CUBO A Mathematical Journal
Vol.11, No¯ 05, (129–172). December 2009

Scattering Theory on Geometrically Finite

Quotients with Rational Cusps

Colin Guillarmou

Département de Mathématiques J.A. Dieudonné,

UMR CNRS no. 6621, Université de Nice Sophia-Antipolis

Parc Valrose, 06108, Nice, France

email : cguillar@math.unice.fr

ABSTRACT

We study Eisenstein functions and scattering operator on geometrically finite hyperbolic man-

ifolds with infinite volume and ‘rational’ non-maximal rank cusps. For both we prove the

meromorphic extension and we show that the scattering operator belongs to a certain class of

pseudo-differential operators on the conformal infinity which is a manifold with fibred bound-

aries.

RESUMEN

Estudiamos funciones de Eisenstein y el operador de dispersión sobre variedades hiperbólicas

geometricamente finitas con volumen infinito y puntas de rango no maximos racionales. Para

ambos probamos las extensiones meromorficas y mostramos que el operador de dispersión

pertence a cierta clase de operadores pseudo-diferenciales sobre la variedade conforme infinita

con fibrados en la frontera.

Key words and phrases: Scattering theory, geometrically finite quotients.

Math. Subj. Class.: 58J50, 35P25.



130 Colin Guillarmou CUBO
11, 5 (2009)

1 Introduction and Results

The purpose of this work is to study the Eisenstein functions and scattering operator on a class of

geometrically finite hyperbolic quotients Γ\Hn+1 with non-maximal rank cusps.

Such problems involving spectral and scattering theory on geometrically finite hyperbolic

quotients have been studied probably since Selberg and lead to many important results. However,

most of the results known are obtained when the group has no parabolic subgroups of non-maximal

rank, in other words when the quotient X = Γ\Hn+1 of hyperbolic space Hn+1 has no cusps of

non-maximal rank. As far as we know, the only results concerning meromorphic extension of the

resolvent or scattering operator for this cases were due, until recently, to Froese-Hislop-Perry [3]

in dimension 3. However, in a preprint, Bunke and Olbrich [1] deal with the meromorphic ex-

tension of the scattering operator in all generality using a very different approach; in particular

they do not study the (pseudo-differential) structure of this operator. We refer the reader to the

introduction of [8] for a more detailed review of works on meromorphic extension of the resolvent

for the Laplacian through the essential spectrum, resonances (i.e. the poles of this extension),

meromorphic continuation of Eisenstein functions and scattering operator for geometrically finite

hyperbolic manifolds, though we do not claim to be complete about references therein.

We consider an infinite volume hyperbolic quotient X := Γ\Hn+1 where Γ is a discrete group

of isometries of Hn+1 which admits a fundamental domain with finitely many sides, X is said

geometrically finite, and such that each rank k parabolic subgroup of Γ fixing a point p ∈ Sn

is generated by k independent translations in the horospheres centered at p. We shall say that

the cusps are rational cusps. For exemple, this last condition is always satisfied in dimension

n + 1 = 3. In general, a rank k parabolic subgroup Γp fixing a point p ∈ S
n gives rise to a

model manifold Γp\H
n+1 which is isometric to R+ × M where M is a flat bundle with basis a

flat compact manifold and with fibers Rn−k; then if the holonomy representation of this bundle

has finite image in O(n−k), there is a finite cover which satisfies our assumptions, in which case

the resolvent, scattering operator and Eisenstein functions are obtained as a finite sum on the

cover. Similarly, elliptic elements of Γ can also be excluded by passing to a finite cover, X is then

a smooth manifold, and since the presence of maximal-rank cusps do not add difficulties, we will

avoid them for simplicity of exposition. The Laplacian on such manifolds have been studied by

Froese-Hislop-Perry [3] in dimension 3 and by Perry [23] in higher dimension. The manifold X

equipped with the hyperbolic metric is complete and the spectrum of the Laplacian ∆X splits into

continuous spectrum [ n
2

4
,∞) and a finite number of L2 eigenvalues included in (0, n

2

4
) which form

the point spectrum σpp(∆X ) (see Lax-Phillips [14]). In [8] we proved that the modified resolvent

R(λ) := (∆X −λ(n−λ))
−1

extends from {ℜ(λ) > n
2
} to C meromorphically with poles of finite multiplicity (i.e. the rank of

the polar part in the Laurent expansion at each pole is finite) from L2comp(X) to L
2
loc(X), these



CUBO
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Scattering Theory on Geometrically Finite Quotients 131

poles are called resonances.

In the present work, we define a Poisson operator, Eisenstein functions, a scattering operator

and we show that they extend meromorphically to C. To explain the main Theorems, we recall

briefly the structure at infinity of the manifold X but in any case, we refer the reader to Section 2 of

Mazzeo-Phillips [19] for a comprehensive description of geometrically finite quotients Γ\Hn+1 (see

also [2, 23, 8]). The first approach is to see X as the interior of a smooth compact manifold with

boundary X̄. If ρ is a boundary defining function of the boundary ∂X̄ and if g is the hyperbolic

metric on X, then ρ2g extends as a smooth non-negative tensor on X̄ which is positive definite

outside some submanifolds of the boundary ∂X̄ where it becomes degenerate. Each one of these

submanifolds arises from a cusp point of X, i.e. a fixed point at infinity of Hn+1 for a parabolic

subgroup of Γ, and is diffeomorphic to a k-dimensional torus T k if the parabolic subgroup has

rank k. If we note c the union of these submanifolds, B = ∂X̄ \ c is a non-compact manifold

which can be thought as the infinity of X; actually B = Γ\Ω where Ω ⊂ Sn is the domain of

discontinuity of Γ. After a real blow-up of these submanifolds in X̄, we obtain a manifold X̄c with

corners of codimension 2 which is the compactification of X defined by Mazzeo-Phillips [19] in the

general case. The topological boundary of X̄c splits into two kind of smooth hypersurfaces with

boundaries, the regular ones whose union is a compactification B̄ of B and the cusp ones which

are diffeomorphic to Sn−k+ ×T
k, Sn−k+ being an n−k dimensional half-sphere with boundary. It

turns out that B has ends diffeomorphic to (Rn−ky \ {|y| < 1}) × T
k, each end arising from a

rank-k parabolic subgroup of Γ fixing a point at infinity of Hn+1. The compactification B̄ of B

corresponds to the radial compactification in the y variable in each end thus B̄ is a fibred boundary

manifold in the sense of Mazzeo-Melrose [18], the fibrations being the projections

Sn−k−1 ×T k → Sn−k−1.

When equipped with the metric h0 := ρ
2g|B, (B,h0) is conformal to an ‘exact Φ-type metric’ near

its infinity as defined in [18], the conformal factor decreasing enough to make the volume of B fi-

nite - the vanishing rate is even stronger than the fibred cusp metrics (see Figure 1 for illustration).

We construct Poisson and scattering operators P(λ),S(λ) by solving a Poisson problem in a

way similar to that introduced on Euclidean manifolds by Melrose and on many other settings

by various authors (see [21] for review). However, in view of the sensitive structure of the metric

near the cusps c, it appears that P(λ),S(λ) do not act naturally on C∞(∂X̄) but on subspaces

related to this structure. We then define the subalgebra C∞
acc

(X̄) of C∞(X̄) of functions which are

asymptotically constant in the cusps, these are the f ∈ C∞(X̄) such that

Z(f|c) = 0, Z((X1 . . .XNf)|c) = 0

for all smooth vector fields X1, . . . ,XN on X̄ (∀N ∈ N) and all smooth vector fields Z on c. In other

words, these are the functions whose restrictions at the cusp submanifolds are locally constant and

similarly for all derivatives. It is actually possible to find a boundary defining function ρ in this



132 Colin Guillarmou CUBO
11, 5 (2009)

subalgebra. Then the volume form dvolg of g can be expressed by ρ
−n−1R2cµX̄ for a function Rc

which is smooth positive in X̄ \ c with R2c ∈ C
∞
acc

(X̄) vanishing at order 2k at each k-dimensional

component of c and where µX̄ is a smooth volume density on X̄. The functions Rc and ρ are not

uniquely determined but we show that the set R−1c C
∞
acc

(X̄) is independent of the choice of R2c,ρ in

C∞
acc

(X̄) (but it certainly depends on the metric). Then we define C∞
acc

(∂X̄) and R−1c C
∞
acc

(∂X̄) by

restriction of C∞
acc

(X̄) and R−1c C
∞
acc

(X̄) at ∂X̄ and B = ∂X̄ \c (here we use the same notation for

Rc and its restriction Rc|∂X̄ ). For any boundary defining function ρ ∈ C
∞
acc

(X̄), one can define the

Poisson operator P(λ) by showing that if ℜ(λ) ≥ n
2

and λ /∈ n
2

+ N, then for all f ∈ R−1c C
∞
acc

(∂X̄)

there exists a unique solution P(λ)f of the following Poisson problem





(∆X −λ(n−λ))P(λ)f = 0

P(λ)f = ρn−λF(λ,f) + ρλG(λ,f)

F(λ,f),G(λ,f) ∈ R−1c C
∞
acc

(X̄)

F(λ,f)|ρ=0 = f

.

The construction of the solution is a consequence of an indicial equation for ∆X and the following

precise mapping property of the meromorphically extended resolvent

R(λ) : Ċ∞(X̄) → ρλR−1c C
∞
acc

(X̄).

where Ċ∞(X̄) is the set of functions in C∞(X̄) vanishing at all order at ∂X̄.

Next we analyze Eisenstein functions. The metric h0 induces an L
2
(B) Hilbert space on B

and we prove

Theorem 1.1. If R(λ; w; w′) denotes the Schwartz kernel of the extended resolvent, then the

Eisenstein function

E(λ; b; w′) := lim
w→b

[ρ(w)−λR(λ; w; w′)], b ∈ B,w′ ∈ X

is a smooth function on B×X if λ is not a resonance. There exists C > 1 such that, for all N > 0,

E(λ; ., .) is the Schwartz kernel of a meromorphic operator

E(λ) : ρNL2(X) → L2(B)

in ℜ(λ) > n
2
− C−1N with poles of finite multiplicity, satisfying P(λ) = (2λ − n)tE(λ) on

R−1c C
∞
acc

(∂X̄). Except possibly at {λ;ℜ(λ) < n
2
,λ(n − λ) ∈ σpp(∆X )}, the set of poles of E(λ)

coincides with the set of resonances.

Using the asymptotic expression of P(λ)f, the scattering operator is then defined (with the

same notations) by

S(λ) :

{
R−1c C

∞
acc

(∂X̄) → R−1c C
∞
acc

(∂X̄)

f → F(λ,f)|ρ=0
.

For ℜ(λ) = n
2
, S(λ) can be extended to L2(B) as a unitary operator and it gives, as usual in

scattering theory, a parametrization of the absolutely continuous spectrum of ∆X . Then, we prove



CUBO
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Scattering Theory on Geometrically Finite Quotients 133

the following result which is expressed in more details in Theorem 6.5, Lemma 6.1, Corollary 6.3

and Proposition 7.1:

Theorem 1.2. The scattering operator S(λ) extends meromorphically to C as a family of pseudo-

differential operators in the full Φ-calculus on the manifold with fibred boundary B̄ in the sense of

Mazzeo-Melrose [18]. In {ℜ(λ) ≤ n
2
,λ(n−λ) /∈ σpp(∆X )}, λ0 is a pole of S(λ) if and only if λ0

is a resonance and it has finite multiplicity. In {ℜ(λ) > n
2
}, S(λ) has only first order poles whose

residue is

Resλ0S(λ) =

{
−

(−1)j+12−2j

j!(j−1)!
Pj + Πλ0 if λ0 =

n
2

+ j,j ∈ N

Πλ0 if λ0 /∈
n
2

+ N

where Pj is the j-th GJMS conformal Laplacian of [6] on (B,h0) and Πλ0 is an operator with rank

dim kerL2 (∆X −λ0(n−λ0)).

Note that the GJMS conformal Laplacians Pj in [6] are well-defined for all j if n ≥ 3 (resp.

for j ≤ 1 if n = 2) if the manifold is locally conformally flat (it is actually done in the compact

setting but they can be extended for non-compact manifolds by using the same local expression in

the curvature tensor), which is the case for B.

The general case of irrational cusps is more technically involved and it is not clear if such

precise results can be obtained, at least the meromorphic extension of the resolvent is carried out

in a forthcoming paper. It is also important to add that this analysis could be used to study the

divisors of Selberg’s zeta function as Patterson-Perry [22] did for convex co-compact hyperbolic

manifolds.

The paper is organized as follows: we first introduce in section 2 the geometric setting, discuss

the compactification X̄ of the manifold X and analyze its infinity B; then in section 3 we define the

class of pseudo-differential operators on B which contains the scattering operator and in section 4

we study the mapping properties and the structure of the resolvent for the Laplacian. In section

5, we construct the Poisson operator and Eisenstein functions using section 4 and in section 6 we

define and describe the scattering operator. To conclude we investigate the relation between the

conformal geometry of B and the scattering theory on X.

Along the paper, we will identify operators with their Schwartz kernel and we consider oper-

ators acting on functions for simplicity of exposition though the correct approach would be to use

half-densities. Consequently the kernels of pseudo-differential operators have to be understood as

tensorized by appropriate half-densities.

Aknowledgements: We thank Rafe Mazzeo, Robin Graham and Jared Wunsch for helpful

discussions. This work was written at Purdue University in 2005 but we are also grateful to the



134 Colin Guillarmou CUBO
11, 5 (2009)

Mathematics Department of Nantes where it was completed. Research was partially supported by

NSF grant DMS0500788.

2 Geometry of the Manifold

2.1 Assumptions on the group

We describe here with more details the assumptions about the cusps discussed roughly in the

introduction; we strongly use Section 2 of Mazzeo-Phillips [19]. Let Γ a discrete subgroup of

orientation preserving isometries of the hyperbolic space Hn+1. Recall that Γ acts also on the

natural compactification H̄n+1 = {m ∈ Rn+1; ||m|| ≤ 1} of Hn+1 and on its boundary Sn; an

element γ is called hyperbolic if it fixes exactly two points on Sn and no point in Hn+1, parabolic

if it fixes one point on Sn and no point in Hn+1, then γ is elliptic if it fixes at least a point of Hn+1.

If Γ contains elliptic elements (other than the identity), there exists a subgroup Γ0 of finite index

of Γ without elliptic elements, thus X is finitely covered by Γ0\H
n+1, the latter being a smooth

manifold. Since we study resolvent of the Laplacian and other related objects, we can always pass

to a finite cover without difficulties: objects on X can indeed be obtained by summing on a finite

set objects on the finite cover. Thus we exclude elliptic elements in Γ. We suppose that Γ is

geometrically finite, which means here that it admits a fundamental domain F with finitely many

sides. Each fixed point p ∈ Sn of a parabolic element of Γ is called a cusp point, and for each cusp

point p, let Γp be the subgoup of Γ fixing p. Actually Γp contains only parabolic elements and it

can be shown that there is a Γp invariant neighbourhood Up of p such that Γ\(F ∩Up) is isometric

to a neighbourhood of p in Γp\(F ∩ Up). The subgroup Γp has a maximal free abelian subgoup

Γa with rank k, the rank of the cusp p is defined to be the integer k. We suppose that k ≤ n− 1

for each p since this case is well known in term of scattering theory. Using now conjugation, it

suffices to look at the case where p = ∞ in the upper half model Hn+1 = R+ × Rn. Section 2 of

[19] (the arguments come from Bieberbach’s analysis of discrete groups of isometries of Euclidean

space) shows that there is an affine subspace Rk ⊂ Rn globally preserved by Γ∞ on which Γa acts

as a group of k translations. This allows to see that every γ ∈ Γ∞ acts as

γ(y,z) = (Ry,Az + b) on Rn−ky ⊕ R
k
z

for some A ∈ O(k),R ∈ O(n−k) and b ∈ Rk; elements in Γa have A = Id. There is a flat compact

manifold N = Γ∞\R
k such that Γ∞\R

n is a flat vector bundle with basis N and T k := Γa\R
k such

that Γa\R
n is a flat bundle over T k. Assuming that the holonomy representation of these bundles

Γ → O(n−k) has finite image implies that the elements R decompose into rotations with rational

angles pπ/q for some p,q ∈ N, then there is a finite cover of this bundle which is T k × Rn−k, T k

being a flat torus. Thus, as we mentionned before, it suffices somehow to study the case where

each rotation R is the identity to get a good description of the analytic objects conidered in the

paper.



CUBO
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Scattering Theory on Geometrically Finite Quotients 135

2.2 Neighbourhoods of infinity, models.

From previous discussions and assumptions on the cusps and using [2, 23, 8] we obtain a covering

of the manifold X by model charts. There exists a compact K of X such that X \K is covered

by a finite number of charts isometric to either a regular neighbourhood (Mr,gr) or a rank-k cusp

neighbourhood (Mk,gk) where

Mr := {(x,y) ∈ (0,∞) × R
n
; x2 + |y|2 < 1,}, gr = x

−2
(dx2 + dy2),

Mk := {(x,y,z) ∈ (0,∞) × R
n−k ×T k; x2 + |y|2 > 1}, gk = x

−2
(dx2 + dy2 + dz2)

for k = 1, . . . ,n− 1 with (T k,dz2) a k-dimensional flat torus.

Note that we could allow maximal rank cusps as in [8] without difficulties but since these

cases are well-known, we restrict ourselves to the non-maximal rank cusps cases for simplicity of

exposition. To avoid too many indices in the exposition, we will assume for simplicity that the

manifold has only one neighbourhood of each type, it will be clear from the analysis which will

follow that it does not change anything in the proofs; we then note Ir, (Ik)k the corresponding

chart isometries. One can also choose the covering such that I−1k (Mk) ∩ I
−1
j (Mj) = ∅ for k 6= j,

possibly by adding regular neighbourhoods.

The model Mk can be considered as a subset of the quotient Xk = Γk\H
n+1 of Hn+1 by a

rank-k parabolic subgroup Γk of Γ which fixes a single point at infinity of H
n+1. Indeed, modulo

conjugation by an isometry, one can suppose that the fixed point is the point at infinity of Hn+1

in the half-space model (0,∞) × Rn. The group Γk is generated by k independent translations

acting on Rn, therefore it is the image of the lattice Zk by a map Ak ∈ GLk(R) and the flat torus

T k := Γk\R
k is well defined. Then Xk is isometric to R

+
x × R

n−k
y ×T

k
z equipped with the metric

gk =
dx2 + dy2 + dz2

x2

dz2 being the flat metric on a k-dimensional torus T k. Therefore Mk is the subset of Xk with

x2 + |y|2 > 1. As a matter of fact it will be often useful to consider R+ × Rn−k as the n−k + 1-

dimensional hyperbolic space Hn−k+1. Hence Xk can be compactified into the compact manifold

with boundary X̄k = H̄
n−k+1 ×T k where H̄n−k+1 is the ball {|w| ≤ 1} in Rn−k+1. Then

ρk(x,y,z) :=
x

|y|2 + x2 + 1
= (2 cosh(dHn−k+1 (x,y; 1, 0)))

−1

is a natural boundary defining function in X̄k (∂X̄k = {ρk = 0} and dρk 6= 0 on ∂X̄k). Let us

define the new coordinates

t :=
x

x2 + |y|2
, u :=

−y

x2 + |y|2
(2.1)

which induce an isometry from (Mk,gk) to

{(t,u,z) ∈ (0,∞) × Rn−k ×T k; t2 + |u|2 < 1}



136 Colin Guillarmou CUBO
11, 5 (2009)

equipped with the metric
dt2 + du2 + (t2 + |u|2)2dz2

t2
(2.2)

and ρk(t,u) = ρk(x,y). These coordinates can be thought as compactification coordinates for Mk,

since t and u extend smoothly to X̄k\{x = y = 0}. The infinity of X in the chart Mk is then given by

{ρk = 0} or equivalently {t = 0}. Also we will call cusp submanifold the submanifold {t = u = 0}

of X̄k it will be denoted by ck and we remark that ck ≃∞×T
k ≃ T k in X̄k where ∞ is the point

at infinity in the half-space model of Hn−k+1. We also have Mk = {w ∈ Xk; t(w)
2

+ |u(w)|2 < 1}

which is a subset of X̄k and we will denote

M̄k := {w ∈ X̄k; t
2
(w) + |u(w)|2 < 1}.

At last we define the manifold

Yk := R
n−k ×T k

which can be viewed as (X̄k \ ck) ∩{x = 0}.

The model Mr is simpler and can be considered as a subset of H
n+1. We define as before

M̄r := {(x,y) ∈ [0,∞) × R
n
; x2 + |y|2 < 1}.

There exist some smooth functions χ,χr,χ1, . . . ,χn−1 on respectively X,Mr,M1, . . . ,Mn−1

which, through the isometric charts Ir,I1, . . . ,In, satisfy

I∗r χ
r
+

n−1∑

k=1

I∗kχ
k

+ χ = 1

with χ having compact support in X. Note that it is possible to choose χk which does not depend

on the variable z ∈ T k.

For what follows we will consider Mk,Mr,M̄k,M̄r as neighbourhoods in X̄ instead of using

the notations I−1k (Mk),I
−1
r (Mr)...

2.3 Compactification, volume densities.

Using the previous discussion about the compactification of the cusp neighbourhoods, one obtains

an obvious compactification of X as a smooth compact manifold with boundary X̄. Moreover, we

can choose a boundary defining function ρ which is equal to the function t in each neighbourhood

M̄k. The boundary ∂X̄ is covered by some charts B1, . . . ,Bn−1,Br induced by M1, . . . ,Mn−1,Mr

by taking

Bk := M̄k ∩∂X̄ ≃{(u,z) ∈ R
n−k ×T k; |u|2 < 1}

Br := M̄r ∩∂X̄ ≃{y ∈ R
n
; |y|2 < 1}.



CUBO
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Scattering Theory on Geometrically Finite Quotients 137

From the discussion above, we see that the metric on X can be expressed by

g =
H

ρ2

with H a smooth non-negative symmetric 2-tensor on X̄ which degenerates at the cusps subman-

ifolds (ck)k=1,...,n−1. Let us define c := (∪kck) ⊂ ∂X̄ ⊂ X̄, and B := ∂X̄ \ c, then the restriction

h0 := H|B = (ρ
2g)|B (2.3)

is a smooth metric on the non-compact manifold B.

We will also need to use functions representing the distance to the cusps submanifolds as

follows: for k = 1, . . . ,n − 1, let rck be a continuous non-negative function in X̄, smooth and

positive in X̄ \ ck which satisfies

Ik∗(rck ) =
√
t2 + |u|2

in M̄k and is equal to 1 in Mj when j 6= k. Then we define the functions

rc :=

n−1∏

k=1

rck, Rc :=

n−1∏

k=1

(rck )
k (2.4)

on X̄ and we will also denote by rck , rc and Rc their restriction to ∂X̄. It can easily be checked

that B equipped with the metric h0 of (2.3) has a volume density dvolh0 which is of the form

dvolh0 = R
2
cµ∂X̄ (2.5)

with µ∂X̄ a smooth non-vanishing density (volume density) on ∂X̄. Similarly the volume density

dvolg on X can be expressed by

dvolg = ρ
−n−1R2cµX̄ (2.6)

for a smooth volume density µX̄ on X̄. In what follows, we will write L
2
(X) and L2(B) for the

Hilbert spaces of square integrable functions on X and B with respect to the volume densities

dvolg and dvolh0 .

2.4 Class of functions.

For a compact manifold M̄ with boundary ∂M̄, we denote by Ċ∞(M̄) the set of smooth functions

on M̄ which vanish at all orders at ∂M̄. Its topological dual is the set of extendible distribution

on M̄, denoted C−∞(M̄) (note that a correct definition would include density bundles).

There will be a special set of smooth functions on X̄,∂X̄ which will play an important role

for what follows, these are the functions which are “asymptotically constant in the cusp variables”.

To give a precise definition we begin by introducing the sets C(TX̄), C(T∂X̄) and C(Tc) of smooth

vector fields on X̄,∂X̄,c. Then we set

C∞
acc

(X̄) := {f ∈ C∞(X̄);∀X1, . . . ,XN ∈ C(TX̄),∀Z ∈ C(Tc),Z(f|c) = 0,Z(X1 . . .XNf|c) = 0}



138 Colin Guillarmou CUBO
11, 5 (2009)

and C∞
acc

(∂X̄),C∞
acc

(X̄k),C
∞
acc

(∂X̄k) are defined similarly by replacing X̄ by ∂X̄,X̄k,∂X̄k. These

functions are constant on each cusp submanifold ck and their derivatives too. In local coordinates

(t,u,z) near the cusp ck = {t = u = 0}, one can check by a Taylor expansion at (0, 0,z) ∈ ck and

Borel Lemma that a function f ∈ C∞
acc

(X̄) can be decomposed locally as a sum

f(t,u,z) = f0(t,u) + O((t
2

+ |u|2)∞) = f0(t,u) + O(r
∞
c ) (2.7)

for some f0 smooth. We remark the following properties, the proofs of which are straightforward:

Lemma 2.1. The set C∞
acc

(X̄) is a subalgebra of C∞(X̄) which is stable under the action of C(TX̄),

and stable by composition with smooth real functions on R.

Observe also that r2c and R
2
c defined by (2.4) are in C

∞
acc

(X̄). Actually this implies that if

ρ̂ ∈ C∞
acc

(X̄) is a boundary defining function of ∂X̄ and R̂2c ∈ C
∞
acc

(X̄) is a non-negative function

vanihing at order 2k at each ck such that dvolg = ρ̂
−n−1R̂2cµ̂X̄ for a smooth volume form on X̄,

then

ρ̂ = F1ρ, R̂
2
c = F2R

2
c, µ̂X̄ = F3µX̄

for some functions F1,F2 ∈ C
∞
acc

(X̄) and F3 ∈ C
∞

(X̄) satisfying F−n−11 F2F3 = 1 and F1 > 0,

F3 > 0. Then necessarily F3 ∈ C
∞
acc

(X̄) and F2 > 0 which shows that R
−1
c C

∞
acc

(X̄) = R̂−1c C
∞
acc

(X̄)

and this space does not depend on the choices of ρ,R2c in C
∞
acc

(X̄). Actually the map f → f|dvolg|
1

2

naturally identifies R−1c C
∞

(X̄) with the space of smooth half-densities C∞(X̄, Γ
1

2

0 ) defined in the

0-calculus of Mazzeo-Melrose [17] (depending only on the C∞ structure of X̄) and the space

R−1c C
∞
acc

(X̄) could then be considered as a subspace of C∞(X̄, Γ
1

2

0 ) (depending on the metric) if

we worked with densities.

We also define the set of smooth functions on X̄k (resp. X̄) vanishing at all order at the cusps

Ċ∞c (X̄) := {f ∈ C
∞

(X̄);∀X1, . . . ,XN ∈ C(TX̄),f|c = 0, (X1 . . .XNf)|c = 0}

and Ċ∞c (∂X̄), Ċ
∞
c (∂X̄k), Ċ

∞
c (∂X̄k) similarly. Remark that there is a natural identification between

Ċ∞c (∂X̄) and Ċ
∞

(B̄) if B̄ is defined as the real blow-up of ∂X̄ around c. By similar arguments,

the spaces C∞
acc

(∂X̄), Ċ∞c (∂X̄), R
−1
c C

∞
acc

(∂X̄) can be defined (here we note again Rc instead of

Rc|B) an they coincide with the restriction of C
∞
acc

(X̄), Ċ∞c (X̄), and R
−1
c C

∞
acc

(X̄) at B = ∂X̄ \ c.

To conclude this part, remark the following inclusions

Ċ∞(X̄) ⊂ Ċ∞c (X̄) ⊂ C
∞
acc

(X̄).

and the same for their restriction at B.

2.5 Model form for the metric.

To use the same ideas than for asymptotically hyperbolic manifolds, we need to choose boundary

defining functions of ∂X̄ in X̄ which induce product decompositions of the metric near infinity.



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Scattering Theory on Geometrically Finite Quotients 139

The different choices of boundary defining functions induce a conformal class of smooth tensors on

∂X̄ which are metrics on B, this is the conformal class [h0] of h0 := ρ
2g|∂X̄ . However, in view of

the presence of the cusps, we need to consider the following smaller class of conformal metrics on

B

[h0]acc := {fh0; f ∈ C
∞
acc

(∂X̄),f > 0}.

Lemma 2.2. For all ĥ0 ∈ [h0]acc, there exists a boundary defining function ρ̂ ∈ C
∞
acc

(X̄) of ∂X̄ in

X̄ such that |dρ̂|ρ̂2g − 1 ∈ Ċ
∞

(X̄) in a collar neighbourhood of ∂X̄ and ρ̂2g|B = ĥ0. Moreover, ρ̂

is uniquely determined modulo Ċ∞(X̄) by ĥ0.

Proof : for ĥ0 ∈ [h0], the construction of a boundary defining function ρ̂ = ρe
ω which satisfies

|dρ̂|ρ̂2g = 1 and ρ̂
2g|B = h0 is equivalent to solving the PDE

2(∇ρ2gρ)(ω) + ρ|dω|
2
ρ2g =

1 −|dρ|2ρ2g
ρ

(2.8)

with initial condition ω|∂X̄ = ω0 where ĥ0 = e
2ω0h0 (see [5, Lem. 2.1]). The construction of

a solution is possible in regular neighbourhoods M̄r and is unique since the equation is non-

characteristic there. In M̄k, we write the equation in coordinates and this gives

2∂tω + t
(
(∂tω)

2
+ |∂uω|

2
+ (t2 + |u|2)−2|∂zω|

2
)

= 0

in view of the form of the metric (2.2) there (recall that ρ = t in M̄k). Taking this equation at t = 0,

we see that ∂tω|t=0 = 0 and by differentiating it N times with respect to t and setting t = 0 we see

by induction that all the values ∂
j
t ω|t=0 in {u 6= 0} are determined by ω|t=0 for j ≤ N + 1. In par-

ticular when j is odd this is 0 (see again [5] for a similar study). Since w0 ∈ C
∞
acc

(∂X̄), we can write

it locally under the form (2.7) which shows by induction that ∂
j
t ω|t=0 ∈ C

∞
acc

(∂X̄); the essential

arguments to use are that the singular term in the equation is killed by |∂zω| = O((t
2
+|u|2)∞) and

the properties of C∞
acc

(∂X̄) discussed previously. By using Borel lemma, we can construct a smooth

function ω in a neighbourhood of ∂X̄ in X with those derivatives, thus ω satisfies (2.8) modulo

O(ρ∞) and this proves that there exists a function ρ̂ which satisfies the Lemma, the uniqueness of

its Taylor expansion with respect to ρ at ∂X̄ is clear from the construction. �

We will now use this function to obtain a certain model form of the metric near ∂X̄. Using

again the same arguments than [5, 9], it suffices to consider the collar neighbourhood [0,ǫ)s ×∂X̄

of ∂X̄ induced by the flow ϕs(m) of the gradient ∇ρ̂2gρ̂ with initial condition ϕ0(m) = m for

m ∈ ∂X̄, that is the diffeomorphism

ϕ : (s,m) → ϕs(m)

from [0,ǫ) × ∂X̄ to its image. We consider the function ω constructed in the proof of previous

Lemma (thus ρ̂ = ρeω) and since ∂sρ̂(ϕs(m)) = 1 + O(ρ
∞

) = 1 + O(s∞), we deduce

ρ = se−ω + O(s∞).



140 Colin Guillarmou CUBO
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Now, we remark that the identity |∇ρ̂2gρ̂|ρ̂2g = 1 + O(s
∞

) implies that s2g can be expressed by

s2ϕ∗g = ds2 + ĥ(s) + O(s∞)

in [0,ǫ) ×∂X̄ where ĥ(s) is a smooth family of tensors on ∂X̄ which are positive for s > 0, with

ĥ(0) = ĥ0 positive on B. We have seen in the proof of last Lemma that, in M̄k, ω is an even function

of ρ = t, thus s is an odd function of t and t is an odd function of s. Let (v,ζ) ∈ Rn−k ×T k some

coordinates on ∂X̄ near ck. We have ϕ0(v,ζ) = (v,ζ) and using the form (2.2) of g

∂sϕs(v,ζ) = ∇ρ̂2gρ̂ = e
−ω

(1 + t∂tω)∂t + te
−ω∂uω.∂u +

te−ω

(t2 + |u|2)2
∂zω.∂z

then the function ϕ(s,v,ζ) = ϕs(v,ζ) can be locally written near ck (in coordinates (t,u,z))

ϕ(s,v,ζ) =
(
t = se−ω + t1,u = v + su1,z = ζ + sz1

)
(2.9)

t1 ∈ Ċ
∞

(X̄), u1 ∈ C
∞
acc

(X̄), z1 ∈ Ċ
∞
c (X̄).

Using that ω is even in s and t odd in s, it is straightforward to verify that u,z are even in s. We

deduce that locally

dt = l1(s,v,ds,dv) + O(r
∞
c ), du = l2(s,v,ds,dv) + O(r

∞
c ), dz = dζ + O(r

∞
c ). (2.10)

for some smooth tensors l1, l2, even in s. We want now to write the metric g in these coordinates

(s,v,ζ). By looking at the expression (2.2) and using (2.9), (2.10) with the properties of C∞
acc

(X̄)

discussed in previous section, we obtain that

ĥ(s) = h1(s,v,dv) + h2(s,v,z,dv,dζ) + e
2ωr4cdζ

2
+ O(s∞) (2.11)

where h1,h2 are smooth tensors, even in s, such that h2 = O(r
∞
c ). Since ρ̂ − s = O(ρ̂

∞
), we

can replace s by ρ̂ in (2.11) and we have the same expression for the metric. Now in a regular

neighbourhood Mr, there exists coordinates (x,y) ∈ (0,ǫ)×R
n such that g = x−2(dx2 +dy2), thus

by writing ρ̂ = xeθ for some θ smooth, we have by mimicking last Lemma that (from (2.8))

2∂xθ + x((∂xθ)
2

+ |∂yθ|
2
) = O(x∞)

with θ|x=0 = θ0 satisfying ĥ0 = e
2θ0dy2. Exactly as before for Mk, this gives that ρ̂ is odd in x,

thus x is odd in s and y even in s, which easily implies that ĥ(s) has an even Taylor expansion in

s at s = 0.

This discussion proves that there exists a collar neighbourhood (0,ǫ)ρ̂ ×∂X̄ of ∂X̄ in X̄ such

that

g =
dρ̂2 + ĥ(ρ̂)

ρ̂2
+ O(ρ̂∞) (2.12)

for a smooth family of symmetric tensors ĥ(ρ̂) on ∂X̄ with an even Taylor expansion in ρ̂ at ρ̂ = 0,

positive for ρ̂ > 0, ĥ(0) = ĥ0 being positive on B and with the local expression (2.11) near the



CUBO
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Scattering Theory on Geometrically Finite Quotients 141

cusps ck. Actually, the evenness of the metric in ρ̂ is a consequence of the constant curvature of

X and is studied in details in [9] more generally for asymptotically hyperbolic manifolds.

Is is quite direct and similar to a result of Graham [5] to check that for two functions ρ̂1, ρ̂2

satisfying Lemma 2.2, then for all j ∈ N

∂
2j
ρ̂1
ρ̂2|∂X̄ = 0, ∂

2j
ρ̂2
ρ̂1|∂X̄ = 0

which will be useful to define renormalized volume in an invariant way.

There is however a very special case of boundary defining function ρ̂ which can be chosen

to put the metric into a simpler form. It is obtained by taking ρ̂ = t in the neighbourhood M̄k

of the cusp ck and extending it to a neighbourhood of ∂X̄ so that it satisfies |dρ̂|ρ̂2g = 1 in this

neighbourhood and ρ̂2g|∂X̄ = h0. To prove the existence of such an extension, it suffices to go

back to the proof of Lemma 2.2 and we see that this amounts to solve the PDE (2.8) without the

error term O(ρ∞) and with initial condition ω|∂X̄ = 0. Since the equation is non-characteristic out

of the cusp c, there exists a unique solution ω in some neighbourhood {ρ < ǫ,δ < rc} (for some

δ,ǫ > 0) of the boundary ∂X̄ avoiding the cusp c, and it is clear that ω = 0 satisfies the equation

in M̄k.

For what follows, we will often work with this boundary defining functions ρ̂ and by convention

we will note it ρ, forgetting the previous choice of function ρ. Then we have in some collar

neighbourhood (0,ǫ)ρ ×∂X̄ of ∂X̄

g =
dρ2 + h(ρ)

ρ2
(2.13)

for some smooth family of symmetric tensors h(ρ) on ∂X̄, depending smoothly on ρ, positive for

ρ > 0, with h(0) = h0 positive on B and satisfying

h(ρ) = du2 + (ρ2 + |u|2)2dz2

in each M̄k.

2.6 Geometry of B

To study the scattering operator and to define the class of pseudo-differential operators which

contains it, we can consider the manifold B as the union of a compact manifold Er (covered by the

charts Br) and n− 1 ends E1, . . . , Ek with Ek diffeomorphic to

{(y,z) ∈ Rn−k ×T k; |y| > 1}⊂ Yk = R
n−k ×T k.

For simplicity, we will consider Ek as this last subset of Yk. By using the radial compactification

in the y variable in each end Ek we see that the manifold B compactifies in a smooth compact

manifold with boundary B̄, the boundary ∂B̄ being a disjoint union on k = 1, . . . ,n−1 of products



142 Colin Guillarmou CUBO
11, 5 (2009)

∂Ek := S
n−k−1 ×T k. A boundary defining function of ∂Ek is given by v = rck = rc = |y|

−1 and

rc is a boundary defining function of ∂B̄. Note that B̄ 6= ∂X̄ but B̄ is actually the blow-up of ∂X̄

around the cusps submanifolds c1, . . . ,cn−1. The structure of the compactified manifold B̄ near

∂Ek is [0, 1)v ×∂Ek and ∂Ek fibers by the projection

φk : S
n−k−1 ×Tk → S

n−k−1. (2.14)

The metric h0 on B is not exactly a fibred cusp metric since too much decreasing at infinity

h0 = dv
2

+ v2dω2 + v4dz2.

For following purposes, it is also quite natural to consider B with the metric h̃0 := r
−4
c h0 conformal

to h0 since this is the flat metric dy
2
+ dz2 on each end Ek. Note that h̃0 in (0, 1)v ×S

n−k−1
ω ×T

k
z

is

h̃0 =
dv2

v4
+
dω2

v2
+ dz2

which is an “exact Φ-metric” in the sense of Mazzeo-Melrose [18]. The volume induced by the

metric h0 on B is finite whereas the volume of B with the metric h̃0 is not finite.

3 Pseudo-Differential Operators at Infinity

There is a natural way to define pseudo-differential operators on B using the euclidean structure

of each end Ek. Recall first from Schwartz theorem that for any continuous linear operator A :

Ċ∞(B̄) → C−∞(B̄) there exists a unique extendible distribution a ∈ C∞(B̄×B̄) (we dropped the

density factor for simplicty), called Schwartz kernel, such that

〈Aφ,ψ〉 = 〈a,ψ ⊗φ〉, ∀φ,ψ ∈ Ċ∞(B̄).

Thus we will identify Schwartz kernel with its associated operator. We can define the space Ψm,l(B)

of pseudo-differential operators of order (m,l) ∈ R2 as the set of linear operators

A : Ċ∞(B̄) → C−∞(B̄) (3.1)

such that in each compact coordinate patch on B (those are the Br of previous section), A has a

distributional Schwartz kernel of the type

A(w; w′) =

∫

Rn

eiξ.(w−w
′)a(w,ξ)dξ (3.2)

with a(w,ξ) a symbol in the coordinate patch, i.e. a(w,ξ) is smooth and

|∂αw∂
β
ξ a(w,ξ)| ≤ Cα,β (1 + |ξ|)

m−|β|,

whereas on the end Ek with coordinates w = (y,z) ∈ R
n−k ×T k, the distributional kernel of A is

of the form (3.2) but with a(w; ξ) smooth and satisfying

|∂αy ∂
β
z ∂

γ
ξ a(y,z,ξ)| ≤ Cα,β,γ (1 + |y|)

−l−|α|
(1 + |ξ|)m−|γ|.



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Scattering Theory on Geometrically Finite Quotients 143

the cusp c = S1

B with the metric h0

The manifold B̄

The manifold ∂X̄

D1

D2

D4

D3

Figure 1: The infinity B of the quotient X = Γ\H3 where Γ is a Schottky group gluing D3 ←→ D4
and D1 ←→ D2; B̄ is a manifold with fibred boundary.

It is not hard to check the mapping property (3.1). One can also define classical (or polyhomo-

geneous) pseudo-differential operators of order m,l ∈ C as operators in Ψℜ(m),ℜ(l)(B) with the

symbol in (3.2) satisfying (for all k)

a(y,z,ξ) = |y|−l|ξ|mã(|y|−1,y/|y|,z, |ξ|−1,ξ/|ξ|) for |ξ| > 1

for some ã ∈ C∞([0, 1) × Sn−k−1 ×T k × [0, 1) ×Sn−k−1), we will use the notation Ψ
m,l
cl (B). In

each end Ek, this corresponds in a sense to the class of pseudo-differential treated by Hörmander

in the y ∈ Rn−k variable (or the Scattering Calculus of Melrose [21]) but with the additional

compact variable z ∈ T k. In particular, an operator A ∈ Ψm,l(B) can be defined in term of its

distributional kernel lifted from B̄ × B̄ to a blown-up version of this product. This is a standard

way due to Melrose to describe in details the various singularities of the kernel: we always have

the usual conormal singularity at the diagonal of X̄ × X̄ (like in the compact setting) but for



144 Colin Guillarmou CUBO
11, 5 (2009)

non-compact manifolds, it is important to include informations in the symbol about the behaviour

at infinity, these can be interpreted as conormal singularities for the kernel on the boundaries of

the compactification X̄ × X̄ (boundary of the compactification = infinity of the manifold). Since

singularities with different nature intesects at the diagonal of the corner ∂X̄×∂X̄, it is convenient

to define a bigger manifold, the blow-up, where the kernel is more readable.

The blow-up here is slightly different from that of Scattering Calculus, it is in a sense the

scattering blow-up defined in [21] but only in y variable. This blow-up corresponding to manifolds

with fibred boundaries is explained in generality by Mazzeo-Melrose in [18], it is achieved in two

essential steps. The principle is to start with the manifold with corners X̄ ×X̄ and to construct a

larger manifold with corners where the phase of (3.2) defines a smooth submanifold (“the diagonal”)

intersecting transversally the boundary of this larger manifold at only one hypersurface.

For what follows, we will use part of the notations of [18]. The manifold B̄ × B̄ has 2n − 2

boundary hypersurfaces Lk := ∂Ek×B̄, Rk = B̄×∂Ek for k = 1, . . . ,n−1 and we have Lk∩Lj = ∅

if j 6= k, the same with Rk and finally Lk ∩Rj = ∂Ek ×∂Ej is a corner of codimension 2. We need

to define the first blow-up of B̄ × B̄ by taking the “b”blow-up

B̄ ×b B̄ := [B̄ × B̄; ∂E1 ×∂E1; . . . ; ∂En−1 ×∂En−1]

which means that we blow-up successively each corner ∂Ek × ∂Ek of Ek × Ek ⊂ B̄ × B̄. This is

done by replacing in B̄ × B̄ the submanifold ∂Ek ×∂Ek by its spherical normal interior pointing

bundle in B̄ × B̄. The blow-down map is denoted

βb : B̄ ×b B̄ → B̄ × B̄.

The manifold B̄×b B̄ has 3n−3 boundary hypersurfaces, the first 2n−2 are the top and bottom

faces

B
′
k := β

−1
b (B ×∂Ek), T

′
k := β

−1
b (∂Ek ×B), k = 1, . . . ,n− 1.

The new ones are called front faces (F′k)k=1,...,n−1 for the b blow-up and F
′
k is the spherical normal

interior pointing bundle of ∂Ek ×∂Ek in B̄× B̄ and is mapped by βb on ∂Ek ×∂Ek. Note that F
′
k

is diffeomorphic to [−1, 1]τ ×∂Ek ×∂Ek using the function τ =
v−v′

v+v′
(see Melrose [20]), thus we

will identify them.

The closure Db := β
−1
b (DB) of the diagonal DB of B×B meets the boundary of B̄×b B̄ only

at the (interior of the) hypersurfaces F′k and it does transversally at a submanifold denoted ∂Db.

The blow-up of B̄×bB̄ along ∂Db would give the blow-up associated to the Scattering Calculus but

it turns out that the second kind of blow-up we need for our purpose are the successive blow-ups

of B̄ ×b B̄ along the submanifolds

Φk = {(0,m,m
′
) ∈ F′k = [−1, 1]τ ×∂Ek ×∂Ek; φk(m) = φk(m

′
)},

with φk the fibration of (2.14), this gives the manifold with corners

B̄ ×Φ B̄ := [B̄ ×b B̄; Φ1; . . . ; Φn−1].



CUBO
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Scattering Theory on Geometrically Finite Quotients 145

The blow-down maps are

B̄ ×Φ B̄
βΦ−b
−→ B̄ ×b B̄

βb
−→ B̄ × B̄, βΦ := βb ◦βΦ−b.

The boundaries of B̄ ×Φ B̄ are the top and bottom faces

Bk = β
−1
Φ (B ×∂B

′
k), Tk = β

−1
Φ (∂B

′
k ×B)

the front faces of the b blow-up

Fk := β
−1
Φ−b(F

′
k \ Φk)

and the front face of the Φ blow-up is the normal spherical interior pointing bundle of Φk in B̄×b B̄

Ik := SN+(Φk; B̄ ×b B̄).

We will denote by ρTk,ρBk,ρFk,ρIk some functions which define the respective hypersurfaces:

{ρTk = 0} = Tk, {ρBk = 0} = Bk, {ρFk = 0} = Fk, {ρIk = 0} = Ik.

The closure DΦ := β
−1
Φ (DB ) meets the topological boundary of B̄ ×Φ B̄ only at (the interior of)

the hypersurfaces Ik and it does transversally. One can thus define (using extension through the

boundary hypersurface) the set Im(B̄ ×Φ B̄; DΦ) of distributions classically conormal of order m

to the submanifold DΦ.

Tk

Bk

DΦ

z −z′
Ik

Fk

Figure 2: The blow-up of Φk in B̄ ×b B̄

The important point is that β∗Φ is a one-to-one map between Ċ
∞

(B̄ × B̄) and Ċ∞(B̄ ×Φ B̄),

this induces a one-to-one map between their respective duals, which allows to indentify continuous



146 Colin Guillarmou CUBO
11, 5 (2009)

operators (3.1) with their Schwartz kernel lifted to B̄×Φ B̄. With this identification, we define the

space

Ψ
m,l
Φ (B̄) := {K ∈ ρ

l
Ik
Im(B̄ ×Φ B̄; DΦ);∀k,K ≡ 0 at Fk, Tk, Bk}

for m,l ∈ C, where ≡ means equality of Taylor series. This forms the (classical) “small Φ-calculus”

and it is not difficult to check that Ψ
m,l
cl (B) = Ψ

m,l
Φ (B̄) with the notations introduced before for the

standard pseudo-differential operators on B. We sketch the proof of the sense Ψ
m,l
cl (B) ⊂ Ψ

m,l
(B̄).

Recall that

v = |y|−1,ω =
y

|y|
,v′ = |y|′,ω′ =

y′

|y′|
,z,z′

give some local coordinates near the corner ∂Ek ×∂Ek on B̄ × B̄ and

s =
v

v′
,v′,ω,ω′,z,z′ with |ω| = |ω′| = 1

give some coordinates on B̄×b B̄ near the front face F
′
k (valid out of B

′
k), in particular Φk = {v

′
=

0; s = 1; ω = ω′}. If A ∈ Ψ
m,l
cl (B), the expression (3.2) with w = (y,z),w

′
= (y′,z′) can be put in

these coordinates

A(w; w′) =

∫
ei(

1

v′
( ω

s
−ω′).ξ1+(z−z

′).ξ2)a
( ω
v′s

,z; ξ1,ξ2

)
dξ1dξ2. (3.3)

It can be checked that ωi
s
−ω′i,ω

′
i,v

′,z,z′ for i = 1, . . . ,n−k give some coordinates near F′k ∩ Φk

and Φk = {
ω
s
−ω′ = 0}. The functions (ωi−sω

′
i)/(sv

′
) lift under βΦ−b to some functions Wi which

are smooth near Ik \ (Ik ∩ Fk) and we have near DΦ ∩ Ik

DΦ = {W1 = · · · = Wn−k = 0; z = z
′}, Ik = {v

′
= 0}

in coordinates W := (W1, . . . ,Wn−k),ω
′,v′,z,z′ with

∑
i ω

′
i
2

= 1. This gives in (3.3)

A(w; w′) =

∫
ei(W.ξ1+(z−z

′).ξ2)a
(
W +

ω′

v′
,z; ξ1,ξ2

)
dξ1dξ2

with {W = 0} = DΦ. This last expression shows that A(w; w
′
) has a classical conormal singularity

at DΦ of order m. Near the front face Ik, that is when v
′ → 0, then v′

−l
a( W +ω

′

v′
,z; ξ) is a

smooth function near DΦ ∩ Ik. Using other systems of coordinates covering Ik ∩ Fk one easily

see that β∗Φ(A) vanishes at all order at Fk (using integration by parts in oscillating integrals and

the “polynomial growth” of a(w,ξ) in |w|) and that ρ−l
Ik
β∗Φ(A) ∈ I

m
(B̄ ×Φ B̄; DΦ). The vanishing

of (3.3) at {v′ = 0; |ω − sω′| > ǫ; 1 > s} comes by integration by parts and shows the vanishing

of β∗Φ(A) at all order at the boundaries near Fk ∩ Tk and the behaviour near Fk ∩ Bk is similar.

Finally the vanishing at Tk and Bk far from Fk is again a consequence of non-stationary phase

(3.2).

The converse Ψ
m,l
Φ (B̄) ⊂ Ψ

m,l
cl (B) is essentially similar.



CUBO
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Scattering Theory on Geometrically Finite Quotients 147

Now one can define the “full Φ-calculus” by considering the set of operators (identifying lifted

kernels and operators)

Ψ
m,l,E
Φ (B̄) := Ψ

m,l
Φ (B̄) +

∏

F =F,I,T,B
k=1,...,n−1

(ρFk )
E(Fk )C∞(B̄ ×Φ B̄) (3.4)

E = {E(T1),E(B1),E(F1),E(I1), . . . ,E(Tn−1),E(Bn−1),E(Fn−1),E(In−1)}, E(Fk) ∈ C

i.e. we allow some classically conormal singularities at all faces. For operators we deal with,

the conormal singularity at the front faces Ik will be of the same order for both terms, that is

l = E(I1) = · · · = E(In−1), hence we will write Ψ
m,E
Φ (B̄) instead of Ψ

m,l,E
Φ (B̄). Finally, a subclass

with much more regularity will appear as error terms in the expression of the scattering operator,

those are operators with kernels of the form

∏

k

(rck )
ak (r′ck )

bkC∞(∂X̄ ×∂X̄).

where ak,bk ∈ C and rck (w,w
′
) := rck (w), r

′
ck

(w,w′) := rck (w
′
). Recall again that ∂X̄ can be

viewed as the smooth compact manifold without boundary obtained from B̄ by collapsing each

∂Ek ≃ S
n−k−1 ×T k to φk(∂Ek) = ck ≃ T

k.

Actually, since we forgot the density factors for the kernels, the orders of such pseudo-

differential operators depend on the density we use to pair two fonctions in Ċ∞(B̄), thus it will be

necessary to precise it.

4 Resolvent

In this section we analyze the meromorphic extension of the modified resolvent

R(λ) := (∆X −λ(n−λ))
−1

and more precisely the necessary informations we shall need to define Eisenstein functions, Poisson

operator and scattering operator. The meromorphic extension of the resolvent is proved in [8] by

parametrix construction. Using also spectral theorem, this can be summarized as follows:

Theorem 4.1. There exists C > 1 such that for all N > 0, the modified resolvent R(λ) on X

extends meromorphically with poles of finite multiplicity from {ℜ(λ) > n
2
} to {ℜ(λ) > n

2
−CN}

with values in the bounded operators from ρNL2(X) to ρ−NL2(X). The only poles of R(λ) in

{ℜ(λ) > n
2
} are first order poles at each λ0 such that λ0(n−λ0) ∈ σpp(∆X ) and with residue

Resλ0R(λ) = (2λ0 −n)
−1

r∑

j=1

φj ⊗φj, φj ∈ ρ
λ0R−1c C

∞
acc

(X̄) ⊂ L2(X)

where (φj )j=1,...,r is an orthonormal basis of kerL2 (∆X −λ0(n−λ0)).



148 Colin Guillarmou CUBO
11, 5 (2009)

Actually the form of φj is a consequence of (4.20) which will be proved in this section.

To construct the Poisson operator, we need more precise information about the mapping

properties of R(λ) and about its Schwartz kernel structure near infinity. One of the main points

is to analyze the Schwartz kernel of the meromorphic extension of the resolvent

RXk (λ) = (∆Xk −λ(n−λ))
−1

for the Laplacian ∆Xk on the model spaces Xk = Γk\H
n+1, and its mapping properties.

Recall that X̄ is a compact manifold with boundary ∂X̄, hence X̄ × X̄ is a manifold with

corners on which we define the functions

ρ(w,w′) := ρ(w), ρ′(w,w′) := ρ(w′), Rc(w,w
′
) := Rc(w), R

′
c(w,w

′
) := Rc(w

′
). (4.1)

Since ρ,Rc are well defined on M̄k via Ik, the functions (4.1) can also be defined on M̄k ×M̄k.

Lemma 4.2. Let θ,θ′ ∈ C∞(X̄k) be functions with support in M̄k and constant near ck, then the

extended resolvent RXk (λ) satisfies

θRXk (λ)θ
′
: Ċ∞(X̄k) → ρ

λR−1c C
∞
acc

(X̄k) (4.2)

for λ /∈ ( k
2
−N0) if n−k + 1 is odd and for λ ∈ C otherwise. If moreover θ,θ

′ are chosen satisfying

supp(θ) ∩ ck = ∅ and θθ
′
= 0 then

θ′RXk (λ)θ ∈ ρ
λρ′

λ
R−1c C

∞
(X̄k × X̄k), θRXk (λ)θ

′ ∈ ρλρ′
λ
R′c

−1
C∞(X̄k × X̄k) (4.3)

Proof : clearly, it is enough to show the lemma with θ,θ′ which are independent of the variable

z ∈ T k. We recall from [8] that the explicit formula for the resolvent on Xk can be obtained by

Fourier analysis on the z ∈ T k variable, RXk (λ) admits a meromorphic continuation to C and its

Schwartz kernel can be written

RXk (λ) =
∑

m∈Zk

eiωm.(z−z
′)Rm(λ) (4.4)

for λ /∈ ( k
2
− N0) if n−k + 1 is odd and for λ ∈ C otherwise, with

Rm(λ; x,y; x
′,y′) := Ck

∫

Rk

eiωm.zRHn+1 (λ; x,y,z; x
′,y′, 0)dz (4.5)

where Ck is a constant, RHn+1 (λ) is the kernel of the resolvent of the Laplacian on H
n+1 and

ωm := 2π
t
(A−1k )m. Note that Rm(λ) can be considered as an operator -a resolvent- on H

n−k+1.

We have seen in [8] that if

τ :=
xx′

r2 + |z|2
, r2 := |y −y′|2 + x2 + x′

2
, d :=

xx′

r2



CUBO
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Scattering Theory on Geometrically Finite Quotients 149

then for all N ∈ N ∪∞ there exists a function FN (λ,τ) smooth in τ ∈ [0,
1
2
) with a conormal

singularity at τ = 1
2

such that

RHn+1 (λ; x,y,z; x
′,y′, 0) = τλ

N−1∑

j=0

αj (λ)τ
2j

+ τλ+2N FN (λ,τ)

for some αj (λ) meromorphic in λ (with only poles at −N0 if n + 1 is even) and if N = ∞,

F∞(λ,τ) = 0 and the sum converges locally uniformly if τ 6=
1
2

(see also [12] and [23, Appendix

A]). Thus by a change of variable w = z/r in (4.5), one has as in [8, Sect. 3.1]

Rm(λ) = d
λrk

N−1∑

j=0

d2jFj,λ(r|ωm|) + d
λ+2N rk

∫

Rk

e−irωm.z
FN (λ,d(1 + |z|

2
)
−1

)

(1 + |z|2)λ+2N
dz (4.6)

with

Fj,λ(u) := Ck,j (λ)|u|
λ− k

2
+2jK

−λ+ k
2
−2j (|u|), Fj,λ(0) := Dk,j (λ)

Ks(z) =
∫
∞

0
cosh(st)e−z cosh(t)dt being the modified Bessel function, Ck,j (λ) some holomorphic

functions and Dk,j (λ) some meromorphic functions in C with only first order poles at
k
2
− N0 if

n−k + 1 is even (in fact we have R0(λ) = (xx
′
)

k
2 RHn−k+1 (λ−

k
2
)). The sum (4.6) with N = ∞ is

locally uniformly convergent in {d < 1
2
, 0 < r}.

We first show (4.3) using these explicit formulae. We will better use the compactification

coordinates (t,u) on Mk, the functions r and d become

d =
tt′

|u−u′|2 + t2 + t′2
, r2 =

t2 + t′
2

+ |u−u′|2

(t2 + |u|2)(t′2 + |u′|2)
. (4.7)

On the support of θRXk (λ)θ
′ we have t2 + t′

2
+ |u−u′|2 > ǫ and d ≤ 1

2
− ǫ for some ǫ > 0 since

θθ′ = 0, thus (4.6) with N = ∞ is absolutely convergent there and r → +∞ when t2 + |u|2 → 0,

that is when we approach the cusp submanifold ck with respect to variables (t,u). Since Bessel’s

function Ks(x) = K−s(x) and all its derivatives with respect to x vanish exponentially when

x →∞, the kernel ∑

m 6=0

θRm(λ)e
iωm .(z−z

′)θ′

is in ρλρ′
λ
Rc

−1C∞({X̄k \ ck}× X̄k) and can be extended to X̄k × X̄k with

∑

m 6=0

θRm(λ)θ
′eiωm.(z−z

′) ∈ ρλρ′
λ
C∞(X̄k × X̄k)

vanishing at all order at (ck ×X̄k)∪(X̄k ×ck). Note that we have used that ρ = t in Mk. For the

term R0(λ), it is clear, using (4.6) and (4.7) that

θR0(λ)θ
′ ∈ ρλρ′

λ
R′c

−1
C∞(X̄k × X̄k)



150 Colin Guillarmou CUBO
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which concludes the proof of (4.3) using the symmetry of the resolvent kernel.

The property (4.2) is more technical since it involves the singularity of RXk (λ) near the

diagonal. Let f ∈ Ċ∞(X̄k), with support in M̄k. We first study for m 6= 0 the function θRm(λ)θ
′fm

in M̄k where fm = 〈f,e
iωm.z〉Tk is the m-th Fourier mode on T

k of f. We clearly have fm ∈

Ċ∞(H̄n−k+1) with

∀l ∈ N, |∂αfm| ≤ Cα,l|ωm|
−l

with Cα,l uniform in m. For simplicity, we consider (4.6) with N = 0 and decompose

F0(λ,τ) = χ(τ)F0(λ,τ) + (1 −χ(τ))F0(λ,τ) =: F0,1(λ,τ) + F0,2(λ,τ)

with χ a C∞0 ([0, 1/4)) which is equal to 1 near τ = 0. The integral

θ(t,u)θ′(t′,u′)rkdλ
∫

Rn−k

e−irωm.z(1 + |z|2)−λF0,1(λ,d(1 + |z|
2
)
−1

)dz

is well defined for ℜ(λ) > k
2

and is equal by integration by parts to

κ1 := θ(t,u)θ
′
(t′,u′)(r|ωm|)

−2Nrkdλ
∫

Rn−k

e−irωm.z∆Nz

(
F0,1(λ,d(1 + |z|

2
)
−1

)

(1 + |z|2)λ

)
dz (4.8)

for all N > 0. In view of the smoothness of F0,1(λ,τ) for τ ∈ R
+, it is straightforward to see that

the integrand in (4.8) satisfies

∣∣∣∣∆
N
z

(
F0,1(λ,d(1 + |z|

2
)
−1

)

(1 + |z|2)λ

)∣∣∣∣ ≤ CN (1 + |z|
2
)
−ℜ(λ)−N

and is a smooth function of d for λ ∈ C\−N0, now integrable with respect to z ∈ R
k if ℜ(λ)+N > k

2
.

Now since fm(t
′,u′) = O(t′

∞
), we have in H̄n−k+1 × H̄n−k+1

|∂αt,u(d/t)∂
βfm| ≤ Cα,β,l|ωm|

−l, |∂αt,ud∂
βfm| ≤ Cα,β,l|ωm|

−l

|∂αt,ur∂
βfm| ≤ Cα,β,l(t

2
+ |u|2)−(1+|α|)/2|ωm|

−l, |∂αt,u(r
√
t2 + |u|2)∂βfm| ≤ Cα,β,l|ωm|

−l

by looking at the expression of d,r in (4.7). For λ /∈−N0 fixed, we take N ≫ 2|ℜ(λ)|, this proves

that

t−λ(t2 + |u|2)−M
∫

Hn−k+1

dλκ1fm(t
′,u′)t′

−n+k−1
(t′

2
+ |u′|2)

k
2 dt′du′

is CN in (t,u) ∈ H̄n−k+1 for 2M ≪ N and all its derivatives of order α with |α| < N are bounded

by Cl,N|ωm|
−l for all l,N,m. Thus for M fixed, by taking N → ∞ we see that this function is

smooth in H̄n−k+1 and its derivatives are rapidly decreasing in |ωm|.

We now have to deal with the integral kernel

κ2 := θ(t,u)θ
′
(t′,u′)rkdλ

∫

Rn−k

e−irωm.z(1 + |z|2)−λF0,2(λ,d(1 + |z|
2
)
−1

)dz



CUBO
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Scattering Theory on Geometrically Finite Quotients 151

and we will show that

f′m(t,u) :=

∫

Hn−k+1

κ2fm(t
′,u′)t′

−n+k−1
(t′

2
+ |u′|2)

k
2 dt′du′

satisfies

f′m ∈ Ċ(H̄
n−k+1

), |∂αt,uf
′
m| ≤ Cα,l|ωm|

−l. (4.9)

First remark that, since d < 1
2
, we have 1 − χ(d(1 + |z|2)−1) = 0 if |z| > C for some C > 0

depending on χ. We use the change of variables s = t/t′,v = (u−u′)/t′ in this last integral. By

elementary computations, it turns out that

d = (2 cosh(dHn−k+1 (t,u; t
′,u′)))−1 = (2 cosh(dHn−k+1 (1, 0Rn−k ; s,v)))

−1

but F0,2(λ,d(1 + |z|
2
)
−1

) is supported in {d > ǫ} for some ǫ > 0 depending on χ thus it is

supported in {(s,v) ∈ K} where K is a euclidean ball included in Hn−k+1 (thus a compact of

H
n−k+1). Moreover in the variables (t,u,s,v),

κ2 = θ(t,u)θ
′

(t
s
,u−

t

s
v
)
rkdλ

∫

|z|<C

e−irωm.z(1 + |z|2)−λF0,2(λ,d(1 + |z|
2
)
−1

)dzRk

and all its derivatives with respect to (t,u) are in L1(K,s−1dsdz), this fact is proved by Perry

[23, Appendix] and is a direct consequence of the conormal singularity of F0(λ,τ) at τ =
1
2
. And

from the expression of r, we see that the derivatives of r or order α are bounded by Cαt
−1−|α| for

(t,u,s,v) ∈ Hn−k+1 ×K. We deduce that

∫

K

κ2fm

(
t

s
,u−

t

s
v

)(
t

s

)n−k+1 ((
t

s

)2
+

∣∣∣∣u−
t

s
v

∣∣∣∣
2
)k

2

s−1dsdv

is in Ċ∞(Hn−k+1) since fm(t,u) = O(t
∞

) and K is compact. In addition, its derivatives of order

α are clearly bounded by Cα,l|ωm|
−l for all α,l. We have thus proved (4.9) and that

∑

m 6=0

Rm(λ)e
iωm .(z−z

′)f ∈ ρλC∞c (X̄k).

It remains now to study θR0(λ)θ
′f0 where f0 := 〈f, 1〉T k is the zeroth Fourier term of f. But

recall from [8] that R0(λ) acting on H
n−k+1 is nothing more than the hyperbolic resolvent

R0(λ; t,u; t
′u′) =

(
tt′

(t2 + |u|2)(t′2 + |u′|2)

)k
2

RHn−k+1

(
λ−

k

2
; t,u; t′,u′

)
.

for λ /∈ ( k
2
−N0) if n−k + 1 is odd and for λ ∈ C otherwise. Using the analysis of [17], we directly

obtain that

θR0(λ)θ
′f0 ∈ ρ

λR−1c C
∞

(H̄
n−k+1

) ⊂ ρλR−1c C
∞
acc(X̄k)

where the inclusion means: consider the function on Xk as constant with respect to z ∈ T
k. As

a conclusion (4.2) is proved and the proof of the lemma is achieved too, at least for λ /∈ −N0.



152 Colin Guillarmou CUBO
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The points at −N0 can in fact be treated by taking N > 0 large in (4.6) and essentially the same

arguments than for N = 0. �

Now we briefly review the construction of a parametrix for R(λ) in [8, Prop 3.1 and 3.5] which

can be continued to infinite order (at least formally, the problem of convergence will be discussed

later). This is obtained by localizing in the neighbourhoods Mk and Mr near infinity. One can

construct some operators Ek
∞

(λ) on Mk (k = 1, . . . ,n− 1) and E
r
∞

(λ) on Mr such that

(∆Mk −λ(n−λ))E
k
∞

(λ) = χk + Kk
∞

(λ),

(∆Mr −λ(n−λ))E
r
∞(λ) = χ

r
+ K

r
∞(λ)

with Kk
∞

(λ), Kr
∞

(λ) having smooth Schwartz kernels Kk
∞

(λ; w,w′) and Kr
∞

(λ; w,w′)) which vanish

at all order when ρ(w) → 0.

The first step of the parametrix construction of Ek
∞

(λ) is to take a smooth function χkL with

support in Mk which is equal to 1 in {x
2

+ |y|2 > 4} such that χkLχ
k

= χk and 1 − χkL can be

chosen as a product (see the construction in [8])

1 −χkL(x,y,z) = ψ
k
L(y)φL(x) (4.10)

independent of the variable on T k; then set

Ek0 (λ) := χ
k
LRXk (λ)χ

k, Kk0 (λ) = [∆Xk,χ
k
L]RXk (λ)χ

k

and we obtain (∆Mk −λ(n−λ))E
k
0 (λ) = χ

k
+Kk0 (λ) as a first parametrix in the neighbourhood Mk

of ∂X̄ in X̄. The next steps of the construction in [8, Prop.3.1] involve only some operators with

Schwartz kernels of the same type than Kk0 (λ) but with additional decay at ∂X̄×X̄ in X̄×X̄. The

part of the parametrix on Mr is done as in the work of Guillopé-Zworski [12] (and more generally

[17]) by using at first step

Er0 (λ) := χ
r
LRHn+1 (λ)χ

r, Kr0 (λ) = [∆Hn+1,χ
r
L]RHn+1 (λ)χ

r

with a function χrL which is equal to 1 on the support of χ
r and which can be expressed as a

product χrL(x,y) = φ
r
L(x)ϕ

r
L(y) in Mr. The other steps of the construction in Mr do not make

more singular kernels than Kr0 (λ) appear.

The previous lemma allows to deduce the following

Proposition 4.3. Let θ,θ′ ∈ C∞(X̄) be constant near c and such that supp(θ′)∩c = ∅ and θθ′ = 0.

Then for λ not a resonance, we have

θR(λ)θ′ ∈ Rc
−1ρ′

λ
ρλC∞(X̄ × X̄), θ′R(λ)θ ∈ R′c

−1
ρλρ′

λ
C∞(X̄ × X̄)

and R(λ) has the mapping property

R(λ) : Ċ∞(X̄) → R−1c ρ
λC∞

acc
(X̄). (4.11)



CUBO
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Scattering Theory on Geometrically Finite Quotients 153

Proof : if we carefully look at the expression of K∞(λ) following [8, Prop. 3.1 and 3.5] and we

use previous lemma, it is not difficult to check that

(Ik)
∗
K

k
∞(λ)(Ik )∗ ∈ ρ

∞ρ′
λ
R′c

−1
C∞(X̄ × X̄), (4.12)

(Ik)
∗
K

r
∞(λ)(Ir )∗ ∈ ρ

∞ρ′
λ
C∞(X̄ × X̄). (4.13)

The second statement is essentially well-known (see [8, 12] for instance) and is a direct consequence

of the explicit formula of RHn+1 (λ). To prove the first one, we essentially use Lemma 4.2. It is not

difficult to check (see again [8]) that [∆Xk ,χ
k
L] is a first order operator with smooth coefficients

supported in {1 < x2 + |y|2 ≤ 4, 0 ≤ x} and vanishing at second order at x = 0. Using the

compactification coordinates (t,u) of (2.1), it is also a first order operator with smooth coefficients

supported in {ǫ < t2 + |y|2 ≤ 1, 0 ≤ t} for some ǫ > 0 and vanishing at second order at t = 0,

moreover its support does not intersect the support of χk. Therefore, using (4.3) in Lemma 4.2 we

easily deduce that

(Ik)
∗
[∆Xk,χ

k
L]RXk (λ)χ

k
(Ik)∗ ∈ ρ

λ+2ρ′
λ
R′c

−1
C∞(X̄ × X̄). (4.14)

Now the iterative construction of [8, Prop. 3.1] corresponds to capture the Taylor expansion of

this term at ρ = 0 and the remaining error terms at each step are like (4.14) but with more decay

in ρ; this finally implies (4.12). The terms appearing in the expression of Ek
∞

(λ) in [8, Prop. 3.1],

are thus χkLRXkχ
k plus some operators whose Schwartz kernels are in ρλ+2ρ′

λ
R′c

−1
C∞(X̄k ×X̄k).

Therefore Ek∞(λ) satisfies exactly the same properties than RXk (λ) described in Lemma 4.2.

By standard pseudo-differential calculus on compact manifolds, we can obtain the compact

part of the parametrix Ei∞(λ) so that

(∆X −λ(n−λ))E
i
∞(λ) = χ + K

i
∞(λ)

with Ki
∞

(λ) having a smooth kernel with compact support in X ×X and Ei
∞

(λ) being a pseudo-

differential operator of order −2 supported in a compact set of X ×X.

Thus we obtain

(∆X −λ(n−λ))E∞(λ) = 1 + K∞(λ)

with

E∞(λ) := E
i
∞

(λ) +
∑

α=1,...,n−1,r

(Iα)
∗
E

α
∞

(λ)(Iα)∗,

K∞(λ) := K
i
∞ +

∑

α=1,...,n−1,r

(Iα)
∗
K

α
∞(λ)(Iα)∗.

Using Lemma 4.2, (4.12), (4.13) and the explicit formulae of the regular terms in Er
∞

(λ) in [8, 12]

it is straightforward to see that

K∞(λ) ∈ ρ
∞ρ′

λ
R′c

−1
C∞(X̄ × X̄) (4.15)

θE∞(λ)θ
′ ∈ Rc

−1ρλρ′
λ
C∞(X̄ × X̄), θ′E∞(λ)θ ∈ ρ

λρ′
λ
R′c

−1
C∞(X̄ × X̄). (4.16)



154 Colin Guillarmou CUBO
11, 5 (2009)

Moreover using Lemma 4.2 for the mapping properties of the cusps terms and [7, Prop. 3.1] for

the mapping properties of the regular terms, we have

E∞(λ) : Ċ
∞

(X̄) → ρλR−1c C
∞
acc

(X̄). (4.17)

We can then write

R(λ) = E∞(λ) − E∞(λ)K∞(λ) + E∞(λ)K∞(λ)(1 + K∞(λ))
−1

K∞(λ) (4.18)

and (1 + K∞(λ))
−1

= 1 + F(λ) with

F(λ) = −K∞(λ) − K∞(λ)F(λ).

This proves that F(λ) is Hilbert-Schmidt on ρNL2(X) for ℜ(λ) > n−1
2

and N large, since K∞(λ)

is. Using that ρ′
n
R′c

−1
is bounded, the composition K∞(λ)F(λ)K∞(λ) has a Schwartz kernel in

the same class than K∞(λ) (and K∞(λ)
2 too). In view of its construction, we see that the range

of K∞(λ) is composed of functions with support in X̄ \ c, thus we can find a smooth function

θ′ ∈ C∞(X̄) with supp(θ′)∩c = ∅ such that θ′K∞(λ) = K∞(λ). Thus if θ is a function in C
∞

(X̄)

such that θ = 1 near c and θθ′ = 0 we have from (4.16), (4.15) that

θE∞(λ)K∞(λ) ∈ ρ
λρ′

λ
R−1c R

′
c
−1
C∞(X̄ × X̄). (4.19)

Now we can for example use Mazzeo’s composition results in [15] to deal with the regular terms

(E
i
∞

(λ) + (Ir )
∗
E

r
∞

(λ)(Ir )∗)K∞(λ) ∈ ρ
λρ′

λ
C∞(X̄ × X̄).

Then (1 − θ)(Ik)
∗
E

k
∞(λ)(Ik)∗K∞(λ) can be studied exactly with the same method than for the

proof of (4.2) in Lemma 4.2 and we see that

(1 −θ)(Ik)
∗
E

k
∞

(λ)(Ik )∗K∞(λ) ∈ ρ
λρ′

λ
R′c

−1
C∞(X̄ × X̄)

and we conclude, using (4.19), that

E∞(λ)K∞(λ) ∈ ρ
λρ′

λ
R−1c R

′
c
−1
C∞(X̄ × X̄)

and the same holds for E∞(λ)K∞(λ)(1 + F(λ))K∞(λ). We have completed the proof in view of

(4.18) and the symmetry of the resolvent kernel.

Moreover we have also proved that

R(λ) − E∞(λ) ∈ (ρρ
′
)
λ
(RcR

′
c)

−1C∞(X̄ × X̄). (4.20)

The mapping property of R(λ) is then easily deduced from (4.18) and (4.17) since K(λ) maps

ρNL2(X) to Ċ∞(X̄) if N ≫|ℜ(λ)| in view of the form (4.15) of its kernel. �

Remark: we did not study the convergence problem of the infinite order parametrix E∞(λ) but

to avoid this problem, it suffices to take the parametrix EN (λ) of [8] for large N and the same proof

actually would show the same results for R(λ) but with CM regularity for some M > N −C|ℜ(λ)|

(with C > 0) instead of C∞ regularity. Since it is true for all N, we get the same results.



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Scattering Theory on Geometrically Finite Quotients 155

5 Poisson Operator, Eisenstein Function

5.1 Poisson operator

Using the product decomposition of the metric in Lemma 2.2, an indicial equation for the Lapla-

cian and the mapping property of the resolvent, we can construct a Poisson operator following the

method of Graham-Zworski [7].

Actually, we now work with the special boundary defining function ρ but every other choice

of boundary defining function ρ̂ ∈ C∞
acc

(X̄) defined in Lemma 2.2 would induce an equivalent

construction for the Poisson operator. We will simply add the necessary arguments when the gen-

eralization is not transparent.

With the metric under the form (2.13), the Laplacian is

∆X = −(ρ∂ρ)
2

+ nρ∂ρ −
1

2
Tr(h−1(ρ).∂ρh(ρ))ρ

2∂ρ + ρ
2
∆h(ρ). (5.1)

In the neighbourhood Mk of the cusp ck this gives

∆X = −(ρ∂ρ)
2
+ nρ∂ρ − 2k(ρ

2
+ |u|2)−1ρ3∂ρ + ρ

2
∆h(ρ)

with h(ρ) = du2 +(ρ2 +|u|2)2dz2 a metric on {0 < |u| < 1}×T kz , and by an elementary computation

we obtain

Rc∆XR
−1
c = −(ρ∂ρ)

2
+ nρ∂ρ + ρ

2
(∆u + (ρ

2
+ |u|2)−2∆z ) (5.2)

where ∆u, ∆z are the flat Laplacians on R
n−k
u ,T

k
z . Similarly with a function ρ̂ of Lemma 2.2 we

have

∆X = −(ρ̂∂ρ̂)
2

+ nρ̂∂ρ̂ −
1

2
Tr(ĥ−1(ρ̂).∂ρ̂ĥ(ρ̂))ρ̂

2∂ρ̂ + ρ̂
2
∆h(ρ̂) + O(ρ̂

∞
).

and in coordinates (ρ̂,v,ζ) near ck, we see from (2.11) that

Rc∆XR
−1
c = −(ρ̂∂ρ̂)

2
+ nρ̂∂ρ̂ + P1 + P2 + ρ̂

2e−2ωr−4c ∆ζ + O(ρ̂
∞

)

for some differential operators

P1 = P1(ρ̂,v, ρ̂
2∂ρ̂, ρ̂∂v), P2 = P2(ρ̂,v,ζ, ρ̂∂v, ρ̂∂ζ ) = O(r

∞
c )

of order 2, with P2 (resp. P1) having smooth coefficents on X̄ (resp. smooth outside ck). By

making the same change of coordinates (2.9) in (5.2), it would give some differential operators

with smooth coefficients at ck except the term with ∆ζ thus P1 has to be smooth at ck.

We now use Graham-Zworski’s construction [7] and we refer the reader to their paper for

additional details. If f ∈ C∞
acc

(∂X̄) we deduce from (5.1) and (5.2) the indicial equation in {ρ < ǫ}

(∆X −λ(n−λ))ρ
n−λ+jR−1c f − j(2λ−n− j)ρ

n−λ+jR−1c f ∈ ρ
n−λ+j+1R−1c C

∞
acc

(X̄). (5.3)



156 Colin Guillarmou CUBO
11, 5 (2009)

Here, the key fact is that the singular term r−4c ∆z applied to f ∈ C
∞
acc(∂X̄) gives a functions in

Ċ∞c (X̄) by (2.7). Therefore for all f ∈ R
−1
c C

∞
acc

(∂X̄) one can construct by induction and Borel

lemma (see again [7]) a function Φ(λ)f ∈ ρn−λR−1c C
∞
acc

(X̄) for λ ∈ C \ 1
2
(n + N) such that

(∆X −λ(n−λ))Φ(λ)f ∈ Ċ
∞

(X̄), ρλ−nΦ(λ)f|ρ=0 = f.

By construction, we have the formal Taylor expansion

Φ(λ)f ∼ ρn−λ
∞∑

j=0

ρ2jcj,λPj,λf, ∀f ∈ C
∞
acc

(∂X̄) (5.4)

where Pj,λ is a differential operator on B which is polynomial in λ and

cj,λ := (−1)
j Γ(λ−

n
2
− j)

22jj!Γ(λ− n
2
)
.

Now we can set for λ /∈ 1
2
(n + N) and λ not a resonance

P(λ)f = Φ(λ)f −R(λ)(∆X −λ(n−λ))Φ(λ)f (5.5)

which satisfies 




(∆X −λ(n−λ))P(λ)f = 0

P(λ)f = ρn−λF(λ,f) + ρλG(λ,f)

F(λ,f),G(λ,f) ∈ R−1c C
∞
acc

(X̄)

F(λ,f)|ρ=0 = f

(5.6)

using Proposition 4.3. We have defined a family of operators

P(λ) : R−1c C
∞
acc

(∂X̄) → ρn−λR−1c C
∞
acc

(X̄) + ρλR−1c C
∞
acc

(X̄)

and we will now prove the uniqueness of an operator satisfying (5.6) in {ℜ(λ) ≥ n
2
}. The principle

is the same than in [7]: if ℜ(λ) > n
2
, λ not a resonance and P1(λ)f, P2(λ)f are two solutions of

(5.6), then the previous indicial equation shows that P1(λ)f − P2(λ)f ∈ ρ
λR−1c C

∞
(X̄) but this

function is in L2(X) using (2.6) so this must be 0; to treat the case ℜ(λ) = n
2
, we use a boundary

pairing Lemma like Proposition 3.2 of [7]:

Lemma 5.1. For i = 1, 2, let ui = ρ
n−λFi + ρ

λGi some functions satisfying

(∆X −λ(n−λ))ui = ri ∈ Ċ
∞

(X̄)

with Fi,Gi ∈ R
−1
c C

∞
(X̄), then we have for ℜ(λ) = n

2
and λ 6= n

2

∫

X

(u1r2 −r1u2) dvolg = (2λ−n)

∫

B

(F1|BF2|B −G1|BG2|B) dvolh0



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Scattering Theory on Geometrically Finite Quotients 157

Proof : we apply Green Lemma in Xǫ = {ρ ≥ ǫ}

∫

Xǫ

(u1r̄2 −u2r̄1) dvolg = ǫ
−n+1

∫

ρ=ǫ

(u1∂ρū2 − ū2∂ρu1) dvolh(ǫ) (5.7)

and we will take the limit as ǫ → 0. Using the asymptotics of u1,u2 we get

u1∂ρū2 − ū2∂ρu1 = (2λ−n)ρ
n−1

(F1F2 −G1G2) + ρ
n
(G1∂ρG2 −G2∂ρG1 + F1∂ρF2 −F2∂ρF1).

Recall from (2.5) that dvolh(ǫ) = Rc(ǫ)
2µ∂X̄ with Rc(ǫ) = (|u|

2
+ ǫ2)

1

2 in the neighbourhood Bk of

the cusp submanifold ck, so the only terms in the right hand side of (5.7) for which the limit are

not apparent are

ǫ

∫

ρ=ǫ

(G1∂ρG2 −G2∂ρG1) dvolh(ǫ), ǫ

∫

ρ=ǫ

(F1∂ρF2 −F2∂ρF1) dvolh(ǫ).

The study of both terms when ǫ → 0 is the same and can be clearly reduced to the limit of

∫

T k

∫

|u|≤1

G1(ǫ,u,z)ǫ∂ǫG2(ǫ,u,z)(|u|
2

+ ǫ2)kduRn−kdzT k (5.8)

when ǫ → 0, Gi(ρ,u,z) being the function Gi in the coordinates of the neighbourhood Bk of ck.

Using that on Gi ∈ R
−1
c C

∞
(X̄), it suffices to show that the limit of

∫

|u|≤1

ǫ∂ǫ[(|u|
2

+ ǫ2)−
k
2 ](|u|2 + ǫ2)

k
2 duRn−k

is 0 when ǫ → 0 to prove that the limit of (5.8) is 0. Now this last integral is equal to

C

∫ 1

0

ǫ2(r2 + ǫ2)−1rn−k−1dr ≤ Cǫ

∫ ∞

0

(1 + r2)−1dr

for a constant C, this finally proves the lemma. �

Now using this lemma with u2 = R(n−λ)ϕ for ϕ ∈ Ċ
∞

(X̄) and u1 = P1(λ)f − P2(λ)f this

proves that 〈u1,ϕ〉 = 0 for all ϕ ∈ Ċ
∞

(X̄), thus u1 = 0. As a conclusion, we have

Proposition 5.2. For ℜ(λ) ≥ n
2
, λ /∈ 1

2
(n + N0), λ(n−λ) /∈ σpp(∆X ) there exists a unique linear

operator

P(λ) : R−1c C
∞
acc

(∂X̄) → ρn−λR−1c C
∞
acc

(X̄) + ρλR−1c C
∞
acc

(X̄)

analytic in λ and solution of the Poisson problem (5.6). It is given by (5.5) and called Poisson

operator.

By (5.5) it admits a meromorphic continuation with poles of finite multiplicity to C\1
2
(n+N0).



158 Colin Guillarmou CUBO
11, 5 (2009)

5.2 Eisenstein functions

In this part, we define Eisenstein functions as a weighted restriction of the Schwartz kernel of the

resolvent at B×X and we prove that they are the Schwartz kernel of the transpose of the Poisson

operator.

As a consequence of Proposition 4.3 and (4.20) we first obtain the

Corollary 5.3. The Eisenstein function E(λ) := (ρ−λR(λ))|B×X is well defined, meromorphic in

λ ∈ C and satisfies

E(λ) ∈ Rc
−1C∞(∂X̄ ×X). (5.9)

Moreover, if Emod(λ) is the ‘model Eisenstein function’ defined by

Emod(λ) := (ρ
−λ

E∞(λ))|B×X

then

E(λ) −Emod(λ) ∈ ρ
′λ

(RcR
′
c)

−1C∞(∂X̄ × X̄). (5.10)

Let EXk (λ) be the Eisenstein function for the model space Xk obtained from (4.4) and (4.6)

(recall that ρ = t = x
x2+|y|2

with our choice in Lemma 2.2)

EXk (λ; y,z; x
′,y′,z′) = |y|2λx′

λ
r−2λ+k

∑

m∈Zk

eiωm.(z−z
′)F0,λ(r|ωm|)

for y 6= 0, where by convention r = (|y−y′|2 + x′
2
)

1

2 denotes here the restriction of r to x = 0. In

the compactification coordinates (t,u) of (2.1) this gives

EXk (λ; u,z; t
′,u′,z′) = t′

λ
r−2λ+k|u|−2λ(t′

2
+ |u′|2)−λ

∑

m∈Zk

eiωm.(z−z
′)F0,λ(r|ωm|) (5.11)

and r is expressed in these coordinates by

r2 =
t′

2
+ |u−u′|2

|u|2(t′2 + |u′|2)
. (5.12)

Similarly let EHn+1 (λ) be the Eisenstein function on H
n+1

EHn+1 (λ; y; x
′,y′) =

π−
n
2 Γ(λ)

(2λ−n)Γ(λ− n
2
)

x′
λ

(|y −y′|2 + x′2)λ
. (5.13)

Using the construction of the parametrix for the resolvent, we can deduce an expression for

the model Eisenstein function

Emod(λ) =
∑

α=1,...,n−1,r

(ια)
∗Eαmod(λ)(Iα)∗ (5.14)



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Scattering Theory on Geometrically Finite Quotients 159

with ια := Iα|ρ=0 and in Mk,Mr

Ekmod(λ; y,z; w
′
) := ψkL(y)EXk (λ; y,z; w

′
)χk(w′),

Ermod(λ; y; w
′
) := ψrL(y)γr(y)

−λEHn+1 (λ; y; w
′
)χr(w′). (5.15)

with ρ(x,y) = xγr(y) + O(x) in Mr for some positive smooth function γr in Br and ψ
α
L defined in

(4.10).

We show that the Eisenstein functions can be viewed as a Schwartz distributional kernel of

an operator, that we also denote E(λ), mapping Ċ∞(X̄) to C−∞(B̄), actually with weighted L2

continuity results.

Lemma 5.4. There exists C > 1 such that for |ℜ(λ) − n
2
| ≤ C−1N,

E(λ) : ρNL2(X) → L2(B)

is a meromorphic family of Hilbert-Schmidt operators with poles of finite multiplicity, included in

the set of resonances. Moreover for ℜ(λ) < 0 and λ not a resonance, (b,w) → ρ(w)−λE(λ; b; w) is

a continuous function on B × (X̄ \ c).

Proof : the terms E(λ) − Emod(λ) and (ιr )
∗Ermod(λ)(Ir )∗ in E(λ) clearly satisfy those two

properties, we thus only have to deal with Ekmod(λ) in Xk. From (5.11) and (5.12) we have

|t′
N
EXk (λ; u,z; t

′,u′,z′)| ≤
t′
ℜ(λ)+N

(|u−u′|2 + t′
2
)

k
2
−ℜ(λ)

|u|k|u′|k

∑

m∈Zk

|F0,λ(r|ωm|)|.

When r|ωm| > 1, the classical estimate |Ks(z)| ≤ Ce
−Cℜ(z) for ℜ(z) > 1 (with C > 0 depending

on s) on Mac Donald’s function shows that |F0,λ(r|ωm|)| ≤ e
−Cr|ωm| thus

∑

|ωm|>1/r

|F0,λ(r|ωm|)| ≤ Cr
−k ≤ Ct′

−k

where C depends on λ. Therefore we get for N > 4|ℜ(λ)|

|t′
N
EXk (λ)| ≤ Ct

′
N
2 |u|−k|u′|−k +

t′
ℜ(λ)+N

(|u−u′|2 + t′
2
)

k
2
−ℜ(λ)

|u|k|u′|k

∑

|ωm|≤1/r

|F0,λ(r|ωm|)|. (5.16)

Now for r|ωm| ≤ 1 we use the definition (6.4) of Mac Donald function Ks(z) to decompose

F0,λ(r|ωm|) under the form

F0,λ(r|ωm|) = c(λ)(ϕ−λ+ k
2

(r2|ωm|
2
) + r2λ−k|ωm|

2λ−kϕλ− k
2

(r2|ωm|
2
))

with ϕs(x) smooth on x ∈ [0,∞) and c(λ) constant depending on λ. The term coming from ϕ−λ+ k
2

is treated exactly as before (the part with r|ωm| > 1) and for the term coming from ϕλ− k
2

we have

∑

|ωm|<1/r

(r|ωm|)
2ℜ(λ)−k|ϕλ− k

2

(r2|ωm|
2
)| ≤

{
C(r−k + r2ℜ(λ)−2k ) if ℜ(λ) − k

2
≤ 0

Cr−k if ℜ(λ) − k
2
> 0



160 Colin Guillarmou CUBO
11, 5 (2009)

for some C > 0 depending on |λ|. In view of (5.16), we conclude that for N > 4|ℜ(λ)| + 2k

|(ιk)
∗t′

N
EXk (λ)(Ik )∗| ≤ Cρ

′
N
2 R−1c R

′
c
−1

and this function is in L2(B×X) if N is large enough using (2.6) (here Rc denotes the restriction

of Rc to B × X). The meromorphic property and the finiteness of the poles multiplicity comes

from the discussion before the Lemma, using the formulae for the model Eisenstein functions and

the fact that the poles of the resolvent have finite multiplicity.

The second statement of the Lemma is essentially treated in the same way. Using that for

ℜ(λ) < 0

r−2λ+kF0,λ(r|ωm|) = c(λ)(r
−2λ+kϕ

−λ+ k
2

(r2|ωm|
2
) + |ωm|

2λ−kϕλ− k
2

(r2|ωm|
2
))

is continuous in (u,t′,u′) ∈ {u 6= 0,u′ 6= 0, t′
2

+ |u′|2 < 1, |u| < 1} (the power in r−2λ+k being

negative) and that the sum
∑

m r
−2λ+kF0,λ(r|ωm|) is locally uniformly convergent in the same set

by previous estimates, we deduce that t′
−λ
EXk (λ; u,z; t

′,u′,z′) is also continous there and this

achieves the proof. �

The transpose tE(λ) is then well-defined from from L2(B) to ρ−NL2(X) for some N depending

on λ and its kernel is E(λ; w,b). Let ϕ ∈ Ċ∞(X̄) and f ∈ Ċ∞c (∂X̄) ≃ Ċ
∞

(B̄), then for ℜ(λ) = n
2

we use Lemma 5.1, identity R(λ) = tR(λ) = R(n−λ)∗ and Lemma 5.4 to deduce

∫

X

ϕ̄(P(λ)f) dvolg = (2λ−n)

∫

B

f(ρλ−nR(n−λ)ϕ)|B dvolh0

= (2λ−n)

∫

B

f(ρ−λR(λ)ϕ̄)|B dvolh0

= (2λ−n)

∫

B

f(E(λ)ϕ̄) dvolh0

which proves

Lemma 5.5. The Schwartz kernel of P(λ) is (2λ−n)E(λ; w; b) ∈ C∞(X ×B).

This also implies that P(λ) admits a meromorphic continuation to C with poles of finite

multiplicity, and in particular it is analytic in {ℜ(λ) > n
2
} except a finite number of poles at points

λ0 such that λ0(n−λ0) ∈ σpp(∆X ). By mimicking the proof of Graham-Zworski [7, Prop. 3.5] it

is straightforward to see that, for f ∈ R−1c C
∞
acc

(∂X̄), P( n
2

+ k)f has log(ρ) terms in the asymptotic

expansion and it is the unique solution of the problem






(∆X −
n2

4
+ k2)P( n

2
+ k)f = 0

P(
n
2

+ k)f = ρ
n
2
−kFk(f) + ρ

n
2
+k

log(ρ)Gk(f)

Fk(f),Gk(f) ∈ R
−1
c C

∞
acc

(X̄)

Fk(f)|ρ=0 = f

(5.17)



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Scattering Theory on Geometrically Finite Quotients 161

The Eisenstein functions are linked to the spectral projectors (via Stone’s formula) of ∆X in

the following sense

Proposition 5.6. If ℜ(λ) = n
2

and λ 6= n
2

then

R(λ; w; w′) −R(n−λ; w; w′) = (n− 2λ)

∫

B

E(λ; b; w′)E(n−λ; b; w) dvolh(b) (5.18)

where h = (ρ2g)|B. Moreover there exists C > 1 such that for N large, we have

R(λ) −R(n−λ) = (2λ−n)tE(n−λ)E(λ)

in the strip |ℜ(λ)| ≤ C−1N as operators from ρNL2(X) to ρ−NL2(X).

Proof : the proof of (5.18) contains nothing more than the proof of Theorem 1.3 of [3] or

Proposition 2.1 of [11] in a simpler case. Note that the convergence of the integral in (5.18) is

insured by (5.9) and (2.5). The second part of the Proposition is a consequence of the mapping

properties of R(λ),E(λ) proved before. �

Combined with Lemma 5.4, this relation implies that E(λ) and R(λ) have same poles, except

possibly at the points λ such that λ(n−λ) ∈ σpp(∆X ).

6 Scattering Operator

Using notations of (5.6), we can define the scattering operator as the linear operator

S(λ) :

{
R−1c C

∞
acc

(∂X̄) → R−1c C
∞
acc

(∂X̄)

f → G(λ,f)|B
(6.1)

for ℜ(λ) ≥ n
2
, λ /∈ 1

2
(n + N) and λ not a resonance. With (5.5), one obtains a meromorphic

continuation of S(λ) to C. Like P(λ), the scattering operator certainly depends on the choice of

boundary defining function (here ρ), but any other choice ρ̂ = eωρ ∈ C∞
acc

(X̄) of Lemma 2.2 induces

an equivalent construction and two corresponding scattering operators S(λ) and Ŝ(λ) are related

by the covariant rule

Ŝ(λ) = e−λω0S(λ)e(n−λ)ω0, ω0 = ω|∂X̄,

this is a trivial consequence of uniqueness of solution of Poisson problem. Therefore it suffices in

this section to deal with the special boundary defining function ρ.

From Lemma 5.5, (5.5) and (6.1), we deduce that for f ∈ Ċ∞c (∂X̄) ≃ Ċ
∞

(B̄) and ℜ(λ) < 0

S(λ)f = lim
ρ→0

[ρ−λ((2λ−n)tE(λ)f − Φ(λ)f)] = (2λ−n) lim
ρ→0

[ρ−λ(tE(λ)f)] (6.2)



162 Colin Guillarmou CUBO
11, 5 (2009)

which is well defined in view of the continuity of E(λ; b; w′) proved in Lemma 5.4. As a consequence

the distributional kernel of S(λ) on B is

S(λ; b; b′) = (2λ−n) lim
w′→b′

(ρ(w′)−λE(λ; b; w′))

which can be rewritten using the symmetry of the resolvent kernel as the restriction

S(λ) = (2λ−n)(ρ−λρ′
−λ
R(λ))|ρ=ρ′ =0 (6.3)

for ℜ(λ) < 0 and λ not resonance. Moreover we deduce from (4.20) that

S(λ) − (ρ−λρ′
−λ

E∞(λ))|ρ=ρ′ =0 ∈ R
−1
c R

′
c
−1
C∞(∂X̄ ×∂X̄)

which is easily seen to be compact on L2(B) in view of (2.5), and this term extends meromorphi-

cally to C with poles of finite multiplicity.

We want to study the structure of the extendible distribution (6.3) on B̄×B̄, which continues

meromorphically to C; it suffices actually to describe the singular part (ρ−λρ′
−λ

E∞(λ))|ρ=ρ′ =0 of

S(λ). To analyze this singular part of S(λ) in the neighbourhood of the cusp submanifolds, it

turns out to be more convenient to work in the neighbourhood Mk with the coordinates (x,y,z)

than in their compactified version (t,u,z). Indeed we will see that, up to conformal factors, the

scattering operator for the model Xk = Γk\H
n+1 is ∆

λ− n
2

Yk
where again Yk = R

n−k ×T k with the

flat metric. This is what Froese-Hislop-Perry used in [3] in dimension 3.

Using Fourier transform in the (y,z) variable on Xk we see that the Laplacian on Xk is

transformed into the one dimensional operator

Pξm = −x
2∂2x + (n− 1)x∂x + x

2|ξm|
2

with ξm = (ξ,ωm). We easily deduce that the resolvent can be expressed by

RXk (λ; w,w
′
) = −(xx′)

n
2

∑

m∈Z

∫

Rn−k

eiξm.(y−y
′,z−z′)Gξm (λ; x,x

′
)dξ

Gξm (λ; x,x
′
) := Kλ− n

2
(|ξm|x)Iλ− n

2
(|ξm|x

′
)H(x−x′) + Kλ− n

2
(|ξm|x

′
)Iλ− n

2
(|ξm|x)H(x

′ −x)

with H the Heaviside function, (w; w′) = (x,y,z; x′,y′,z′) the coordinates on Xk × Xk and

Iν (z),Kν (z) the modified Bessel functions. Therefore using that ρ =
x

x2+|y|2
and

Iν (z) =
2
−νzν

νΓ(ν)
+ O(zℜ(ν+2)), Kν (z) = −

ν

2
Γ(ν)Γ(−ν)(Iν (z) − I−ν (z)) (6.4)

as z → 0, we obtain for ℜ(λ) < 0 (using {ρ = 0} = {x = 0} on B)

EXk (λ; y
′,z′; w) =

−|y′|2λ2
n
2
−λ

Γ(λ− n
2

+ 1)
x

n
2

∑

m∈Z

∫

Rn−k

eiξm .(y−y
′,z−z′)|ξm|

λ− n
2 Kλ− n

2
(|ξm|x)dξ (6.5)



CUBO
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Scattering Theory on Geometrically Finite Quotients 163

and

SXk (λ; y,z; y
′,z′) := (2λ−n)[ρ(x,y)−λEXk (λ; y

′,z′; x,y,z)]|x=0

= 2
n−2λ

Γ(
n
2
−λ)

Γ(λ− n
2
)
|y|2λ|y′|2λ

∑

m∈Z

∫

Rn−k

eiξm.(y−y
′,z−z′)|ξm|

2λ−ndξ

where this last sum-integral is understood (by splitting the term with ωm = 0 and the terms with

ωm 6= 0) as the function on R
n−k
y ×T

k
z × R

n−k
y′ ×T

k
z′

2
2λ−nπ−

n−k
2 Γ(λ− k

2
)

Γ(
n
2
−λ)

|y −y′|−2λ+k +
∑

m 6=0

∫

Rn−k

eiξm .(y−y
′,z−z′)|ξm|

2λ−ndξ

which is continuous on {y 6= 0,y′ 6= 0}. This last function continues meromorphically to λ ∈ C in

the distribution sense thus

Skmod(λ; y,z; y
′,z′) := [ρ(x′,y′)

−λ
Ekmod(λ; y,z; x

′,y′,z′)]|x=0 = ψ
k
L(y)SXk (λ; y,z; y

′,z′)ψk(y′)

(6.6)

continues meromorphically to C as a distribution. Note that the measure dvolh0 on Yk is

dvolh0 = |y|
−2ndydz.

To work on Yk = R
n−k
y ×T

k
z with the natural measure dydz corresponding to the flat metric h̃0,

we have to multiply the kernel of SXk (λ) by |y|
−n|y′|−n, thus (6.6) can be rewritten, acting on

L2(Yk,dydz)

Skmod(λ) = c(λ)ψ
k
L|y|

2λ−n
∆

λ− n
2

Yk
|y|2λ−nψk with c(λ) := 2n−2λ

Γ(
n
2
−λ)

Γ(λ− n
2
)
. (6.7)

Note that it has poles at λ = n
2

+ j (with j ∈ N) with residue the differential operator on Yk

Res n
2
+j (S

k
mod(λ)) =

(−1)j+12−2j

j!(j − 1)!
ψkL|y|

2j
∆

j
Yk
|y|2jψk on L2(Yk,dydz).

For the singularity of the kernel of S(λ) in the regular neighbourhood Br on L
2
(Br, dvolh0 )

(to see it acting on L2(Br, dvolh̃0 ) it suffices to multiply the kernel by (rcr
′
c)

n) we define the model

scattering operator using (5.13)

SHn+1(λ; y; y
′
) := (2λ−n)[x′

−λ
EHn+1 (λ; y; x

′,y′)]|x′=0 =
π−

n
2 Γ(λ)

Γ(λ− n
2
)
|y −y′|−2λ

and we get from (5.15)

Srmod(λ; y; y
′
) := [ρ(x′,y′)

−λ
Ermod(λ; y; x

′,y′)]|x′=0 =
ψrL(y)ψ

r
(y′)

γr(y)λγr(y′)λ
SHn+1 (λ; y; y

′
), (6.8)

which continues meromorphically to C with poles at n
2

+ j (with j integers) and residue

Res n
2
+j (S

r
mod(λ)) =

(−1)j+12−2j

j!(j − 1)!
ψrLγ

−
n
2
−j

r ∆
j
Rn
γ
−

n
2
−j

r ψ
r.



164 Colin Guillarmou CUBO
11, 5 (2009)

With notations of (6.8), (6.6) we can now define the model scattering operator

Smod(λ) :=
∑

α=1,...,n−1,r

(ια)
∗Sαmod(λ)(ια)∗ (6.9)

and we have

S(λ) −Skmod(λ) ∈ R
−1
c R

′
c
−1
C∞(∂X̄ ×∂X̄)

which is a compact operator on L2(B). From this study, it is straightforward to check that S(λ)

is a bounded operators on L2(B) in {ℜ(λ) ≤ n
2
} (and λ not resonance).

We summarize this discussion in the following

Lemma 6.1. S(λ) is meromorphic in C as an operator acting on R−1c C
∞
acc

(∂X̄), with Schwartz

kernel the meromorphic continuation from {ℜ(λ) < 0} to C of the distribution

(2λ−n)(ρ−λρ′
−λ
R(λ))|B×B ∈ C

−∞
(X̄ × X̄).

Its poles in {ℜ(λ) ≤ n
2
} are included in the set of resonances and have finite multiplicity, whereas

the poles in {ℜ(λ) > n
2
} are first order poles with residue

Resλ0S(λ) =

{
−

(−1)j+12−2j

j!(j−1)!
Pj + Πλ0 if λ0 =

n
2

+ j,j ∈ N

Πλ0 if λ0 /∈
n
2

+ N

where Pj is the differential operator on (B,h0) with principal symbol σ0(Pj ) = |ξ|
2j
h0

, defined by

[Res n
2
+jρ

−λ
Φ(λ)]|ρ=0 =

(−1)j2−2j

j!(j − 1)!
Pj

and Πλ0 is a finite-rank operator with Schwartz kernel 2j
(
(ρρ′)−λ0 Resλ0R(λ)

)
|B×B satisfying

rank Πλ0 = dim kerL2 (∆X −λ0(n−λ0)).

Proof : the meromorphic property of S(λ) and its Schwartz kernel have been discussed, the

statement about the poles outside {ℜ(λ) ≤ n
2
} is also clear by (5.5) . For the case of a pole λ0

with ℜ(λ0) >
n
2
, the proof is the same than [7, Prop 3.6]. The fact about the rank of Πλ0 is quite

straightforward by mimicking the proof of [10, Th. 1.1]: we only need the indicial equation (5.3)

and that there is no solution of (∆X − λ0(n − λ0))u = 0 with u ∈ Ċ
∞

(X), this last fact being

already proved by Mazzeo [16]. �

Note that this Lemma also holds for any boundary defining function ρ̂ ∈ C∞
acc

(X̄). The oper-

ators Pj will be discussed in next section.

We now give functional relations for Eisenstein functions and scattering operator:



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Scattering Theory on Geometrically Finite Quotients 165

Proposition 6.2. If ℜ(λ) < 0, we have for w ∈ X, b′ ∈ B,

E(λ; b′; w) = −

∫

B

S(λ; b′; b)E(n−λ; b; w) dvolh0 (b)

and there exists C > 1 such that for N large the meromorphic identity

E(λ) = −S(λ)E(n−λ) (6.10)

holds true in the strip −C−1N < ℜ(λ) ≤ n
2

as operators from ρNL2(X) to L2(B).

Proof : if for w ∈ X fixed and ℜ(λ) < 0 we multiply (5.18) by ρ(w′)−λ and take the limit

w′ → b′ ∈ B, then we obtain the first result using the symmetry of the resolvent kernel (which also

induces the symmetry of the kernel of S(λ)). The next part is just a meromorphic continuation

using mapping properties of E(λ) and S(λ). �

We deduce easily from this Proposition and Proposition 5.6 the

Corollary 6.3. If λ0 is such that ℜ(λ0) ≤
n
2
, λ0(n−λ0) /∈ σpp(∆X ) and S(λ) holomorphic at λ0,

then λ0 is not a resonance.

Here is another inmportant property of S(λ):

Proposition 6.4. For ℜ(λ) = n
2
, S(λ) is invertible on L2(B) and we have

S(λ)−1 = S(n−λ) = S(λ)∗

Proof : the unitarity of S(λ) on the critical line comes directly from the density of Ċ∞(B̄) ⊂

C∞
acc

(∂X̄) in L2(B) and Lemma 5.1 whereas the equation S(λ)−1 = S(n−λ) is a consequence of

the definition of S(λ) and again the density of C∞
acc

(∂X̄) in L2(B). �

We give a description of the scattering operator as a pseudo differential in the class defined

in Section 3 and characterized by the type of singularity of its Schwartz kernel on the blown-up

manifold B̄ ×Φ B̄.

Theorem 6.5. Let λ 6∈ n
2

+ N and λ not a resonance, then with definition (3.4), the scattering

operator S(λ) is a Φ-pseudo-differential operator on B̄ of order

S(λ) ∈ Ψ
2λ−n,Eλ
Φ (B̄) + (RcR

′
c)

−1C∞(∂X̄ ×∂X̄)

with respect to volume density dvolh0 , where for k = 1, . . . ,n− 1

Eλ(Fk) = −2λ−k, Eλ(Ik) = −4λ, Eλ(Tk) = Eλ(Bk) = −k.



166 Colin Guillarmou CUBO
11, 5 (2009)

Proof : for technical reasons, we begin by working with the density dvol
h̃0

and it will suffice

to multiply by the correct factors at the end. If η ∈ C∞0 ([0,∞)) is a function which is equal to 1

in a small neighbourhood of 0, we can decompose (6.7) as

Skmod(λ) = c(λ)ψ
k
L|y|

2λ−n
(
η(∆y)∆

λ− n
2

y + (1 −η(∆Yk ))∆
λ− n

2

Yk

)
ψk|y|2λ−n

on L2(Yk,dydz). The first term has a kernel

ψkL(y)ψ
k
(y′)|y|2λ−n|y′|2λ−n

∫

Rn−k

eiξ.(y−y
′)|ξ|2λ−nη(|ξ|)dξ

which is smooth for y,y′ in Rn−k and since it is the Fourier transform of a distribution classically

conormal to 0, it is straightforward to check that it can be expressed by

ψkL(y)ψ
k
(y′)|y|2λ−n|y′|2λ−nFλ(

√
1 + |y −y′|2) (6.11)

with Fλ(x) smooth on [0,∞) and having an expansion

Fλ(x) ∼ x
−2λ+k

∞∑

j=0

aj (λ)x
−j (6.12)

when x → ∞. To describe the singularity of this kernel on the manifold B̄, we use near infinity

the polar coordinates v = |y|−1,ω = y/|y|,v′ = |y′|−1,ω′ = y′/|y′|. Since |y − y′| = | ω
v′
− ω

′

v
| we

deduce that the kernel (6.11)

ψkL(
ω

v
)ψk(

ω′

v′
)v−2λ+nv′

−2λ+n
Fλ



√

1 +

∣∣∣∣
ω

v′
−
ω′

v

∣∣∣∣
2

 .

First, it is clearly smooth in B ×B. By lifting | ω
v′
− ω

′

v
|,v,v′ on B̄ ×Φ B̄ we have that

βΦ
∗




√

1 +

∣∣∣∣
ω

v′
−
ω′

v

∣∣∣∣
2


ρTkρBkρFk ∈ C
∞

(B̄ ×Φ B̄) (6.13)

does not vanish on Fk, Bk, Tk and

βΦ
∗
(vv′)ρ−1

Tk
ρ−1

Bk
ρ−2

Fk
ρ−2

Ik
∈ C∞(B̄ ×Φ B̄) (6.14)

does not vanish on Tk, Bk, Fk, Ik. From this and (6.12) it is straightforward to check that

ψkL|y|
2λ−nη(∆y)∆

λ− n
2

y ψ
k|y|2λ−n ∈ (ρTkρBk )

n−kρ2n−2λ−k
Fk

ρ−4λ+2n
Ik

C∞(B̄ ×Φ B̄). (6.15)

To deal with the term ψkL|y|
2λ−n

(1 −η(∆Yk ))∆
λ− n

2

Yk
ψk|y|2λ−n, we first analyze the operator

A(λ) := ψkL|y|
2λ−n

(1 + ∆Yk )
λ− n

2 ψk|y|2λ−n.



CUBO
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Scattering Theory on Geometrically Finite Quotients 167

For that we can begin to use a partition of unity (θi)i associated to a covering by some euclidian

ball on T k and some functions θ′i ∈ C
∞
0 (T

k
) such that θ′i = 1 on the support of θi, then it is

standard to see that for s ∈ C \ [0,∞)

(∆Y k + 1 −s)
−1

=

∑

i

θ′i(∆Rn + 1 −s)
−1θi + κ(s) (6.16)

κ(s) := (∆Y k + 1 −s)
−1
∑

i

[∆z,θ
′
i](∆Rn + 1 −s)

−1θi.

The kernel κ(s; y,z; y′,z′) of κ(s) can be written as the composition

κ(s; y,z; y′′,z′′) = (∆Yk + 1 −s)
p

∫

Yk

κ1(s; y −y
′,z −z′)κ2(s; y

′ −y′′,z′,z′′)dy′dz′ (6.17)

with

κ1(s; Y,Z) :=
∑

m∈Z

∫

Rn−k

ei(ξ.Y +ωm.Z)(1 + |ξ|2 + |ωm|
2
)
−1−pdξ

κ2(s; y
′ −y′′,z′,z′′) :=

∑

i

[∆z′,θ
′
i(z

′
)](∆Rn + 1 −s)

−1
(y′,z′; y′′,z′′)θi(z

′′
).

Since for some ǫ > 0 we have [∆z′,θ
′
i(z

′
)]θi(z

′′
) = 0 for |z − z′′| < ǫ, it suffices to use the explicit

formula of the resolvent kernel of ∆Rn with Bessel functions to see that κ2(s) is smooth and satisfies

the estimate

|∂αY,z′,z′′κ2(s; Y,z
′,z′′)| ≤ Cα exp(−Cα

√
ℜ(s)(1 + |Y |2))

for ℜ(s) ≥ 1
2

and some constant Cα > 0. The kernel κ1(s) is continuous and uniformly bounded if

p is large enough, moreover it satisfies for all N > 0 the estimate

|∂αY κ2(s; Y,Z)| ≤ Cα,N (1 + |Y |)
−N

for some constant Cα,N > 0. Therefore, using all these estimates and change of variables y
′
= u+y

in (6.17), it is straightforward to check that κ(s; w; w′) is smooth and satisfies the estimate for all

N > 0

|∂αw,w′κ(s; w; w
′
)| ≤ Cα,Ne

−C′
α
ℜ(s)

(1 + |y −y′|)−N. (6.18)

for some constant Cα,N,C
′
α > 0 and using the notation w = (y,z),w

′
= (y′,z′).

Let Γ be the oriented contour in C defined by

Γ = {
1

2
+ rei

π
4 ;∞ > r > 0}∪{

1

2
re−i

π
4 ; 0 < r < ∞}.

As a consequence of (6.16) and using Cauchy formula, the kernel of A(λ) is (with the notation

w = (y,z),w′ = (y′,z′))

A(λ; w; w′) = A1(λ; w,w
′
) + A2(λ; w; w

′
),

A1(λ; w; w
′
) := ψkL(y)|y|

2λ−nψk(y′)|y′|2λ−n
∑

i

θ′i(z)θi(z
′
)

∫

Rn

eiξ.(w−w
′)
(1 + |ξ|2)λ−

n
2 dξ,



168 Colin Guillarmou CUBO
11, 5 (2009)

A2(λ; w; w
′
) := ψkL(y)|y|

2λ−nψk(y′)|y′|2λ−n
∫

Γ

sλ−
n
2 κ(s; w; w′)ds.

To analyze A1(λ), we use the polar coordinates v = |y|
−1,ω = y/|y|,v′ = |y′|−1,ω′ = y′/|y′| in

the y,y′ variables and we have w −w′ = ( ω
v′
− ω

′

v
,z −z′) which vanishes only (and at first order)

on the lifted interior diagonal DΦ of B̄ ×Φ B̄. From the Fourier representation of A1(s; w; w
′
),

we deduce that A1(s; w; w
′
) is a distribution which is polyhomogeneous conormal to DΦ of order

2λ−n, vanishes at all order on the boundaries Tk, Bk, Fk of B̄×Φ B̄ and has a conormal singularity

of order −4λ + 2n at Ik (this last one coming from the term |y|
2λ−n|y′|2λ−n as before):

β∗ΦA1(λ) ∈ ρ
−4λ+2n
Ik

I2λ−n(B̄ ×Φ B̄; DΦ).

The behaviour of A2(λ) comes directly from (6.18) using the polar coordinates and (6.13) and

(6.14) as before: we see that

β∗ΦA2(λ) ∈ ρ
∞
Tk
ρ∞

Bk
ρ∞

Fk
ρ−4λ+2n

Ik
C∞(B̄ ×Φ B̄)

thus

β∗ΦA(λ) ∈ ρ
−4λ+2n
Ik

I2λ−n(B̄ ×Φ B̄; DΦ). (6.19)

For N > ℜ(λ) − n
2
, we have

Skmod(λ) = c(λ)ψ
k
L|y|

2λ−n
(
η(∆y )∆

λ− n
2

y + (1 + ∆YK )
λ− n

2 + (1 + ∆Yk )
Nϕ(1 + ∆Yk )

)
ψk|y|2λ−

n
2

with

ϕ(x) = x−N
(
(1 −η(x− 1))(x− 1)λ−

n
2 − (1 −η(x))xλ−

n
2

)

which is a symbol in (0,∞) of order λ− n
2
−N −1 in the sense that it has a support in [ǫ,∞) for

some ǫ > 0, it is smooth and satisfies

|∂lxϕ(x)| ≤ Cl(1 + x)
ℜ(λ)− n

2
−1−N−l.

Hence following the method of Helffer-Robert [13], we have

ϕ(1 + ∆Yk ) =
1

2πi

∫ i∞

−i∞

M[ϕ](s)(1 + ∆Yk )
−sds

where M[ϕ](s) is the Mellin transform of ϕ defined by

M[ϕ](s) :=

∫
∞

0

ts−1ϕ(t)dt

and which is rapidly decreasing on iR. From the previous study of (1+∆YK )
λ− n

2 and using Mellin’s

transform, we deduce that if B(λ) is the operator

B(λ) := ψkL|y|
2λ−n

(1 + ∆Yk )
Nϕ(1 + ∆YK )ψ

k|y|2λ−n



CUBO
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Scattering Theory on Geometrically Finite Quotients 169

then its kernel satisfies

B(λ; w; w′) = B1(λ; w; w
′
) + B2(λ; w; w

′
)

B1(λ; w,w
′
) := ψkL(y)|y|

2λ−nψk(y′)|y′|2λ−n
∑

i

θ′i(z)θi(z
′
)

∫

Rn

eiξ.(w−w
′)
(1 + |ξ|2)Nϕ(1 + |ξ|2)dξ

B2(λ; w; w
′
) := ψkL(y)|y|

2λ−nψk(y′)|y′|2λ−n
(1 + ∆w)

N

2πi

∫ i∞

−i∞

M[ϕ](s)

∫

Γ

τs−
n
2 κ(τ; w,w′)dτds.

In view of the estimate (6.18) on κ(τ; w; w′) and its smoothness, we easily obtain that the kernel

B2(λ; w; w
′
), when lifted on B̄ ×Φ B̄, has exactly the same properties than A2(λ; w,w

′
). For the

term B1(λ; w; w
′
) we can proceed as for A1(λ; w,w

′
) and it finally shows that

βΦ
∗B(λ) ∈ ρ−4λ+2n

Ik
I2λ−n−1(B̄ ×Φ B̄; DΦ).

Combined with (6.15), (6.19), this proves the Theorem after multiplying by the lift of (rcr
′
c)

−n to

return with the correct density. �

Remark: As a consequence, we can obtain quite general mapping properties for S(λ) (i.e. the

actions of S(λ) on extendible distributions on B̄ conormal to ∂B̄) using general theory for those

operators, see for exemple Vaillant [26, Section 2.2].

7 Conformal Operators on the Boundary

As explained by Graham-Zworski [7], there is a strong connection between scattering theory on

Einstein conformally compact manifolds (in particular convex co-compact hyperbolic quotients)

and conformal theory of its boundary. Here similar results hold in this degenerate case.

First recall from Lemma 2.2 that for any ĥ0 := e
2ω0h0 ∈ [h0]acc, there exists a boundary

defining function ρ̂ = eωρ ∈ C∞
acc

(X̄), unique up to Ċ∞(X̄), such that ω|∂X̄ = ω0 and which put

the metric under the almost product form (2.12). This gives a way to identify special boundary

defining functions of Lemma 2.2 with representatives of the subconformal class [h0]acc. Moreover

we saw that the scattering operators S(λ), Ŝ(λ) obtained by solving Poisson problem respectively

with ρ and ρ̂ (i.e. for conformal representatives h0 and ĥ0) are related by

Ŝ(λ)f = e−λω0S(λ)e(n−λ)ω0f. (7.1)

In this sense, S(λ) is a conformally covariant operator and by looking at the residues we have the

rule

P̂j = e
(− n

2
−j)ω0Pje

( n
2
−j)ω0

which also makes this differential operator being conformally covariant.



170 Colin Guillarmou CUBO
11, 5 (2009)

Let us now give a few words about conformal GJMS Laplacians. In [6], Graham-Jenne-

Manson-sparling defined, on any n-th dimensional Riemannian compact manifold (M,h0), a family

of “natural” conformally covariant differential operators (Pj )j with principal symbol ∆
j
h0

. We call

Pj the j-th GJMS Laplacian. They are defined for j ∈ N if n is odd and for j ≤ n/2 integer if

n is even and natural in the sense that they can be written in terms of covariant derivatives and

curvature of h0 and conformally covariant in the sense that the operator P̂j obtained with the same

expression than Pj but with a conformal metric ĥ0 = e
2ω0h0 is related to Pj by the identity

P̂j = e
−( n

2
+j)ω0Pje

( n
2
−j)ω0.

Moreover P1 is Yamabe’s Laplacian and P2 is Paneitz operator. If h0 is locally conformally flat

and n > 2 is even, it is also proved in [6] that the Pj can be constructed without obstruction for

any j ∈ N, this is the case in particular of the conformal infinity of a convex co-compact hyperbolic

quotients. Note that, since the expression of Pj is local with respect to the metric, these operators

can also be defined on non-compact Riemannian manifolds. Graham and Zworski [7] show that on

asymptotically Einstein manifolds (X,g) of dimension n + 1 (with X̄ the conformal compactifica-

tion), the residue Res n
2
+jS(λ) of the scattering operator obtained by solving the Poisson problem

with boundary defining function x is Pj on the conformal infinity (∂X̄,x
2g|T ∂X̄ ) for any j integer

if n is odd (resp. for j ≤ n
2

if n is even). Actually, we learnt from Robin Graham that this also

holds for any j if n > 2 is even and if (X,g) has negative constant curvature outside a compact

set, where in this case the conformal infinity is locally conformally flat. The reason, given in [4],

which makes this special case working is that there is no obstruction to construct a hyperbolic

conformally compact metric g on (0,ǫ]x × M with conformal infinity (M ≃ {x = 0},h0) for any

(M,h0) locally conformally flat compact manifold, and actually g is necessarily given by

g = x−2(dx2 + h0 −x
2P + x4(

1

4
Ph−10 P)) (7.2)

where P = (n− 2)−1(Ric − (2n− 2)−1Kh0) is the Schouten tensor of h0, with K, Ric the scalar

and Ricci curvatures of h0. This is a consequence of the constant curvature equation.

Since in our case the metric on X = Γ\Hn+1 is also hyperbolic, the curvature equation (which

is local) implies again that the tensor ĥ(ρ̂) in (2.12) has all its Taylor expansion with respect to ρ̂

at ρ̂ = 0 determined by ĥ0 = ĥ(0) if n > 2: the expression of ĥ(ρ̂) is explicit and, like (7.2),

ĥ(ρ̂) = ĥ0 − ρ̂
2P + ρ̂4(

1

4
Pĥ−10 P)

with P is the Schouten tensor of ĥ0.

If n > 2, we saw that the expression of Res n
2
+jS(λ) is obtained from the construction of Φ(λ)

exactly like in the convex co-compact case (the construction is local in term of ĥ(ρ̂) thus in term

of ĥ0). By equivalence of the construction of Φ(λ) in [7] and in our case, it is clear that



CUBO
11, 5 (2009)

Scattering Theory on Geometrically Finite Quotients 171

Proposition 7.1. The operator Pj of Lemma 6.1 is the j-th conformal GJMS Laplacian defined in

[6] on locally conformally flat compact manifolds in the sense that it has the same local expression

in term of the metric h0.

Received: March, 2009. Revised: May, 2009.

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