June 10, 1998

NORMAL FORMS AND INVERSE SPECTRAL THEORY

STEVE ZELDITCH

Introduction

Our purpose in this report is to survey some relatively new results on the classical inverse spectral problem: can one determine a compact Riemannian manifold (M, g) (with or without boundary) from the spectrum Sp(Δ_{g}) of its Laplacian Δ_{g}? More precisely, is the correspondence (M, g) → Sp(Δ_{g}) injective as (M, g) ranges over a class M of compact Riemannian manifolds? It is well-known that the answer is ‘no’ if M is the entire class of (M, g) (see [Go]), and there are few positive inverse results for any reasonably broad class M. Recently, however, a new approach to the inverse spectral problem has developed which offers hope of obtaining positive results for at least some special classes of metrics or domains. To give it a name for this article, we will call it the ‘strategy of normal forms’. The basic idea (due essentially to V.Guillemin, but with antecedents in the work of Colin de Verdiere [CV.1], Melrose [M][M.M] and others) is that for generic (M, g), Sp(Δ_{g}) should determine the (quantum) Birkhoff normal form of Δ_{g} around each closed geodesic γ. The inverse spectral problem then reduces to the problem of determining the metric from the normal forms. This is still a difficult problem but for certain classes M one expects that it can be solved. In particular, we will describe a special class M of metrics, namely the analytic simple surfaces of revolution, where the strategy has succeeded to a reasonable degree [Z.3]. This class of surfaces is in some ways analogous to (but much simpler than) the class of real analytic plane domains, where the connection between Laplace spectrum and Birkhoff normal forms was first made in [CV.1][M.M]. We will not discuss domains here but it is hoped that the strategy of normal forms extends to them and we sometimes include them in our general remarks.

In view of the considerable number of counterexamples to the inverse spectral problem, and of the wide gap between the many negative and the few positive results, it may help orient the reader if we begin by discussing the restrictions we will place on the metrics (or domains). First, we will always assume that the length spectrum Lsp(M, g) is simple in the sense that the length functional on the loop space takes on distinct values on distinct components of its critical point set. E.g., in the typical case where all geodesics are non-degenerate, simplicity means that (up to time reversal) there is just one closed geodesic of each given length in Lsp(M, g). This is a generic condition which is necessary to separate out terms in the trace formula ( §1). As far as the author knows, all known examples of non-isometric isospectral pairs have multiple length spectra, so this condition appears to eliminate the known obstacles to inverse spectral results. Secondly, we will often assume that the metrics (or domains) are real analytic. This is because we do not know how to combine information from distinct closed geodesics, so we must determine the metric from information on its germ at just one closed geodesic. Third, it is natural at this stage to restrict to classes M of metrics (or domains) where the unknown has just one functional degree of freedom (i.e. is a function of one variable). Our experience with surfaces of revolution (and symmetric plane domains) seems to indicate that the spectral invariants coming from just one closed geodesic just suffices to determine this amount of the unknown object.

Partially supported by NSF grant #DMS-9703775.

We now divide up our discussion of results according to the relevant class of metrics.

0.1. Compact Riemannian manifolds with non-degenerate closed geodesics. A closed geodesic γ will be called ‘non-degenerate’ if 1 is not an eigenvalue of its linear Poincare map P_{γ}, and ‘strongly non-degenerate’ if the eigenvalues of P_{γ} are independent (with π) over the rational numbers (§1). The main result for this class of Riemannian manifolds is the following:

Theorem 1. ([G.1-2]; see also [Z.1-2]) Suppose that (M, g) is a compact Riemannian manifold without boundary, and suppose that γ is a non-degenerate closed geodesic whose length L_{γ} is of multiplicity one in Lsp(M, g). Then the quantum Birkhoff normal form of \sqrt{Δ}at γ is a spectral invariant.

The definition of ‘quantum Birkhoff normal form’ will be given in §2. Roughly speaking, it is the expression of \sqrt{Δ} as a function F (D_{s}, Î_{1}, …, Î_{n}) of (micro-)locally defined transversal ‘action operators’ Î_{j} and of the tangential derivative D_{s} = \frac{1}{i} \frac{∂}{∂s} along γ. In the elliptic case, Î_{j} = \frac{1}{2}(D_{y_{j}}^{2} + y_{j}^{2}). Since the classical Birkhoff normal form of the symbol

H(x, ξ) = \sqrt{∑_{ij} g^{ij}(x)ξ_{i}ξ_{j}} is the principal symbol of the quantum normal form, it follows that the classical BNF of the metric Hamiltonian H at γ is also a spectral invariant.

The question then arises to what degree a metric (or domain) is determined by the normal form of \sqrt{Δ}at one (or all) closed geodesic(s)? In particular, to what degree does the classical Birhoff normal form of the geodesic flow near γ determine the metric? The normal form coefficients were determined in [Z.1,2] and are very messy metric invariants even in dimension two (§2). Hence one looks for simple model examples where the normal forms have simple geometric or ‘quantum’ interpretations.

0.2. Simple analytic surfaces of revolution. The simplest situation is that of real analytic surfaces of revolution of ‘simple type’. Roughly, ‘simple type’ means that there is just one circle (geodesic) of points of critical distance to the axis of rotation, which we will call the equator, and that its associated Poincare map is of twist type (§3). Such surfaces were studied in [CV.2] where it was shown that there exist global action operators Î_{1}, Î_{2} and a (polyhomogeneous) function F such that \sqrt{Δ}= F (Î_{1}, Î_{2}).

The inverse result above applies to the normal form of \sqrt{Δ} at the equator, but it is simpler to take advantage of the complete integrability of the geodesic flow (and wave group) and prove an inverse normal form result in the integrable setting:

Theorem 2. ([Z.3]) Let (S^{2}, g) be an analytic simple surface of revolution with simple length spectrum. Then the normal form F (I_{1}, I_{2}) is a spectral invariant.

In this example, both the classical and quantum Birkhoff normal forms have simple expressions in terms of the metric. We have:

Theorem 3. ([Z.3]) Suppose that g_{1}, g_{2} are two real analytic metrics on S^{2} such that (S^{2}, g_{i}) are simple surfaces of revolution with simple length spectra. Then Sp(Δ_{g_{1}}) = Sp(Δ_{g_{2}}) implies g_{1} = g_{2}.

Here, equality means isometry of metrics. An obvious shortcoming of this result is that both metrics were assumed to belong to R. One would like to know if it suffices to assume just that g_{1} ∈ R. In other words, is a metric g ∈ R spectrally determined? At the present time, the only metric on S^{2} which is known to be spectrally determined is the round metric g_{can}, so we do not expect to fully answer the question for general g ∈ R. However, in a work in progress with G.Forni, we prove some partial results in the direction that g_{2} has completely integrable geodesic flow. At the present time, our result is:

Theorem 4. Let g_{1} ∈ R^{∗} and suppose that g_{2} is a real analytic metric on S^{2} with Sp(Δ_{g_{1}}) = Sp(Δ_{g_{2}}). Then:

(a) Up to time reversal, g_{2} has precisely one elliptic non-degenerate closed geodesic;

(b) The Poincare map for this orbit is completely integrable in the C^{0} sense.

In other words, the Poincare map (and geodesic flow near the orbit) preserve a continuous foliation by tori, at least near the ‘equator’. Even if one could improve this result to show that the geodesic flow of g_{2} was completely integrable, it would still not be enough to show that g_{2} ∈ R^{∗}; indeed, at this time, the completely integrable metrics on S^{2} have not been classified.

1. Preliminaries

We begin with some definitions.

1.1. Jacobi fields and Poincare map for non-degenerate closed geodesics. The Jacobi equation along a closed geodesic γ(t) is the equation

\frac{D^{2}}{Dt^{2}} J + R(T, J)T = 0

where \frac{D}{dt} denotes covariant differentiation, T = γ′(t) and R(·, ·) is the curvature tensor. A solution is called a Jacobi field, and a solution J⊥T is called an orthogonal Jacobi field. We let J_{γ}^{⊥} ⊗ C denote the space of complex normal Jacobi fields along γ. It is a symplectic vector space of (complex) dimension 2n (n = dim M − 1) with respect to the Wronskian

ω(X, Y) = g(X, \frac{D}{ds}Y) − g(\frac{D}{ds}X, Y).

The linear Poincaré map P_{γ} is then the linear symplectic map on J_{γ}^{⊥} ⊗ C defined by P_{γ}Y (t) = Y (t + L_{γ}).

A closed geodesic is called non-degenerate if det(I −P_{γ}) ≠ 0. Since P_{γ} is symplectic, its eigenvalues come in 4-tuples (inverses and complex conjugates, which may coincide), i.e. the spectrum of P_{γ}) has the form: Sp(P_{γ}) = {e^{±μ_{j}±iα_{j}}, j = 1, ..., n}. P_{γ} is called elliptic if all of its eigenvalues have modulus one, in which case Sp(P_{γ}) = {e^{±iα_{j}}, j = 1, ..., n}.

It is (real) hyperbolic if the eigenvalues come in pairs Sp(P_{γ}) = {e^{±μ_{j}}, j = 1, ..., n}. For the sake of simplicity we will generally assume that P_{γ} is elliptic.

The Jacobi eigenvectors of P_{γ} will be denoted \{Y_{j}, \overline{Y_{j}}, j = 1, ..., n\}. In the elliptic case they may be normalized to satisfy:

P_{γ}Y_{j} = e^{iα_{j}}Y_{j} P_{γ} \overline{Y}_{j} = e^{−iα_{j}} \overline{Y}_{j} ω(Y_{j}, \overline{Y}_{k}) = δ_{jk}.

We introduce a fixed parallel normal frame e(s) := (e_{1}(s), ..., e_{n}(s)) along γ and write the eigenvectors in the form Y_{j}(s) = ∑_{k=1}^{n} y_{jk}(s)e_{k}(s).

1.2. Wave trace invariants in the non-degenerate case on a manifold without boundary. The wave group of a compact Riemannian manifold is the unitary group U(t) = e^{it\sqrt{Δ}}. As is well-known, it has a distribution trace TrU(t) which is a Lagrangean distribution on IR with singularities at lengths of closed geodesics [D.G]. In the case of a ‘bumpy’ riemannian manifold (M, g), i.e. one for which all closed geodesics are non-degenerate, the wave trace has the singularity expansion

Tr U(t) = e_{0}(t) + ∑_{L∈Lsp(M,g)} e_{L}(t)  (1)

with

e_{0}(t) = a_{0,−n}(t + iO)^{−n} + a_{0,−n+1}(t + iO)^{−n+1} + ⋯  (2)

\begin{aligned}e_{L}(t) & = & a_{L,−1}(t − L + iO)^{−1} + a_{L,0} log(t − (L + iO)) \\ & + & a_{L,1}(t − L + iO) log(t − (L + iO)) + ⋯ \\ a_{L,−k} & = & ∑_{γ:L_{γ}=L} a_{γ,−k}, \end{aligned}  (3)

where ⋯ refers to homogeneous terms of ever higher integral degrees ([DG]). The coefficients a_{L,−k} and a_{γ,−k} are known as wave invariants. The coefficients a_{0,k} at t = 0 are essentially the same as the heat coefficients, i.e. the coefficients of the small t expansion of Tre^{−tΔ} and are well-known to be given by integrals over M of curvature polynomials. The question arises of similarly characterizing the wave trace invariants associated to closed geodesics.

The principal wave invariant at t = L_{γ} was determined in [D.G]. It is given by

a_{γ,−1} = exp^{\frac{iπmγ}{4}} L_{γ}^{#}|det(I − P_{γ})|^{−1/2},

where L_{γ}^{#} is the primitive period (once around), where m_{γ} is the Maslov (= Morse) index, and (as above) where P_{γ} is the linear Poincaré map.

The following describes the wave trace invariants associated to a non-degenerate closed geodesic. Undefined terminology is reviewed below.

Theorem 5. Let γ be a strongly non-degenerate closed geodesic. Then a_{γk} = ∫_{γ} I_{γ;k}(s; g)ds where:

(i) I_{γ;k}(s; g) is a homogeneous Fermi–Jacobi–Floquet polynomial of weight −k − 1 in the data {y_{ij}, ẏ_{ij}, D_{s,y}^{m}g} with m = (m_{1}, …, m_{n+1}) satisfying |m| ≤ 2k + 4 ;

(ii) The degree of I_{γ;k} in the Jacobi field components is at most 6k + 6; (iii) At most 2k + 1 indefinite integrations over γ occur in I_{γ;k};

(iv) The degree of I_{γ;k} in the Floquet invariants β_{j} is at most k + 2.

Let us define the term ‘Fermi-Floquet-Jacobi polynomial’. First, we write the metric coefficients g_{ij} relative to Fermi normal coordinates (s, y) along γ. The vector fields \frac{∂}{∂s}, \frac{∂}{∂y_{j}} and their real linear combinations will be referred to as Fermi normal vector fields along γ and contractions of tensor products of the ∇^{m}R’s with these vector fields will be referred to as Fermi curvature polynomials. The m-th jet of g along γ is denoted by j_{γ}^{m}g, the curvature tensor by R and its covariant derivatives by ∇^{m}R. Such polynomials will be called invariant if they are invariant under the action of O(n) in the normal spaces. Invariant contractions against \frac{∂}{∂s} and against the Jacobi eigenfields Y_{j}, \overline{Y}_{j}, with coefficients given by invariant polynomials in the components y_{jk}, are called Fermi–Jacobi polynomials. We will also use this term for functions on γ given by repeated indefinite integrals over γ of such FJ polynomials. Finally, FJ polynomials whose coefficients are given by polynomials in the Floquet invariants β_{j} = (1−e^{iα_{j}})^{−1} are Fermi–Jacobi–Floquet polynomials.

The role of the Floquet invariants should be particularly emphasized. The fact that the wave invariants are polynomials in the β_{j}’s is a crucial ingredient in Guillemin’s (and subsequent) proof(s) of the inverse result.

The ‘weights’ referred to above describe how the various objects scale under g → ϵ^{2}g. Thus, the variables g_{ij}, D_{s,y}^{m}g_{ij} (with m := (m_{1}, …, m_{n+1})), L := L_{γ}, α_{j}, y_{ij}, ẏ_{ij} have the following weights: wgt(D_{s,y}^{m}g_{ij}) = −|m|, wgt(L) = 1, wgt(α_{j}) = 0, wgt(y_{ij}) = \frac{1}{2}, wgt(ẏ_{ij}) = −\frac{1}{2}. A polynomial in this data is homogeneous of weight s if all its monomials have weight s under this scaling. Finally, τ denotes the scalar curvature, τ_{ν} its unit normal derivative, τ_{νν} the Hessian Hess(τ)(ν, ν); Y denotes the unique normalized Jacobi eigenfield, Ẏ its time-derivative and δ_{j0} the Kronecker symbol (1 if j = 0 and otherwise 0).

For instance, in dimension 2 (where there is only one Floquet invariant β) the residual wave invariant a_{γ0} is given by:

a_{γ0} = \frac{a_{γ,−1}}{L^{#}} [B_{γ0;4}(2β^{2} − β − \frac{3}{4}) + B_{γ0;0}]  (4)

where:

(a) a_{γ,−1} is the principal wave invariant;

(b) L^{#} is the primitive length of γ; σ is its Morse index; P_{γ} is its Poincaré map;

(c) B_{γ0;j} has the form:

B_{γ0;j} = \frac{1}{L^{#}} ∫_{o}^{L^{#}}[a |Ẏ |^{4} + b_{1} τ|Ẏ · Y |^{2} + b_{2} τRe (Ȳ Ẏ)^{2} + c τ^{2}|Y |^{4} + d τ_{νν}|Y |^{4} + e δ_{j0}τ]ds

+\frac{1}{L^{#}} ∑_{0≤m,n≤3;m+n=3} C_{1;mn} \frac{sin((n − m)α)}{|(1 − e^{i(m−n)α})|^{2}} |∫_{o}^{L^{#}} τ_{ν}(s)Ȳ^{m} · Y^{n}(s)ds|^{2}

+\frac{1}{L^{#}} ∑_{0≤m,n≤3;m+n=3} C_{2;mn} Im \{∫_{o}^{L^{#}} τ_{ν}(s)Ȳ^{m} · Y^{n}(s) [∫_{o}^{s} τ_{ν}(t)Ȳ^{n} · Y^{m}(t)dt] ds\}

for various universal (computable) coefficients.

The wave invariants (and normal form coefficients) are obviously very messy. The beauty of the normal forms strategy is that it organizes these coefficients into a potentially meaningful invariant, namely the classical (and quantum) Birkhoff normal form.

1.3. Wave trace expansion on a simple surface of revolution. The wave trace expansion on a simple surface of revolution is quite different since the geodesic flow is completely integrable and closed geodesics (other than the equator) fill out tori in the cosphere bundle.

To explain this, let us be more precise about the definition of ‘simple surface of revolution.’ First, we will assume that there is an effective action of S^{1} by isometries of (S^{2}, g). The two fixed points will be denoted N, S and (r, θ) will denote geodesic polar coordinates centered at N, with θ = 0 some fixed meridian γ_{M} from N to S. The metric may then be written in the form g = dr^{2} + a(r)^{2}dθ^{2} where a : [0, L] → IR^{+} is defined by a(r) = \frac{1}{2π} |S_{r}(N)|, with |S_{r}(N)| the length of the distance circle of radius r centered at N. We define the class R of simple analytic surfaces of revolution as follows:

Definition 6. R is the moduli space of metrics of revolution (S^{2}, g) with the properties:

(i) g (equivalently a) is real analytic; (ii) a has precisely one non-degenerate critical point r_{o} ∈ (0, L), with a′′(r_{o}) < 0, corresponding to an ‘equatorial geodesic’ γ_{E}; (iii) the (non-linear) Poincare map P_{γ_{E}} for γ_{E} is of twist type. We denote by R^{∗} ⊂ R the subset of metrics with ‘simple length spectra’ in the sense above.

From a geometric (or dynamical) point of view, the principal virtue of metrics in R is described by the following:

Proposition 7. Suppose that g ∈ R. Then the Hamiltonian |ξ|_{g} := \sqrt{∑ g^{ij}ξ_{i}ξ_{j}} on T^{∗}S^{2} is completely integrable and possesses global real analytic action-angle variables.

By an action variable we mean a homogeneous function on T^{∗}S −0 whose Hamiltonian flow is 2π-periodic. There is an obvious action variable given by the Clairaut integral p_{θ}(v) := ⟨v, \frac{∂}{∂θ} ⟩. Since the Poisson bracket \{p_{θ}, |ξ|_{g}\} = 0, the geodesics are constrained to lie on the level sets of p_{θ}. With the assumption on a, the level sets are compact and the only critical level is that of the equatorial geodesics γ_{E}^{±} ⊂ S_{g}^{∗}S^{2} (traversed with either orientation). The other level sets consist of two-dimensional tori.

The second action variable is less familiar but is obtained in a standard way by integrating the action form over a homology basis of the invariant tori. The formula for it is given by (cf. [CV.2, §6])

I_{2}(I_{1}, E) = \frac{1}{π} ∫_{r_{−}}^{r_{+}} \sqrt{E^{2} − \frac{I_{1}^{2}}{a(r)^{2}}} dr + |I_{1}|

where r_{±} are the extremal values of r on the annulus π(F_{I_{1},E}) (with π : S_{g}^{∗}S^{2} → S^{2} the standard projection and F_{I_{1},E} the torus given by |ξ| = E, p_{θ} = I_{1}).

The pair I := (I_{1}, I_{2}) generate a global Hamiltonian torus (S^{1} ×S^{1})-action commuting with the geodesic flow. The singular set of I equals Z := {I_{2} = ±p_{θ}}, corresponding to the equatorial geodesics. The map

I : T^{∗}S^{2} − Z → Γ := {(x, y) ∈ IR × IR^{+} : |x| < y}

is a trivial torus fibration. Henceforth we write T_{I} for the torus I^{−1}(I) with I ∈ Γ. Using the product structure we may equip each T_{I} with angle variables ϕ_{j} which are symplectically dual to the action variables.

The metric Hamiltonian may be written in the form H(I_{1}, I_{2}). The vector ω_{I} = ∇_{I}H is called the frequency vector. The equations of motion for a geodesic on T_{I} are given by İ_{j} = 0, ϕ_{j} = ω_{j}(I).

We now come to the definition of length spectrum and simple length spectrum for a completely integrable geodesic flow.

Definition 8. (a) A torus T_{I} is a periodic torus if all the geodesics on it are closed.

(b) The period L of the periodic torus is then the common period of its closed geodesics.

(c) The length spectrum L of the completely integrable system is the set of these lengths.

(d) The completely integrable system has a simple length spectrum if there exist a unique periodic torus (up to time reversal) of each length L ∈ L.

Consider now the wave trace formula for a metric on a general Riemannian manifold of dimension n with completely integrable geodesic flow. The dimension of each periodic torus T of period L equals dimT = n and the wave trace has the form:

T rU(t) = e_{o}(t) + ∑_{T} e_{T} (t)

where the sum runs over the periodic tori in S^{∗}M and where

e_{T} (t) = a_{T ;− \frac{n+1}{2}} (t − L + i0)^{− \frac{n+1}{2}} + a_{T ;− \frac{n+1}{2} +1}(t − L + i0)^{− \frac{n+1}{2} +1} + ….

More precisely, it takes this form if n is even; if n is odd, the positive powers of (t−L+i0) should be multiplied by log(t − L + i0). Thus, if dimM = 2, the wave trace expansion at a torus T has the form

e_{T} (t) = a_{T ,− \frac{3}{2}} (t − L + i0)^{− \frac{3}{2}} + a_{T ,− \frac{1}{2}} (t − L + i0)^{− \frac{1}{2}} + …

while if dimM = 3, the wave trace expansion at T has the form

e_{T} (t) = a_{T ,−2}(t − L + i0)^{−2} + a_{T ,−1}(t − L + i0)^{−1} + a_{T ,0}log(t − L + i0) + ….

2. Wave invariants as non-commutative residues

In connecting the wave invariants to the normal form, it will prove useful to interpret the wave invariants as non-commutative residues of the wave group and its time derivatives. We digress to recall the definitions and basic results.

The non-commutative residue of a Fourier Integral operator is an extension of the well-known non-commutative residue of a pseudodifferential operator A on a compact manifold M , defined by

res(A) = 2 Res_{s=0} ζ(s, A)

where

ζ(s, A) = Tr AΔ^{−s/2} (Re s >> 0) .

Let A be a Fourier Integral operator in I^{m}(M ×M, Λ) for some homogeneous canonical relation Λ ⊂ T^{∗}(M × M)\0 and m ∈ ZZ and let diag(X × X) denote the diagonal in X × X. We have:

Theorem 9. Let A ∈ I^{m}(M × M, Λ), where Λ is a homogeneous canonical relation intersecting diag(T^{∗}M × T^{∗}M) cleanly. Then ζ(s, A, P) := Tr AP^{−s} is analytic for Re s large for any P ∈ Ψ^{1} positive elliptic. It admits a meromorphic extension to C with simple poles only among the points s = m+1+ \frac{e_{o}−1}{2} −j, with e_{o} = dimΛ∩diag(S^{∗}M ×S^{∗}M), and with j = 0, 1, 2, ... The residues are given by local invariants of the germ of the complete symbol of a and Λ near the intersection with the diagonal.

In the case of the wave group we then have:

Theorem 10. Let Δ be the Laplacian for a metric g on M, and let U_{t} = exp it\sqrt{Δ} be the wave group. Let the eigenvalues of \sqrt{Δ} be denoted 0 = λ_{o} < λ_{1} < .... Then the zeta function

ζ(s, t) := ∑_{j=0}^{∞} e^{itλ_{j}}λ_{j}^{−s}

has a meromorphic continuation to C with poles only among s = 1 + 1/2(dim SFix(G^{t}) − 1) − j.

The non-commutative residue of the Fourier integral operator A is then defined by:

Definition 11.

res(A) := Res_{s=0} ζ(s, A)

The residue res(A) has the properties:

- it is independent of the choice of Δ;

- if either A or B is associated to a local canonical graph, then res(AB) = res(BA)

- there is a local formula for res(A).

Corollary 12. In the case of a non-degenerate closed geodesic we have:

a_{L,k} = res(D_{t}^{k}U(t)|_{t=L})

(D_{t} = \frac{1}{i} \frac{d}{dt}).

In the case of a completely integrable system, we recall, the dimension of each periodic torus T of period L equals e_{o} = dimT = n. Hence we have:

Corollary 13.

∑_{±} a_{±T ,−( \frac{n+1}{2} )+k} = res(\sqrt{Δ}^{− \frac{n+1}{2} +k}U(t)|_{t=L})

Thus, in the case of simple surface of revolution the first wave trace invariants at a periodic torus T have the form:

∑_{±} a_{±T ,− \frac{3}{2}} = res\sqrt{Δ}^{− \frac{3}{2}} e^{iL\sqrt{Δ}}, ∑_{±} a_{±T ,− \frac{1}{2}} = res\sqrt{Δ}^{− \frac{1}{2}} e^{iL\sqrt{Δ}}, …

The wave invariants for a closed geodesic γ (or periodic torus T_{L}) are exactly the same as for its time reversal, hence the same residue formulae also give the individual wave invariants.

3. Normal forms

In this section we review the normal forms of \sqrt{Δ} around closed geodesics in the three rather different cases. For detailed discussions see [G.1-2], [Z.1-4].

3.1. Quantum Birkhoff normal forms: non-degnerate case. Roughly speaking, to put \sqrt{Δ} into microlocal normal form around γ is to conjugate it into a maximal abelian subalgebra A of pseudodifferential operators on the normal bundle N_{γ} ∼ S^{1} × IR^{n} of γ. The appropriate algebra A is determined by the spectrum of P_{γ}. In the tangential direction, one takes D_{s} where D_{x} = \frac{1}{i}∂/∂x. In the transverse directions IR^{n} to γ, one chooses a generating set of action operators Î_{j}, that is, quadratic Hamiltonians in the transverse variables y_{j}, D_{y_{j}}. Here, n = dim M − 1, and the coordinates (s, y_{j}) are the Fermi normal coordinates around γ.

In the elliptic directions, Î_{j} will be a Harmonic oscillator α_{j}(D_{y_{j}}^{2} + y_{j}^{2}). In the real hyperbolic directions, Î_{j} = μ_{j}(y_{j}D_{y_{j}} + D_{y_{j}}y_{j}). In the complex hyperbolic directions, Î_{j} involves two y-variables y_{j}, y_{j+1} and has the form μ_{j}(y_{j}D_{y_{j}} +y_{j+1}D_{y_{j+1}})+α_{j}(y_{j+1}D_{y_{j}} − y_{j}D_{y_{j+1}}).

The normal form theorem in the non-degenerate case is the following:

Theorem 14. Assume that γ is a non-degenerate closed geodesic and that its Floquet exponents {α_{j}, μ_{j}}, together with π are independent over Q. Then there exists a microlocally elliptic Fourier Integral operator W from a conic neighborhood V of IR^{+}γ in T^{∗}N_{γ} − 0 to a conic neighborhood of T_{+}^{∗}S^{1} in T^{∗}(S^{1} × IR^{n}) such that:

W \sqrt{Δ_{ψ}}W^{−1} :≡ D ≡ D_{s}+\frac{1}{L}H_{α}+\frac{p̃_{1}(Î_{1}, …, Î_{n}, L)}{D_{s}}+\frac{p̃_{2}(Î_{1}, …, Î_{n}, L)}{D_{s}^{2}}+⋯+\frac{p̃_{k+1}(Î_{1}, …, Î_{n}, L)}{D_{s}^{k+1}}+…

where the numerators p̃_{j}(Î_{1}, ..., Î_{n}, L) are polynomials of degree j + 1 in the variables Î_{1}, ..., Î_{n}, where W^{−1} denotes a microlocal inverse to W in V. The k-th remainder term lies in the space ⨁_{j=0}^{k+2} O_{2(k+2−j)}Ψ^{1−j}.

Here, O_{m}Ψ^{k} is the space of transverse pseudodifferential operators of order k (in the y-variables) which vanish to order m at y = 0. Thus the error term is bigraded by pseudodifferential order and by order of vanishing. The remainder is small if in some combination it has a low pseudodifferential order or a high vanishing order at γ. This is a useful remainder estimate since a given wave invariant a_{γ} only involves a finite part of the jet of the metric and only a finite part of the complete symbol of \sqrt{Δ}.

3.2. Normal form for \sqrt{Δ} on a simple surface of revolution. As mentioned in the introduction, \sqrt{Δ}on a simple surface of revolution has a global Birhoff normal form. The reason for this is that the wave group is integrable on the quantum level and in fact commutes with a Fourier integral torus action. Let us first define this notion:

Definition 15. The wave group e^{it\sqrt{Δ}} of a compact, Riemannian n-manifold (M, g) is quantum torus integrable if there exists a unitary Fourier-Integral representation

τ̂ : T^{n} → U(L^{2}(M)), τ̂_{(t_{1},...,t_{n})} = e^{i(t_{1}Î_{1}+...t_{n}Î_{n})}

of the n-torus and a symbol Ĥ ∈ S^{1}(IR^{n} − 0) such that \sqrt{Δ} = Ĥ(I_{1}, …, I_{n}).

The generators Î_{j} are first order commuting elliptic action operators with the property that e^{2πiÎ_{j}} = C_{j}Id for some constant C_{j} of modulus one. Hence their joint spectrum consists of lattice points in the image of the classical moment map (I_{1}, …, I_{n}) : T^{∗}M − 0 → IR^{n} − 0. More correctly, the lattice can be shifted by a Maslov index.

Since Ĥ is a first order elliptic symbol on IR^{n} − 0 it has an asymptotic expansion in homogeneous functions of the form:

Ĥ ∼ H_{1} + H_{o} + H_{−1} + …, H_{j}(rI) = r^{j}H_{j}(I).

Thus we have:

Proposition 16. ([CV.2]) Suppose g ∈ R. Then \sqrt{Δ_{g}} is quantum torus integrable and hence there exists a polyhomogeneous function Ĥ such that \sqrt{Δ_{g}} = Ĥ(Î_{1}, Î_{2}). Moreover, H_{o} = 0.

The spectrum of \sqrt{Δ} is therefore given as the values \{Ĥ(m, ℓ + \frac{1}{2}) : |m| ≤ ℓ, ℓ > 0\} of Ĥ at lattice points in the convex polyhedral cone Γ (shifted by (0, \frac{1}{2}).

To break the homogeneity of the normal form coefficients, we fix a point I^{o} ∈ {H = 1}, let ω^{o} denote the common frequency vector of the ray of torii IR^{+}T_{I^{o}} and put

I · ω^{o} := ∑_{k=1}^{2} ω_{k}^{o}I_{k}.

The equation of the tangent line to {H = 1} at I^{o} in the action cone Γ_{o} is then given by ω^{o} · I = 1. The conic neighborhood will be parametrized in the following way: we fix a basis (i.e. a non-zero vector) v of the line I · ω^{o} = 0, and define the map

(ρ, ξ) → ρ(I^{o} + ξv), ξ ∈ (−ϵ, ϵ).

For sufficiently small ϵ, this map sweeps out a conic neighborhood W_{o} of I^{o} with inverse given by

ρ = ω^{o} · I, ξv_{j} := \frac{I_{j}}{I · ω^{o}} − I_{j}^{o}.)

Since H(I_{1}, I_{2}) = (ω^{o} · I) H(\frac{I_{1}}{ω^{o}·I} , \frac{I_{2}}{ω^{o}·I}) and since (\frac{I_{1}}{ω^{o}·I} , \frac{I_{2}}{ω^{o}·I}) ∈ \{ω^{o} · I = 1\} we may write

H(I_{1}, I_{2}) = ρh_{I^{o}}(ξ)

where h_{I^{o}} is the function on W_{o} ∩ {ω^{o} · I = 1} defined by

h_{I^{o}}(ξ) := H(I^{o} + ξv).

The C^{∞} Taylor expansion of h_{I^{o}}(ξ) around ξ = 0 is then a symplectic invariant of H.

Similary, we write the higher homogeneous terms as

H_{j}(I_{1}, I_{2}) = (ω^{o} · I)^{j} H_{j}(\frac{I_{1}}{ω^{o} · I} , \frac{I_{2}}{ω^{o} · I}) := (ω^{o} · I)^{j} h_{j}(ξ).

The Taylor expansion of h_{j}(ξ) around ξ = 0

h_{j}(ξ) = ∑_{α≥0} h_{j}^{α}(0)ξ^{α}

then defines the quantum Birkhoff normal form coefficients.

4. Normal form from spectrum

4.1. Inverse results for non-degenerate manifolds. Let us now sketch the proof of Theorem 1. The following is somewhat schematic; we refer to [Z.1,2] for the detailed proof.

Using the residue description, we have

a_{γk} = res D_{t}^{k}e^{it\sqrt{Δ}} := Res_{z=0}Tr D_{t}^{k}e^{it\sqrt{Δ}}\sqrt{Δ}^{−z}.  (5)

Since res is invariant under conjugation by (microlocal) unitary operators, and depends on only a finite jet of the Laplacian near γ, it may be calculated by conjugating to the normal form, and only depends on a finite part of the normal form. Applying D_{t}^{k} and formally exponentiating the terms of order ≤ −1 in D_{s} we get

res D_{t}^{k}e^{itD}|_{t=L} = res e^{i2πLD_{s}}e^{iH_{α}}D^{k}(I + iL\frac{p̃_{1}(I_{1}, ..., I_{n}, L)}{D_{s}} + ...).  (6)

Since e^{i2πD_{s}} = I on IR/2πLZZ the Fourier Integral factor in (6) is just e^{iH_{α}}. This is an operator in the metapletic representation and its trace is the character of this representation. Recall that for elements of the metaplectic group Mp(n, IR) not having 1 as an eigenvalue, the character is given by

Ch(x) = \frac{i^{σ}}{\sqrt{|det(I − x)|}}

where σ is a certain Maslov index. For non-degenerate x with p pairs of eigenvalues e^{±iα_{j}} of modulus one, q pairs of positive real eigenvalues e^{±λ_{j}} and c quadruplets of eigenvalues e^{±(μ_{j}±iν_{j})}, Ch(x) is therefore given (up to a Maslov factor) by

(5.6) T (α, λ, (μ, ν)) = Π_{j=1}^{p} \frac{e^{\frac{1}{2} iα_{j}}}{1 − e^{iα_{j}}} · Π_{j=1}^{q} \frac{e^{\frac{1}{2} λ_{j}}}{1 − e^{λ_{j}}} · Π_{j=1}^{c} \frac{e^{\frac{1}{2} (μ_{j}+iν_{j})}}{1 − e^{μ_{j}+iν_{j}}} \frac{e^{\frac{1}{2} (μ_{j}−iν_{j})}}{1 − e^{μ_{j}−iν_{j}}} .

Here we have selected one eigenvalue ρ from each symplectic pair ρ, ρ^{−1}. The ambiguity is fixed by the Maslov factor i^{σ}, which can (and will) be ignored below for the sake of brevity. For simplicity let us assume that γ is elliptic.

The next observation is that the residue is the trace of the term of order D_{s}^{−1}. This is formally obvious and the detailed justification is given in [Z.1, 2]. Therefore the wave invariant has the form

a_{γk} = T rF_{k,−1}(Î)e^{iαÎ}

for a certain polynomial F_{k,−1} which can easily be determined from the p_{j}(Î)’s. We then observe that

T rF_{k,−1}(Î)e^{iαÎ} = F_{k,−1}(D_{α_{1}}, ..., D_{α_{n}})T re^{iαÎ}.

Hence we get the result:

a_{γk} = F_{k,−1}(D) · Ch(x)|_{x=P_{γ}} .

The coefficients of the polynomials F_{k,−1} are evidently polynomials in the quantum Birkhoff normal form invariants. For the inverse result, one observes (with Guillemin) that the differentiation process produces polynomials in the β_{j}’s. Moreover, it is important to observe that the expressions for the wave invariants at iterates γ^{k} of γ involve the same operator F_{k,−1} applied to Ch(kα). Determining the Birkhoff normal form coefficients from the wave invariants is then equivalent to determining the coefficients of these polynomials from their special values at β_{j} = (1 − e^{ikα_{j}})^{−1} corresponding to γ and its iterates. Under the irrationality condition, the points (e^{iα_{1}}, …, e^{iα_{n}}) are dense on the torus and hence the special values determine the polynomial.

4.2. Inverse results for simple surfaces of revolution. We now prove that the wave trace invariants of a metric g ∈ R^{∗} determine its quantum normal form Ĥ. Below ψ_{L} is a smooth cutoff to a neighborhood of the periodic torus T_{L} of period L in the ‘action space’. Also, by the vector of winding numbers one means the homology class of a closed geodesic on T_{L} relative to a fixed basis (it may be fixed independently of the torus here).

Proposition 17. Let g ∈ R^{∗} and let L ∈ L. Then we have:

∑_{±} a_{±T_{L},− \frac{1}{2} +k} = Res_{s=0} ∫_{Γ} ψ_{L}(I +μ)e^{i⟨M_{L},I⟩}e^{−iLĤ(I+μ)}(Ĥ(I +μ))^{− \frac{1}{2} +k}(ω_{L}·(I +μ))^{−s}dI

where as above M_{L} is the vector of winding numbers of T_{L} and ω_{L} is the frequency vector of T_{L}.

Proof: Since \sqrt{Δ} = Ĥ(Î_{1}, Î_{2}) and since ψ̂_{L} is a function of the action operators, we have that

∑_{±} a_{±T_{L},− \frac{1}{2} +k} = Res_{s=0} ∑_{N∈ZZ^{2}} ψ_{L}(N + μ)e^{iLĤ(N+μ)}(Ĥ(N + μ))^{− \frac{1}{2} +k}(ω_{L} · (N + μ))^{−s}.

We then apply the Poisson summation formula for Re s >> 0 to replace the sum over N ∈ ZZ^{2} by

Res_{s=0} ∑_{M∈ZZ^{2}} J_{L,M,k}(s)

where

J_{L,M,k}(s) := ∫_{Γ} ψ_{L}(I + μ)e^{−i⟨M,I⟩}e^{iLĤ(I+μ)}(Ĥ(I + μ))^{− \frac{1}{2} +k}(ω_{L} · (I + μ))^{−s}dI.

By simplicity of the length spectrum, only the term with M = M_{L} has a pole at s = 0. Indeed, only in this term does the phase −⟨M, I⟩ + LH(I) have a critical point, since M = L∇_{I}H(I_{M}) implies that the torus with actions I_{M} is periodic of period L.

Now let use calculate the residue using the Taylor coefficients of the normal form at the point I^{o} = (0, 1), corresponding to the torus of meridians between the poles. Some computation based on the residue formula above shows that

a_{T_{L},− \frac{3}{2}} := c_{L} = \frac{1}{\sqrt{2πiαL}}e^{i⟨M_{L},μ⟩}

and that the higher wave invariants a_{T_{L},− \frac{1}{2} +k} are given by c_{L} times polynomials in L and in the derivatives of h, h^{−1}, h_{−1}, … at ξ = 0. For instance, the subprincipal wave invariant in dimension 2 is given given in terms of universal coefficients C_{ijk} by:

a_{T_{L},− \frac{1}{2}} = c_{L}[C_{004}∂_{ξ}^{4}h(ξ)|_{ξ=0} + C_{010}L^{2}h_{−1}(0) + C_{002}∂_{ξ}^{2}h(ξ)^{− \frac{1}{2}} |_{ξ=0}].

The key observation now is that the different terms decouple under iteration of the closed geodesics. That is:

Proposition 18. We have

a_{L,− \frac{1}{2} +k} = c_{L}P_{k}(L, h^{(2)}(0), …, h^{(2k+4)}(0), h_{−1}(0), …, h_{−1}^{(2k)}(0), …, h_{−k}(0), h_{−k}^{2}(0), h_{−k−1}(0))

where P_{k} is a polynomial with the following properties:

(i) It involves only the first 2k + 4 Taylor coefficients of h at 0, the first 2k of h_{−1},..., the first 2k + 2 − 2n of h_{−n} ..., the first 2 of h_{−k} and the 0th of h_{−k−1}.

(ii) It is of degree 1 in the variables h_{−k−1}(0), h_{−k}^{(2)}(0), …, h^{(2k+4)}(0), and each occurs in precisely one term.

(iii) The L-order of the monomials containing these terms is respectively L^{k+2}, L^{k+1}, …, L^{0}.

By using the joint ρ, L → ∞ asymptotics one can recover the complete Taylor expansions of all the h_{j}’s from the wave invariants of the meridian torus and its iterates. A key role is played by the fact that each h_{j} is even under reflection thru the I_{1}-axis, stemming from the fact that the Laplacian is invariant under complex conjugation. Hence all of the odd Taylor coefficients of h_{j} vanish and the even ones are separated by different powers of L.

5. Metric from normal form

The final step is to try to determine the metric from the normal form. At this time the only results pertain to the cases of surfaces of revolution and axisymmetric plane domains [CV.1]. We restrict our discussion to surfaces of revolution.

5.1. Simple surfaces of revolution. To complete the proof of the Theorem 3, we need to show that Ĥ determines a metric in R. The proof is basically to write down explicit expressions for H and H_{−1} in terms of the metric (i.e. in terms of a(r)) and then to invert the expressions to determine a(r). The first step is therefore to construct an initial part of the quantum normal form explicitly from the metric.

We begin from the fact that H(I_{1}, I_{2}) is a known function. Set H = E and solve for I_{2}. We know that I_{2} = |I_{1}| + ∫ (E − \frac{I_{1}}{a(r)^{2}} )dr and hence the function ∫ (E − \frac{I_{1}}{a(r)^{2}} )dr is a spectral invariant. We may write the integral in the form

∫_{IR}(E − x)+^{\frac{1}{2}}dμ(x)

where μ is the distribution function of \frac{1}{a^{2}} , i.e.

μ(x) := |\{r : \frac{1}{a(r)^{2}} ≤ x\}|

with | · | the Lebesgue measure. The above integral is an Abel transform and as is well known it is invertible. Hence

dμ(x) = ∑_{r: \frac{1}{a(r)^{2}} =x} |\frac{d}{dr} \frac{1}{a(r)^{2}} |^{−1}dx

is a spectral invariant. It follows that the function

J(x) := ∑_{r:a(r)=x} \frac{1}{|a′(r)|}

is known from the spectrum. By the simplicity assumption on a, there are just two solutions of a(r) = x; the smaller will be written r_{−}(x) and the larger, r_{+}(x). Thus, the function

J(x) = \frac{1}{|a′(r_{−}(x))|} + \frac{1}{|a′(r_{+}(x))|}

is a spectral invariant.

Clearly, J(x) does not determine a(r). Hence we must go into the H_{−1} expression. For various universal constants C_{1}, C_{2}, C_{3} it takes the form

[C_{1}∂_{E}^{2} ∫ \frac{(a′)^{2}}{a^{6}} (E − \frac{1}{a^{2}} )_{+}^{− \frac{1}{2}} dr + C_{2}∂_{E} ∫ \frac{(a′)^{2}}{a^{4}} (E − \frac{1}{a^{2}} )_{+}^{− \frac{1}{2}} dr

+C_{3} ∫ \frac{(a′)^{2}}{a^{2}} (E − \frac{1}{a^{2}} )_{+}^{− \frac{1}{2}} dr]|_{E=_{\frac{H(I_{1},I_{2})}{I^{2}_{1}}}} .

By a change of variables, we may rewrite this in the form

[C_{1}∂_{E}^{2} ∫ K(x)x^{\frac{3}{2}} (E − x)_{+}^{− \frac{1}{2}} dx + C_{2}∂_{E} ∫ K(x)x^{\frac{1}{2}} (E − x)_{+}^{− \frac{1}{2}} dx
+C_{3} ∫ K(x)x^{− \frac{1}{2}} (E − x)_{+}^{− \frac{1}{2}} dx]|_{E=_{\frac{H(I_{1},I_{2})}{I^{2}_{1}}}}

where

K(x) = |a′(r_{−}(x))| + |a′(r_{+}(x))|.

All values of E which occur as ratios \frac{H(I_{1},I_{2})}{I_{1}^{2}} give spectral invariants.

We now claim that K is a spectral invariant. To determine it we rewrite the above in terms of the fractional integral operators

I_{α}f(E) = f ∗ \frac{x_{+}^{α−1}}{Γ(α)} (E) = \frac{1}{Γ(α)} ∫_{0}^{E} f(y)(E − y)^{α−1}dy

on the half-line IR^{+}. These operators satisfy

I_{α} ∘ I_{β} = I_{α+β}, I_{−k} = (\frac{d}{dx})^{k}.

Thus the above equals L(K) where L is the fractional integral operator

L := C_{1}I_{−3/2}x^{\frac{3}{2}} + C_{2}I_{− \frac{1}{2}} x^{\frac{1}{2}} + C_{3}I_{\frac{1}{2}} x^{− \frac{1}{2}} .

To solve for K we apply I_{− \frac{1}{2}} to LK to get

C′_{1} \frac{d}{dx}^{2}(x^{\frac{3}{2}} K(x)) + C′_{2} \frac{d}{dx}(x^{\frac{1}{2}} K) + C′_{3}x^{− \frac{1}{2}} K = I_{− \frac{1}{2}} LK.

This equation determines K up to a solution f of the associated homogeneous equation, essentially an Euler equation. But also K = 0 on [0, a(r_{o})^{−2}] Since no homogeneous solution can have this property, K is uniquely determined by this boundary condition.

Given two real numbers a, b one knows that from a+b and \frac{1}{a} + \frac{1}{b} one can determine the pair (a, b). Hence a′(r) is determined from the spectrum. Since a(0) = 0 this determines a and hence the surface.

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Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218