Quantum ergodicity for Eisenstein functions C.R. Acad. Sci. Paris, Ser.I

(1)

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Partial differential equations

Quantum ergodicity for Eisenstein functions

Ergodicité quantique des fonctions d’Eisenstein

Yannick Bonthonneau

^a

, Steve Zelditch

^b

^,

¹

aCIRGET,UQÀM,201,av.Président-Kennedy,Montréal,Québec,H2X3Y7,Canada bDepartmentofMathematics,NorthwesternUniversity,Evanston,IL60208,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received21December2015 Acceptedafterrevision28June2016 Availableonline5August2016 PresentedbytheEditorialBoard

A newproof is givenofQuantumErgodicityfor EisensteinSeries for cuspedhyperbolic surfaces. This result is also extended to higher dimensional examples, with variable curvature.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

r é s um é

Ondonne unenouvellepreuvede l’ergodicitéquantiquedessériesd’Eisensteinpour les surfacesde Riemannàpointes.Onétend aussicerésultatenplusgrandedimension,en autorisantlacourburevariable.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

1. Introduction

ThepurposeofthisnoteistoproveaquantumergodicitytheoremforEisensteinseriesonarathergeneralcuspmanifold withergodicgeodesicflow. Inthespecialcaseoffinitearea hyperbolicsurfaceswithcusps, asimilar quantumergodicity theoremwasprovedin[10].Inthisnote, wegiveanew,simplerandmoregeneralproof ofthat result,whichgeneralizes toanycuspedmanifoldwithergodicgeodesicflow.Inparticular,themanifoldsmayhavevariablenegativecurvature.

Tostate theresult,we introducesome notationandterminology.A

(

d+¹

)

-dimensionalcuspmanifoldisaRiemannian manifold M thatdecomposesasa compactmanifold M0 whoseboundariesaretorii,anda ﬁnitenumber

κ

oftopological half-cylinders Z₁,

. . .

, Zκ,suchthatthemetricontheseZ_i’sis

ds²

=

^dy²

+

d

θ

²

y²

,

(1)

E-mailaddresses:[email protected](Y. Bonthonneau),[email protected](S. Zelditch).

1 ResearchpartiallysupportedbyNSFgrantDMS-1541126.

http://dx.doi.org/10.1016/j.crma.2016.06.006

1631-073X/©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

(2)

where y isthe vertical coordinate,starting atai

>

0,and

θ

is thehorizontal coordinate,livingin R^d

/

i,where

i is a unimodular lattice. Onsuch a manifold, we can pick a globalcoordinate y_M,that coincides with thevertical coordinate suﬃcientlyhighinthecusps,andissomepositiveconstantinM0.

Forsuch amanifold,the spectrumoftheLaplace–Beltramioperator −

actingon L²

(

M

,

g

)

decomposesintothepure spectrumpart(

μ

0=⁰

< μ

1≤

. . .

)possiblyﬁnite,possiblyinﬁnite,andtheabsolutelycontinuousspectrum[^d²

/

4

,

+∞)^.

Witheach eigenvalue

μ

,we canassociateaneigenfunction uμ (repeatingeventual multipleeigenvalues).Forthecon- tinuousspectrum,werecalltheexistenceofmeromorphicfamilies[3,5]ofsmooth,non-L² eigenfunctions E_i

(

s

)

thatsatisfy thefollowing.Foreachi, E_i decomposesas

E_i

(

s

) =

^Iiy^s

+

^Gi

(

s

)

(2)

where Ii isthecharacteristicfunctionof Zi,andGi

(

s

)

isafunctioninL²

(

M

)

whenever^s

>

d

/

2.Wealsorequirethatfor all s,

(− −

^s

(

d

−

^s

))

E_i

(

s

) =

⁰

.

(3)

Such a collectionis unique, andthey are calledthe Eisenstein functions. The E_i’s do not havepoles on {^s=^d

/

2}^. ^We denote E

(

s

)

=

(

E1

(

s

), . . . ,

Eκ

(

s

))

.The scatteringmatrix is deﬁnedinthe following way.Let f

(

s

,

y

)

be the square matrix whosei lineisthezeroFouriercoeﬃcientincusp Zi,ataﬁxedheight y

>

ai,ofE;let Ibetheidentitymatrix.Then

f

(

s

,

y

) =

y^sI

+ φ (

s

)

y¹⁻^s

.

(4)

Thematrix

φ (

s

)

iscalledthescatteringmatrix,and

ϕ (

s

)

,itsdeterminant,isthescatteringdeterminant.When^s=^d

/

2,

φ

is unitary,and|

ϕ

_|₌1.

TheWeyllawofW.Muellerforacuspmanifoldstates[5],

1 4

π

h⁻¹

−h⁻¹

− ϕ

ϕ

d 2

+

^ir

dr

+

h√μ≤¹

1

=

^vol

(

B^∗M

) (

2

π)

^d⁺¹ ^h⁻

d−¹

+

^o

(

h⁻^d⁻¹

).

(5)

Mueller obtained it using the small-time asymptotics of the heat kernel. In [1], a semi-classical machinery is used to generalizetheWeyllawtoalocalWeyllaw,whichgivesanapproximationofthetraceofpseudo-differentialoperators(see Proposition2.3in[1]).Forsemi-classicalpseudo-differentialoperatorswithcompactlysupportedsymbols

σ

:Ash→^0,

1 4

π

R

Op_h

( σ )

E

d

2

+

^ir

,

E

d

2

+

^ir

dr

+

μ

Op_h

( σ )

u_μ

,

u_μ

∼

¹

(

2

π

^h

)

^d⁺¹

T^∗M

σ .

(6)

Combining(5)and(6),weget

h^d⁺¹ 4

π

R

Op_h

( σ )

E

d

2

+

^ir

,

E

d

2

+

^ir

+ ϕ

ϕ

d 2

+

^ir

λS^∗M

σ

dr

+

^h^d⁺¹

μ

Op_h

( σ )

u_μ

,

u_μ

−

h√μS^∗M

σ _→

0

.

(7)

ThelocalWeyllawgivestheaveragevalueofthematrixelementsofOp_h

( σ )

overthespectrum.Themainresultofthis noteis“themeanabsolutedeviation”goestozero:

Theorem1.1.LetM beacuspmanifoldwhosegeodesicﬂowisergodic.Then,forallcompactlysupportedsymbol

σ

,wehavethe followingconvergence,ash→^0.

h^d⁺¹ 4

π

R

Op_h

( σ )

E

d

2

+

ⁱ

λ

h

,

E

d

2

+

ⁱ

λ

h

+ ϕ

ϕ

d

2

+

ⁱ

λ

h

λS^∗M

σ

^d

λ

h

+

^h^d⁺¹

μ

Op_h

( σ )

u_μ

,

u_μ

−

h√μS^∗M

σ →

⁰

.

(8)

(themeasureonS^∗M istheusualnormalizedLiouvillemeasure).

Remark1.Henceforthwelet

ε ( σ ,

h

)

denotethequantityintheLHSof(8).

(3)

Notethatwe onlysumabsolutevaluesandnot squaresasinthenow-standardquantum variances(see [11] forbackground). Thisis because Eisenstein functionsare not L²,so that absolutevalues are easierto control than squares.The theoremthusstatesthat“themeanabsolutedeviation”goestozero.

Bycomparison,Theorem5.1of[10] statesthat,foraﬁniteareahyperbolicsurface X=H²

/

(withr=^λ_h^),^whenever Op

( σ )

isacompactlysupportedhomogeneousorder0 pseudo-differentialoperator,

h² 4

π

|^r|≤^h⁻¹

Op_h

( σ )

E

1

2

+

^ir

,

E

1

2

+

^ir

+ ϕ

ϕ

1 2

+

^ir

rhS^∗M

σ

^dr

+

^h²

μ:√μ≤h⁻¹

Op_h

( σ )

u_μ

,

u_μ

−

h√μS^∗M

σ →

⁰

.

(9)

Here, H² ^is ^the^upperhalf-plane,

⊂^{P S L}

(

2

,

R

)

isa discreteco-ﬁnite subgroupand

σ

_∈C₀^∞

(

S^∗X

)

.Theorem 6.1of[10]

extendstheresulttomoregeneralhomogeneoussymbols,includingEisensteinseries.

In thisarticle,we use semi-classical operators. The asymptotic(8) isa priori strongerthan (9) since itinvolves inte- grals/sums ofpositive quantities over thefull spectrum withno changein thepower ofh. However, if

σ

_∈C^∞₀

(

T^∗M

)

is supportedfor2|ξ|

< λ

0,thenthecontributionfromthe

λ

’sandh√

μ

’slargerthan

λ

0 isO(^h^∞

)

.Indeed,byresultsof[1], Op

( σ )1(

−^h²

> λ

²₀

)

isa negligibleoperator —i.e.smoothingandO(h^∞

)

—thus itstrace norm(afterremoving thezero- Fouriermodecoeﬃcientinthecusps)isO(^h^∞

)

(againfrom[1]).Hencetheresultisessentiallythesameifoneintegrates overR^or^over[−^Ch⁻¹

,

Ch⁻¹]^.În^the^following,^we^will^mostly^consider^theîntegralôver[−

λ

0h⁻¹

, λ

0h⁻¹]^.

Forcompactmanifolds,variance(orabsolutesum)decayestimatessuchasTheorem 1.1implythatthereisasubsequence ofdensityoneoftheeigenvalues,sothattheindividualtermstendtozeroforeveryobservableOp_h

( σ )

.Wereferto[4],[9]

and[11]forbackground.Thepresenceofcontinuousspectrumcomplicatesthisconclusionbecausetherelativedensitiesof discreteandcontinuous spectrumisitselfnotknown.Asdiscussedintheremarksonpage38of[10],onemaydeﬁnethe densityofafamilyofEisensteinseriesandcuspformsby usingthemeasure M

(

T

)

:= −₄¹_π ₋^T_T ^ϕ_ϕ

(

¹₂+^ir

)

dr onintervals ofthecontinuousspectrumandthecountingmeasure N

(

T

)

fordiscreteeigenvalues.Onecouldthendeﬁneanormalized densityon intervalsof the unionof thediscrete and continuous spectrum by dividingthe sumof the two measures by M

(

T

)

+^N

(

T

)

. Inthis sense, there should exist a density-one subset of thespectrum so that individual integrandsor summandsshouldtendtozeroforevery

σ

.Sinceweworkinaverygeneralcontext,weomitfurtherdiscussionofsifting outdensityonesubsets.

Sincethesymbol

σ

isassumedtobecompactlysupported,theresultisindependentofthechoiceofquantizationOp.

Especiallyconvenientquantizationsarethehyperbolicpseudo-differentialonesof[8]thatwereusedin[10]andtherelated quantizationdeﬁnedin[1].Tokeepthisarticlebrief,we willusethelatter;wedonotreview themanyconstructionsand deﬁnitionsin[10]and[1],butreferthereadertothosearticlesforfurtherbackground.

ThequantizationprocedureOp thatwasgivenin[1]enablesonetoquantizeaclassofsymbolsincludingthecompactly supportedfunctionsof

(

x

, ξ )

∈^T^∗^M,^the^constants,ând^the^functionsôf^theênergy ^f

(

|

ξ

|²

)

,with f∈^C^∞c

(

R

)

.

2. Overviewoftheproof

We followtheorganization ofthe proof ofTheorem 5.1in[10].There are two partsofthe proof. Theﬁrst stepis to provethefollowinglemma.

Lemma2.1.TheresultofTheorem 1.1holdsifthesymbol

σ

hasmeanvaluezeroineveryenergylevel{|ξ|=^constant}^.

Asthesymbolhasmeanvaluezero,andhascompactsupport,theproofofthislemmaisverysimilartotheproofofthe compactcase[4,9].Theproof isgivenindetailin[10] forhyperboliccusp surfaces,andtheproof isessentiallythesame forallcuspmanifolds.Henceweomittheproof.Themaincontributionofthisnoteistoprovethesecondpartoftheproof wherethesymboldoesnothavemeanzero.In[10],thispartoftheproofisgiveninpp. 39–43andisquitecomplicated.

Theproofwenowgiveismuchsimplerandmoregeneral.

Letusexplainnowwhythelemmaisuseful.Let

σ

_∈C^∞_c

(

T^∗M

)

.Then,onecandeﬁneaC^∞

(

R⁺

)

functionby

σ (λ) =

|ξ|=λ

σ .

Also,asabove,let

ε ( σ ,

h

)

bethequantityintheLHSof(8).

Ifthe averageof

χ

_∈C^∞_c

(

M

)

is 1 overM, then

σ

and

σ

˜ _:

(

x

, ξ )

→

χ (

x

) σ (

|

ξ

|

)

havethesame averagein each energy layer.Hence,toprovethetheorem,separate

σ

into

σ

˜ and

σ

− ˜

σ

.Thelattercanbetreatedusingthelemma.Thenweﬁnd, usingthetriangularinequality

(4)

lim sup

h→⁰

ε ( σ ,

h

) =

^{lim sup}

h→⁰

ε ( σ ˜ ,

h

).

(10)

Now,

σ

˜ hasaveryconvenientstructure,evenmoresoifwepickcarefullythebaseproﬁle

χ

,aswewillseeinthenext section.

However, before we get to the next step, we recall some facts that will be very useful. Let

^∗_y be the projector on functionswhosezero-Fouriermodevanishes for y_M

>

y.WestartwiththefamousMaass–Selbergrelation([7,section 2], or[2]),whichisacleverapplicationoftheStokesformula

yM≤y

^E

d 2

+

i

λ

h

²

=

2

κ

logy

− ϕ

ϕ

d 2

+

i

λ

h

+

Try²ⁱ^λ/^h

φ

^∗

−

^y⁻²ⁱ^λ/^h

φ

2i

λ/

h

+

yM>y

|

^∗_yE

|

²

.

(11)

WeusethefollowingPoissonformula[6]:

ϕ

d 2

+

^ir

=

^Q

(

r

) +

ρpole ofϕ

2

ρ ₋

d

( ρ ₋

d

/

2

)

²

+ (

r

− ρ )

² ⁽¹²⁾

forsome polynomial Q ofpowerlessorequalto2^d

/

2^.^Notice âll^butâ^finite^numberôf^poles ôf

ϕ

areontherightof {^s

<

d

/

2}^.

Lastly, inthe sum/integraldeﬁningthe quantity

( σ ,

h

)

,considerthecontributionfromthe

λ

’sandh√

μ

’slarger than

λ

0where

σ

issupportedfor2|

ξ

|

< λ

0.AsmentionnedaboveO

(

h^∞

)

becauseOp

( σ )1(

−^h²

> λ

²₀

)

isanegligibleoperator— i.e.smoothingandO(^h^∞

)

—henceitstracenorm(afterremovingthezero-Fouriermodecoeﬃcient)isO(^h^∞

)

(againfrom [1]).

3. Thecaseofnon-zeromeanvalue

Now, we concentrate on

σ

˜

(

x

, ξ )

=

χ (

x

) σ (

|

ξ

|

)

. We can stop considering altogether the L² eigenfunctions. Indeed, in their contribution

pp the LHS of (8), we can write

σ (

h√

μ

j

)

=

σ (

h√

μ

j

)

uj

,

uj^. ^Up ^to ^a O(h

)

error term, this is

Op_h

( σ (|ξ|))

^uj

,

u_j^,^so^that^the^main^termⁱⁿ pp is h^d⁺¹

μ

|

Op_h

( σ − σ )

u_μ

,

u_μ

|.

(13)

With

σ

=

σ

₋

σ

,wearebacktothecase

σ

=^0,ând^the^proofôf^{Lemma 2.1}âpplies.

Now, by the localization of eigenfunctions — see [11] — E

(

d

/

2+ⁱ

λ/

h

)

is semi-classically microlocally supported on {|ξ|∼

λ}

^;^as^aconsequence,Op_h

( σ

˜

)

E

(

d

/

2+ⁱ

λ/

h

)

∼

χ

×^E

(

d

/

2+ⁱ

λ/

h

)

×

σ (λ)

.Thisstatementcanbemadepreciseby

Op_h

( σ ˜ )

E

d

2

+

ⁱ

λ

h

,

E

d

2

+

ⁱ

λ

h

=

M

χ

E

d

2

+

ⁱ

λ

h

²

σ (λ) +

O

(

h

)

^E

²_{_χ₌₀_}

.

CallR1 thecontributionoftheremainderintheRHSto

ε ( σ

˜

,

h

)

.Letusﬁrstdealwiththemainterm.Let

λ

0 besuchthat

σ

issupportedfor2|ξ|

< λ

0.Themaincontributionto

ε ( σ

˜

,

h

)

willbe h^d

λ0

−λ0

| σ (λ)| ×

M

χ

E

d

2

+

ⁱ

λ

h

²

+ ϕ

ϕ

d 2

+

ⁱ

λ

h

^d

λ

Now,wechoosetheshape

χ

inthefollowingway.Wesuppose that

χ

isconstantin M0,andthatinthecusps,itwrites ascν

χ

0

(

y

ν )

,where

χ

0∈^C^∞c

(

R

)

equals 1 closetozero,

ν

isasmallparameterandcν∈R⁺ ^is^anormalizationconstant.It ischosensothat

χ

hasmeanvalue1,i.e. _M

χ

=^vol

(

M

)

.Write

M

χ

E

d

2

+

ⁱ

λ

h

²

= −

^cν

νχ

₀

(

y

ν )

⎧ ⎨

⎩

y_M≤^y

|

^E

|

²

⎫ ⎬

⎭

^dy

.

For

ν

smallenough,thisiswelldeﬁned,andwecanusetheMaass–Selbergformula(11):

M

χ

E

d

2

+

ⁱ

λ

h

²

= −

^cν

νχ

₀

(

y

ν )

2

κ

logy

− ϕ

ϕ ⁺

^Tr

y²ⁱ^λ/^h

φ

^∗

−

y⁻²ⁱ^λ/^h

φ

2i

λ/

h

dy

+

M

(

1

− χ )

^∗₀E

²

IntheRHS,thethirdtermishighlyoscillating.ItwillonlycontributeforO(^h^∞

)

totheﬁnalresult,bynon-stationaryphase.

Thesecondwillcontributeby−^cν

ϕ

/ ϕ

.Thefourthisanintegralsupportedinthecusps.Wearelefttoprovethat

(5)

h^d λ0

−λ0

| σ (λ)| ×

²

κ

c_ν

νχ

₀

( ν

y

)·

logydy

+

(

1

−

c_ν

) ϕ

ϕ

d 2

+

i

λ

h

d

λ

+

^h^d⁺¹^Tr

^∗₀

(

1

− χ ) | σ _| ( −

^h²

)

,

goes to 0 withh

.

Adirectcomputationsshowsthisfortheﬁrstterm.LetuscallthethirdtermR₂.Thesecondtermgives

h^d

|

1

−

c_ν

|

λ0

−λ0

h^d

σ (λ) ϕ

ϕ

d

2

+

i

λ

h

.

Inthedecomposition(12),thesumoverthe resonancesisafunction thatisalways negative forbigvaluesof

λ

,andthe polynomialpartisexplicit,soweknowthat

λ0

−λ₀

ϕ

d 2

+

ⁱ

λ

h

+ ϕ

ϕ

d

2

+

ⁱ

λ

h

d

λ =

O

(

h⁻^d

).

CombiningthiswiththeWeyllaw(5),wearriveat

|

¹

−

^cν

|

λ0

−λ0

h^d

σ (λ) ϕ

ϕ

d

2

+

ⁱ

λ

h

= |

¹

−

^cν

|

O

(

1

).

Now,wecomebackto R1andR2.First,R1 canbeboundedbysomeO(^h

)

Tr Op_h

( σ

)

,with

σ

compactlysupported(on asetslightlybiggerthan

σ

˜).Then,wecanapplythelocalWeylLaw(6).ForR2,wecanalsoapplyformula(6)(thesymbol thistimeisnotcompactlysupported,butitalsoisinthequantizableclassof[1]).Hence,weﬁndthat R₁andR₂contribute tothelim sup by

O

(

1

)

vol

{

^yM

≥

¹

/ ν }.

Atlast,letusobservethatwhen

ν

→^0,^the^assumption^that

χ

hasmeanvalue1 impliesthatcν→^1.^Actually,^this^can^be mademoreprecise,as1−^cν=O(^vol{^y

>

1

/ ν

})^.^We^deduce^that,^for^some^constant^C

>

0,

lim sup

h→⁰

ε ( σ ˜ ,

h

) ≤

^C^vol

{

^y

≥

¹

/ ν _} .

Asthisdoesnotdependon

ν

,wecantake

ν

→^0,^and^this^ends^the^proof.

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