= T r{A_{0}[χ_{U}, P]P^{−1}f(P) − A_{0}[χ_{U}, P_{0}]P_{0}^{−1}f(P_{0})} + R_{4},  (33)

where the error term R_{4} is given by

R_{4} = T r{R_{1}(1 − χ_{U})P^{−1}f(P))}
+T r{R_{1}[A_{0}, χ_{U}]P^{−1}f(P)}
+T r{[A_{0}, R_{1}(1 − χ_{U})]P^{−1}f(P)}
+T r{R_{N}(1 − χ_{U})(P_{0}^{−1}f(P_{0})I − P^{−1}f(P))}.  (34)

The operator A_{0} is the conjugate operator for P_{0} constructed in the next section.

In order to derive this representation, we use the fact that R_{1}(1 − χ_{U}) = (P − P_{0})(1 − χ_{U}), so we can write a localized commutator relation for P similar to (28). This relation has the form,

[P_{0}, A_{0}](1 − χ_{U}) = [P, A_{0}](1 − χ_{U}) + (P − P_{0})[A_{0}, χ_{U}] + [A_{0}, R_{1}(1 − χ_{U})]. (35)

Simple manipulations lead to the expression

T r{P (1 − χ_{U})g(P) − I^{∗}P_{0}(1 − χ_{U})g(P_{0})I}
= T r{P_{0}(1 − χ_{U})g(P) − I^{∗}P_{0}(1 − χ_{U})g(P_{0})I} + T r{R_{1}(1 − χ_{U})g(P)}
= T r{[P_{0}, A_{0}](1 − χ_{U})g(P) − I^{∗}[P_{0}, A_{0}](1 − χ_{U})g(P_{0})I} + R_{2}
= T r{[P, A_{0}](1 − χ_{U})g(P) − I^{∗}[P_{0}, A_{0}](1 − χ_{U})g(P_{0})I} + R_{4},  (36)

where the remainder term R_{4} is given in (36). As for the first term in (53), we use the trick introduced by Robert [23] in order to rewrite this term as the trace of an operator localized to a compact region,

T r{[P, A_{0}](1 − χ_{U})g(P) − I^{∗}[P_{0}, A_{0}](1 − χ_{U})g(P_{0})I}
= T r{A_{0}[χ_{U}, P]g(P) − A_{0}[χ_{U}, P_{0}]g(P_{0})}.  (37)

The form of the exterior term given above follows from this.

¿From standard trace estimates as used in the estimate for the interior term, we derive the estimate, |T r{A_{0}[P, χ_{U}]g(P) − A_{0}[P_{0}, χ_{U}]g(P_{0})I}| ≤ C_{0}|λ|^{n}. The remainder R_{4} consists of four pieces. The first three terms are controlled by the rapid decay of R_{1} as the boundary coordinate x → 0. Finally, we consider the term T r{I^{∗}R_{N}(1 − χ_{U})(g(P_{0})I − g(P))}. The operator R_{N} has arbitrary large negative order (see the next section). Using the functional calculus, we extract the decay of (P − P_{0}) from (g(P) − g(P_{0})). The operator R_{N}(1 − χ_{U})(P − P_{0}) is trace class, and the trace is bounded above, independent of λ. The proof of Theorem 4.1 now follows from these estimates.

5. Escape Function and Commutator Estimates

We mention how to compute a conjugate operator A_{0} for the operator P_{0}, as in (28), and prove that the remainder R_{N} has the correct properties. We begin by studying the classical system with Hamiltonian h(r, ξ) = ξ^{2} + e^{−2r}p + c_{n}^{2}, where c_{n} = (n−1)/2, and p > 0 is a real parameter. The Hamiltonian function h generates the geodesic flow on the cotangent bundle T^{∗}IR ∼ IR × IR.

Proposition 5.1. Any function a_{0}(r, ξ), with (r, ξ) ∈ IR^{2}, satisfying the relation {h, a_{0}} = h, has the form

a_{0}(r, ξ) = \frac{\sqrt{h}}{4\sqrt{h − c_{n}^{2}}} log (\frac{\sqrt{h − c_{n}^{2}} + ξ}{\sqrt{h − c_{n}^{2}} − ξ} ) + Φ(h),  (38)

for an arbitrary function Φ of h.

Proof. It is easy to check that a_{0}(r, ξ) has the desired property. Note that since h−c_{n}^{2} > ξ^{2}, for p ≠ 0, the function a_{0}(r, ξ) is well-defined. To show that any function a_{0} satisfying {h, a_{0}} = h has this form, we study the dynamical system generated by Hamiltonian h. The equations of motion are

ṙ(t) = {h, r} = 2ξ(t)  (39)
ξ̇(t) = {h, ξ} = 2pe^{−2r(t)}.  (40)

The Hamiltonian is a constant of the motion. Let the initial point at t = 0 be (r_{0}, ξ_{0}), and define h_{0} to be the energy associated with the geodesic. We also define ϕ_{0} ≡ \sqrt{h_{0} − c_{n}^{2}}. Then, the solutions are

r(t) = log(cosh{2tϕ_{0} + C_{1}}) + C_{2},  (41)
ξ(t) = ϕ_{0} tanh{2tϕ_{0} + C_{1})},  (42)

where the constants C_{1} and C_{2} are determined by the initial conditions, tanhC_{1} = ξ_{0}ϕ_{0}^{−1}, and C_{2} = − log{ϕ_{0}p^{−1/2}}. We now determine a_{0}(r, ξ) through the identity

a_{0}(t) − a_{0}(0) = ∫_{0}^{t} ȧ_{0}(s)ds = ∫_{0}^{t}{h, a_{0}}ds = ∫_{0}^{t} h(r(s), ξ(s))ds = h_{0}t,  (43)

that follows from Hamilton’s equations of motion, the Poisson bracket condition, and the conservation of energy. It is important to note that the equations for the trajectory (r(t), ξ(t)) can be solved explicitly for t as a function of the initial point. □

Given this consrtuction, it is easy to verify that for coordinates (r, y) on IR×∂X, the escape function a_{0}((r, y), (ξ, η)) has the identical form as in (38) where now the Hamiltonian is

h((r, y), (ξ, η)) = ξ^{2} + e^{−2r} ∑_{i,j=1}^{n−1} h_{ij}(y)η_{i}η_{j} + c_{n}^{2}.  (44)

We now want to construct operator A_{0} so that the commutator [P_{0}, A_{0}] is equal to P_{0} modulo a remainder R_{N} ∈ OP S^{−N}. To this end, we first construct the Weyl quantization of the symbol a_{0}. The Weyl quantization yields the operator A′_{0} acting on u ∈ C_{0}^{∞},

A′_{0}u(r, y)
= ∫ e^{i(r−r′)ξ+i(y−y′)·η} a_{0}((\frac{r + r′}{2} , \frac{y + y′}{2} ), (ξ, η))u(r′, y′)dx′dy′dξdη. (45)

Weyl quantization is useful because the commutator has a remainder term that is one order lower than in the usual quantization (althought this is not essential). In particular, the usual Weyl calculus yields

[P_{0}, A′_{0}] = P_{0} + R_{0},  (46)

acting on functions localized near the boundary at infinity, where R_{0} ∈ OP S^{0}.

We need to improve this error estimate through iterations. For simplicity, we write w = (r, y) and ψ = (ξ, η). The Weyl calculus yields an asymptotic expansion for the symbol of R_{0},

σ(R_{0})((r, y), (ξ, η)) ∼ ∑_{2j+1,j≥1} \frac{1}{(2j + 1)!} \{h, a_{0}\}_{2j+1}((r, y), (ξ, η)).  (47)

The repeated Poisson bracket {h, a_{0}}_{2j+1} is defined by

{h, a_{0}}_{2j+1}(u, v)
= 2 (\frac{−i}{2})^{2j+1} (∂_{u} · ∂_{ψ} − ∂_{w} · ∂_{v})^{2j+1}h(w, ψ)a_{0}(u, v)|_{w=u,ψ=v}.  (48)

We now continue the construction of the escape function. We look for a symbol a_{−1} ∈ S^{−1} so that

{h, a_{−1}} = σ_{pr}(R_{0}).  (49)

Here, we denote the principal symbol of R_{0} by σ_{pr}(R_{0}). The explicit formulas for the geodesic flow insure that we can solve this first-order, linear, nonhomegenous equation by integrating along the characteristics. Let A_{−1} be the Weyl quantization of the symbol a_{−1}. It then follows that

[P_{0}, A′_{0} − A_{−1}] = P_{0} + R_{−2},  (50)

where R_{−2} ∈ OP S^{−2}. We now repeat this iteration j-times, where j > n/2 + 1.

6. Bounds on Singular Values

The final major component in the proof of the upper bound on the resonance counting function is the proof of bounds on the function F (k) = det(1+T (k)) in the upper-half plane. Let us recall that the form of the inverse of the absolute S-matrix for the model problem S_{0}(k) can be expressed as

S_{0}(k)^{−1} = (R̃_{∞} + 1)c_{0}(k)^{−1}Δ_{h_{0}}^{ik},  (51)

where the smoothing operator R̃_{∞}(k) has an explicit spectral decomposition on L^{2}(∂X, dV_{h_{0}}) in terms of modified Bessel functions as in (23). The coefficient c_{0}(k) is defined before (20). Given this formula, and the formula (18) for the absolute S-matrix, we can write

T (k) = (S_{0}(k)^{−1}S(k) − 1) = ∑_{i=1}^{3} T_{i}(k),  (52)

where the operator-valued functions T_{i}(k) have the form:

T_{1}(k) = c_{0}(k)^{−1}Δ_{h_{0}}^{ik}R_{n}(k),  (53)
T_{2}(k) = R̃_{∞}(k)c_{0}(k)^{−1}Δ_{h_{0}}^{ik}R_{n}(k),  (54)
T_{3}(k) = R̃_{∞}(k).  (55)

The singular values of T_{2}(k) and T_{3}(k) are the most easy to treat because of the explicit formulas given in section 3.2 for the eigenvalues of R̃_{∞}(k), and the discussion of R_{n}(k) given below. Consequently, we will not discuss these terms any further.

In order to estimate the singular values of T_{1}(k), the main term in (32), we need an explicit formula for R_{n}(k). As in [6], one can obtain an explicit formula for R_{n}(k) in terms of the analogue of the Eisenstein functions and the model resolvent. To see this, we write a localization formula expressing R(k) in terms of R_{0}(k). Let χ_{i}, for i = 1, 2, be two smooth functions localized near ∂X, with support in X\U, and smooth up to the boundary. We can write

χ_{i}R(k)χ_{j} = χ_{i}R_{0}(k)χ_{j}
−R_{0}(k)[χ_{i}, P_{0}]R_{0}(k)[P_{0}, χ_{j}]R_{0}(k)
+R_{0}(k)[χ_{i}, P_{0}]R(k)[P_{0}, χ_{j}]R_{0}(k)
R(k).  (56)

The remainder term R(k) depends explicitly on the difference R_{1} and the resolvents. Consequently, the coefficients of this term vanish more rapidly than the others in (56), and they will not affect the asymptotics. We won’t consider R(k) further but refer to [4]. The boundary-values of the resolvents are controlled by a limiting absorption principle proved via the Mourre theory using the conjugate operstor A_{0} constructed in section 5. We can now take weighted limits of the terms in (56). We obtain a representation of the generalized eigenfunctions for P in terms of those for P_{0}, and proceed as in [6].

We present a result that can be applied to more general situations than those discussed here. In particular, this result can be used to prove the sectorial bounds on the resonance counting function mentioned in the introduction. The following result holds for certain integral operators in IR^{n} that are obtained by localizing to a coordinate patch. It is then seen to be sufficient to prove bounds on the singular values of operators on L^{2}(IR^{n}) of the form C_{0}(k)χ_{0}Δ_{h_{0}}^{ik}χ_{1}Kχ_{2}. Here, k is a complex number with ℑk > 0, and the functions χ are cutoff functions, and K is an integral operator (depending on k) whose kernel satisfies the bounds given below. The Laplacian here is the Laplacian Δ_{h_{0}} extended smoothly to IR^{n}.

Lemma 6.1 Let k ∈ I C with ℑk > 0 and k ∉ i(n/2 + ZZ). Let χ_{0},χ_{1} and χ_{2} be cutoff functions on IR^{n}, n ≥ 2, where χ_{0} and χ_{2} have compact support, and χ_{1} is identically one in a neighbourhood of ∂X. Assume that the cutoff functions are analytic to order 2p, where p = max{n, [ℑk]}, ([l] denotes the integer part of l) that is,

|D^{α}χ| ≤ C(Cp)^{|α|},  (57)

for all multiindices α with |α| ≤ 2p. Let K(y, z; k) be an integral kernel satisfying the bounds

|D_{y}^{α}K(y, z; k)| ≤ |C_{1}(k)|⟨y⟩^{−2Imk−n−|α|}  (58)

for y ∈ supp(χ_{1}), z ∈ supp(χ_{2}) and |α| ≤ 2n+2. Let q be an integer with q+ℑk−p ≥ 0. The singular values of C_{0}(k)χ_{0}Δ^{ik}χ_{1}Kχ_{2} satisfy the bound

μ_{j}(C_{0}(k)χ_{0}Δ^{ik}χ_{1}Kχ_{2}) ≤ C |\frac{Γ(ik − q + n/2)}{Γ(−ik + q)}C_{0}(k)C_{1}(k)| (\frac{C⟨k⟩}{j^{1/n}})^{2p} .  (59)

It follows from this estimate, and Weyl’s convexity theorem relating eigenvalues to singular values (cf. [25]), that we can prove an exponential bound on F (z) away from the real axis. Sketch of the Proof. Let Δ_{D} denote the Laplacian with Dirichlet boundary conditions on some ball containing the support of χ_{0}. Then

μ_{j}(χ_{0}Δ^{ik}χ_{1}Kχ_{2}) = μ_{j}(Δ_{D}^{−p}Δ_{D}^{p}χ_{0}Δ^{ik}χ_{1}Kχ_{2})
≤ μ_{j}(Δ_{D}^{−p})∥Δ^{p}χ_{0}Δ^{ik}χ_{1}Kχ_{2}∥
≤ C(Cj)^{−2p/n}∥Δ^{p}χ_{0}Δ^{ik}χ_{1}Kχ_{2}∥.  (60)

The constants depend only on the diameter of the support of χ_{0}. We must now bound the norm on the right of (60). We write

∥Δ^{p}χ_{0}Δ^{ik}χ_{1}Kχ_{2}∥ ≤ ∥Δ^{p}χ_{0}Δ^{ik−q}⟨y⟩^{−2ℑk−2q}∥∥⟨y⟩^{2ℑk+2q}Δ^{q}χ_{1}Kχ_{2}∥. (61)

To bound the second term on the right in (61), we commute Δ^{q} past χ_{1} giving two terms. The first one is ⟨y⟩^{2ℑk+2q}χ_{1}Δ^{q}Kχ_{2}. The assumptions on the integral kernel for K imply that this first operator has an integral kernel that is Hilbert Schmidt, and hence satisfies the bound, ∥⟨y⟩^{2ℑk+2q}χ_{1}Δ^{q}Kχ_{2}∥ ≤ CC_{1}(k). The second term arising from the commutation is ⟨y⟩^{2ℑk+2q}[Δ^{q}, χ_{1}]Kχ_{2}. Since the derivatives of χ_{1} have compact support, the norm of this term is bounded by CC_{1}(k). Combining the preceding two estimates yields ∥⟨y⟩^{2ℑk+2q}Δ^{q}χ_{1}Kχ_{2}∥ ≤ CC_{1}(k). Now we estimate the first term on the right of (61). For k ∉ i(n/2 + ZZ) the operator whose norm we must bound has integral kernel

(Δ^{p}χ_{0}Δ^{ik−q}⟨y⟩^{−2ℑk−2q})(x, y)
= C2^{2ik}g(k)Δ_{x}^{p}χ_{0}(x)|x − y|^{−2ik+2q−n}⟨y⟩^{−2ℑk−2q},  (62)

where g(k) = Γ(ik − q + n/2) Γ(−ik + q)^{−1}. This kernel has to be analysed carefully near and away from the diagonal. The off-diagonal piece is easily seen to be Hilbert Schmidt. It is in the analysis of the kernel near the diagonal the we use the partial analyticity of the cut-off functions. □

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Mathematics Department, University of British Columbia, Vancouver, BC Canada

rfroese@math.ubc.ca

Mathematics Department, University of Kentucky, Lexington, KY 40506-0027 USA

hislop@ms.uky.edu