YAIZA CANZANI AND BORIS HANIN

C SCALING ASYMPTOTICS FOR THE SPECTRAL PROJECTOR OF THE LAPLACIAN YAIZA CANZANI AND BORIS HANIN Abstract. This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n. 0. Introduction Let (M, g) be a compact, smooth, Riemannian manifold without boundary. We assume throughout that the dimension of M is n and write g for the non-negative Laplace-Beltrami operator. Denote the spectrum of g by 0 = λ 0 < λ 1 λ. This article concerns the behavior of the Schwarz kernel of the projection operators E I : L (M) λ j I ker( g λ j), where I [0, ). Given an orthonormal basis {ϕ j } j=1 of L (M, g) consisting of real-valued eigenfunctions, g ϕ j = λ jϕ j and ϕ j L = 1, (1) the Schwarz kernel of E I is E I (x, y) = ϕ j (x)ϕ j (y). λ j I () The study of E [0,λ] (x, y) as λ has a long history, especially when x = y. For instance, it has been studied notably in [7, 8, 9, 10] for its close relation to the asymptotics of the spectral counting function #{j : λ j λ} = E [0,λ] (x, x)dv g (x), (3) M where dv g is the Riemannian volume form. An important result, going back to Hörmander [8, Thm 4.4], is the pointwise Weyl law (see also [4, 18]), which says that there exists ε > 0 so that if the Riemannian distance d g (x, y) between x and y is less than ε, then E [0,λ] (x, y) = 1 (π) n ξ gy <λ e i exp 1 y (x), ξ dξ + R(x, y, λ). (4) gy 1

Y. CANZANI AND B. HANIN The integral in (4) is over the cotangent fiber Ty M and the integration measure is the quotient of the symplectic form dξ dy by the Riemannian volume form dv g = g y dy. In Hörmander s original theorem, the phase function exp 1 y (x), ξ is replaced by any so-called adapted phase function and one still obtains that j x k yr(x, y, λ) = O(λ n 1+j+k ) (5) d g(x,y)<ε as λ, where denotes covariant differentiation. The estimate (5) for j = k = 0 is already in [8, Thm 4.4], while the general case follows from the wave kernel method (e.g. as in 4 of [16] see also [3, Thm 3.1]). Our main technical result, Theorem, shows that the remainder estimate (5) for R(x, y, λ) can be improved from O(λ n 1+j+k ) to o(λ n 1+j+k ) under the assumption that x and y are near a non self-focal point (defined below). This paper is a continuation of [4] where the authors proved Theorem for j = k = 0. An application of our improved remainder estimates is Theorem 1, which shows that we can compute the scaling limit of E (λ,λ+1] (x, y) and its derivatives near a non self-focal point as λ. Definition 1. A point x M is non self-focal if the loopset L x := {ξ S xm : t > 0 with exp x (tξ) = x} has measure 0 with respect to the natural measure on T x M induced by g. Note that L x can be dense in S xm while still having measure 0 (e.g. for points on a flat torus). Theorem 1. Let (M, g) be a compact, smooth, Riemannian manifold of dimension n, with no boundary. Suppose x 0 M is a non self-focal point and consider a non-negative function r λ satisfying r λ = o(λ) as λ. Define the rescaled kernel E x 0 (λ,λ+1] (u, v) := λ (n 1) E (λ,λ+1] (exp x0 ( u λ Then, for all k, j 0, ( j u v k E x 1 0 (λ,λ+1] (u, v) (π) n u, v r λ S x 0 M ), exp x0 ( v λ e i u v,ω dω) )). = o(1) as λ. The inner product in the integral over the unit sphere S x 0 M is with respect to the flat metric g(x 0 ) and dω is the hypersurface measure on S x 0 M induced by g(x 0 ). Remark 1. Theorem 1 holds for Π (λ,λ+δ] with arbitrary fixed δ > 0. The difference is that the limiting kernel is multiplied by δ and the rate of convergence in the o(1) term depends on δ. Remark. One can replace the shrinking ball B(x 0, r λ ) in Theorem 1 by a compact set S M in which for any x, y S the measure of the set of geodesics joining x and y is zero (see Remark 3 after Theorem ). In normal coordinates at x 0, Theorem 1 shows that the scaling limit of E x 0 (λ,λ+1] in the C topology is E1 Rn (u, v) = 1 (π) n e i u v,ω dω, S n 1

SCALING ASYMPTOTICS 3 which is the kernel of the frequency 1 spectral projector for the flat Laplacian on R n. Theorem 1 can therefore be applied to studying the local behavior of random waves on (M, g). More precisely, a frequency λ monochromatic random wave ϕ λ on (M, g) is a Gaussian random linear combination ϕ λ = a j ϕ j a j N(0, 1) i.i.d, λ j (λ,λ+1] of eigenfunctions with frequencies in λ j (λ, λ + 1]. In this context, random waves were first introduced by Zelditch in [0]. Since the Gaussian field ϕ λ is centered, its law is determined by its covariance function, which is precisely E (λ,λ+1] (x, y). In the language of Nazarov-Sodin [11] (cf [6, 14]), the estimate (6) means that frenquency λ monochromatic random waves on (M, g) have frequeny 1 random waves on R n as their translation invariant local limits at every non self-focal point. This point of view is taken up in the forthcoming article [5]. Theorem. Let (M, g) be a compact, smooth, Riemannian manifold of dimension n, with no boundary. Let K M be the set of all non self-focal points in M. Then for all k, j 0 and all ε > 0 there is a neighborhood U = U(ε, k, j) of K and constants Λ = Λ(ε, k, j) and C = C(ε, k, j) for which R(x, y, λ) C k x (U) C j y(u) ελn 1+j+k + Cλ n +j+k (6) for all λ > Λ. Hence, if x 0 K and U λ is any sequence of sets containing x 0 with diameter tending to 0 as λ, then R(x, y, λ) C k x (U λ ) C j y(u λ ) = o(λn 1+j+k ). (7) Remark 3. One can consider more generally any compact S M such that all x, y S are mutually non-focal, whic means L x,y := {ξ S xm : t > 0 with exp x (tξ) = y} has measure zero. Then, combining [1, Thm 3.3] with Theorem, for every ε > 0, there exists a neighborhood U = U(ε, j) of S and constants Λ = Λ(ε, j, S) and C = C(ε, j, S) such that x,y S j x j yr(x, y, λ) ελ n 1+j + Cλ n +j. We believe that this statement is true even when the number of derivatives in x, y is not the same but do no take this issue up here. Our proof of Theorem relies heavily on the argument for Theorem 1 in [4], which treated the case j = k = 0. That result was in turn was based on the work of Sogge- Zelditch [18, 19], who studied j = k = 0 and x = y. This last situation was also studied (independently and significantly before [4, 18, 19]) by Safarov in [1] (cf [13]) using a somewhat different method. The case j = k = 1 and x = y is essentially Proposition.3 in [0]. We refer the reader to the introduction of [4] for more background on estimates like (6).

4 Y. CANZANI AND B. HANIN 1. Proof of Theorem Let x 0 be a non-self focal point. Let I, J be multi-indices and set Ω := I + J. Using that S e i u,w dw = (π) n/ J n 1 n ( u ) u n for all u R n, we have ( ) 1 (π) n e i exp 1 y (x), ξ dξ λ ξ gy <λ gy = µ n 1 J n (µd g (x, y)) 0 (π) n dµ. (8) (µd g (x, y)) n Choose coordinates around x 0. We seek to show that there exists a constant c > 0 so that for every ε > 0 there is an open neighborhood U ε of x 0 and a constant c ε so that we have ( ) λ I x y J µ n 1 J n (µd g (x, y)) E λ (x, y) x I J 0 (π) n y dµ (µd g (x, y)) n c ελn 1+Ω +c ε λ n +Ω. (9) Let ρ S(R) satisfy p (ρ) ( inj(m, g), inj(m, g)) and ρ(t) = 1 for all t < 1 inj(m, g). (10) We prove (9) by first showing that it holds for the convolved measure ρ I x J y E λ (x, y) and then estimating the difference ρ I x J y E λ (x, y) I x J y E λ (x, y) in the following two propositions. Proposition 3. Let x 0 be a non-self focal point. Let I, J be multi-indices and set Ω = I + J. There exists a constant c so that for every ε > 0 there exist an open neighborhood U ε of x 0 and a constant c ε so that we have ( ) λ ρ I x y J µ n 1 J n (µd g (x, y)) E λ (x, y) x I J 0 (π) n y dµ (µd g (x, y)) n c ελn 1+Ω + c ε λ n +Ω, for all x, y U ε. Proposition 4. Let x 0 be a non-self focal point. There exists a constant c so that for every ε > 0 there exist an open neighborhood U ε of x 0 and a constant c ε so that for all multi-indices I, J we have ρ x I y J E λ (x, y) x I y J E λ (x, y) c ελ n 1+Ω + c ε λ n +Ω. The proof of Proposition 4 hinges on the fact that x 0 is a non self-focal point. Indeed, for each ε > 0, Lemma 15 in [4] (which is a generalization of Lemma 3.1 in [18]) yields the existence of a neighborhood O ε of x 0, a function ψ ε C c (M) and operators B ε, C ε Ψ 0 (M) ported in O ε satisfying both: p(ψ ε ) O ε and ψ ε = 1 on a neighborhood of x 0, (11) B ε + C ε = ψ ε. (1) The operator B ε is built so that it is microlocally ported on the set of cotangent directions that generate geodesic loops at x 0. Since x 0 is non self-focal, the construction can be carried so that the principal symbol b 0 (x, ξ) satisfies b 0 (x, ) L (B x M) ε for

SCALING ASYMPTOTICS 5 all x M. The operator C ε is built so that U(t)C ε is a smoothing operator for 1 inj(m, g) < t < 1 ε. In addition, the principal symbols of B ε and C ε are real valued and their sub-principal symbols vanish in a neighborhood of x 0 (when regarded as operators acting on half-densities). In what follows we use the construction above to decompose E λ, up to an O(λ ) error, as E λ (x, y) = E λ B ε (x, y) + E λ C ε (x, y) (13) for all x, y sufficiently close to x 0. This decomposition is valid since ψ ε 1 near x 0. 1.1. Proof of Proposition 3. The proof of Proposition 3 consists of writing ρ I x J y E λ (x, y) = λ 0 µ (ρ I x J y E µ (x, y)) dµ, and on finding an estimate for µ (ρ x I y J E µ (x, y)). Such an estimate is given in Lemma 5, which is stated for the more general case µ (ρ x I y J E µ Q (x, y)) with Q {Id, B ε, C ε } that is needed in the proof of Proposition 4. Lemma 5. Let (M, g) be a compact, smooth, Riemannian manifold of dimension n, with no boundary. Let Q {Id, B ε, C ε } have principal symbol D Q 0. Consider ρ as in (10), and define Ω = I + J. Then, for all x, y M with d g (x, y) 1 inj(m, g), all multi-indices I, J, and all µ 1, we have µ (ρ x I y J E µ Q )(x, y) ( = µn 1 (π) n I x y J ( ) e iµ exp 1 y (x),ω gy D Q Sy M 0 (y, ω) + µ 1 D Q 1 (y, ω) ) dω gy + W I,J (x, y, µ). (14) Here, dω is the Euclidean surface measure on SyM, and D Q 1 is a homogeneous symbol of order 1. The latter satisfy D 1 Bε (y, ) + DCε 1 (y, ) = 0 y O ε, (15) where O ε is as in (11). Moreover, there exists C > 0 so that for every ε > 0 e i exp 1 y (x),ω gy D Q dω 1 (y, ω) C ε. (16) gy x,y O ε S ym Finally, W I,J is a smooth function in (x, y) for which there exists C > 0 such that for all x, y satisfying d g (x, y) 1 inj(m, g) and all µ > 0 W I,J (x, y, µ) Cµ n +Ω ( d g (x, y) + (1 + µ) 1). (17) Remark 4. Note that Lemma 5 does not assume that x, y are near an non self-focal point.

6 Y. CANZANI AND B. HANIN Remark 5. We note that Lemma 5 is valid for more general operators Q. Indeed, if Q Ψ k (M) has vanishing subprincipal symbol (when regarded as an operator acting on half-densities), then (14) holds with D Q 0 (y, ω) substituted by µk D Q k (y, ω) and with µ 1 D Q 1 (y, ω) substituted by µk 1 D Q k 1 (y, ω). Here, DQ k is the principal symbol of Q and D Q k 1 is a homogeneous polynomial of degree k 1. In this setting, the error term satisfies W I,J (x, y, µ) Cµ n+k +Ω ( d g (x, y) + (1 + µ) 1). Proof of Lemma 5. We use that µ (ρ EQ )(x, y, λ) = 1 π + e itλ ρ(t) U(t)Q (x, y)dt, (18) where Q Ψ(M) is any pseudo-differential operator and U(t) = e it g is the halfwave propagator. The argument from here is identical to that of [4, Proposition 1], which relies on a parametrix for the half-wave propagator for which the kernel can be controlled to high accuracy when x and y are close to the diagonal. The main corrections to the proof of [4, Proposition 1] are that x I y J gives an O(µ n 3+Ω ) error in equations (54) and (60), and gives an O(µ n 1 ) error in (59). We must also take into account that x Θ(x, y) 1/ and y Θ(x, y) 1/ are both O(d g (x, y)). Proof of Proposition 3. Following the technique for proving [4, Proposition 7], we obtain Proposition 3 by applying Lemma 5 to Q = Id (this gives D0 Id = 1 and DId 1 = 0) and integrating the expression in (14) from µ = 0 to µ = λ. One needs to choose U ε so that its diameter is smaller than ε, since this makes λ 0 W I,J(x, y, µ)dµ = O(ελ n 1+Ω + λ n +Ω ) as needed. One also uses identity (8) to obtain the exact statement in Proposition 3. 1.. Proof of Proposition 4. As in (13), E λ (x, y) = E λ B ε (x, y) + E λ C ε (x, y) + O ( λ ) for all x, y sufficiently close to x 0. Proposition 4 therefore reduces to showing that there exist a constant c independent of ε, a constant c ε = c ε (I, J, x 0 ), and a neighborhood U ε of x 0 such that and x I y J E λ Bε (x, y) ρ x I y J E λ Bε (x, y) c ελ n 1+Ω + c ε λ n +Ω, (19) x I y J E λ Cε (x, y) ρ x I y J E λ Cε (x, y) c ελ n 1+Ω + c ε λ n +Ω. (0) Our proofs of (19) and (0) use that these estimates hold on diagonal when I = J = 0 (i.e. no derivatives are involved). This is the content of the following result, which was proved in [18] for Q = Id. Its proof extends without modification to general Q Ψ 0 (M). Lemma 6 (Theorem 1. and Proposition. in [18]). Let Q Ψ 0 (M) have realvalued principal symbol q. Fix a non-self focal point x 0 M and write σ sub (QQ ) for

SCALING ASYMPTOTICS 7 the subprincipal symbol of QQ (acting on half-densities). Then, there exists c > 0 so that for every ε > 0 there exist a neighborhood O ε and a constant C ε making QE λ Q (x, x) = (π) n ( q(x, ξ) + σ sub (QQ )(x, ξ) ) dξ gx + R Q(x, λ), with for all λ 1. ξ gx <λ R Q (x, λ) c ελ n 1 + C ε λ n x U We prove relation (19) in Section 1..1 and relation (0) in Section 1... 1..1. Proof of relation (19). Define g I,J (x, y, λ) := I x J y E λ B ε (x, y) ρ I x J y E λ B ε (x, y). Note that g I,J (x, y, ) is a piecewise continuous function. We aim to find c, c ε and U ε so that x 0 U ε and g I,J (x, y, λ) c ελ n 1+Ω + c ε λ n +Ω. (1) By [4, Lemma 17], which is a Tauberian Theorem for non-monotone functions, relation (1) reduces to checking the following two conditions: F λ t (g I,J )(x, y, t) = 0 for all t < 1 inj(m, g), () g I,J (x, y, λ + s) g I,J (x, y, λ) c ελ n 1+Ω + c ε λ n +Ω. (3) By construction, F λ t ( λ g I,J )(x, y, t) = (1 ρ(t)) I x J y U(t)B ε (x, y) = 0 for all t < 1 inj(m, g). Hence, since F λ t(g I,J ) is continuous at t = 0, we have (). To prove (3) we write g I,J (x, y, λ + s) g I,J (x, y, λ) x I y J E (λ,λ+s] Bε (x, y) + ρ x I y J E (λ,λ+s] Bε (x, y). (4) The second term in (4) is bounded above by the right hand side of (3) by Lemma 5. To bound the first term, use Cauchy-Schwartz to get I x y J [E (λ,λ+s] Bε (x, y)] = xϕ I j (x) y J B ε ϕ j (y) λ j (λ,λ+1] λ j (λ,λ+1] [( B ε J y + [ J y, B ε ] ) ϕ j (y) ] I x ϕ j (x). Write b 0 for the principal symbol of B ε. By construction, for all y in a neighborhood of x 0, we have y b 0 (y, ξ) = 0. Therefore, σ J 1 ( [ J y, B ε ] ) = i J {ξ J, b 0 (y, ξ)} = 0 and

8 Y. CANZANI AND B. HANIN we conclude that [ y J, B ε ] Ψ J. Thus, by the usual pointwise Weyl Law (e.g. [19, Equation (.31)]), x I y J [E (λ,λ+s] Bε (x, y)] B ε y J ϕ j (y) xϕ I j (x) + O(λ n 3+Ω ) λ j (λ,λ+1] Next, define for each multi-index K N n the order zero pseudo-differential operator P K := K K / g. Using Cauchy-Schwarz and that K ϕ j = λ K j P K ϕ j, we find B ε y J ϕ j (y) xϕ I j (x) λ j (λ,λ+1] (λ + 1) Ω [(B ε P J )E (λ,λ+1] (B ε P J ) (y, y)] 1 [PI E (λ,λ+1] P I (x, x)] 1. Again using the pointwise Weyl Law (see [19, Equation (.31)]), we have [P I E (λ,λ+1] PI (x, x)] 1 is O(λ n 1 ). Next, since according to the construction of B ε we have b 0 (x, ) L (B ε xm) x U ε and x b 0 (x, ξ) = 0 for x in a neighborhood U ε of x 0, we conclude that σ sub (B ε P J (B ε P J ) )(x, ) L (Bx x U M) ε. ε Proposition 6 therefore shows that there exists c > 0 making (B ε P J )E (λ,λ+1] (B ε P J ) (y, y) 1 cελ n 1. (5) This proves (3), which together with () allows us to conclude (1). 1... Proof of relation (0). Write I x J y E λ C ε (x, y) = λ j λ λ Ω j (P I ϕ j (x)) (C ε P J ϕ j (y)) + λ j λ λ I j (P I ϕ j (x)) ([ J, C ε ]ϕ j (y) ). As before, [ J, C ε ] Ψ J. Hence, by the usual pointwise Weyl law, the second term in (6) and its convolution with ρ are both O(λ n +Ω ). Hence, x I y J E λ Cε (x, y) ρ x I y J E λ Cε (x, y) = V (x, y, λ) ρ V (x, y, λ) where we have set + O ( λ n+ω ), V (x, y, λ) := I E λ (C ε J ) (x, y) = λ j λ λ Ω j (P I ϕ j (x)) (C ε P J ϕ j (y)). (6)

Define α I,J (x, y, λ) := V (x, y, λ) + 1 SCALING ASYMPTOTICS 9 λ j λ β I,J (x, y, λ) := ρ V (x, y, λ) + 1 λ Ω j λ j λ ( PI ϕ λj (x) + Cε P J ϕ λj (y) ) (7) λ Ω j ( P I ϕ λj (x) + C ε P J ϕ λj (y) ). (8) By construction, α I,J (x, y, ) is a monotone function of λ for x, y fixed, and α I,J (x, y, λ) β I,J (x, y, λ) = V (x, y, λ) ρ V (x, y, λ). So we aim to show that α I,J (x, y, λ) β I,J (x, y, λ) c ελ n 1+Ω + c ε λ n +Ω. (9) We control the difference in (9) applying a Tauberian theorem for monotone functions [4, Lemma 16]. To apply it we need to show the following: There exists c > 0 and c ε > 0 making λ+ε λ ε µ β I,J (x, y, µ) dµ cελ n 1+Ω + c ε λ n +Ω. (30) For all N there exists M ε,n so that for all λ > 0 λ (α I,J (x, y, ) β I,J (x, y, )) φ ε (µ) M ε,n (1 + λ ) N. (31) In equation (31) we have set φ ε (λ) := 1 ε φ ( ) λ ε for some φ S(R) chosen so that p φ ( 1, 1) and φ(0) = 1. Relation (30) follows after applying Lemma 6 to the piece of the integral corresponding to the second term in (8) and from applying Lemma 5 together with Remark 5 to ρ V = ρ I E λ Q, where Q := C ε J has vanishing subprincipal symbol. To verify (31) note that p(1 ρ) {t : t inj(m, g)/} and p( φ ε ) {t : t 1 ε }. Observe that ) λ (α I,J (x, y, ) β I,J (x, y, ) ( φ ε (λ) = F 1 t λ (1 ρ(t)) φ ) ε (t) I U(t)( J C ε ) (x, y) (λ). By construction U(t)Cε is a smoothing operator for 1 inj(m, g) < t < 1 ε. Thus, so is I U(t) ( J ) C ε which implies (31). This concludes the proof of relation (0). References [1] P. Bérard. On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z. 155 (1977), 49-76. [] M. Berger, P. Gauduchon, and E. Mazet. Le spectre d une variété Riemannienne. Lecture Notes in Math. Springer (1971), 51p. [3] B. Xu. Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold. Anal. of Global. Anal. and Geom., (004) 31 5. [4] Y. Canzani, B. Hanin. Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law. Analysis and PDE, 8 (015) no. 7, 1707 1731. [5] Y. Canzani, B. Hanin. Local Universality of Random Waves. In preparation. [6] Y. Canzani, P. Sarnak. On the topology of the zero sets of monochromatic random waves. Preprint available: arxiv:141.4437. [7] J. Duistermaat and V. Guillemin. The spectrum of positive elliptic operators and periodic bicharacteristics. Inventiones Mathematicae, 9 (1975) no. 1, 39 79.

10 Y. CANZANI AND B. HANIN [8] L. Hörmander. The spectral function of an elliptic operator. Acta mathematica, 11.1 (1968), 193-18. [9] V. Ivrii. Precise spectral asymptotics for elliptic operators. Lecture Notes in Math. 1100, Springer, (1984). [10] V. Ivrii. The second term of the spectral asymptotics for a Laplace Beltrami operator on manifolds with boundary. (Russian) Funksional. Anal. i Prolzhen. (14) (1980), no., 5 34. [11] F. Nazarov and M. Sodin. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions. Preprint: http://arxiv.org/abs/1507.0017. [1] Yu. Safarov. Asymptotics of the spectral function of a positive elliptic operator without a nontrapping condition. Funktsional. Anal. i Prilozhen. (1988), no. 3, 53-65, 96 (Russian). English translation in Funct. Anal. Appl. (1988), no. 3, 13-3 [13] Y. Safarov and D. Vassiliev. The asymptotic distribution of eigenvalues of partial differential operators. American Mathematical Society, 1996. [14] P. Sarnak and I. Wigman Topologies of nodal sets of random band limited functions. Preprint: http://arxiv.org/abs/1510.08500 (015). [15] C. Sogge, J. Toth, and S. Zelditch. About the blowup of quasimodes on Riemannian manifolds. J. Geom. Anal. 1.1 (011), 150-173. [16] C. Sogge. Fourier integrals in classical analysis. Cambridge Tracts in Mathematics 105, Cambridge University Press, (1993). [17] C. Sogge. Hangzhou lectures on eigenfunctions of the Laplacian Annals of Math Studies. 188, Princeton Univ. Press, 014. [18] C. Sogge and S. Zelditch. Riemannian manifolds with maximal eigenfunction growth. Duke Mathematical Journal 114.3 (00), 387 437. [19] C. Sogge and S. Zelditch. Focal points and -norms of eigenfunctions. Preprint: http://arxiv.org/abs/1311.3999. [0] S. Zelditch. Real and complex zeros of Riemannian random waves. Contemporary Mathematics 14 (009), 31. (Y. Canzani) Department of Mathematics, Harvard University, Cambridge, United States. E-mail address: canzani@math.harvard.edu (B. Hanin) Department of Mathematics, MIT, Cambridge, United States. E-mail address: bhanin@mit.edu