spectral analysis of sub-riemannian laplacians, weyl...

sR Laplacian heat kernels canonical sR measures Weyl law quantum limits, QE

Spectral analysis of sub-RiemannianLaplacians, Weyl measures

Yves Colin de Verdiere Luc Hillairet Emmanuel Trelat1

1Univ. Pierre et Marie Curie (Paris 6), Labo. J.-L. Lions

Seminaire d’Analyse de Bordeaux, 26 janvier 2017

Projet ANR SRGI (Sub-Riemannian Geometry and Interactions)

Y. Colin de Verdiere, L. Hillairet, E. Trelat Spectral analysis of sub-Riemannian Laplacians


Sub-Riemannian Laplacian

(M,D, g) sub-Riemannian (sR) structure:

M (smooth) connected manifold of dimension n

D ⊂ TM subsheaf (horizontal distribution)

g Riemannian metric on D

Locally: D = Span(X1, . . . ,Xm),

∀q ∈ M ∀v ∈ Dq gq(v , v) = inf

( mXi=1

u2i

˛v =

mXi=1

ui Xi (q)

)

(gq : positive definite quadratic form on Dq)



Sub-Riemannian Laplacianµ: arbitrary smooth volume on M

Definition

−4sR = selfadjoint nonnegative operator on L2(M, µ) defined as the Friedrichsextension of the Dirichlet integral

Q(φ) =

ZM‖dφ‖2

g∗ dµ, φ ∈ C∞(M)

g∗: cometric associated with g g∗(ξ, ξ) = maxv∈Dq\0

〈ξ, v〉2

gq (v, v)

Equivalent definitions:




Definition


Q(φ) =

ZM‖dφ‖2

g∗ dµ, φ ∈ C∞(M)


〈ξ, v〉2

gq (v, v)


4sRφ = divµ (∇sRφ) ∀φ ∈ C∞(M)

where

divµ defined by LX dµ = divµ(X) dµ ∀X vector field on M

horizontal gradient ∇sR defined by gq(∇sRφ(q), v) = dφ(q).v ∀v ∈ Dq

note that ‖dφ‖g∗ = ‖∇sRφ‖g




Definition


Q(φ) =

ZM‖dφ‖2

g∗ dµ, φ ∈ C∞(M)


〈ξ, v〉2

gq (v, v)


If (X1, . . . ,Xm) is a local g-orthonormal frame of D then ∇sRφ =Pm

i=1(Xiφ)Xi and

4sR = −mX

i=1

X∗i Xi =mX

i=1

“X 2

i + divµ(Xi )Xi

”

Setting hX (q, p) = 〈p, X(q)〉, note that σP (−4sR ) =mX

i=1

(hXi)2 and that σsub(−4sR ) = 0



Sub-Riemannian Laplacian

4 = 4sR = −mX

i=1

X∗i Xi =mX

i=1

“X 2

i + divµ(Xi )Xi

”

For M compact, under Hormander’s assumption Lie(D) = TM, the operator −4 ishypoelliptic (and even subelliptic), has a compact resolvent and thus a discretespectrum

0 = λ1 6 λ2 6 · · · 6 λk → +∞

Let (φk )k∈IN∗ be an orthonormal eigenbasis of L2(M, µ).

Objectives

Establish heat kernel asymptotics.

Derive a (micro-)local Weyl law for4.

Identify (micro-)local Weyl measure.

Establish Quantum Ergodicity (QE) properties.



More generally: X0 smooth vector field on M, c smooth bounded function on M.

X = (X0,X1, . . . ,Xm)

4X =mX

i=1

X 2i + X0 + c id

→ operator on L2(M, µ).

Remark: 4X is selfadjoint⇔ X0 =Pm

i=1 divµ(Xi )Xi .



sR flag

Sequence of subsheafs Dk ⊂ TM:

D0 = 0D1 = D = Span(X1, . . . ,Xm)

Dk+1 = Dk + [D,Dk ] for k > 1

sR flag at q: 0 = D0q ⊂ Dq = D1

q ⊂ D2q ⊂ . . . ⊂ Dr(q)−1

q ( Dr(q)q = TqM

r(q): degree of nonholonomy at q

ni (q) = dim Diq (nr(q)(q) = n = dim M)

w1(q) = · · · = wn1 (q) = 1wn1+1(q) = · · · = wn2 (q) = 2

.

.

.wnr−1+1(q) = · · · = wnr (q) = r

Q(q) =rX

i=1

i(ni (q)− ni−1(q)) =nX

i=1

wi (q)

(= Hausdorff dimension of the sR structure at q if q regular)

q is said regular if the flag at q is regular.The sR structure is said equiregular if all points are regular.



sR flag

Sequence of subsheafs Dk ⊂ TM:

D0 = 0D1 = D = Span(X1, . . . ,Xm)

Dk+1 = Dk + [D,Dk ] for k > 1

sR flag at q: 0 = D0q ⊂ Dq = D1

q ⊂ D2q ⊂ . . . ⊂ Dr(q)−1

q ( Dr(q)q = TqM

Example: 3D contact case

X1 = ∂x , X2 = ∂y + x∂z

[X1,X2] = ∂z

→ equiregular

n1 = 2, n2 = 3

Q = 4

Example: 3D Martinet case

X1 = ∂x , X2 = ∂y + x2

2 ∂z

[X1,X2] = x∂z , [X1, [X1,X2]] = ∂z

→ singular at x = 0 (and contact outside)

n1(0) = n2(0) = 2, n3(0) = 3

Q(0) = 5



Nilpotentization

Nilpotentization of the sR structure (M,D, g) at q0 ∈ M:(intrinsic) sR structure (bMq0 , bDq0 , bgq0 ), wherebMq0 smooth connected manifold of dimension n bMq0 ∼ IRnbDq0 = Span(bX q0

1 , . . . , bX q0m )

sR metric bgq0 is defined, accordingly, by

∀x ∈ bMq0 ∀v ∈ bDq0x bgq0

x (v, v) = inf

8<:mX

i=1

u2i | v =

mXi=1

uibXq0

i (x)

9=; ,(equivalence class under the action of sR isometries on sR structures)

If q0 is regular then bMq0 ' Gq0 (nilpotent Lie group of diffeomorphims of IRn generated by the exp(tbXq0i )) is a

Carnot group, endowed with a left-invariant sR structure.

—————–In a local chart ψq0 : Uq0 ⊂ M → V0 ⊂ IRn of privileged coordinates:

Dilation δε(x) =“εw1(q0)x1, . . . , ε

wn(q0)xn

”Nilpotentization at q0 of the vector fields: bX q0

i = limε→0

εδ∗ε (ψq0 )∗Xi

Nilpotentization at q0 of the measure: bµq0 = limε→0

1εQ(q0)

δ∗ε (ψq0 )∗µ = Cst(q0) dx



Nilpotentization

Example:

X1 = ∂x , X2 = ∂y + x∂z + x2∂zbX 01 = ∂x , bX 0

2 = ∂y + x∂z

Example:

X1 = ∂x , X2 = ∂y + x2

2 ∂z + x3∂z

bX 01 = ∂x , bX 0

2 = ∂y + x2

2 ∂z

Nilpotentization bX q0i of Xi = “first-order” terms of the “nonholonomic Taylor expansion”

(bMq0 , bDq0 , bgq0 ) = tangent space (in the metric sense of Gromov)

(Carnot group if q0 is regular)

Carnot groups are to sR geometry as Euclidean spaces are to Riemannian geometry.

In contrast to Riemannian geometry where all tangent spaces are isometric, two nilpotentizations at q0 and q1 may

not be sR-isometric.



Heat kernel

Heat kernels of4 and of b4q0 :

D(4) = f ∈ L2(M, µ) | 4f ∈ L2(M, µ).

(4,D(4)) generates (et4)t>0 on L2(M, µ).

Let e = e4,µ : (0,+∞)×M ×M → IR be the heat (Schwartz) kernel.

Nilpotentized sR Laplacian: b4q0 =Pm

i=1

“bX q0i

”2

D(b4q0 ) = f ∈ L2(bMq0 , bµq0 ) | b4q0 f ∈ L2(bMq0 , bµq0 ).

→ semigroup (et b4q0 )t>0 on L2(bMq0 , bµq0 ).

Let be = e b4q0 ,bµq0 : (0,+∞)× bMq0 × bMq0 → IR be the heat (Schwartz) kernel.

Note that (homogeneity)

e(t , x , x ′) = εQ(q0) be(ε2t , δε(x), δε(x ′))



Heat kernel asymptotics

Fundamendal lemma

q0 ∈ M arbitrary, and µ arbitrary smooth measure on M.In a local chart of privileged coordinates at q0:

∀k ∈ IN tQ(q0)/2 e(t , δ√t (x), δ√t (x ′))

= be(1, x , x ′) + t a1(x , x ′) + · · ·+ tk ak (x , x ′) + o(tk ),

as t → 0+, in C∞(M ×M), with aj smooth.

q0 need not be regular.

If q0 is regular, then the above convergence and asymptotic expansion are alsolocally uniform with respect to q0.

Still valid for4X =Pm

i=1 X 2i + X0 + c id, with an expansion in tk/2, provided

that:either X0 smooth section of D;or X0 smooth section of D2, and then replace b4q0 with b4q0 + bX q0

0



Heat kernel asymptotics

Fundamendal lemma

q0 ∈ M arbitrary, and µ arbitrary smooth measure on M.In a local chart of privileged coordinates at q0:

∀k ∈ IN tQ(q0)/2 e(t , δ√t (x), δ√t (x ′))

= be(1, x , x ′) + t a1(x , x ′) + · · ·+ tk ak (x , x ′) + o(tk ),

as t → 0+, in C∞(M ×M), with aj smooth.

Taking x = x ′ = 0, we get the expansion of the kernel along the diagonal, and

e(t , q0, q0) dµ(q0) ∼be(1, 0, 0)

tQ(q0)/2dµ(q0) = be(t , 0, 0) dµ(q0) = be(t , 0, 0) dν(q0) ∀ν

→ useful to derive the local Weyl law.Generalization of results by Metivier (1976), Ben Arous (1989).

estimations near the diagonal→ microlocal Weyl law and singular sR structures.



General question in sR geometry: define an intrinsic volume in a sR manifold.

[Agrachev Barilari Boscain 2012]

Hausdorff (standard or spherical) Mitchell 1985, Gromov 1996

Popp Montgomery 2002

a new one: Weyl



Popp measure

Let q ∈ M, regular or not.

Popp volume (Montgomery 2002): canonical smooth volume form associated with thesR metric g and the flag structure, defined by

|dP(q)| = Φ∗|ν1 ∧ · · · ∧ νr |

with the canonical isomorphismΛn(T?q M)

Φ−→ Λn„ r(q)M

k=1

Dkq/Dk−1

q

«?

and with νk = canonical volume form on Dkq/Dk−1

q induced by the Euclidean structure coming from the surjection

D⊗kq → Dk

q/Dk−1q (take Lie brackets modulo Dq ).

P is invariant under (local) sR isometries

P commutes with nilpotentization: cPMq

= PbMq ⇒ P is “doubly intrinsic”

P is smooth near regular points

Explicit expression in [Barilari Rizzi 2013] (in the equiregular case)



(Micro-)local Weyl measure−4φk = λkφk , (φk )k∈IN∗ orthonormal eigenbasis of L2(M, µ), λ1 6 λ2 6 · · · 6 λk → +∞

Spectral counting function: N(λ) = #n | λk 6 λ M compact

Local Weyl measure = probability measure w4 on M defined (if the limit exists) byZM

f dw4 = limλ→+∞

1N(λ)

Xλk 6λ

ZM

f |φk |2 dµ ∀f ∈ C0(M)

i.e.,

w4 = weak limλ→+∞

1N(λ)

Xλk 6λ

|φk |2 µ (Cesaro mean)

If it exists:w4 does not depend on the choice of (φk )k∈IN∗

w4 is invariant under sR isometries of M

In the equiregular case: w4 exists, is smooth (cf further), does not depend on µ,and commutes with nilpotentization: dw4q

= “w b4q ”



(Micro-)local Weyl measure−4φk = λkφk , (φk )k∈IN∗ orthonormal eigenbasis of L2(M, µ), λ1 6 λ2 6 · · · 6 λk → +∞

Spectral counting function: N(λ) = #n | λk 6 λ M compact

Local Weyl measure = probability measure w4 on M defined (if the limit exists) byZM

f dw4 = limλ→+∞

1N(λ)

Xλk 6λ

ZM

f |φk |2 dµ ∀f ∈ C0(M)

Microlocal Weyl measure = probability measure on S?M = S(T?M) defined (if the limitexists) byZ

S?Ma dW4 = lim

λ→+∞

1N(λ)

Xλk 6λ

˙Op(a)φk , φk

¸L2(M,µ)

∀a ∈ S0(S?M)

Note that π∗W4 = w4 with π : T∗M → M canonical projection.

Lemma: We always have supp(W4) ⊂ SΣ, where Σ = D⊥ (annihilator of D).

General problem: identify these measures, compare them with Hausdorff, Popp.



(Micro-)local Weyl measure

Defining the averaged correlation of eigenfunctions by

C(x , y) = limλ→+∞

1N(λ)

Xλk 6λ

φk (x + y/2)φk (x − y/2),

for all (x , y) ∈ M ×M (when the limit exists), the correlation is the Fourier transformwith respect to ξ of the microlocal Weyl density, i.e.,

C(x , y) =

Ze−iy.ξ dW4(x , ξ)

dx



Local Weyl law in the equiregular case

Assume that (M,D, g) is equiregular⇒ Q(q) = Q constant.

∀q ∈ M, consider the sR structure (bMq , bDq , bgq), that is the nilpotentization of(M,D, g) at q. It induces the sR Laplacian b4q =

Pmi=1(bX q

i )2.

beq = e b4q ,bPq = heat kernel on (0,+∞)× bMq × bMq associated with b4q and with

the Popp measure bPq on bMq .

Since (along the diagonal)

tQ/2e(t , q, q) dµ(q) −→ beq(1, 0, 0) dP(q),

as t → 0+, we get

TheoremZM

e4X ,µ(t , q, q)f (q) dµ(q) =

RMbeq(1, 0, 0)f (q) dP(q)

tQ/2+ o

„1

tQ/2

«∀f ∈ C0

c (M)




By inverse Laplace transform (Karamata), we get:

Corollary

Assumptions: M compact, equiregular. Then:

Xλk 6λ

ZM

f |φk |2 dµ =

1

Γ(Q/2 + 1)λQ/2

ZMbeq (1, 0, 0)f (q) dP(q) + o(λQ/2) ∀f ∈ C0

c (M)

as λ→ +∞. In particular:

N(λ) ∼R

Mbeq(1, 0, 0) dP(q)

Γ(Q/2 + 1)λQ/2

and the local Weyl measure w4 exists, is smooth, and

dw4(q) = beq(1, 0, 0) dP(q)

Remark: Since w4 is smooth, it differs in general from HS (which is not smooth ingeneral for n > 5, see [AgrachevBarilariBoscain 2012])




Equiregular case: dw4(q) = beq(1, 0, 0) dP(q)

Hence: w4 = P (up to some multiplying scalar; i.e., Weyl = Popp)⇔ beq(1, 0, 0) = Cst ⇔ all nilpotent sR structures (bMq , bDq , bgq) are sR-isometric⇔ IsosR(M) of sR isometries of M acts transitively on M

This is so in the following cases:

The sR structure on M is free

The sR structure on M is nilpotent and equiregular, and dim M 6 5, with:dimension 3: growth vector (2, 3) (Heisenberg)dimension 4: growth vector (2, 3, 4) (Engel) and (3, 4) (quasi-Heisenberg)dimension 5: growth vector (2, 3, 5) (Cartan), (2, 3, 4, 5) (Goursat rank 2), (3, 5) (corank 2),(3, 4, 5) (Goursat rank 3)

[Agrachev Barilari Boscain 2012]

Remark: In the 3D contact case, we have beq(1, 0, 0) = 1/16.

Remark: w4 6= P in general (even in the equiregular case; in which both are smooth)



Microlocal Weyl law in the equiregular case

Define Σi = (Di )⊥ ⊂ T?M (annihilator of Di ) for i = 1, . . . , r .

Theorem

Assumptions: M compact, equiregular.For every A ∈ Ψ0(M), we have

Xλk 6λ

〈Aφk , φk 〉 =C

Γ(Q/2 + 1)λQ/2

ZSΣr−1

a dW4 + o(λQ/2),

where SΣ = (Σ \ 0)/IR+. In particular:

supp(W4) ⊂ SΣr−1

and

dW4(q, p) =1

(2π)nR

M beq (1, 0, 0) dP(q)

0@ZT∗q M/Σ

r−1q

Kq

1A δΣr−1 (q, p),

with Kq (p) =

ZM

eiq′.p beq (1, q′, q) dP(q′) = F“

q′ 7→ beq (1, q′, q)”

(p)



Singular sR structuresMartinet case (without singularity):

M compact 3D manifold, endowed with a smooth measure µD = kerα, with α ∧ dα vanishing on S (Martinet surface), D transverse to S.

⇔ D2 6= D3, D2 = TM outside of S, D3 = TM along S, D ∩ TS line bundle over S.

Local model: α = dz − x2

2 dy , S = x = 0.

In M \ S, we have dP = 1|x|dx dy dz = 1

|x|dx ⊗ dν with ν smooth.

Theorem

We have (π|Σ : SΣ→ M double covering)

N(λ) ∼ν(S)

32λ2 lnλ, w4 =

ν

ν(S), W4 =

12π∗|Σw4

“Spectral concentration occurs on the Martinet (singular) surface.”Two-terms expansion:

ZM

e4,µ(t , q, q)f (q) dµ =1t2

„“ | ln t |16

+ A”ZS

f dν + P.V.Z

Mfh dP + o(1)

«



Singular sR structures

Grushin case: almost-Riemannian structure on a surface

M compact 2D manifold, endowed with a smooth measure µ

Locally, X and Y generate TM except along a 1D submanifold S

Local model: X = ∂x , Y = x∂y . −4 = X∗X + Y∗Y

In M \ S, we have dP = 1|x|dx dy = 1

|x|dx ⊗ dν with νsmooth.

Theorem

We have (π|Σ : SΣ→ M double covering)

N(λ) ∼ν(S)

4πλ lnλ, w4 =

ν

ν(S), W4 =

12π∗|Σw4

Two-terms expansion:

ZM

e4,µ(t , q, q)f (q) dµ =1

4πt

„(| ln t |+ γ + 4 ln 2)

ZS

f dν + P.V.gZ

Mf dxg + o(1)

«



Singular sR structures

Generalization to any singular structure: ongoing.

limλ→+∞

ln N(λ)

lnλ= (sR spherical) Hausdorff dimension

This may a priori allow N(λ) ∼ λα lnλ ln lnλ (open question).



Quantum Limits (QL)

M locally compact space, endowed with a Radon measure µ

T selfadjoint nonnegative operator on L2(M, µ), with compact resolvent

discrete spectrum λ1 6 λ2 6 · · · 6 λj 6 · · · → +∞

(φj )j∈IN∗ orthonormal eigenbasis of L2(M, µ)

A quantum limit (on the base) is a (weak) limit of the sequence of probability measures |φ2j | dx .

More generally (pseudo-diff. version), a QL is a probability measure on S?M, closure point ofthe sequence of measures µj (a) = 〈Op(a)φj , φj 〉 (a: symbol of order 0)

- Does the energy equidistribute at highfrequencies (i.e., µj uniform measure)

- Or, at the contrary, may the energy concentrate? (for instance, µj Dirac)→ insight on the way eigenfunctions concentrate

General question in quantum physics, quantum chaos: what are the possible QLs?



Examples of quantum limits for the Dirichlet-Laplacian

1D case Ω = (0, π): φj (x) =q

2π

sin(jx), sin2(jx) → 1/2⇒ Quantum Unique Ergodicity (QUE)

2D square Ω = (0, π): φj,k (x) = 2π

sin(jx) sin(ky) → many QLs

2D disk: Bessel functions

Whispering galleriesphenomenon

All quantum limits in the disk: [Hillairet Privat Trelat]



Integrable systems:

Chaotic systems:

QLs must be invariant under thegeodesic flow.

On the sphere: every invariantmeasure is a QL (Zelditch).



Quantum Ergodicity (QE)

Water wave in a stadium-shaped basin:

Laser wave in an optic fiber:



We say we have QE for (T , (φn)n∈IN∗ ) if there exist a probability measure ν on M and asubsequence (nj )j∈IN∗ of density one such that

|φnj |2 dµ dν as j → +∞

More generally (pseudo-diff. version),

DOp(a)φnj , φnj

EL2(M,µ)

→Z

Σa d ν ∀a ∈ S0



Quantum Ergodicity

Historical example of QE, by A. Shnirelman in 1974 (incomplete proof):

On (M, g) compact Riemannian manifold, if the geodesic flow is ergodic, then we haveQE for any orthonormal basis of eigenfunctions of the Laplace-Beltrami operator4,with dν = normalized Riemannian volume.

(ergodicity for instance if the curvature of (M, g) is < 0)

More generally, in pseudo-diff. version: ν = Liouville measure on S∗M

M compact without boundary: S. Zelditch (1987), Y. Colin de Verdiere (1985)(semi-classical version) B. Helffer, A. Martinez, D. Robert (1987)

M compact with (smooth) boundary: P. Gerard, E. Leichtnam (1993), S. Zelditch, M. Zworski (1996)

discontinuous metrics: D. Jakobson, Y. Safarov, A. Strohmaier, Y. Colin de Verdiere (2015)

Other existing results (Schrodinger operators, graphs),but always for elliptic operators and never in hypoelliptic situations.



Quantum chaos

QUE conjecture (Rudnick-Sarnak 1994): every compact manifold having negativesectional curvature satisfies QUE.

If QUE fails, we may have scars: energy concentration phenomena(exceptional subsequences may converge to other invariant measures, like, forinstance, measures carried by closed geodesics)



The 3D contact case

M compact connected 3D manifold (without boundary), arbitrary smooth measure µ,Riemannian contact structure⇒ sub-Riemannian Laplacian4SR

(φn, λn)n∈IN∗ orthonormal eigenbasis of L2(M, µ)

Theorem: If the Reeb flow is ergodic on M for the Popp measure, then we have QE.

We identify S?M = U?M ∪ SΣ, with U?M = g? = 1 (cylinder bundle).

Theorem: Without any ergodicity assumption,1 ∀β quantum limit associated with (φn)n∈IN∗ ,

β = β0 + β∞ with β0 ⊥ β∞ and

supp(β0) ⊂ U?M, and β0 invariant under the sR geodesic flow

supp(β∞) ⊂ SΣ, and β∞ invariant under the (lift to SΣ of the) Reeb flow

2 ∃(nj )j∈IN∗ of density one s.t. ∀β quantum limit associated with (φnj )j∈IN∗ , wehave supp(β) ⊂ SΣ (i.e., β0 = 0)



Proposition: In the 3D Heisenberg flat case:

If γ is a QL for the flat torus IR2/√

2πZ2, then β = γ ⊗ |dz| ⊗ δpz =0 is a QL for4 (with β∞ = 0).

Any probability measure β on Σ that is invariant under the Reeb flow is a QL for4 (with β0 = 0).

There exist QLs β0 and β∞ for4, with β0 invariant under the sR flow and β∞ supported on Σ andinvariant under the Reeb flow, such that, for every a ∈ [0, 1], βa = aβ0 + (1− a)β∞ is also a QL for4.

Model: compact Heisenberg flat case

Heisenberg group G = IR3 with product rule (x, y, z) ? (x′, y′, z′) = (x + x′, y + y′, z + z′ − xy′)

contact form α = dz + x dy

vector fields X = ∂x and Y = ∂y − x∂z

)left-invariant on G

Γ = (x, y, z) ∈ G | x, y ∈√

2πZ, z ∈ 2πZ discrete co-compact subgroup of G

M = Γ\G, D = kerα endowed with g s.t. (X , Y ) g-orthonormal frame of D

Reeb vector field Z = −[X , Y ] = ∂z

Lebesgue volume dµ = dx dy dz = dP Popp volume

4 = X2 + Y 2

Spec(−4sR ) = λ`,m = (2` + 1)|m| | m ∈ Z \ 0, ` ∈ IN[µj,k = 2π(j2 + k2) | j, k ∈ Z,

where λ`,m is of multiplicity |m|



A general path towards QE

(see Zelditch)

N(λ) = #n | λn 6 λ

First step: establish a microlocal Weyl law

(and identify the invariant measure ν)

E(A)def= lim

λ→∞

1N(λ)

Xλn6λ

〈Aφn, φn〉 = a =

ZST?M

a dW4

∀A ∈ Ψ0 with a = σP(A).

(E(A) = Cesaro mean)

→ Cesaro convergence property, under weak assumptions (without ergodicity):

〈(A− a id)φn, φn〉 → 0 in Cesaro mean




(see Zelditch)

N(λ) = #n | λn 6 λ

Second step: prove a variance estimate

V (A− a id)def= lim

λ→∞

1N(λ)

Xλn6λ

|〈(A− a id)φn, φn〉|2 = 0.

i.e.|〈(A− a id)φn, φn〉|2 → 0 in Cesaro mean

→ Combine the microlocal Weyl law with ergodicity properties of some associatedclassical dynamics and with an Egorov theorem.




(see Zelditch)

End of the proof of QE:

Lemma (Koopman and Von Neumann)

Given a bounded sequence (un)n∈IN of nonnegative real numbers:

1n

n−1Xk=0

uk −→n→+∞

0 ⇐⇒ ∃S ⊂ IN of density one s.t. uk −→k→+∞

k∈S

0

(density one meaning that 1n #k ∈ S | k 6 n − 1 −→

n→+∞1)

Hence, there exists a density-one sequence (nj )j∈IN∗ s.t.

limj→+∞

DAφnj , φnj

E= a.

Conclusion with a diagonal argument, using the fact that S0 admits a countabledense subset.



The 3D contact caseNormal form in the Heisenberg flat case:

−4sR = RΩ = ΩR

withR =

√Z?Z acting by multiplication by |m| on the functions eimz f (x , y)

Ω = U2 + V 2, U = 1i R−

12 X = OpW

“hX√|hZ |

”, V = 1

i R−12 Y = OpW

“hY√|hZ |

”harmonic oscillator (pseudo of order 1/2 in the cone Cc = p2

z > c(p2x + p2

y ))

[U,V ] = ±id, [R,Ω] = 0, e2iπΩ = id

In terms of symbols,

σ(−4sR) = h2X + h2

Y = |hZ ||z0BB@

hXp|hZ |

!2

+

hYp|hZ |

!2

| z 1CCA

↓ ↓R U2 + V 2 = Ω



The 3D contact case

General case: Birkhoff normal form

(M,4sR) arbitrary 3D contact case ∼ (MH ,4H ) 3D Heisenberg flat case

Theorem (classical normal form, see also Melrose 1984)

∀q ∈ M ∃ Cq conical neighbourhood of Σ±q∃ χ : Cq → MH homogeneous symplectic diffeo., with χ(Σ± ∩ Cq ) ⊂ ΣH , s.t.

σP(−4H ) χ = σP(−4sR) + OΣ(∞)

Proof by symplectic reductions, Darboux-Weinstein lemma, cohomological equations. By quantization:

Corollary (quantum normal form)

In a microlocal neighborhood of Σq : −4sR = RΩ + V0 + OΣ(∞)

V0 ∈ Ψ0, R,Ω ∈ Ψ1 selfadjoint

σP(R) = |hZ |+ OΣ(2), σP(Ω) =h2

X|hZ |

+h2

Y|hZ |

+ OΣ(3)

[R,Ω] = 0 mod Ψ−∞, exp(2iπΩ) = id mod Ψ−∞



Perspectives, open questions

3D contact case: are the Reeb periods spectral invariants?

5D contact case: two harmonic oscillators, hence resonances

⇒ failure of Birkhoff normal form at infinite order along Σ

QE (and QLs) in more general cases:Grushin: we have QE if the singular curve is connected.

Martinet: ergodicity of the singular flow (in Martinet surface)⇒ QE?

Quasi-contact in dim 4 (equiregular):magnetic lines = projections of singular geodesics.ergodicity of the magnetic vector field⇒ QE?

Existence of w4 and W4 in general (singular cases)

Controllability, observability of subelliptic heat or wave equations.


spectral analysis of sub-riemannian laplacians, weyl...

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