Annales de l’institut Fourier (AIF)

Lin, Samuel

Schmidt, Benjamin

Sutton, Craig

Geometric structures and the Laplace spectrum, Part I

Brunault, François

Guilloux, Antonin

Mehrabdollahei, Mahya

Pengo, Riccardo

Limits of Mahler measures in multiple variables

Saito, Takahiro

The Hodge filtration of a monodromic mixed Hodge module and the irregular Hodge filtration

Pavel, Mihai

Moduli spaces of slope-semistable pure sheaves

Haiech, Mercedes

Deformations of solutions of algebraic differential equations

Diez, Tobias

Janssens, Bas

Neeb, Karl-Hermann

Vizman, Cornelia

Induced differential characters on nonlinear Graßmannians

Mangoubi, Dan

Weller Weiser, Adi

Université Grenoble Alpes
ANNALES DE
L’INSTITUT FOURIER
DanMangoubi & Adi Weller Weiser
A note on the semiclassical measure at singular points of
the boundary of the Bunimovich stadium
Tome 74, n
o
1 (2024), p. 367-375.
https://doi.org/10.5802/aif.3601
Article mis à disposition par ses auteurs selon les termes de la licence
Creative Commons attribution – pas de modification 3.0 France
http://creativecommons.org/licenses/by-nd/3.0/fr/
C EN T R E
MER S ENN E
Les Annales de l’Institut Fourier sont membres du
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Ann. Inst. Fourier, Grenoble
74, 1 (2024) 367-375
A NOTE ON THE SEMICLASSICAL MEASURE AT
SINGULAR POINTS OF THE BOUNDARY OF THE
BUNIMOVICH STADIUM
by Dan MANGOUBI & Adi WELLER WEISER (*)
Abstract. — An argument by Hassell proving the existence of a Bunimovich
stadium for which there are semiclassical measures giving positive mass to the sub-
manifold of bouncing ball trajectories uses a notion of non-gliding points. However,
this notion is defined only for domains with C2-boundaries. The purpose of this
note is to clarify the argument.
Résumé. — Un argument de Hassell montrant l’existence d’un stade de Buni-
movich sur lequel on trouve une mesure semiclassique chargeant la sous-variété des
trajectoires de boules rebondissantes fait appel à la notion de points non-gliding.
Pourtant, cette notion suppose un domaine lisse C2. Le but de cette note est de
clarifier ce point.
1. Introduction
In a celebrated paper [4] Hassell proves the existence of quantum er-
godic manifolds which are not quantum uniquely ergodic. Furthermore,
Hassell shows that there is a Bunimovich stadium, M , with a semiclas-
sical measure giving positive mass to the submanifold of bouncing ball
trajectories. For this refinement, in order to rule out formations of certain
non-uniform semiclassical measures, Hassell applies a theorem of Burq and
Gérard [2] showing that a boundary semiclassical measure cannot put any
mass on the set of non-gliding points, a special subset of S∗∂M where
the classical trajectories passing through are not affected by the bound-
ary (see Appendix). However, the definition of non-gliding points requires
Keywords: Bunimovich’s stadium, quantum unique ergodicity, non-smooth boundary,
non-gliding.
2020 Mathematics Subject Classification: 58J51, 37D50, 81Q50.
(*) This research was partially supported by ISF Grant No. 681/18.
368 Dan MANGOUBI & Adi WELLER WEISER
at least C2-smoothness of the boundary, while the boundary of the Buni-
movich stadium is only C1,1-smooth. Moreover, in a neighborhood of a
singular point p of the boundary, one has points in S∗∂M which are far
from being non-gliding, since the curvature jumps there from zero to a pos-
itive constant, and it is not a-priori clear from Burq and Gérard’s argument
whether one should regard the points of S∗p∂M as gliding or non-gliding.
The aim of this note is to clarify Hassell’s argument, by showing (in Theo-
rem 3.1) that indeed the semiclassical measure on the boundary gives zero
mass to the portion of S∗∂M lying above the closure of the straight part
of ∂M . Our result is an instance where one can relax the smoothness as-
sumption in [2] and hints that in two dimensions the C1,1 assumption may
be sufficient to conclude that the semiclassical measure of the boundary
vanishes on the closure of the subset of S∗∂M where the classical trajec-
tories passing through are not reflected as a replacement for non-gliding
points in the smooth case. In this paper, we treat the Bunimovich stadium
case. The proof, like in [2], uses Gérard–Leichtnam’s transport equation
from [3], recalled here in Section 2, and follows similar lines of microlocal-
ization on non-gliding points. However, unlike in [2], we do not make any
change of variables, allowing us to treat the singular points of the boundary
as well.
Acknowledgments
This paper arose from a discussion with Patrick Gérard. Patrick showed
us an initial ad hoc argument explaining the application of [2] in [4]
(Section 3.1). We are very grateful to Nicolas Burq, Patrick Gérard and
Stéphane Nonnenmacher for their encouragement to write this note and for
many hours of discussions on topics of Quantum Chaos. D. M. especially
thanks Stéphane for the invitation to spend a sabbatical leave in Paris-Sud
XI in a very pleasant research atmosphere, during which this work started.
A. W. appreciates her welcome in the course of a visit to Paris-Sud XI. Both
authors thank Elon Lindenstrauss for his guidance. The financial supports
of LabEx Mathématiques Hadamard and the French government are grate-
fully acknowledged. This paper is part of A. W.’s research towards a Ph.D.
dissertation, conducted at the Hebrew University of Jerusalem.
2. Background: Gérard–Leichtnam’s transport equation
By Egorov’s theorem, in the case of a closed manifold M a semiclassical
measure µ is invariant under the geodesic flow; or equivalently ξ · ∂xµ = 0,
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL MEASURE AT SINGULAR BOUNDARY POINST 369
where (x, ξ) are canonical coordinates on T ∗M . In the presence of a C1,1-
boundary a semiclassical measure µ evolves under the geodesic flow accord-
ing to Gérard–Leichtnam’s equation. It expresses the distribution ξ ·∂xµ in
terms of a corresponding semiclassical measure ν on T ∗∂M , and Snell’s law
is manifested in the symmetry of this equation. Here we explain the notions
of a corresponding semiclassical measure on T ∗∂M , hyperbolic point and
glancing point and we are then able to present the equation.
Let M ⊂ R2 be a bounded domain with C1,1-boundary, and {uj} an
orthonormal basis of Laplace eigenfunctions of M with Dirichlet’s boundary
conditions and with increasing positive eigenvalues {Ej} respectively. For
the definition below we consider {uj} as a sequence in L2(R2).
Definition 2.1. — M -Semiclassical measures on T ∗R2.
Let {ukj } be a subsequence such that for all a ∈ C∞c
(
T ∗R2
)
there exists
the limit
lim
j→∞
〈
Ophkj (a) ukj , ukj
〉
where hkj = E
−1/2
kj
and Oph (a) u (x) = 1(2π)2
∫
a (x, hξ) eix·ξû (ξ) dξ is the
standard quantization. Then we define the distribution µ by
⟨µ, a⟩ = lim
j→∞
〈
Ophkj (a) ukj , ukj
〉
.
We call µ an M -semiclassical measure associated to the subsequence {ukj }.
Remark 2.2. — The distribution in the above definition is actually a
positive measure supported on T ∗M (see [3, p. 565]).
To define the corresponding boundary semiclassical measures recall that
in [3] Gérard and Leichtnam showed that the standard quantization pro-
cedure may be extended to the following class of symbols defined on any
closed C1-manifold N
Σc (T ∗N) = {b ∈ Cc (T ∗N)|
b has continuous vertical derivatives up to order n + 1}
where n is the dimension of N , while keeping the resulting operators Oph (b)
bounded on L2 (N).
In our case, this extension allows to define below a notion of semiclassical
measure on T ∗∂M . For the sake of this definition we also recall that due
to our assumption that ∂M is C1,1-smooth, the normal derivatives on the
boundary ∂nuk exist in L2 (∂M), and moreover, the sequence hk∂nuk is
bounded in L2 (∂M) (see e.g. [3, Lemma 2.1]).
TOME 74 (2024), FASCICULE 1
370 Dan MANGOUBI & Adi WELLER WEISER
Definition 2.3 (Corresponding semiclassical measures on T ∗∂M for a
Dirichlet problem). — For an M -semiclassical measure µ associated to the
subsequence {ukj }, assume that for all b ∈ Σc (T ∗∂M) there exists the limit
lim
j→∞
〈
Ophkj (b) hkj ∂nukj , hkj ∂nukj
〉
Then, define a positive measure ν on T ∗∂M by
⟨ν, b⟩ = lim
j→∞
〈
Ophkj (b) hkj ∂nukj , hkj ∂nukj
〉
.
We call ν a boundary semiclassical measure corresponding to µ.
Remark 2.4. — For any given subsequence {ukj } there is a subsequence
{ukjl } for which the limits in Definitions 2.1 and 2.3 exist (see [3, p. 565,
576]).
Next, points in T ∗∂M are classified according to their dynamical be-
haviour [5] (see also [1]). The definitions below are only for the case of the
billiard dynamics.
Definition 2.5. — Glancing points
G = {(x, ξ) ∈ T ∗∂M | |ξ| = 1} = S∗∂M
Definition 2.6. — Hyperbolic points
H = {(x, ξ) ∈ T ∗∂M | |ξ| < 1}
The Gérard–Leichtnam transport equation is
Theorem 2.7 ([3]). — For ν a boundary semiclassical measure corre-
sponding to an M -semiclassical measure µ we have for all a ∈ C∞c
(
T ∗R2
)
(2.1) −
∫
T ∗R2
ξ · ∂xa dµ =
∫
H∪G
a (x (ρ) , ξ+ (ρ)) − a (x (ρ) , ξ− (ρ))
⟨ξ+ (ρ) − ξ− (ρ) , n (x (ρ))⟩ dν (ρ)
where n (x (ρ)) is the inward pointing normal at x (ρ), and ξ+ (ρ) is the co-
vector in S∗x(ρ)R2 such that its orthogonal projection on T ∗x(ρ)∂M is equal
to ρ’s co-vector and so that ⟨ξ+ (ρ) , n (x (ρ))⟩ ⩾ 0. The co-vector ξ− (ρ) is
similarly defined with ⟨ξ− (ρ) , n (x (ρ))⟩ ⩽ 0.
3. Zero mass on the set of glancing points of the straight
part of the boundary
It is shown in [2] that in the case that the boundary is smooth the
set of non-gliding points is of boundary semiclassical measure zero. We
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL MEASURE AT SINGULAR BOUNDARY POINST 371
extend the proof to the case of the Bunimovich stadium and show that
the set of glancing points lying above the closure of the straight part of the
boundary is of boundary semiclassical measure zero. The main new point is
the avoidance of change of variables which allows us to handle non-smooth
points of ∂M .
Let M = ([−a, a] × (0, 1))∪
{
(x ± a)2 +
(
y − 12
)2
< 14
}
. Decompose ∂M
as S ∪ C where S is the straight part of the boundary and C is its curved
part. To be precise, we let S = S0 ∪ S1 where Si = [−a, a] × {i} and put
C = (∂M) \S. Let π : T ∗∂M → ∂M and πR2 : T ∗R2 → R2 denote the
projection maps.
Theorem 3.1. — Let ν be a semiclassical measure on T ∗∂M corre-
sponding to an M -semiclassical measure. Let G be the set of glancing points
of T ∗∂M . Then ν
(
G ∩ π−1 (S)
)
= 0.
Proof. — Let µ be an M -semiclassical measure to which ν corresponds.
Fix a point ρ0 ∈ G ∩ π−1 (S0) and consider
aε (x, ξ) = ξ2b
(
x1,
x2
ε
,
ξ22
ε
)
φ (ξ1)
where b ∈ C∞c
(
R3
)
is nonnegative with b (x1 (ρ0) , 0, 0) > 0 and where the
cutoff function φ is in C∞c (R) with |φ| ⩽ 1 and φ (ξ1) = 1 for |ξ1| ⩽ 1.
In equation (2.1) we make the substitution a = aε and take the limit as ε
tends to zero.
For calculating the left hand side of (2.1) we note that
ξ · ∂xaε = ξ1ξ2φ (ξ1) (∂1b)
(
x1,
x2
ε
,
ξ22
ε
)
+ ξ22φ (ξ1)
1
ε
(∂2b)
(
x1,
x2
ε
,
ξ22
ε
)
We claim that ξ · ∂xaε −→ 0 as ε → 0. Indeed, if ξ2 = 0 then ξ · ∂xaε = 0.
If ξ2 ̸= 0, then for all ε small enough we have that ∂ib
(
x1,
x2
ε ,
ξ22
ε
)
≡ 0
since b is compactly supported. Furthermore, ξ · ∂xaε is dominated by a
constant independent of ε ⩽ 1. In fact, we have∣∣∣∣ξ1ξ2φ (ξ1) (∂1b)(x1, x2ε , ξ22ε
)∣∣∣∣ = ∣∣∣∣ξ1φ (ξ1) ξ2√ε (∂1b)
(
x1,
x2
ε
,
ξ22
ε
)√
ε
∣∣∣∣
⩽ sup
x′1,x
′
2,y,ξ
′
1
∣∣y∂1b (x′1, x′2, y2)∣∣ |ξ′1φ (ξ′1)|
while ∣∣∣∣ξ22ε φ (ξ1) (∂2b)
(
x1,
x2
ε
,
ξ22
ε
)∣∣∣∣ ⩽ sup
x′1,x
′
2,y
|y∂2b (x′1, x′2, y)| .
TOME 74 (2024), FASCICULE 1
372 Dan MANGOUBI & Adi WELLER WEISER
Since µ has compact support (supp µ ⊆ S∗R2 ∩ π−1R2
(
M
)
, see [3]) we may
conclude by the Lebesgue Dominated Convergence Theorem that
(3.1)
∫
T ∗R2
ξ · ∂xaε dµ −→
ε→0
0 .
Next, we calculate the right hand side of (2.1). We set
bε (x1, x2, ξ2) := b
(
x1,
x2
ε
,
ξ22
ε
)
.
For y ∈ ∂M , let n∗ (y) ∈ S∗yR2 be the co-vector such that ⟨n∗ (y) , n (y)⟩ = 1
and ⟨n∗ (y) , γ̇⟩ = 0 where γ̇ ∈ Sy∂M . Note that for ρ ∈ H ∪ G
ξ± (ρ) = ξ (ρ) ±
√
1 − |ξ (ρ)|2n∗ (x (ρ)) .
It will be convenient to write
ξ± (ρ) = ξ±1 (ρ) dx1 + ξ
±
2 (ρ) dx2 .
The numerator of the integrand in the right hand side of (2.1) is
aε
(
x (ρ) , ξ+ (ρ)
)
− aε
(
x (ρ) , ξ− (ρ)
)
|ξ±1 |⩽1= ξ+2 (ρ) bε
(
x (ρ) , ξ+2 (ρ)
)
− ξ−2 (ρ) bε
(
x (ρ) , ξ−2 (ρ)
)
= fε
(
x (ρ) , ξ+2 (ρ)
)
− fε
(
x (ρ) , ξ−2 (ρ)
)
where fε (x, ξ2) := ξ2bε (x, ξ2). Hence we can write the integrand in the
right hand side of (2.1) as
(3.2) Aε (ρ) =
fε
(
x (ρ) , ξ+2 (ρ)
)
− fε
(
x (ρ) , ξ−2 (ρ)
)
∥ξ+ (ρ) − ξ− (ρ) ∥
= ξ
+
2 (ρ) − ξ
−
2 (ρ)
∥ξ+ (ρ) − ξ− (ρ) ∥ ·
1
ξ+2 (ρ) − ξ
−
2 (ρ)
∫ ξ+2 (ρ)
ξ−2 (ρ)
∂ξ2fε (x (ρ) , y) dy
= ξ
+
2 (ρ) − ξ
−
2 (ρ)
∥ξ+ (ρ) − ξ− (ρ) ∥
∫ 1
0
∂ξ2fε
(
x (ρ) , tξ+2 (ρ) + (1 − t) ξ
−
2 (ρ)
)
dt .
To take the limit notice first that
(3.3) |∂ξ2fε (x1, x2, ξ2)| =
∣∣∣∣bε (x1, x2, ξ2) + 2ξ22ε (∂3b)
(
x1,
x2
ε
,
ξ22
ε
)∣∣∣∣
⩽ sup
y
∣∣∣b(x1, x2
ε
, y
)∣∣∣+ 2 sup
y
∣∣∣y∂3b(x1, x2
ε
, y
)∣∣∣
and
(3.4)
∣∣ξ+2 (ρ) − ξ−2 (ρ)∣∣
∥ξ+ (ρ) − ξ− (ρ) ∥ ⩽ 1 .
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL MEASURE AT SINGULAR BOUNDARY POINST 373
As x2 (ρ) ̸= 0 when ρ ∈ π−1 (C ∪ S1) and b is with compact support we
can infer that limε→0 Aε (ρ) = 0 in this case. For ρ ∈ π−1 (S0) notice that
we have ξ+2 (ρ) = −ξ
−
2 (ρ) =
√
1 − |ξ (ρ)|2, and since bε (x, ξ2) = bε (x, −ξ2)
we get
Aε (ρ) =
ξ+2 (ρ) bε
(
x (ρ) , ξ+2 (ρ)
)
− ξ−2 (ρ) bε
(
x (ρ) , ξ−2 (ρ)
)
ξ+2 (ρ) − ξ
−
2 (ρ)
= bε
(
x1 (ρ) , 0,
√
1 − |ξ (ρ)|2
)
= b
(
x1 (ρ) , 0,
1 − |ξ (ρ)|2
ε
)
.
We conclude that
(3.5) Aε (ρ) −→
ε→0
{
b (x1 (ρ) , 0, 0) , if π (ρ) ∈ S0 and |ξ (ρ)| = 1,
0, otherwise.
We verify that Aε is bounded independently of ε. Indeed, from the integral
expression (3.2) and the bounds (3.3) and (3.4) it follows that
|Aε (ρ)| ⩽ sup |b| + 2 sup
x′1,x
′
2,y
|y∂3b (x′1, x′2, y)| .
From (3.5) we have by the Lebesgue Dominated Convergence Theorem
the convergence of the integral
(3.6)
∫
H∪G
Aε (ρ) dν (ρ) −→
ε→0
∫
G∩π−1(S0)
b (x1 (ρ) , 0, 0) dν (ρ) .
Comparing (3.1) and (3.6) we learn that
0 =
∫
G∩π−1(S0)
b (x1 (ρ) , 0, 0) dν (ρ) .
Since b (x1 (ρ) , 0, 0) ⩾ 0 and b (x1 (ρ0) , 0, 0) > 0 we have that ρ0 /∈ supp (ν).
Because ρ0 ∈ G ∩ π−1 (S0) is arbitrary, we see that ν
(
G ∩ π−1 (S0)
)
= 0.
Similarly ν
(
G ∩ π−1 (S1)
)
= 0. □
3.1. An argument under a vanishing assumption on the curved
part
In [4] the zero mass of non-gliding points theorem from [2] is applied in a
case where it is known that the boundary semiclassical measure ν on T ∗∂M
vanishes on the portion lying above the curved part of ∂M , π−1 (C). In this
circumstance one does not need the full power of Theorem 3.1 and we bring
here an ad hoc approach which was explained to us by Gérard.
TOME 74 (2024), FASCICULE 1
374 Dan MANGOUBI & Adi WELLER WEISER
Since ν vanishes on π−1 (C), it follows from Gérard–Leichtnam’s equation
that if ν corresponds to an M -semiclassical measure µ, then µ is supported
on the portion of T ∗R2 lying above the closed rectangular part of the
billiard table. As a result, one may replace the singular billiard table by
an infinite strip, and then apply the analysis in [2] on a smooth domain in
order to conclude that ν vanishes on G ∩ π−1(S).
Appendix. Non-Gliding points in the case of a two
dimensional billiard with C2-boundary
We recall the definition of a special subset Gng ⊂ G ⊂ T ∗∂M which is
known as the set of non-gliding points. For a two dimensional C2-billiard M ,
one can first positively orient ∂M , and then define
Definition A.1. — Non-gliding points for M ⊂ R2 with billiard dy-
namics.
Gng = {(x, ξ) ∈ G | The curvature of ∂M at x is nonpositive}
Remark A.2. — In [2] these points are called non-strictly-gliding, a term
which we avoid due to grammatical ambiguity.
More generally [5], the notion of non-gliding points is defined in the con-
text of a Hamiltonian dynamical system on a manifold with boundary M ,
and in any case it requires that the normal vector to ∂M be C1, or, equiva-
lently, that the boundary be C2. The set of non-gliding points is the union
of the set of diffractive points and the set of high order glancing points
as defined in [5]. In the case of two dimensional billiards the definition
simplifies to the one above.
BIBLIOGRAPHY
[1] K. G. Andersson & R. B. Melrose, “The propagation of singularities along gliding
rays”, Invent. Math. 41 (1977), no. 3, p. 197-232.
[2] N. Burq & P. Gérard, “Condition nécessaire et suffisante pour la contrôlabilité
exacte des ondes”, C. R. Math. Acad. Sci. Paris 325 (1997), no. 7, p. 749-752.
[3] P. Gérard & É. Leichtnam, “Ergodic properties of eigenfunctions for the Dirichlet
problem”, Duke Math. J. 71 (1993), no. 2, p. 559-607.
[4] A. Hassell, “Ergodic billiards that are not quantum unique ergodic”, Ann. Math.
171 (2010), no. 1, p. 605-618, with an appendix by the author and Luc Hillairet.
[5] R. B. Melrose, “Microlocal parametrices for diffractive boundary value problems”,
Duke Math. J. 42 (1975), no. 4, p. 605-635.
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL MEASURE AT SINGULAR BOUNDARY POINST 375
Manuscrit reçu le 3 juillet 2022,
accepté le 23 septembre 2022.
Dan MANGOUBI
Einstein Institute of Mathematics
Edmond J. Safra campus
The Hebrew University of Jerusalem
Jerusalem 9190401 (Israel)
dan.mangoubi@mail.huji.ac.il
Adi WELLER WEISER
Einstein Institute of Mathematics
Edmond J. Safra campus
The Hebrew University of Jerusalem
Jerusalem 9190401 (Israel)
adi.weller@mail.huji.ac.il
TOME 74 (2024), FASCICULE 1


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