Table Of ContentEquidistribution of phase shifts in trapped scattering
Maxime Ingremeau
December 6, 2016
6
1
0 Abstract
2 WeconsidersemiclassicalscatteringforcompactlysupportedperturbationsoftheLaplacian
c andshowequidistributionofeigenvaluesofthescatteringmatrixat(classically)non-degenerate
e energy levels. The only requirement is that sets of fixed points of certain natural scattering
D relations have measure zero. This extends the result of [GRHZ15], where the authors proved
the equidistribution result under a similar assumption on fixed points but with the condition
3
that there is no trapping.
]
h
1 Introduction
p
-
h
Consider a Riemannian manifold (X,g) which is Euclidean near infinity, in the sense that there
t
a exist compact sets X ⊂X and K ⊂Rd such that (X\X ,g) and (Rd\K ,g ) are isometric.
m 0 0 0 0 eucl
Let us consider an operator P :=−h2∆ +V, where V ∈C∞(X) has its support in X . It is
h g c 0
[ well-known (see for example [Mel95, §2] or [DZ, §3.7, §4.4]), that for any φ ∈C∞(Sd−1), there is
3 a unique solution to (cid:0)Ph−1(cid:1)u=0 satisfying, for all x∈(X\X0)∼=(Rd\Ki0n):
v
1 u(x)=|x|−(d−1)/2(cid:0)e−i|x|/hφin(ω)+ei|x|/hφout(−ω)(cid:1)+O(|x|−(d+1)/2).
4
1 We define the scattering matrix1 Sh :C∞(Sd−1)−→C∞(Sd−1), which depends on h, by
0
0 Sh(φin):=eiπ(d−1)/2φout.
.
2 The factor eiπ(d−1)/2 is taken so that the scattering matrix is the identity operator when (X,g)=
0
(Rd,g ) and V ≡0.
6 Eucl
1 For each h ∈ (0,1], Sh can be extended by density to a unitary operator acting on L2(Sd−1).
: S −Id is then a trace class operator. Therefore, S admits a sequence of eigenvalues of modulus
v h h
i 1, which converge to 1, and which we denote by (eiβh,n)n∈N.
X Our aim in this paper will be to study the behaviour of (eiβh,n) in the limit where h → 0. To
r do this, we define a measure µ on S1 by
h
a
(cid:88)
(cid:104)µh,f(cid:105):=(2πh)d−1 f(eiβh,n),
n∈N
for any continuous f :S1 −→C. This measure is not finite, but (cid:104)µ ,f(cid:105) is finite as soon as 1 is not
h
in the support of f.
Let us now state the assumptions we make on the manifold X and on the potential V.
1whichisnotamatrixassoonasd>1!
1
The scattering map Wedenotebyp(x,ξ)=|ξ|2+V(x):T∗X −→RtheclassicalHamiltonian,
g
which is the principal symbol of P . Let us write E for the energy layer of energy 1:
h
E ={(x,ξ)∈T∗X; p(x,ξ)=1}. (1)
We denote by Φt(ρ) the Hamiltonian flow for the Hamiltonian p. The outgoing and incoming
sets are defined as
Γ± :={ρ∈E, such that Φt(ρ) remains in a compact set for all ∓t≥0}.
The trapped set is the compact set
K =Γ+∩Γ−.
Since,awayfromX ,thetrajectoriesbyΦtarejuststraightlines,wehavethatforanyω ∈Sd−1,
0
and η ∈ω⊥ ⊂Rd, there exists a unique ρ ∈E such that
ω,η
π (cid:0)Φt(ρ )(cid:1)=tω+η for t<−T , (2)
X ω,η 0
whereπ :T∗X →X denotestheprojectiononthebasemanifoldX,andwhereT islargeenough,
X 0
so that K ⊂ B(0,T ). Here, ω is the incoming direction, and η is the impact parameter. In the
0 0
sequel, we will identify
{(ω,η); ω ∈Sd−1,η ∈ω⊥}∼=T∗Sd−1.
We define the interaction region as
I :={(ω,η)∈T∗Sd−1;∃t∈R such that π (cid:0)Φt(ρ )(cid:1)∈X }.
X ω,η 0
By compactness of X , I is compact.
0
If ρ ∈/ Γ−, then there exists ω(cid:48) ∈Sd−1, η(cid:48) ∈(ω(cid:48))⊥ ⊂Rd and t(cid:48) ∈R such that for all t≥T ,
ω,η 0
π (cid:0)Φt(ρ )(cid:1)=ω(cid:48)(t−t(cid:48))+η(cid:48).
X ω,η
The (classical) scattering map is then defined as κ(ω,η)=(ω(cid:48),η(cid:48)), as represented on Figure 1.
The assumptions from [GRHZ15] The main assumption in [GRHZ15] is that
Γ± =∅. (3)
Under this assumption, κ : T∗Sd−1 → T∗Sd−1 is well defined. One can actually show that κ
is a symplectomorphism for the canonical symplectic structure on T∗Sd−1, and in particular, it is
invertible (see for example [Gui77]).
It is easy to see that
κ(I)=I, κ(Ic)=Ic and κ| =Id .
Ic→Ic Ic→Ic
The results in [GRHZ15] require a diversion hypothesis which concerns the periodic points of κ
in the interaction region.
For any l∈Z\{0}, denote the set of periodic points of κ of period l by
F :={(ω,η)∈I; κl(ω,η)=(ω,η)}.
l
2
Figure 1: The scattering map κ.
The diversion hypothesis says that
∀l∈N\{0}, Vol(cid:0)F (cid:1)=0, (4)
l
where Vol denotes the Liouville measure on T∗Sd−1.
This hypothesis roughly says that most of the classical trajectories in E which interact with the
potentialortheperturbationoftheEuclideanmetricareindeeddiverted. In[GRHZ15],theauthors
work in the setting where (X,g) ≡ (Rd,g ), and with X = supp(V), and they conjecture that
Eucl 0
this hypothesis holds for generic potentials.
The main result in [GRHZ15] is the following.
Theorem ([GRHZ15]). Suppose that the manifold (X,g) and the potential V are such that (3) and
(4) are satisfied. Let f :S1 −→C be a continuous function such that 1∈/ suppf. Then we have
Vol(I)(cid:90) 2π
lim(cid:104)µ ,f(cid:105)= f(eiθ)dθ.
h→0 h 2π 0
Our objective in this paper is to show that this theorem remains true if the incoming and
outgoing sets are non-empty. We define the incoming set at infinity as
Γ˜− :={(ω,η)∈T∗Sd−1 such that ρ ∈Γ−}. (5)
ω,η
Similarly, we define the outgoing set at infinity Γ˜+ ⊂T∗Sd−1 by:
(ω(cid:48),η(cid:48))∈Γ˜+ ⇔∃(x,ξ)∈Γ+;Φt(x,ξ)=tω(cid:48)+η(cid:48) for t large enough. (6)
3
NotethatΓ˜± arecompactsubsetsofT∗Sd−1,sinceifη islargeenough,atrajectorywithimpact
parameter η will not meet the interaction region, and therefore cannot be trapped.
Instead of supposing (3), we will make the following assumption.
Hypothesis 1.
Vol(Γ˜±)=0. (7)
This hypothesis is very mild: as we will see in the next section, it is satisfied as soon as the
energy level E is non-degenerate in the sense that
dp (cid:54)=0. (8)
|E
We will also make an assumption which is the analogue of (4). In the case when (3) does not
hold, thisassumptionisslightlymoretechnicaltowrite, sinceκl isnotwell-definedonallofI. We
will therefore postpone the precise statement of this assumption to Hypothesis 2 in section 2.
Statement of the results Under these hypotheses, we may state our result.
Theorem 1. Suppose that the manifold (X,g) and the potential V are such that Hypotheses 1 and
2 are satisfied. Let f :S1 −→C be a continuous function such that 1∈/ suppf. Then we have
Vol(I)(cid:90) 2π
lim(cid:104)µ ,f(cid:105)= f(eiθ)dθ.
h→0 h 2π 0
Remark 1. For simplicity, we shall only state and prove this result for smooth potentials, but it
should still be true for less regular potentials, as long as the Hamiltonian dynamics is well defined.
The proof should work without many changes for a potential V ∈C1(X;R).
c
As in [GRHZ15], we may deduce the following corollary.
Corollary 1. Let 0 < φ < φ < 2π be angles, and let N (φ ,φ ) be the number of eigenvalues
1 2 h 1 2
eiβh,n of Sh with φ1 ≤βh,n ≤φ2 modulo 2π. Then we have
φ −φ
lim(2πh)d−1N (φ ,φ )=Vol(I) 2 1.
h→0 h 1 2 2π
The proof of Corollary 1 is exactly the same as that of Corollary 1.2 in [GRHZ15]: we simply
approach uniformly the indicator function 1 by continuous functions and use Theorem 1. We
[φ1,φ2]
refer to [GRHZ15] for more details.
Relation to other works The distribution of the eigenvalues of the scattering matrix has been
studied since the eighties ([BY82], [BY84], [SY85]). More recently, in the (non-semiclassical) high-
energy limit, it was studied in [BP12], and extended to more general Hamiltonians in [BP13] and
[Nak14]. For related topics in the physics literature for obstacle scattering, see [DS92].
In the semiclassical setting, equidistribution of phase shifts was first observed in [DGRHH14]
for spherically symmetric potentials, and in [GRHZ15] for more general non-trapping potentials.
It was also studied in [GRH15] for long-range potentials, without any assumption on the classical
dynamics. In [ZZ99], the authors obtain much finer results on the distribution of phase shifts in
the semiclassical limit for a family of surfaces of revolution.
4
Just as in [GRHZ15], the main tool in the proof of the equidistribution of phase shifts is the
fact that the scattering matrix is a Fourier Integral Operator associated to the scattering map
microlocally away from the incoming and outgoing directions. This was proven in [Ale05], and also
in [HW08] in a geometric non-trapping setting.
The scattering map is trivial outside of the interaction region, while it can be very complicated
inside the interaction region. This mixed behaviour is somehow similar to the situation described
in [MO05], where the authors prove a Weyl law for general systems for which the phase space can
be separated into a part where the classical dynamics is periodic, and another where its is ergodic.
Acknowledgements The author would like to thank St´ephane Nonnenmacher for supervising
thisprojectandforusefuldiscussions. HewouldalsoliketothankJesseGell-Redman,AndrewHas-
sellandSteveZelditchfortheircommentswhichhelpedimprovethefirstversionofthemanuscript.
Lastbutnotleast,theauthorwouldliketothanktheanonymousrefereewhohelpedimprovemany
aspects of this paper, in particular by suggesting Lemma 1 and some of the estimates in section 5.
The author is partially supported by the Agence Nationale de la Recherche project GeRaSic
(ANR-13-BS01-0007-01).
2 Classical dynamics
Recall that, if (ω,η)∈T∗Sd−1, ρ was defined in (2), and that we defined the sets Γ˜± in (5) and
ω,η
(6).
Althoughwewillnotuseitinthesequel,letusprovenowthefactannouncedintheintroduction
that (8) implies Hypothesis 1. The proof is standard (it is very similar to that of [DZ, Proposition
6.5] or [GS87, Proposition A.3]), but we recall it for the reader’s convenience.
Lemma 1. Suppose that p:T∗X →R is such that dp (cid:54)=0. Then we have Vol(Γ˜±)=0.
|E
Proof. Suppose that dp (cid:54)= 0. E is then a smooth manifold, which can be equipped with the
|E
Liouville measure µ. This measure is invariant by the Hamiltonian flow (Φt).
Note that, outside of X , this measure is just the Lebesgue measure on S∗(X\X ), so that, if
0 0
we define for r large enough and r >r the annulus
0 1 0
C :=(B(0,r )\B(0,r ))⊂(Rd\K )(cid:39)(X\X ),
r0,r1 1 0 0 0
we have
Vol(Γ˜±)=0 ⇔ ∃ 0<r <r large enough such that µ(cid:0)Γ±∩S∗C (cid:1)=0.
0 1 r0,r1
Suppose for contradiction that we may find 0<r <r such that µ(cid:0)Γ±∩S∗C (cid:1)>0.
0 1 r0,r1
Since the motion of a point in Γ± ∩S∗C as ±t ≥ 0 is just a straight line, we may find a
time t0 =t0(r0,r1) such that for any j ≥1, Φr±0,jrt10(cid:0)Γ±∩S∗Cr0,r1(cid:1)∩(cid:0)Γ±∩S∗Cr0,r1(cid:1)=∅. Since Φt0
is a diffeomorphism, we then have that for all j,j(cid:48) ∈N with j (cid:54)=j(cid:48),
(cid:16)Φ∓jt0(cid:0)Γ±∩S∗C (cid:1)(cid:17)∩(cid:16)Φ∓j(cid:48)t0(cid:0)Γ±∩S∗C (cid:1)(cid:17)=∅.
r0,r1 r0,r1
5
Since µ is invariant by the Hamiltonian flow, we have that
∞ ∞
µ(cid:16) (cid:91) Φ∓jt0(cid:0)Γ±∩S∗C (cid:1)(cid:17)=(cid:88)µ(cid:16)Φ∓jt0(cid:0)Γ±∩S∗C (cid:1)(cid:17)
r0,r1 r0,r1
j=0 j=0
∞
=(cid:88)µ(cid:0)Γ±∩S∗C (cid:1)
r0,r1
j=0
=+∞,
tbhyeabssausemppotiionnts. aBruetefiothrearllinjX≥00,,oΦr∓injt0B(cid:0)(Γ0±,r∩1)S⊂∗CRr0d,.r1(cid:1)Hebnecloen,gwsetmo uasctohmapveacµt(cid:16)re(cid:83)g∞jio=n0Φof∓Ejt,0(cid:0)wΓh±er∩e
S∗C (cid:1)(cid:17)<+∞, a contradiction.
r0,r1
If ρ ∈/ Γ−, then there exists ω(cid:48) ∈ Sd−1, η(cid:48) ∈ (ω(cid:48))⊥ ⊂ Rd and t(cid:48) ∈ R such that for all t large
ω,η
enough,
π (cid:0)Φt(ρ )(cid:1)=ω(cid:48)(t−t(cid:48))+η(cid:48).
X ω,η
We may then define the (classical) scattering map
κ:T∗Sd−1\Γ˜− −→T∗Sd−1\Γ˜+
by (ω(cid:48),η(cid:48))=κ(ω,η). κ is then a symplectomorphism.
We define the “good” sets (G+)⊂T∗Sd−1 and (G−)⊂T∗Sd−1 by induction for k ∈N, by
k k
G+ :=T∗Sd−1\Γ˜−, G+ :={(ω,η)∈G+ such that κ(ω,η)∈G+}
0 k+1 k 0 (9)
G− :=T∗Sd−1\Γ˜+, G− :={(ω,η)∈G− such that κ−1(ω,η)∈G−}.
0 k+1 k 0
Thescatteringmapmaythenbeiteratedandinverted,toobtainforanyk ≥1symplectomorphisms
κk :G+ −→G− ,
k−1 k−1
κ−k :G− −→G+ ,
k−1 k−1
or, written in a more condensed way, we may define for k ∈ Z\{0}, κk : G(cid:15)(k) → G−(cid:15)(k), where
|k|−1 |k|−1
(cid:15)(k) is the sign of k.
We also define, for k ∈Z\{0}
B :=T∗Sd−1\G(cid:15)(k) . (10)
k |k|−1
B is hence the “bad” set where κk is not well-defined.
k
Lemma 2. Suppose Hypothesis 1 is satisfied, and let k ∈ Z\{0}. Then B has zero Liouville
k
measure.
Proof. By assumption, Γ˜± has zero Liouville measure. Since κ preserves the Liouville measure, we
see from (9) that G± has full measure.
k
6
For l∈Z\{0}, we define the set of l-periodic interacting points as
F :={(ω,η)∈I∩G(cid:15)(l) ;κl(ω,η)=(ω,η)}, (11)
l |l|−1
where (cid:15)(l) is the sign of l. Note that this set is closed.
Our diversion hypothesis is the following.
Hypothesis 2. For any l∈Z\{0}, the Liouville measure of F is 0.
l
We conjecture that if V ≡ 0 and if (Int(X ),g) has (not uniformly) negative curvature, where
0
Int(X ) denotes the interior of X then this hypothesis holds.
0 0
Note that, since κl preserves the volume, this Hypothesis is equivalent to the seemingly weaker
statement that for any l∈N\{0}, the Liouville measure of F is 0.
l
Note also that this hypothesis implies that
Vol(∂I)=0. (12)
Indeed, a point in the boundary of I is in I because I is closed, and it is fixed by κ.
Before proving Theorem 1, we need to recall a few facts and definitions from semiclassical
analysis.
3 Refresher on semiclassical analysis
3.1 Pseudodifferential calculus
Let Y be a compact manifold (Y will often be Sd−1 in the sequel). We shall say that a function
a(x,ξ;h)∈C∞(T∗Y ×(0,1]) is in the class Scomp(T∗Y) if it can be written as
(cid:16)(cid:16) h (cid:17)∞(cid:17)
a(x,ξ;h)=a˜ (x,ξ)+O ,
h (cid:104)ξ(cid:105)
where a˜ ∈ C∞(T∗Y), with supp(a˜ ) ⊂ Ω for some bounded open set Ω independent of h, and
h c h
where a˜ is bounded in any Ck(Ω) norm independently of h.
h
We associate to Scomp(T∗Y) the algebra of pseudodifferential operators Ψcomp(Y), through a
h
surjective quantization map
Op :Scomp(T∗Y)−→Ψcomp(Y).
h h
This quantization map is defined using coordinate charts, and the standard Weyl quantization on
Rd. It is therefore not intrinsic. However, the principal symbol map
σ :Ψcomp(Y)−→Scomp(T∗Y)/hScomp(T∗Y)
h h
is intrinsic, and we have
σ (A◦B)=σ (A)σ (B)
h h h
and
σ ◦Op:Scomp(T∗Y)−→Scomp(T∗Y)/hScomp(T∗Y)
h
is the natural projection map.
7
Formoredetailsonallthesemapsandtheirconstruction,wereferthereaderto[Zwo12,Chapter
14].
For a∈Scomp(T∗Y), we say its essential support is equal to a given compact K(cid:98)T∗Y,
ess supp a=K(cid:98)T∗Y,
h
if and only if, for all χ∈C∞(T∗Y),
c
supp(χ)⊂(T∗Y\K)⇒χa∈h∞Scomp(T∗Y).
For A∈Ψcomp(Y),A=Op (a), we define the wave front set of A as:
h h
WF (A)= ess supp a,
h h
noting that this definition does not depend on the choice of the quantization.
3.2 Lagrangian states and Fourier Integral Operators
Inthissection, wewillrecallthedefinitionofFourierIntegralOperatorswithnotationsinspiredby
[DG14]. We refer to this paper and to the references therein for the classical proofs we omit.
Phasefunctions Letφ(y,θ)beasmoothreal-valuedfunctiononsomeopensubsetU ofY×RL,
φ
for some L ∈ N. We call x the base variables and θ the oscillatory variables. We say that φ is a
nondegenerate phase function if the differentials d(∂ φ)...d(∂ φ) are linearly independent on the
θ1 θL
critical set
C :={(y,θ);∂ φ=0}⊂U .
φ θ φ
In this case
Λ :={(y,∂ φ(y,θ));(y,θ)∈C }⊂T∗Y
φ y φ
is an immersed Lagrangian manifold. By shrinking the domain of φ, we can make it an embedded
Lagrangian manifold. We say that φ generates Λ .
φ
Lagrangian states Given a phase function φ and a symbol a ∈ Scomp(U ), consider the h-
φ
dependent family of functions
(cid:90)
u(y;h)=h−L/2 eiφ(y,θ)/ha(y,θ;h)dθ. (13)
RL
We call u=(u(h)) a Lagrangian state, (or a Lagrangian distribution) generated by φ.
Definition 1. Let Λ⊂T∗Y be an embedded Lagrangian submanifold. We say that an h-dependent
family of functions u(y;h) ∈ C∞(Y) is a (compactly supported and compactly microlocalized) La-
c
grangian state associated to Λ, if it can be written as a sum of finitely many functions of the form
(13), for different phase functions φ parametrizing open subsets of Λ, plus an O(h∞) remainder in
the C∞(Y) topology. We will denote by Icomp(Λ) the space of all such functions.
8
Fourier integral operators Let Y,Y(cid:48) be two manifolds of the same dimension d, and let κ
be a symplectomorphism from an open subset of T∗Y to an open subset of T∗Y(cid:48). Consider the
Lagrangian
Λ ={(y,ν;y(cid:48),−ν(cid:48));κ(y,ν)=(y(cid:48),ν(cid:48))}⊂T∗Y ×T∗Y(cid:48) =T∗(Y ×Y(cid:48)).
κ
A compactly supported operator T : D(cid:48)(Y) → C∞(Y(cid:48)) is called a (semiclassical) Fourier integral
c
operator associated to κ if its Schwartz kernel K (y,y(cid:48)) lies in h−d/2Icomp(Λ ). We write T ∈
T κ
Icomp(κ). Note that such an operator is automatically trace class. The h−d/2 factor is explained as
follows: thenormalizationforLagrangianstatesischosensothat(cid:107)u(cid:107) (cid:16)1,whilethenormalization
L2
for Fourier integral operators is chosen so that (cid:107)T(cid:107) (cid:16)1.
L2(Y)→L2(Y(cid:48))
Note that if κ◦κ(cid:48) is well defined, and if T ∈ Icomp(κ) and T(cid:48) ∈ Icomp(κ(cid:48)), then T ◦T(cid:48) ∈
Icomp(κ◦κ(cid:48)).
ThemainpropertywewilluseaboutFIOsisthefollowing,whichisaneasyversionof[GRHZ15,
Proposition 2].
Lemma 3. Let κ:T∗Y ⊃U →V ⊂T∗Y have no fixed point, and let T ∈Icomp(κ). Then
Tr(T)=O(h∞)
Proof. (Sketch) By definition, the integral kernel of T can be written as a finite sum of terms of
the form
(cid:90)
(2πh)−L eiφ(y,y(cid:48);θ)/ha(y,y(cid:48),θ,h)dθ,
RL
where φ locally parametrises Λ in the sense that in some open subset U ⊂T∗(Y ×Y(cid:48)), we have
κ
Λ ∩U ={(y,∂φ (y,y(cid:48),θ),y(cid:48),−∂ φ(y,y(cid:48),θ));(y,y(cid:48),θ) such that ∂ φ(y,y(cid:48),θ)=0}.
κ y(cid:48) y θ
The trace is then given by a sum of terms of the form
1 (cid:90) (cid:90)
eiφ(yh,y;θ)a(y,y,θ,h)dθdy.
(2πh)L+d−1
Y RL
Thefactthatκhasnofixedpointimpliesthatif(y,y,θ)aresuchthat∂ φ(y,y,θ)=0, wehave
θ
∂ (cid:2)φ(y,y,θ)(cid:3) = (cid:2)∂ φ(y,y(cid:48),θ)+∂ φ(y,y(cid:48),θ)(cid:3) (cid:54)= 0. Then, by non-stationary phase, we obtain
y y y(cid:48) y=y(cid:48)
the result.
3.3 The scattering matrix as a FIO
The main result we will use about the scattering matrix in this paper is [Ale05, Theorem 5], which
can be rephrased as follows.
Theorem (Alexandrova 2005). (i) Let (ω,η) ∈ G+. If U is an open neighbourhood of (ω,η)
0
contained in G+ and A∈Ψcomp(Sd−1) is such that WF (A)⊂U, then we have S A∈Icomp(κ| ).
0 h h h U
(ii) S is microlocally equal to the identity away from the interaction region in the following
h
sense. If a∈Scomp(Sd−1) is such that a≡1 near I, then we have
(cid:107)(S −Id)(Id−Op (a))(cid:107) =O(h∞). (14)
h h L2(Sd−1)→L2(Sd−1)
9
4 Trace formula
Our aim in this section will be to prove the following proposition, which is the cornerstone of the
proof in [GRHZ15].
Proposition 1. Suppose that the manifold (X,g) and the potential V are such that Hypotheses 1
and 2 are satisfied, and let k ∈Z\{0}. Then we have
Tr(cid:0)Sk−Id(cid:1)=− Vol(I) +o(h−(d−1)). (15)
h (2πh)d−1
Proof. To prove this proposition, we fix k ∈Z\{0}, and build an adapted partition of unity.
Partition of unity RecallthatB wasdefinedin(10),andisthesetwhereκk isnotwell-defined.
k
We will write
P :=B ∪F ⊂T∗Sd−1,
k k k
where F are as in (11) This set is closed, has zero Liouville measure by Lemma 2 and Hypothesis
k
2, and the map κk is well-defined and has no fixed points in I\P .
k
Since P is closed with zero Liouville measure, by outer regularity of the Liouville measure,
k
we may find for each ε > 0 a cut-off function χk ∈ C∞(T∗Sd−1;[0,1]) such that χk(ω,η) = 1 if
ε c ε
(ω,η) ∈ P , such that the support of χk is contained in an ε-neighbourhood of I, and such that
k ε
the Liouville measure of the support of χk is smaller than ε:
ε
Vol(supp(χk))≤ε.
ε
We denote by Op (χk) the Weyl quantization of χk, as defined in section 3.1.
h ε ε
Wealsotakeψ1 ∈C∞(T∗Sd−1;[0,1])suchthatψ1 =1nearI andψ1(ω,η)=0ifd((ω,η),I)≥
ε c ε ε
ε and ψ2 ∈ C∞(T∗Sd−1;[0,1]) such that ψ2 = 0 outside of I, and ψ2 = 1 outside of an ε-
ε c ε ε
neighbourhood of T∗Sd−1\I (see Figure 2).
Note that we have for all (ω,η) ∈ T∗Sd−1, ψ1(ω,η) ≥ ψ2(ω,η), and that (cid:107)ψ1−ψ2(cid:107) = O((cid:15))
ε ε ε ε L1
thanks to (12).
We have
1=(1−ψ1)+ψ2(1−χk)+ψ2χk+(ψ1−ψ2). (16)
ε ε ε ε ε ε ε
The first term corresponds to points outside of the interaction region. The second term cor-
responds to points in the interaction region which are neither trapped nor fixed, while the last
two terms have a support of a size O(ε). We shall compute the trace of (Sk − Id) using this
h
decomposition.
Traceinsidetheinteractionregion ByAlexandrova’sTheorem(seeSection3.3),wehavethat
SkOp (ψ2(1−χk))isaFourierintegraloperatorassociatedtoκk microlocallynear(I\κk(P ))×
h h ε (cid:15) |I\Pk k
(I\P ).
k
Since, by definition of P , κk has no fixed points in I\P , Lemma 3 tells us that
k k
Tr(SkOp (ψ2(1−χk))=O(h∞).
h h ε (cid:15)
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