Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Partial Differential Equations

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Vesselin Petkov

, Latchezar Stoyanov

aUniversité Bordeaux I, Institut de mathématiques, 351, cours de la Libération, 33405 Talence, France bUniversity of Western Australia, School of Mathematics and Statistics, Perth, WA 6009, Australia

Received 24 September 2007; accepted 8 October 2007 Available online 7 November 2007

Presented by Gilles Lebeau

Abstract

Lets₀<0 be the abscissa of absolute convergence of the dynamical zeta function Z(s)for several disjoint strictly convex compact obstaclesK_i⊂R^N, i=1, . . . , κ₀, κ₀3 and letR_χ(z)=χ (−_D−z²)⁻¹χ , χ∈C^∞₀ (R^N), be the cut-off resolvent of the Dirichlet Laplacian−_DinΩ=R^N\κ0

i=1K_i. We prove that there existsσ₂< s₀such thatZ(s)is analytic for(s) σ₂ and the cut-off resolventRχ(z)has an analytic continuation for(z) <−iσ2, |(z)|C.To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Résumé

Prolongement analytique de la résolvante du Laplacien et de la fonction zeta dynamique.Soits₀<0 l’abscisse de conver- gence absolue de la fonciton zeta dynamiqueZ(s)pour des obstacles compacts, disjoints et strictement convexesK_i⊂R^N, i= 1, . . . , κ₀, κ₀3 et soitRχ(z)=χ (−D−z²)⁻¹χ , χ∈C₀^∞(R^N), la résolvante tronquée du Laplacien de Dirichlet−Ddans Ω=R^N\κ0

i=1K_i. On prouve qu’il existeσ₂< s₀tel queZ(s)est analytique pour(s)σ₂et la résolvante tronquéeR_χ(z) admet un prolongement analytique pour(z) <−iσ2, |(z)|C.Pour citer cet article : V. Petkov, L. Stoyanov, C. R. Acad. Sci.

Paris, Ser. I 345 (2007).

©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Version française abrégée

SoitK=K₁∪K₂∪· · ·∪K_k0,oùK_i⊂R^N, N2,sont des domaines compacts, disjoints et strictement convexes ayant des frontières Γ_i =∂K_i et κ₀3. SoitΩ =R^N\K etΓ =∂K. On suppose que K satisfait la condition suivante :

(H) Pour chaque coupleK_i, K_j de différentes composantes connexes deKl’enveloppe convexe deK_i∪K_j n’a pas de points communs avec les autres composantes connexes deK.

E-mail addresses:petkov@math.u-bordeaux1.fr (V. Petkov), stoyanov@maths.uwa.edu.au (L. Stoyanov).

1631-073X/$ – see front matter ©2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.crma.2007.10.019

Étant donné un rayon périodique réfléchissantγ⊂Ω avecmγ réflexions, soitTγ la période primitive (longueur) de γ et soit Pγ l’application de Poincaré linéaire associée à γ (cf. [8]). Notons par λi,γ, i =1, . . . , N−1, les valeurs propres dePγ telles que|λi,γ|>1 et désignons parP l’ensemble de rayons primitifs périodiques. Soitδγ=

−¹₂log(λ1,γ· · ·λ_N−1,γ),r_γ=0 sim_γ est pair etr_γ=1 sim_γ est impair. On considère la fonction zeta dynamique Z(s)=

∞ m=1

1 m

γ∈P

(−1)^mrγe^{m(−sTγ+δγ)}.

On voit facilement qu’il existe une abscisse de convergence absolues₀∈Rtelle que pour(s) > s₀la sérieZ(s) est absolument convergente. D’autre part, en utilisant la dymanique symbolique et les résultats de [7], on conclut queZ(s)est méromorphe pour(s) > s₀−a, a >0 (cf. [4]). En suivant les résultats récents (cf. [9] pourN=2 et [10] pourN=3 sous certaines conditions) on sait qu’il existe 0< < atel que la fonction zeta dynamiqueZ(s) admet un prolongement analytique pour(s) > s₀−. On considère maintenant pour(s) <0 la résolvante tronquée R_χ(z)=χ (−_D−z²)⁻¹χ:L²(Ω)→L²(Ω), oùχ∈C₀^∞(R^N), χ=1 surKet−_Dest le Laplacien de Dirirchlet dansΩ=R^N\K. La résolvanteR_χ(z)admet un prolongement mémororphe dansCpourNimpair et dansC\{iR⁺} pourN pair avec des pôlesz_j, (z_j) >0. On se propose d’étudier la liaison entre les prolongements analytiques de Z(s)etR_χ(z). Le cass₀>0 est plus facile car on sait que pour−is0(z) <0 la résolvante tronquéeR_χ(z)est analytique [6]. Dans la suite on suppose ques₀<0. Sous l’hypothèse s₀<0, Ikawa [3] a démontré que pour tout >0 il existeC>0 tel queRχ(z)est analytique pour(z) <−i(s0+),|(z)|C. Un résultat similaire pour un problème du contrôle a été établi par Burq [1]. La fonction zeta dynamiqueZ(s)est liée aux périodes des rayons périodiques et formellementZ(s)ne contient pas une information sur la dynamique de rayons dans un voisinage de l’ensemble ‘non-wandering’ (captif). Dans cette Note on examine le cas(s) < s0 en exploitant les propriétés spectrales de l’opérateur de RuelleLs(cf. Section 2). Notre résultat principal est le suivant :

Théorème 0.1.Soits₀<0. Supposons que l’opérateur de RuelleL_ssatisfait les estimations(6). Alors il existeσ₂< s₀ tel queZ(s)est analytique pour(s) > σ₂et la résolvante tronquéeR_χ(z)est analytique pour

(z) <−iσ2, (s)C.

Les estimations (6) sont un analogue aux estimations de Dolgopyat [2]. Ces estimaitons ont été démontrées pour N=2 dans [9] et pour N 3 sous certaines conditions dans [10]. On espère que (6) sont valables pour N3 sans aucune restriction. Notons qu’il y a quelques ans, Ikawa [5] a annoncé un résultat concernant le prolongement analytique deR_χ(z)dans un domaine−iD,α, où

D,α=

s∈C: (s) > s0−(s)^−α,(s)C,0< α <1

en imposant des conditions fortes sur le comportement de la fonction proprewde l’opérateur de Ruelle associée à la valeur propre maximale et un prolongement analytique deZ(s)dansD,α. De plus, il suppose qu’on ait l’estimation

|Z(s)||s|¹⁻, 0< <1, s∈D,α. A notre connaissance la preuve de ce résultat n’a pas été publiée ailleurs.

1. Introduction

LetK be a subset ofR^N, N2, of the formK=K₁∪K₂∪ · · · ∪K_k0, whereK_i are compact strictly convex disjoint domains inR^NwithC^∞boundariesΓ_i=∂K_i andk₀3. SetΩ=R^N\KandΓ =∂K. We assume thatK satisfies the following (no-eclipse) condition:

(H) For every pairK_i,K_j of different connected components ofKthe convex hull ofK_i∪K_jhas no common points with any other connected component ofK.

With this condition, the billiard flowφ_t defined on the cosphere bundleS^∗(Ω)in the standard way is called an open billiard flow. Given a periodic reflecting rayγ ⊂Ω withm_γ reflections, denote byT_γ the primitive period (length) ofγ and byP_γ the linear Poincaré map associated toγ(see [8]). Letλ_i,γ, i=1, . . . , N−1, be the eigenvalues ofP_γ

with|λi,γ|>1 and letP be the set of primitive periodic rays. Forγ ∈P setδγ = −¹₂log(λ1,γ· · ·λN−1,γ),rγ=0 if mγ is even andrγ =1 ifmγ is odd. Consider the dynamical zeta function

Z(s)= ∞ m=1

1 m

γ∈P

(−1)^mrγe^{m(−sTγ+δγ)}.

It is easy to show that there exists an abscissa of absolute convergences0∈Rsuch that for(s) > s0the seriesZ(s)is absolutely convergent. On the other hand, using symbolic dynamics and the results of [7], we deduce thatZ(s)is mero- morphic for(s) > s0−a, a >0 (see [4]). According to some recent results (see [9] forN=2 and [10] forN3 un- der some additional conditions) there exists 0< < aso that the dynamical zeta functionZ(s)admits an analytic con- tinuation for(s)s0−. For(z) <0 consider the cut-off resolventR_χ(z)=χ (−_D−z²)⁻¹χ:L²(Ω)→L²(Ω), whereχ∈C₀^∞(R^N), χ =1 onKand−_Dis the Dirichlet Laplacian inΩ=R^N\K. The cut-off resolventR_χ(z) has a meromorphic continuation inCforN odd and inC\ {iR⁺}forN even with polesz_j such that(z_j) >0. The analytic properties and the estimates ofR_χ(z)play a crucial role in many problems related to the local energy decay, distribution of the resonances etc. We study the link between the analytic continuations ofZ(s)andR_χ(z). The case s₀>0 is much easier, since we know that for−is0(z) <0 the cut-off resolventR_χ(z)is analytic (see [6]).

In the followingwe assume thats₀<0. The problem is to examine the link between the analyticity ofZ(s)for (s) > s₀and the behavior ofR_χ(z)for 0(z) <−is0. Assumings₀<0, Ikawa [3] proved that for every >0 there existsC>0 so that the cut-off resolventR_χ(z)is analytic for(z) <−i(s0+),|(z)|C. A similar result for a control problem has been established by Burq [1]. The proofs in [3] and [1] are based on the construction of an asymptotic solutionU_M(x, s;k)with boundary datam(x;k)=e^ikϕ(x)h(x), k∈R, k1, whereϕis a phase function (∇ϕ =1) andh∈C^∞(Γ )has a small support. More precisely,UM(·,s;k)isC^∞(Ω)-valued holomorphic function¯ inD0= {s∈C: (s) > s0}, and we have

(−s²)U_M(·,s;k)=0 forx∈Ω,(s) > s₀, (1)

U_M(·,s;k)∈L²(Ω) if(s) >0, (2)

U_M(x, s;k)=m(x, k)+r_M(x, s;k) onΓ, (3)

where, forrM(x, s;k)ands∈D0, |s+ik|1, we have the estimates r_M(·,s;k)

C^p(Γ )C_pk^−M+p

∇ϕ_CM2+p+2(Γ )+1 h_CM2+p+2(Γ ), ∀p∈N. (4) The functionU_M(x, s;k)is given by a finite sum of series having the form

∞ n=0

|j|=n+2,j_n+2=l

M

q=0

e^−sϕj(x) 2q ν=0

a_j,q,ν(x, s;k)(s+ik)^ν (ik)^−q, (5)

wherej=(j₁, . . . , j_m), j_i ∈ {1, . . . , κ0}are configurations,|j| =m, ϕ_j(x)are phase functions and the amplitudes a_j,q,ν(x, s;k)depend ons∈Candk∈R. The main difficulty is to establish the summability of these series and to obtain for(s) > s₀suitableC^pestimates of their traces onΓ. The absolute convergence ofZ(s)makes it possible to establish the absolute convergence of the series in (5) and to get crude estimates leading to (1)–(4). The dynamical zeta functionZ(s)is related to the periods of periodic rays and formally fromZ(s)we get no information about the dynamics of all rays in aneighborhood of the non-wandering(trapped)set. In this Note we study the case(s) < s₀ by means of the Ruelle operatorL_s (see Section 2 for the definition). Our main result is the following:

Theorem 1.1.Lets0<0. Assume that for the Ruelle operatorLs the estimates(6)hold. Then there existsσ2< s0

such thatZ(s)is analytic for(s) > σ2and the cut-off resolventRχ(z)is analytic for (z) <−iσ2, (s)C.

The estimates (6) are analogous to Dolgopyat’s estimates in [2]. For open billiard flows (6) have been established in [9] forN=2 and under some conditions in [10] forN3. We expect that (6) hold forN3 without any restrictions.

Several years ago, Ikawa [5] announced a result concerning an analytic continuation ofRχ(z)in a domain−iD,α, where

D,α=

s∈C: (s) > s0−(s)^−α,(z)C,0< α <1

assuming some strong conditions on the behavior of the eigenfunctionwof the corresponding Ruelle operator re- lated to its maximal eigenvalue as well as an analytic continuation of Z(s) in D,α combined with an estimate

|Z(s)||s|¹⁻, 0< <1, s∈D,α. To our best knowledge a proof of the above result of Ikawa has not been published anywhere.

2. Ruelle operator Introduce the spaces

Σ_A=

(. . . , η_−m, . . . , η₋₁, η₀, η₁, . . . , η_m, . . .): 1η_jκ₀, η_j∈N, η_j=η_j+1for allj∈Z , Σ_A⁺=

(η₀, η₁, . . . , η_m, . . .): 1η_jκ₀, η_j∈N, η_j=η_j+1for allj1 . We define the operatorσ:Σ_A⁺→Σ_A⁺by(σ ξ )_i=ξ_i+1, i∈N. Givenξ∈Σ_A, let

. . . , P₋₂(ξ ), P₋₁(ξ ), P₀(ξ ), P₁(ξ ), P₂(ξ ), . . .

be the successive reflection points of the unique billiard trajectory in the exterior ofKsuch thatPj(ξ )∈Kξ_j for all j∈Z. Setf (ξ )= P0(ξ )−P1(ξ ). Following [4], one constructs a sequence{ϕξ,j}^∞_j=−∞ of phase functions such that for eachj,ϕ_ξ,j is defined and smooth in a neighborhoodU_ξ,j of the segment[P_j(ξ ), P_j+1(ξ )]inΩ and

(i) ∇ϕ_ξ,j(P_j(ξ ))=^P_P^j+1_j+1^{(ξ )}_{(ξ )}⁻₋^P_P^j_j^{(ξ )}_{(ξ )}, (ii) ϕ_ξ,j =ϕ_ξ,j+1onΓ_ξj+1∩U_ξ,j∩U_ξ,j+1,

(iii) for eachx∈Uξ,j the surfaceCξ,j(x)= {y∈Uξ,j:ϕξ,j(y)=ϕξ,j(x)}is strictly convex with respect to its normal field∇ϕξ,j. For anyy∈Uξ,j denote byGξ,j(y)theGauss curvatureofCξ,j(x)aty.

Now defineg:Σ_A→Rby g(ξ )= 1

N−1 lnGξ,0(P1(ξ )) G_ξ,0(P₀(ξ )).

By Sinai’s Lemma, there existf ,˜ g˜depending on future coordinates only andχ₁, χ₂such that f (ξ )= ˜f (ξ )+χ₁(ξ )−χ₁(σ ξ ), g(ξ )= ˜g(ξ )+χ₂(ξ )−χ₂(σ ξ ), ξ∈Σ_A.

Settingr(ξ, s)˜ = −sf (ξ )˜ + ˜g(ξ )+iπ, we define theRuelle transfer operator Ls:C(Σ_A⁺)→C(Σ_A⁺)byLsu(ξ )=

σ η=ξe^r(η,s)˜ u(η) for any continuous (complex-valued) function uon Σ_A⁺ and any ξ ∈Σ_A⁺. For our analysis the Dolgopyat type estimates [2] for the norms of Lⁿ_s play a crucial role. Following the results in [9,10] there exist constantsC >0, σ0< s₀and 0< ρ <1 so that fors=τ +it withτ σ₀andn=p[log|t|] +l, p∈N, 0l [log|t|] −1, we have

Lⁿ_s∞Cρ^p[log|t|]e^{lPr(−τf˜+ ˜g)}, |t|t₀, (6) Pr(G)being the topological pressure of the functionG(see [7]).

3. Idea of the proof of Theorem 1.1

Fixl∈ {1, . . . , κ0}. Given a phase functionϕ(x)and an amplitudeh(x)∈C^∞(Γ ), we wish to construct an asymp- totic solutionU_M(x, s;k)which is a holomorphic function for(s)σ₂andU_M has properties similar to (1)–(4).

The first approximation ofU_Mis an infinite sum w_0,l(x,−is)=^∞

n=0

|j|=n+2.j_n+2=l

u_j(x,−is),

whereuj(x,−is)=(−1)^m−1e^−sϕj(x)aj(x)is related to a configurationj= {j1, . . . , jm}by using successive phase functionsϕj(x)and amplitudes aj(x)determined by the transport equation (see [5]). To justify the convergence of this series, we need to compare the general term with a suitable composition of operators related to the dynamics. Let μ=(μ0=1, μ1, . . .)∈Σ_A⁺. It follows from [3] that there exists a unique pointy(μ)∈Γ1such that the rayγ (y, ϕ) issued from a pointy(μ)in direction∇ϕ(y(μ))follows the configurationμ. LetQ0(μ)=y(μ), Q1(μ), . . . ,be the consecutive reflection points of this ray. Definef_j⁺(μ)= Qj(μ)−Qj+1(μ), and

g_j⁺(μ)= 1

N−1lnG^ϕ_μ,j(Q_j+1(μ)) G^ϕ_μ,j(Q_j(μ)) <0,

whereG^ϕ_μ,j(y)denotes theGauss curvatureof the surfaceC^ϕ_μ,j(x)= {z: ϕ(μ₀,μ₁,...,μ_j)(z)=ϕ(μ₀,μ₁,...,μ_j)(x)}aty. We define an extensione:Σ_A⁺→Σ_A. Fors∈Candξ∈Σ_A⁺withξ₀=1, following [5], set

φ⁺(ξ, s)= ∞ n=0

−s f

σⁿe(ξ ) −f_n⁺(ξ ) +

g

σⁿe(ξ ) −g_n⁺(ξ ) .

Formally, we defineφ⁺(ξ, s)=0 whenξ₀=1, thus obtaining a functionφ⁺:Σ_A⁺×C→C. Setχ (ξ, s)= −sχ₁(ξ )+ χ₂(ξ )and for anys∈Cdefine the operatorGs:C(Σ_A⁺)→C(Σ_A⁺)by

Gsv(ξ )=

σ η=ξ, η₀=1

e^{−φ+(η,s)+χ (e(η),s)−sf (η)˜ + ˜g(η)}v(η), v∈C(Σ_A⁺), ξ∈Σ_A⁺.

Fix an arbitraryl∈ {1, . . . , κ0}and an arbitrary pointx₀∈Γ_l. Define the functionφ⁻(x₀; ·,·):Σ_A×C→C (depending onlas well) as follows. First, setφ⁻(x₀;η, s)=0 ifη₀=l. Next, assume thatη∈Σ_A satisfiesη₀=l.

There exists a unique billiard trajectory inΩwith successive reflection pointsP˜_j(x₀;η)∈∂K_ηj (−∞< j0) such thatx0= ˜P₋1(x0;η)+t∇ϕ_η−(P˜₋1(x0;η))for somet >0. In general the segment[ ˜P₋1(x0;η), x0]may intersect the interior ofK_l. If this is the case, set againφ⁻(x₀;η, s)=0. Otherwise, denoteP˜₀(x₀;η)=x₀and for anyj <0 set

f_j⁻(x0;η)=P˜j+1(x0;η)− ˜Pj(x0;η), g_j⁻(x0;η)= 1

N−1 lnG_η,j(P˜_j+1(x₀;η)) G_η,j(P˜_j(x₀;η)) , and defineφ⁻(x0;η, s)= −s_−∞

j=−1[f (σ^j(η))−f_j⁻(x0;η)] +_−∞

j=−1[g(σ^j(η))−g_j⁻(x0;η)]. Next, similarly to [5], introduce the operatorMn,s(x₀):C(Σ_A⁺)→C(Σ_A⁺)by

Mn,s(x₀)v (ξ )=

σ η=ξ

e^{−φ−(x0;σn+1e(η),s)−χ (σn+1e(η),s)−sf (η)˜ + ˜g(η)}v(η), v∈C(Σ_A⁺), ξ∈Σ_A⁺. Introduce the functionv_s(ξ )=e^{−sϕ(Q0(ξ ))}h(Q₀(ξ ))ifξ₀=1 andv_s(ξ )=0 otherwise and define the norms

fΓ ,p=max

x∈Γ max

a⁽¹⁾,...,a^(p)∈TxΓ

(D_a(1)· · ·D_a(p)f )(x), fΓ ,(p)= max

0jpfΓ ,j

wherea^{(j )} =1 for allj =1, . . . , p.

Theorem 3.1.There exist global constantsC >0, c >0, a∈(0,1]andθ∈(0,1)depending only onKsuch that for any choice ofl∈ {1, . . . , κ0}the following holds:For any integersp1andn1, anyξ ∈Σ_A⁺withξ₀=land any s∈Cwith(s)s₀−awe have

Lⁿ_sMn,s(·)Gsv˜_s (ξ )−

|j|=n+2,j_n+2=l

u_j(·,−is) Γ ,p

C(θ+ca)ⁿe^{C[|(s)|(1+ϕΓ ,0)+∇ϕΓ ,(1)]} p

j=0

|s|∇ϕΓ ,j+ ∇ϕΓ ,j+1 j+1

hΓ ,p−j. (7)

A similar estimate holds forp=0; in this case the sum in the right-hand side of (7) has to be replaced by[(|s| +

∇ϕΓ ,(1))hΓ ,0+ hΓ ,(1)]. Applying the casep=0, we reduce the convergence ofw_0,lto the summability of the series_∞

n=0Lⁿ_s. On the other hand, forτσ₀,|t|2 the estimates (6) yield ∞

n=0

Lⁿ_s∞ C 1−ρ^[log|t|]

[log|t|]−1 j=0

e^{jPr(−τf˜+ ˜g)}C1max

log|t|,|t|^{Pr(−τf˜+ ˜g)} .

Moreover, forσ0sufficiently close tos0there exists 0< β <1 such thatLⁿ_sMn,sGΓ ,0C|t|^1+β and we conclude that w0,l(x,−iτ +t )Γ_l,0B|t|^1+β. Exploiting the case p1, we obtain similar estimates for w0,l(x,−iτ + t )Γ_l,p, p1 and we get the first approximation. Repeating this procedure, we complete the construction ofUM. References

[1] N. Burq, Controle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France 55 (1993) 126.

[2] D. Dolgopyat, On decay of correlations in Anosov flows, Ann. Math. 147 (1998) 357–390.

[3] M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier 2 (1988) 113–146.

[4] M. Ikawa, Singular perturbations of symbolic flows and the poles of the zeta function, Osaka J. Math. 27 (1990) 281–300.

[5] M. Ikawa, On zeta function and scattering poles for several convex bodies, in: Conf. EDP, Saint-Jean de Monts, SMF, 1994.

[6] M. Ikawa, On scattering by several convex bodies, J. Korean Math. Soc. 37 (2000) 991–1005.

[7] W. Parry, M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188 (1990).

[8] V. Petkov, L. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, John Wiley & Sons, 1992.

[9] L. Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math. 123 (2001) 715–759.

[10] L. Stoyanov, Spectra of Ruelle transfer operators for contact flows on basic sets, Preprint, 2007.