THE BLOWUP ALONG THE DIAGONAL OF THE SPECTRAL FUNCTION OF THE LAPLACIAN LIVIU I. NICOLAESCU

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1 THE BLOWUP ALONG THE DIAGONAL OF THE SPECTRAL FUNCTION OF THE LAPLACIAN LIVIU I. NICOLAESCU ABSTRACT. We formulate a precise conjecture about the universal behavior near the diagonal of the spectral function of the Laplacian of a smooth compact Riemann manifold. We prove this conjecture when the manifold and the metric are real analytic, and we also present an alternate proof when the manifold is the round sphere. CONTENTS. Introduction. The Universality Conjecture in the real analytic case 3 3. The spectral function of a round sphere 5 Appendix A. Sharp elliptic estimates 9 References 3. INTRODUCTION Suppose that M, g is a compact, connected m-dimensional Riemannian manifold, and Ψ n n 0 is a complete orthonormal basis of L M, g consisting of eigenfunctions of g g Ψ n = λ n Ψ n, 0 = λ 0 < λ λ λ n. For every L > 0 we define the spectral function E L : M M R by E L p, q = Ψ n pψ n q. λ n L Equivalently, E L is the Schwartz kernel of the orthogonal projection onto H L := λ L kerλ. This shows that as L the spectral function E L converges in the sense of distributions to the Dirac-type distribution supported by the diagonal D M = { p, q M M; p = q }. The goal of this paper is to describe a universal law governing the behavior of E L as L in an infinitesimal neighborhood of the diagonal. Here are the specifics. Date: Started on February, 0. Completed on March, 0. Latest revision January, Mathematics Subject Classification. 58J50, 35J5, 33C45, 3C05. Key words and phrases. Riemann manifolds, Laplacian, eigenfunctions, spectral function, real analytic manifolds, harmonic polynomials. This work was partially supported by the NSF grant, DMS

2 LIVIU I. NICOLAESCU We denote by N the normal bundle of the diagonal embedding. For any p M we denote by N p the fiber over p, p D M. Also we let r : N R denote the radial distance function along the fibers of N, and we set N R := r [0, R, N R p := N R N p, R > 0, p M. In other words, N R N is the associated bundle of normal disks of radius R. If is sufficiently small, then the exponential map induces a diffeomorphism from N onto an open neighborhood U of the diagonal. Fix once and for all such a. We denote by E L the pullback of E L U to N. For every positive real number λ we denote by M λ : N N the rescaling map described on N p by N p v λ v N p. We define Ē L : N L/ R, Ē L = L m M E L. L For any p M we denote by ĒL,p the restriction of ĒL to the fiber N p. The Universality Conjecture. There exists ρ > 0 such that for any p M the functions Ē L,p : N L p R converge as L in the topology of C N ρ p to the smooth function E : N p R, E u = π u m where J ν denotes the Bessel function of the first kind and order ν. J m u, Remark.. a The limit function E u has a more suggestive description, namely E u = J π u m m u = π m e iξ,u dξ, B m r := { ξ R m ; ξ r }.. B m We denote by E L the spectral function of the Laplacian on R m corresponding to generalized eigenvalues L. We then have see [6, Eq..] or [, Eq..3] E L x, y = π m e iξ,x y dξ. This shows that B m L E u = E u, 0. b We ought to elaborate on the crux of the Universality conjecture. First of all, let us point out that known local Weyl trace formulæ [], [4, 7.5], [3, Thm..8.5], [5, 7,8] imply that the family ĒL L is precompact in the C -topology and as detailed in see Section, any limit point Ē is asymptotically equivalent to E at u = 0, i.e., for any N > 0, Ē,p u = E u + O u N as u 0. The Universality Conjecture makes the stronger claim that the family ĒL L has a unique limit point in C -topology and moreover, that limit point is not just asymptotically equivalent, but equal to the function E. c Recently, Lapointe, Polterovich and Safarov [7] have described a global type of relationship between E L and E L as L.

3 THE BLOWUP OF THE SPECTRAL FUNCTION 3 The main result of this paper states that the Universality Conjecture is true in the case when M and g are real analytic. We achieve this in Section by relying on some sharp a priori estimates that we prove in Appendix A, Theorem A.. A weaker version of these estimates were used by Donnelly and Fefferman [3, 7] in their investigation of nodal sets of eigenfunctions on real analytic Riemannian manifolds. We believe that Theorem A. will find many other uses. In Section 3 we give an alternate proof of the conjecture in the special case when M, g is the round sphere. Acknowledgments. I want to thank Steve Zelditch for a most illuminating discussion on spectral geometry.. THE UNIVERSALITY CONJECTURE IN THE REAL ANALYTIC CASE Theorem.. The Universality Conjecture is true when M and g are real analytic. Proof. Since both M and g are real analytic we deduce that the spectral function E L is real analytic; see [8, 9]. The Cauchy-Kowaleskaya theorem, [5], implies that the exponential map is also real analytic so that E L is real analytic. Fix a point p 0 in M and normal coordinates x = x,..., x m at p 0 defined on an open neighborhood O of p 0. With these choices we can regard the restriction of E L to O O as a real analytic function E L x, y defined on a neighborhood of 0, 0 in R m R m. Via the exponential map exp p0,p 0 : T p0,p 0 M M M M we can identify the space { x, y R m R m ; x + y = 0 } with the fiber N p0. The results of [] show that as L we have α+β x α y β E Lx, y x=y=0 = L m+ α + β where C α,β = 0 if α β Z m, while if α β Z m we have C α,β = α β 4π m α + β Γ + m + α + β C α,β + O L,. m α i + β i!!. In fact we can say a bit more. More precisely, according to [4, Thm ], for any α, β there exists a constant K α,β > 0 that depends on the geometry of M, g but it is independent of p 0 such that α+β x α y β E Lx, y K α,βl m+ α + β, x, y.. A priori, the constants K α,β can grow really fast as α + β. Our next result provides a key upper bound on this growth. To keep the flow of arguments uninterrupted we deferred its rather sneaky proof to Appendix A. Lemma.. There exist constant C, T > 0 such that for any L > and any multi-indices α, β we have x,y α+β E L p, q CT α + β α + β!l m+ α + β,.3 sup p,q M M where x, y denote normal coordinates at p, q. In other words, in. we can choose constants K α,β satisfying where K and T are independent of α, β and L. K α,β KT α + β α + β!,.4 i=

4 4 LIVIU I. NICOLAESCU Introduce new coordinates η i := x i y i, τ j := x j + y j, i, j =,..., m. The fiber N p0 is described by the equations τ j = 0, j =,..., m. Note that x j = η j + τ j, y j = τ j η j. Along N p0 we can use the functions η j as orthonormal coordinates. We have r = i η i, and We set E L,p0 = E L η, η, ĒL,p0 η = L m EL L η, η L C α L := α η α Ē L,p0 η η=0. From.4 we deduce that there exist constants K, T > 0 such that for any multi-index α and any L > we have C α L K α!t α..5 Using. we deduce that l η iēlx, y x=y=0 = C l + O L, where C l = 0 if l is odd, while if l is even we have l C l = l l k l k k k x i y iēlx, y x=y=0 = l l!! 4π m Γ + m + l. More generally, and k=0 C α L := O L if α Z,.6 C α L = α 4π m Γ + m + α m α j!! + O L..7 The estimates.5,.6 and.7 show that there exists ρ > 0 such the family { Ē L of real }L analytic functions converges in the topology of C { η < ρ} to a function E that is real analytic on the ball { η < ρ}. Moreover, η α E 0 = α m 4π m Γ + m + α α j!!..8 We deduce that for η < ρ we have = n 0 E η = α j= j= α! α η E 0η α = n 4π m Γ + m n 0 + n n 4π m Γ + m + n α =n α α! ηα = n 0 α =n j α j! j α η α j! n 4π m n n!γ + m + n α =n n! j α j! ηα

5 THE BLOWUP OF THE SPECTRAL FUNCTION 5 = n 4π m n 0 n n!γ + m + n η i n = i n 0 r = πr m m n r n!γ + m n 0 + n n r n 4π m n!γ + m + n. } {{ } =:J m r 3. THE SPECTRAL FUNCTION OF A ROUND SPHERE To appreciate the complexity involved in the Universality Conjecture we believe that it is instructive to give an alternate proof of the conjecture in the special case when M is the round sphere S d R d. The spectrum of the Laplacian on S d is see [0] and We set λ n = nn + d, n = 0,,,... dim kerλ n = µ n = n + d n + d H m := kerλ m, U n = n + d d n H m. As is well known, the space H k coincides with the space of restrictions to S d of harmonic polynomials of degree k in d variables. Fix an is an orthonormal basis Ψ k,m α µm of H m and set n µ m E n p, q := E λn p, q = Ψ k,m pψ k,m q. m=0 k= The addition formula for spherical harmonics [0] implies that µ m k= Ψ k,m pψ k,m q = k=0. µ m σ d P m,d p q, p, q S d, 3. where p q is the canonical inner product of p, q R d, σ d is the area of S d, σ d = π d Γ, 3. d and P n,d is the Legendre polynomial of degree n and order d, P n,d t := n n [n + d 3 ] t d 3 d n n dt n t d 3 n+, [x] k := xx x k Γx + = Γx + k. The collection P n,d n 0 is an orthogonal family of polynomials with respect to the measure wtdt = t d 3 dt on [, ]. More precisely we have, [0, ], P n,d tp m,d twtdt = h n δ n,m, h n = σ d σ d µ n.

6 6 LIVIU I. NICOLAESCU Observe that we can rephrase 3. as µ m k= Ψ k,m pψ k,m q = P m,dp q σ d h m. 3.3 We denote by k n the leading coefficient of P n,d. Its precise value is known, [0, Eq. 7.6], but all we need in the sequel is the equality [0, Eq. 7.7] k n k n = n + d 3 n + d 4. We recall the Christoffel-Darboux formula, [, 5.], n P m,d sp m,d t = k n P n+,d tp n,d s P n+,d sp n,d t 3.4 h m k n+ t sh n m=0 In 3.4 we let s =. Using the equality P m,d =, m, we deduce n P m,d t = k n P n+,d t P n,d t. 3.5 h m k n+ h n t m=0 Suming 3.3 for m = 0,..., n we deduce from 3.5 that Hence E n p, q = E n p, q = E n ϕ := k n P n+,d t P n,d t = µ n k n+ σ d h n t σ d µ n k n P n+,d cos ϕ P n,d cos ϕ σ d k n+ cos ϕ k n P n+,d t P n,d t k n+ t., cos ϕ = p q. 3.6 Observe that ϕ is the geodesic distance between p and q. Now set r n := λ n and define Ē n ϕ := r d n E n ϕ r n = µ n σ d r d n k n P n+,d cos ϕ r n P n,d cos ϕ r n k n+ r n cos ϕ r n Taking into account Remark., we see that the Universality Conjecture will follow from the equality Observe first that Ē n ϕ = n+α n P n,d t = πϕ d P α,α J d ϕ + On, uniformly for ϕ π n t, α = d 3, n + α = n Γn + α + Γα + Γn +, 3.8 where P n α,β are the Jacobi polynomials defined in [4, 4.]. To prove 3.7 we will use the Hilb type asymptotic estimate for Jacobi polynomials., [4, Eq. 8..7] or [, 9]. Here are the details. Set θ := ϕ r n. Observe that = On α. 3.9 n+α n

7 THE BLOWUP OF THE SPECTRAL FUNCTION 7 The Hilb type estimate [4, Eq. 8..7] coupled with 3.8 and 3.9 yields θ α P n,d cos θ = Γα + sin θ a α n sin α θ J αa n θ + θα+ sin α θ O θ = Γα + sin θ θ α sin α θ a n θ α J αa n θ + θα+ sin α θ O, where α is as in 3.8 and a n := n + d. We set F α x := x α J αx and we deduce that θ θ α P n,d cos θ = Γα + sin θ sin α θ F αa n θ + θα+ sin α θ O. Hence P n+,d cos θ P n,d cos θ = Γα + P n+,dcos θ P n,d cos θ r n cos θ r n sin θ θ θ α Fα a n+ θ F α a n θ θ α+ = Γα + sin θ sin α θ r n sin θ + r n sin α+ θ O. Observe that a n+ θ = a n θ + θ and we deduce F α a n+ θ F α a n θ = F αa n θθ + Oθ. The classical identity involving Bessel functions, [, Eq. 4.6.], d x ν J ν x = x ν J ν+ x, dx implies that F αz = x α J α+ x. Now observe that r n sin θ = r nθ Oθ. Since Oθ = Orn = On we deduce θ α+ θ α = P n+,d cos θ P n,d cos θ r n cos θ = α+ ϕ F αϕ + On = r F αa n θ sin θ sin α + On θ r n θ α+ Jα+ r + On. Hence µ n k n r α+ Ē n ϕ = Γα + Jα+ σ d rn d r + On. k n+ The equality 3.7 now follows from the estimates µ n rn d = d! + On, k n k n+ = + On, the equality 3., and the doubling formula πγx = x ΓxΓ x +.

8 8 LIVIU I. NICOLAESCU FIGURE. A depiction of E 00 0 E 00ϕ, 0 ϕ π 8. Remark 3.. In the special case M = S we can visualize the blowup behavior of the spectral function. In this case the eigenvalues of the Laplacian are λ n = nn + and Ē n ϕ = ϕ E n λn. λ n The function E n ϕ has a peak at ϕ = 0, E n 0 = n + 4π 4π λ n as n. The oscillations in the graph depicted in Figure are pushed at by the rescaling in ϕ, and the resulting behavior becomes rather tame, Figure. FIGURE. A depiction of E 00 0 E 00ϕ/00, 0 ϕ π ϕ 8. The rescaling ϕ 00 pushes the oscillations outside the screen.

9 THE BLOWUP OF THE SPECTRAL FUNCTION 9 APPENDIX A. SHARP ELLIPTIC ESTIMATES The main goal of this appendix is to provide a proof of the estimates.3. We will follow the same strategy employed in the proof of [4, Thm ]. We will reach a conclusion stronger than [4, Thm ] because we will rely on sharp a priori elliptic estimates. We obtain these sharp estimates from the precise a priori estimates for elliptic operators with real analytic coefficients in [8, Chap. 8]. For the reader s convenience we first synchronize our notations with those in [8]. For any nonnegative integers s we set M s := s!. Observe that [8, Chap. 8, Eq..0] becomes M t+s d t+s M t M s, s, t Z 0, d :=. Denote by the Laplacian operator defined by a real analytic metric g on the ball B 4, where B R = { x R m ; x < R }. A. For any R > 0 we denote by AB R the space of functions real analytic on the ball B R. We set u k,r = x α ux dx. A. B R α =k Theorem A.. Then there exist constants K, T > 0, ρ 0 0, that depend only on the real analytic metric g and satisfying the following properties. If u AB and Z 0, L 0 > are such that then k u 0, Z 0 L k 0M k, k 0, u j,ρ0 Z 0 T L0 jmj, j 0. A.3 A.4 α x u0 KZ 0 L m 4 0 T L0 jmj, j 0, α = j. A.5 We want to emphasize that K and T are independent of u, L 0, Z 0. Proof. We follow closely the approach in [8, Chap. 8, Sec. ]. Fix a smooth function χ : R R such that {, t 0 χt = 0, t. For ρ, δ > 0 we define x ρ ϕ ρ,δ : R m R, ϕ ρ,δ x = χ. δ Note that supp ϕ ρ,δ B ρ,δ, ϕ ρ,δ on B ρ and for all p Z >0 there exists γ p > 0 such that D p ϕ ρ,δ x γ p δ p, x R m. From [8, Chap. 8, Sec., Thm..]. we deduce the following. Lemma A.. There exist ρ, C > 0, ρ <, such that if 0 < ρ < ρ + δ < ρ and any u AB we have m u,ρ C ϕ ρ,δ u 0,ρ+δ + δ m l u l,ρ+δ. A.6 l=0 Arguing as in the proof of [8, Chap. 8, Lemma.5] we obtain the following result.

10 0 LIVIU I. NICOLAESCU Lemma A.3. Suppose that a AB satisfies sup α ax L r M r, r 0. X B 3 α =r Then for every r, s Z 0, 0 < ρ < ρ + δ < and u AB we have β [ϕ ρ,δ a α u α au 0,ρ+δ C s where C s depends only on s. s l=0 β =s α =r s l δ l t=0 r L r+t M r+t i=0 L i M i u s l t+i,ρ+δ, A.7 Using Lemma A. and A.3 we deduce as in [8, Chap. 8, Lemma.6] the following result. Lemma A.4. Suppose that ρ, C are as in Lemma A.. For any ε > 0 there exists γε > 0 that depends only on the metric and ε such that for any u AB, k > 0 and ρ < ρ + δ < ρ we have { k= M k s=0 u k+,ρ C γε k s u s+,ρ+δ + M k M s δ where the constant C depends only on the metric g. k s= u k,ρ+δ + ε u k+,ρ+δ } γε k s u s+,ρ+δ, M s+ A.8 For every λ, R > 0 such that R < ρ we set σ k u, λ, R := M k λ k+ sup R ρ k+ u k+,ρ. R/ ρ<r Using Lemma A.4 as in the proof of [8, Chap. 8, Lemma.7] we obtain the following result. Lemma A.5. There exists λ > 0 such that for any R < ρ, λ λ, k 0 and every u AB we have σ k u, λ, R M k 4M k σ k u, λr + 4 k s= σ s u, λ, R. A.9 An argument identical to the one used in the proof of [8, Chap. 8, Thm..] and based on Lemma A.5 shows that if u AB satisfies A.3, then for λ, R as in Lemma A.5 we have σ k u, λ, R M k+ M k Z 0 4L 0 + k+. A.0 If we let λ = λ and R 0 = 4 ρ in the above inequality we deduce λ k+ sup R 0 ρ k+ u k+,ρ M k+ Z 0 4L 0 + k+ R 0 / ρ<r 0

11 THE BLOWUP OF THE SPECTRAL FUNCTION In particular, if ρ 0 = 3R 0 /4 and µ := λ we deduce 4µ k+ u k+,ρ0 c 0 M k+ 4L 0 + k+ = Z 0 M k+ζ k+, ζ := 4µ 4L0 +. R 0 R 0 We can assume that ζ >. Using the interpolation inequality [8, Chap. 8, Eq..7] we deduce that u k+,ρ0 ζ u k+,ρ0 + c m ζ u k,ρ0, where c m > 0 is a constant that depends only on the dimension m. Hence u k+,ρ0 Z 0 M k+ ζ k+ + c m Z 0 M k ζ k+ = Z 0 M k+ ζ k+ k + + c m Z 0 M k+ + cm ζ k+. k + We deduce that for any k 0 we have This proves A.4 with u k,ρ0 Z 0 M k Z k, Z = + c m ζ. T = Z L0 +. A. Using the Sobolev lemma [9, Thm. 3.5.] we deduce that there exists a constant K m > 0 that depends only on m such that for any v C B r/ we have v0 K p m r j r m j! v j,r + rp p! v p,ρ 0 m, p = p m := +. j=0 We deduce that for any multi-index α such that α = k and any r such that r/ < ρ 0 we have α u0 K p m r j r m j! v j+k,ρ 0 + rp p! u p+k,ρ 0 j=0 A. K mz 0 p r j k + j! Z k+j + rp k + p! Z k+p r m j! p! = K mz 0 k!z k r m j=0 p k + j j j=0 rz j + p Recall that Z = T L 0 +. Choose r of the form r = B L 0 + < B. where B is sufficiently large so that r = B L 0 + < B < ρ 0, and rz = T B. k + p rz p. p We have to warn the reader that the Sobolev norms j v,r defined in [9] do not coincide with the ones defined in A. but they are equivalent. The constants implied by this equivalence of norms depend only on j and m.

12 LIVIU I. NICOLAESCU p α u0 K m B m Z 0 L 0 + m 4 Z k k! k + j k + p j + p j p j=0 p m K m B m Z 0 L 0 + m 4 Z k k! This proves A.5 with K = p m K m B m. k + j j j j=0 }{{} = k+ = p m K m B m Z 0 L 0 + m 4 T L0 + k k! p Remark A.6. The above arguments extend with no changes to the case when the metric belongs to a Gevrey space, or more general to one of the spaces D Mk B 4 of [8], where the weights M k are subject to the constraints.6-. in [8, Chap.8]. Theorem A. has the following immediate consequence. Corollary A.7. Suppose N is a compact real analytic manifold of dimension n and g is a real analytic metric on N with injectivity radius rn. Denote by g the Laplace operator of the metric g and by AN the space of real analytic functions on N. Then there exist constants K, T > 0 and 0 < ρ 0 < rn depending only on g with the following property: for any Z 0, L 0 > and any u AN such that, if k gu L N,g Z 0 L k 0k!, k 0, A. then x α up KZ 0 L m 4 jj!, 0 T L0 α = j, p M, A.3 where x = x,..., x m are normal coordinates at p. Proof of.3. We follow the same strategy used in the proof of [4, Thm ]. Fix L > and denote by P L the orthogonal projection onto the space H L := λ L kerλ. Fix j, l 0 points p 0, q 0 in M, and normal coordinates x at p 0 and y at q 0. Let f L M and set f L = P L f. Then k f L L M L k f, k 0. Using Corollary A.7 we deduce that α x f L p 0 KL m 4 + j T j j! f, p M, α = j, where K, T are independent of f, k, α and L. Now observe that α x f L p 0 = f, g 0 L M, g 0 q = α x E L p 0, q = λ n L The above discussion shows that for every f L M we have f, g 0 L M KL m 4 + j T j j! f, α x Ψ n p 0 Ψ n q. so that g 0 L M KL m 4 + j T j j!.

13 Observe that g 0 H L and thus for any k 0 we have THE BLOWUP OF THE SPECTRAL FUNCTION 3 k g 0 L M L k g 0 L M and Corollary A.7 implies that for any β = l we have α x β y E L p 0, q 0 = β y g 0 q 0 KL m 4 + l 4T l l! g 0 K L m + j + l T j+l j!l! The inequality.3 follows by observing that j + l j!l! j + l! j+l j + l!. j REFERENCES [] G.E. Andrews, R. Askey, R. Roy: Special Functions, Encyclopedia of Math. and its Appl., vol 7, Cambridge University Press, 006. [] X. Bin: Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold, Ann. Global. Analysis an Geometry, 6004, 3-5. [3] H. Donnely, C. Fefferman: Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math , [4] L. Hörmander: The Analysis of Partial Differential Operators III, Springer Verlag, 994. [5] S.G. Krantz, H. R. Parks: A Primer of Real Analytic Functions, nd Edition Birkhäuser, 00. [6] P.T. Lai: Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatif au Laplacien, Math. Scand. 4898, [7] H. Lapointe, I. Polterovich, Yu. Safarov: Average growth of the spectral function on a Riemannian manifold, Comm. Part. Diff. Eqs., 34009, [8] J.L. Lions, E. Magenes: Non-Homogeneous Boundary Value Problems and Applications. Volume III, Springer Verlag 973. [9] C.B. Morrey: Multiple Integrals in the Calculus of Variations, Springer Verlag, 966. [0] C. Müller: Analysis of Spherical Symmetries in Euclidean Spaces, Appl. Math. Sci. vol. 9, Springer Verlag, 998. [] J. Peetre: Remark on eigenfunction expansions for elliptic operators with constant coefficients, Math. Scand. 5964, 83-9 [] H. Rau: Über eine asymptotische Darstellung der Jacobischen Polynome durch Besselsche Funktionen, Math. Z., 40936, [3] Yu. Safarov, D. Vassiliev: The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Translations of Math. Monographs, vol. 55, Amer. Math. Soc., 997. [4] G. Szegö: Orthogonal Polynomials, Colloquium Publ., vol 3, Amer. Math. Soc., 003. [5] S. Zelditch: Local and global analysis of eigenfunctions, Handbook of geometric analysis. No., , Adv. Lect. Math. ALM, 7, Int. Press, Somerville, MA, 008. arxiv: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN address: nicolaescu.@nd.edu URL: lnicolae/