Spectral Geometry of Eigenvalues and Resonances: a Retrospective on the Work of Robert Brooks

SPECTRAL GEOMETRY OF EIGENVALUES AND RESONANCES: A RETROSPECTIVE ON THE WORK OF ROBERT BROOKS

PETER PERRY

Dedicated to the memory of Robert W. Brooks

Abstract. Spectral geometry has its roots in the study of the geometric content of eigenvalues on a compact surface or the Dirichlet eigenvalues of a bounded planar domain. On the one hand, explicit constructions of discrete and continuous families of manifolds with the same spectrum show that the spectrum does not completely determine the geometry of a manifold; on the other, compactness theorems show that the spectrum does place strong constraints on the geometries allowed by a given spectrum. Here we will explore the techniques used in each of these two approaches and discuss their recent application to the spectral geometry of resonances.

1. Introduction Many years ago, Mark Kac posed a model inverse problem which has attracted the attention and energy of many mathematicians: do the Dirichlet eigenvalues of a bounded domain determine its geometry? Exercising their penchant for generaliza- tion, mathematicians recast the question in a mathematically natural setting: does the spectrum of the Laplacian on a compact Riemannian manifold X (with suitable boundary conditions if ∂X = ) determine its geometry? Call two compact Rie- 6 ∅ mannian manifolds X1 and X2 isospectral if the Laplacians on functions have the same spectrum, and deﬁne the isospectral set of X to be the set of all Riemannian manifolds with for which the spectrum of the Laplacian on functions equals that of X. The problem is to characterize the isospectral set of a given Riemannian manifold X. A similar problem can be posed for non-compact manifolds for which the Lapla- cian has a well-deﬁned scattering theory: examples quotients of real hyperbolic two-dimensional space by a discrete group of isometries, exterior domains in Eu- clidean space, and compactly supported metric perturbations of Euclidean space. In all of these cases, the resolvent of the Laplacian admits an analytic continuation whose poles are the scattering resonances, and there is a natural function, the scattering phase, which behaves much like the counting function for eigenvalues. Two such manifolds are said to be isopolar if they have the same scattering poles, and isophasal if their scattering phases are the same. The problem is then to characterize the isopolar or isophasal set of a given manifold.

Date: January 2003. Supported in part by NSF Grant DMS-0107051. 1 2 PETER PERRY

There are two natural approaches to this problem. First, one can form spectral invariants such as the heat trace, the wave trace, or the determinant of the Lapla- cian and compute them geometrically in order to obtain geometric invariants of the spectrum: this approach has its roots in Selberg’s trace formula for a compact surface and its natural expression in such key developments as Duistermaat-Guillemin trace formula [19] and the computation of heat invariants by Gilkey and others (see, for example, [21]). A second and complementary approach is to use techniques of group theory or Lie theory to construct families of manifolds with the same spectrum but distinct geometry. A key development in this approach was the celebrated paper of Sunada [30] in which he showed how to reduce the construction of isospectral manifolds to an exercise in group theory. Robert Brooks made fundamental contributions to both of these lines of research. Today I will discuss two (among many) themes of Brooks’ research: the Sunada technique for constructing examples of manifolds with the same spectrum, and the use of spectral constraints on isoperimetric constants to prove compactness theorems. A number of his recent papers concerned the extension of these ideas and techniques to geometric scattering theory, a development to which I will return later.

2. The Sunada Construction In the seminal paper [30], Sunada showed that pairs of isospectral manifolds with a common Riemannian cover could be constructed by ﬁnding a geometric model for the following group-theoretic situation: G is a ﬁnite group with normal subgroups H1 and H2 which are not conjugate in G but satisfy the Sunada condition: for each G-conjugacy class [g],

(2.1) # ([g] H1) = # ([g] H2) . ∩ ∩ Sunada [30] proved the following theorem:

Theorem 2.1. Suppose that (G, H1,H2) satisfy the condition (2.1), let M be a compact Riemannian manifold, and let φ : π1(M) G be a surjective homo- → 1 morphism. If M1 and M2 are the coverings of M with π1(M1) = φ− (H1) and 1 π1(M2) = φ− (H2) then H1 and H2 are isospectral. A full discussion of Sunada’s theorem may be found in the survey article [4]. Sunada’s original proof uses a finite trace formula which shows that the heat traces exp( tL1) and exp( tL2), where L1 and L2 are the positive Laplacians on M1 − − and M2, are equal, by calculating them in terms of the heat trace of the Laplacian 1 on the common cover M with π1(M) = φ− (G). Observe that M1 and M2 are trivially isometric if H1fand H2 aref actually conjugate in G. Thus, to apply the Sunada method, one seeks nonconjugate subgroups satisfying the condition (2.1), and a geometric model in which one can show that the resulting manifolds M1 and M2 are not isometric. It is well worth noting that Sunada’s proof extends to any elliptic differential operator which is natural with respect to finite coverings. Thus, M1 and M2 have the same spectrum for the Laplacian on differential p-forms as well as on functions. An important explicit example of a Sunada triple is the following (see [10] for examples of isospectral surfaces and [5], [6] for applications to inverse scattering). SPECTRAL GEOMETRY OF EIGENVALUES AND RESONANCES 3

Let G = PSL(3, Z/2) and let H1 and H2 be the subgroups 0 0  ∗ ∗ ∗   ∗  H1 = 0 ,H2 = . ∗ ∗ ∗ ∗ ∗  0    ∗ ∗ ∗ ∗ ∗ 1 t The outer automorphism A (A− ) takes H1 to H2, and also takes elements in 7→ H1to G-conjugate elements. For applications to Riemann surfaces, it is important to note that PSL(3, Z/2) is isomorphic to PSL(2, Z/7). This simple Sunada triple gives rise to a number of important examples. Brooks and Tse used these Sunada triples to construct isospectral surfaces of small genus. In particular, they showed: Theorem 2.2. There are isospectral pairs of Riemann surfaces of genus 4 and 6 having constant curvature, and isospectral pairs of surfaces of genus 3 having variable curvature. For later use, we note that these ﬁnite groups have the following important property: Lemma 2.1. [10] (a) There are elements A and B of G so that A, B, and AB are all of order 7, and A and B generate G. (b) There are elements A0 and B0 of G so that A0 and B0 generate G, and the commutator [A0,B0] is of order 7. Since 7 is prime, the elements A, B, and AB each have one cyclic orbit in the coset spaces G/H1 and G/H2. The Sunada construction has been used to generate many examples of isospectral metrics and potentials. It has also proved robust enough to generate examples of non-compact manifolds with the same scattering data. To describe the scattering data, let us consider one of the several classes of non-compact manifolds for which the Laplacian has a well-deﬁned scattering theory: the set of non-compact manifolds X which are Euclidean outside a compact set, i.e., X is the union of a compact manifold with smooth boundary glued to an exterior domain in Euclidean Rn (see Melrose [25] and [26] for a discussion of manifolds which include this class). The Laplacian ∆X on these manifolds has absolutely continuous spectrum in [0, ) and may have no L2-eigenvalues! The continuous spectrum is spanned by ‘generalized∞ 2 eigenfunctions,’ i.e., solutions of the eigenvalue equation (∆X λ )u = 0 (here λ > 0) having the asymptotic form −

(1 n)/2 u(rω) r − [exp(iλr)f+(ω) + exp( iλr)f (ω) + (1/r)] ∼ − − O n 1 as r , where f ∞(S − ). It can be shown that there is a unique such u → ∞ ± ∈ C given the asymptotic data f , so that the mapping f f+ is a well-deﬁned linear n 1 − − 7→ mapping from ∞(S − ) to itself. This linear mapping is the absolute scattering operator S(λ).C In case X = Rn it is given by

(S0(λ)f)(ω) = cnf( ω). − If, on the other hand, X is a compactly supported perturbation of Rn, the operator 1 S(λ)S0(λ)− is determinant class and the scattering phase σ(λ) is a continuous function with 1 log det(S(λ)S0(λ)− ) = 2πiσ(λ). Two manifolds are isophasal if they have the same scattering phase. 4 PETER PERRY

1 It is not diﬃcult to prove that the resolvent (∆X z)− , viewed as a map from − 0∞(X) to ∞(X), has a meromorphic continuation to the Riemann surface of √z ifC n is odd,C and to the logarithmic plane if n is even (see [26] for discussion and references); the poles of the meromorphically continued resolvent operator are called scattering resonances. Two manifolds are isopolar if they have the same scattering resonances. We can now use the Sunada construction to constructing isoscattering surfaces which are Euclidean outside a compact set as follows [6]. First, let S0 be the sphere with three singular points of order 7. S0 is an orbifold and if π1(S0) is the orbifold fundamental group there is a homomorphism φ : π1(S0) G given by → φ(X1) = A

φ(X2) = B 1 φ(X3) = (AB)− where the Xi correspond to loops going around the singular points of S0. The finite 1 1 coverings with fundamental groups φ− (H1) and φ− (H2) are surfaces of genus 3. If we pick one of the singular points and replace it with a cone of opening angle 2π/7, the covering spaces then become Euclidean at infinity, and the Sunada construction produces a linear mapping which intertwines the resolvents and scattering operators. Non-isometry can be assured by choosing a “bumpy” metric on the orbifold S0. An analogous construction beginning with the torus with one singular point gives a pair of surfaces of genus 4 which are Euclidean at infinity and isophasal. More recently, Brooks and his student Orit Davidovich [5] used Sunada techniques to construct a number of examples of Riemann surfaces of infinite area with the same scattering phase. (See [23] for an analysis of scattering on these manifolds which also includes examples of Sunada pairs of surfaces with the same scattering phase). In addition to finding further examples of isophasal surfaces of infinite area, they prove the following striking result. Recall that a congruence subgroup of PSL(2, Z) is a subgroup Γ which also contains a subgroup Γk of the form a b 1 0 Γk = = mod k . c d ∼ ± 0 1 A congruence surface is a surface of the form Γ H2 where Γ is a congruence subgroup of PSL(2, Z). \

Theorem 2.3. There exist congruence surfaces S1 and S2 which are isoscattering. That this should be so is surprising since the congruences surfaces have a rich structure of eigenvalues embedded in the continuous spectrum, while the congruence subgroups have a rigid algebraic structure.

3. Cheeger’s Constant, Cheeger Finiteness, and Isospectral Manifolds Cheeger’s celebrated inequality [14] relates the first non-trivial eigenvalue of a compact manifold to an isoperimetric constant, the Cheeger constant, defined as follows: let X be a closed Riemannian manifold (compact, no boundary) and let S be a hypersurface dividing X into two parts, A and B. Then area(S) h(X) = inf S min (vol(A), vol(B)) SPECTRAL GEOMETRY OF EIGENVALUES AND RESONANCES 5 where the infimum runs over all such hypersurfaces S. Let λ1 denote the first nontrivial eigenvalue of X. Cheeger proved: Theorem 3.1. Let X be a closed Riemannian manifold. Then

1 2 λ1 h . ≥ 4 This bound is remarkable for its universal character. An important motivating example is the ‘dumbbell’ domain with a long thin rod connecting its two ends, for which λ1 can be made very small. It is natural to ask whether there is an upper bound for the ﬁrst eigenvalue in terms of Cheeger’s constant. With an added hypothesis on curvatures, such an upper bound was proved by Buser [12]: Theorem 3.2. [12] Suppose that X is a smooth Riemannian manifold with Ricc(X) c. ≥ − Then there are constants c1 and c2 depending on c so that 2 λ1 c1h + c2h ≤ It was Bob’s insight to recognize that the relationship between the spectrum and Cheeger’s constant indicated a broader relationship between the spectrum and isoperimetric constants, and his further insight that this relationship could be ex- ploited to make progress on diﬃcult problems in inverse spectral geometry. To appreciate the importance of his insight it is helpful to recall the pathbreaking result of Osgood, Sarnak, and Phillips on isospectral sets of surfaces.

Theorem 3.3. [27] Let Xk be a sequence of closed Riemannian surfaces with { } the same spectrum. Then the Xk are diffeomorphic and there is a subsequence of metrics converging in the ∞ topology to a nondegenerate limiting metric. C Their proof relies on two key spectral invariants: the heat invariants, which yield an infinite sequence of nonlinear Sobolev norms of the Gauss curvature, and the determinant of the Laplacian, which controls possible degenerations of the surface which the heat invariants cannot detect. Recall that the heat invariants are the coefficients of the small-t asymptotic expansion for the trace of the heat operator:

n/2 ∞ j Tr (exp( t∆X )) (4πt)− ajt − ∼ X j=0 and are, by their very deﬁnition, spectral invariants. The ﬁrst few heat invariants are

a0 = vol(X), 1 a1 = Z Scal(X) dg 6 X 1 2 2 2 a2 = Z h5 Scal(X) 2 Ricc(X) + 2 Riem(X) i dg 360 X | | − | | | | where Scal(X), Ricc(X), and Riem(X) are respectively the scalar, Ricci, and Rie- mann curvatures. Note that, by the Gauss-Bonnet Theorem, the heat invariant a1 ﬁxes the genus of X, and therefore its diﬀeomorphism class, if n = 2. 6 PETER PERRY

The determinant of the Laplacian is deﬁned by showing that the spectral zeta function ∞ ζ(s) = λ s X j− j=1 has a meromorphic continuation to the complex plane and is regular at s = 0, so that one may deﬁne

ζ0(0) det(∆X ) = e− . To understand the crucial role of the determinant, note that in the category of constant curvature 1 metrics, all of the heat invariants but a0 and a1 are trivial (this is an easy consequence− the explicit formula for the heat kernel on H2). On the other hand, there are deformations which take the length of a given closed geodesic to zero while preserving the hyperbolic structure. Thus, the heat invariants do not by themselves guarantee any compactness of isospectral metrics. The work of Osgood, Phillips, and Sarnak relies on techniques which are very special to two dimensions. First, the heat invariants fix the diffeomorphism class of the manifold, which is not true in higher dimensions. Secondly, the Laplacian is a conformally covariant operator and so its determinant is well-behaved under conformal deformation, yielding the Polyakov formula. Thus its behavior as a function on the space of metrics may be analyzed in terms of its conformal variation in a conformal class and its behavior across conformal classes–using the well-known parameterization of these conformal classes available from Teichmüllertheory. In three dimensions one confronts a much harder problem. There is no determinant to control degenerations, and there is no control over the diffeomorphism class of isospectral manifolds coming from the heat invariants. First results (see Brooks- Perry-Yang [7] and Chang-Yang [16], [17]) concerned isospectral metrics within a conformal class (so that the ‘inverse problem’ was to determine the conformal factor relative to a reference metric); these analyses used the conformal deformation equation in an essential way and the study of isospectral conformal metrics on S2 ([16] and [17]) involved particularly sophisticated and delicate analysis. There remained the challenge of removing the restriction to conformal classes. Independently, Michael Anderson [1] and Brooks, Perry, and Petersen [8] realized that the key to a more global compactness result was to control the Sobolev isoperimetric constant spectrally. They also realized that one can do so by controlling the length of the shortest closed geodesic. One can then use the Cheeger-Gromov compactness theorem and spectral invariants to show that the isospectral class of such manifolds contains finitely many diffeomorphism types, and extract the necessary information from the heat invariants to conclude that the metrics in a given diffeomorphism class are compact (modulo isometries) in the ∞ topology on metrics. The Duistermaat-Guillemin trace formula [19] (see also ColinC de Verdière[18]and Chazarain [13]) shows that, for manifolds of strictly negative sectional curvature (among other possible sufficient conditions), the spectrum controls the length of the shortest closed geodesic. This follows because the wave trace has singularities at the lengths of closed geodesics, and the curvature assumption guarantees that the coefficient of the leading singularity is nonzero. The Ck-version of the Cheeger-Gromov compactness theorem (see [15] and [22]) states the following compactness result. Let be the set of ∞ Riemannian metrics on a fixed smooth manifold X. M C SPECTRAL GEOMETRY OF EIGENVALUES AND RESONANCES 7

Theorem 3.4. The space of all n-dimensional Riemannian manifolds (X, g) satisfying the bounds j Riem(X) Λ (0 j k), vol(X) v > 0, and ∇ C0 ≤ ≤ ≤ ≥ diam(X, g) D consists of only finitely many diffeomorphism types, and is pre- compact in the≤ k+1,α-topology on for any α < 1. C M As already noted, the heat invariants fix the volume and integrals of the form

j j 2 j 2 aj+2 = ( 1) Z αj Scal(X) + βj Ricc(X) + (lower orders) dg − h ∇ ∇ i X where αj and βj are positive constants with the same sign and the “lower orders” are nonlinear functions of Scal(X), Ricc(X), and their ﬁrst j 1 covariant derivatives. The key to unlocking this information is the “analytic” Sobolev− constant:

S df L2 (3.1) C (X) = inf inf k k : f ∞(X) . a R f + a 2n/(n 2) ∈ C ∈ k kL − As is well-known1, this constant is controlled by the isoperimetric constant area(S) (3.2) C (X) = inf ( ) S 1 1/n min (vol(A), vol(B)) − where the inﬁmum runs over hypersurfaces S that divide X into two parts A and S B. More precisely, a lower bound on CS implies an upper bound on C . Thus, to make the information in the heat invariants usable, one must ﬁnd a way to bound the Sobolev isoperimetric constant spectrally. Anderson [1] proved: Theorem 3.5. The space of compact isospectral 3-manifolds X for which the length of the shortest closed geodesic is bounded below,

(3.3) `X ` > 0 ≥ is compact in the ∞ topology. In particular, there are only finitely many diffeomorphism types ofC isospectral manifolds satisfying (3.3). Anderson’s proof uses the fact that the length of the shortest closed geodesic controls the Sobolev constant. A sufficient condition for (3.3) to hold is that the manifolds have strictly negative sectional curvatures, so that the wave trace controls `X . Cheeger’s inequality and the fact that Cheeger’s constant is a kind of isoperimetric constant strongly suggests that one should be able to use the eigenvalue spectrum to control the Sobolev constant. In particular, a “small” Sobolev isoperimetric constant should imply the existence of a large number of “small” eigenvalues. In [9], this idea is applied in the following way. Fix a smooth 3-manifold with- out boundary and let S be the surface which minimizes the isoperimetric Sobolev constant (the existence and regularity of such a surface is a result of the calculus of variations). One can then use results of Gallot [20] to estimate the volume of ‘tubes’

1Explicitly, one has the estimate

Z ( f)(x) CS f n n 1 X | ∇ | ≥ k k − for all f 1(X), from which it follows that ∈ C0 f np C(p) f n p p k k − ≤ k∇ k for all p with 1 p n, all f 1(X), and a constant C(p) depending on p, a lower bound for ≤ ≤ ∈ C0 CS (X), and vol(X). See, for example, [8], 2 for details. § 8 PETER PERRY constructed about this minimizing hypersurface and build test functions for the low- est k eigenvalues of the Laplacians which are localized in ‘slices’ of X bounded by hypersurfaces at diﬀerent ﬁxed distances from S. One obtains the following compactness theorem [9]. To state it, recall that in three dimensions the Riemann curvature tensor is determined by the scalar curvature and the Ricci tensor, and there is an orthogonal decomposition (in the space of symmetric two-tensors)

Riem(X) = Scal(X) Ricct(X). ⊕ where Ricct is the traceless Ricci tensor. The reduced Riemann tensor, Riemr(X), is the tensor 1 Scal(X) Z Scal(X) dg 1 Ricct(X). − vol(X) X · ⊕

Notice that Riemr(X) vanishes for a metric of constant curvature. Theorem 3.6. For n = 2 and n = 3 there are constants Q(n) and K(n, k) so that if

λk > Q(n) Z Scal(X) dg X and X Riemr(X) λk > K(k)R | | vol(X) p then the set of all manifolds isospectral to (X, g) is compact in the ∞ topology on metrics. C The important point of this theorem is that there is no restriction on conformal classes. Previous results on compactness of isospectral sets of 3-manifolds (some of which required very deep analysis) involved compactness within a conformal class. It is not diﬃcult to compute the asymptotics of K(k) by examining the proof. One ﬁnds that, as k , K(k) k2 whereas Weyl’s law for eigenvalues of Laplacians in three dimensions→ ∞ suggests∼ that K(k) k2/3 would be optimal. The behavior of K(k) is an artifact of the “one-dimensional”∼ nature of the proof. It would be very interesting to see if one could control the Sobolev isoperimetric constant using test functions for eigenvalues constructed in balls rather than slices. The key would be to use the Sobolev constant and Lp-estimates on volumes of balls to control the norms of the test functions in terms of CS.

4. Conclusion I hope that these few examples of Bob’s work give a sense of his inventiveness and the importance of his results. I would like to close and address another of Bob’s contributions for which he will be much missed and gratefully remembered. Writing in a 1997 review paper on “Inverse Spectral Geometry” [3], Brooks wrote: Finally, I would like to take this opportunity to thank my colleagues, including those present at the conference and those who could not attend, for making spectral geometry a truly pleasant and exciting area in which to work. While it is my hope that the picture presented here will induce some to join this area of research, I think that a far greater inducement would be the opportunity SPECTRAL GEOMETRY OF EIGENVALUES AND RESONANCES 9

to get to know, and to be a part of, the community which occupies itself with these questions. No one exemplified the spirit of that community more than Robert Brooks. In September, longtime colleague and friends Carolyn Gordon and David Webb wrote the following remembrance: Perhaps Brooks’ most enduring mathematical legacy, however, is his influence upon his students and colleagues. He had a vision of mathematics as a cooperative, collaborative endeavor, and his generosity in sharing his ideas and in collaborating widely has had a profound impact upon an entire generation of spectral geometers. I hope, in turn, that the problems presented here will induce some to join this area of research, and that our community will continue to uphold the values of cooperative, collaborative endeavor which were so finely displayed in Bob’s life and work.

References

[1] Anderson, Michael T. Remarks on the compactness of isospectral sets in low dimensions. Duke Math. J. 63 (1991), no. 3, 699–711. [2] Brooks, Robert. Constructing isospectral manifolds. Am. Math. Monthly 95 (1988), no. 9, 823–839. [3] Brooks, Robert. Inverse spectral geometry. Progress in inverse spectral geometry, 115–132, Trends Math., Birkhäuser,Basel, 1997. [4] Brooks, Robert. The Sunada method. Tel Aviv Topology Conference: Rothenberg Festschrift (1998), 25–35, Contemp. Math., 231, Amer. Math. Soc., Providence, RI, 1999. [5] Brooks, Robert; Davidovich, Orit. To appear in J. Geom. Analysis. [6] Brooks, Robert; Perry, Peter A. Isophasal scattering manifolds in two dimensions. Comm. Math. Phys. 223 (2001), no. 3, 465–474. [7] Brooks, Robert; Perry, Peter A.; Yang, Paul C. Isospectral sets of conformally equivalent metrics. Duke Math. J. 58 (1989), 131–150. [8] Brooks, Robert; Perry, Peter A.; Petersen V, Peter. Compactness and finiteness theorems for isospectral manifolds. J. Reine. Angew. Math. 426 (1992), 67–89. [9] Brooks, Robert; Perry, Peter A.; Petersen V, Peter. Spectral geometry in dimension 3. Acta. Math. 173 (1994), 283–305. [10] Brooks, Robert; Tse, Richard. Isospectral surfaces of small genus. Nagoya Math. J. 107 (1987), 13–24. [11] Brooks, Robert; Tse, Richard. Correction to: “Isospectral surfaces of small genus”. Nagoya Math. J. 117 (1990), 227. [12] A note on the isoperimetric constant. Ann. Sci. Ec.´ Norm. Sup. 15 (1982), 213–230. [13] Chazarain, J. Formule de Poisson pour les variétésRiemanniennes. Invent. Math. 24 (1974), 65–82. [14] Cheeger, Jeff. A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. [15] Cheeger, Jeff. Finiteness theorems for Riemannian manifolds. Amer. J. Math. 92 (1970), 61–74. [16] Chang, S.-Y. A.; Yang, Paul C. The conformal deformation equation and isospectral sets of conformal metrics. Recent developments in geometry (Los Angeles, California, 1987), 165–178, Contemp. Math. 101, American Math. Soc., Providence, Rhode Island, 1989. [17] Chang, S.-Y. A.; Yang, Paul C. Isospectral conformal metrics on 3-manifolds. J. Amer. Math. Soc. 3 (1990), 117–145. [18] Colin de Verdière,Yves. Spectre du Laplacien et longeurs des geodesiques periodiques, II. Compositio. Math. 27 (1973), 159–184. [19] Duistermaat, J. J.; Guillemin, V. W. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), no. 1, 39–79. 10 PETER PERRY

[20] Gallot, Sylvestre. Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque 157-158 (1988), 54–63. [21] Gilkey, Peter. Leading terms in the asymptotics of the heat equation. Geometry of Random Motion (Ithaca, N.Y., 1987), 79–85, Contemp. Math. 73, Amer. Math. Soc., Providence, R. I., 1988. [22] Gromov, Mikhail. Structures métriques pour les variétés Riemanniennes. Cedic- Fernand/Nathan, Paris, 1981. [23] L. Guillopé,M. Zworski. Scattering asymptotics for Riemann surfaces. Ann. of Math. 145 (1997), 597–660. [24] Melrose, Richard. Isospectral drumheads are ∞. Preprint, Math. Sci. Res. Inst. Pub. [25] Melrose, Richard. Spectral and scattering theoryC for the Laplacian on asymptotically Eu- clidean spaces. Spectral and Scattering Theory (Sanda, 1992), 85–130, Lecture Notes in Pure andd Appl. Math. 161, Dekker, New York, 1994. [26] Melrose, Richard. Geometric Scattering Theory. Stanford Lectures. Cambridge University Press, Cambridge, 1995. [27] Osgood, Brad; Phillips, Ralph; Sarnak, Peter. Extremals of determinants of Laplacians. J. Funct. Anal. 80 (1988), 148–211. [28] Osgood, Brad; Phillips, Ralph; Sarnak, Peter. Compact isospectral sets of surfaces. J. Funct. Anal. 80 (1988), 212–234. [29] A. Selberg. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), , 47–87. [30] Sunada, Toshikazu. Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 121 (1985), no. 1, 169–186. [31] Zelditch, Steven. Kuznecov sum formulae and Szegölimit formulas on manifolds. Comm. P. D. E. 17 (1992), 221–260.

Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506– 0027 E-mail address: [email protected] URL: http://www.uky.edu/~perry