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BIRKHOFF NORMAL FORMS IN SEMI-CLASSICAL INVERSE PROBLEMS 2 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI 0 0 2 n a 1. Introduction J Thepurposeofthisnoteistoapplytherecentresultsonsemi-classicaltraceformulæ[17],and 1 2 on quantum Birkhoff normalforms for semi-classicalFourier Operators [12] to inverse problems. We show how the classical Birkhoff normal form can be recovered from semi-classical spectral ] invariants. In fact the full quantum Birkhoff normal form of the quantum Hamiltonian near a P closed orbit, and infinitesimally with respect to the energy can be recovered. This generalizes S . recent results of Guillemin [7] and Zelditch [19],[20], [21] obtained in the high energy setting (a h special case of semi-classicalasymptotics). t a Wewillillustratetheresultsinanewsettingtowhichtheyapply. LetP(h)beasemi-classical m Schrödinger operator: [ (1.1) P(h)= h2∆+V(x) E, V ∞(Rn+1;R), 1 − − ∈C v with the principal symbol p(x,ξ)=ξ2+V(x) E. We make the following assumptions: 0 − 9 (1.2) ǫ>0, p−1([ ǫ,ǫ])⋐T∗Rn+1, dp=0 on p−1(0). 1 ∃ − 6 1 The first assumption guarantees that P(h) has a discrete spectrum near 0, and the second one, 0 that the energy surface p−1(0) is smooth. 2 The semi-classical inverse problem can be formulated as follows: 0 / What information about the energy surface, ξ2+V(x) = E, can be recovered from h the asymptotics of the spectrum of h2∆+V(x) in a small fixed neighbourhood of at E, as h 0 ? − m → Thisisverynaturalandcanbeconsideredasamathematicalformulationofthe generalprob- : v lemarisinginspectroscopy. Inconcretephysicalsituations,therelationbetweentheHamiltonian i and the spectrum is investigated using spectroscopic Hamiltonians which are essentially Hamil- X tonians in Birkhoff normal forms – see [18] for one of the first uses of Birkhoff normal forms in r a theoretical chemistry, and [13] for recent applications using experimental spectral data. The energy surface, p−1(0) T∗Rn has a natural bicharactistic foliation, coming from the Hamilton vector field, H = ⊂n ∂ p ∂ ∂ p ∂ . Since it describes classical dynamics, p j=1 ξj xj − xj ξj its properties are a natural target of a recovery procedure based on quantum information, such P as the spectrum. A na¨ıve intuition suggests that the quantum bound states (corresponding to eigenvalues) should correspond to the closed orbits of the classical dynamics. The simplest symplectic invariants associated to such closed orbits, γ, are the actions, S = ξdx, γ Zγ theeigenvaluesofthelinearPoincarémap,P ,andtheMaslov index,ν Z . WerecallthatP γ γ 4 γ ∈ isthedifferentialofthe(non-linear)Poincarémap,κ ,definedasfollows: letU beahypersurface γ 1 2 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI inp−1(0),transversaltoγ,locallydefinednearsomepointonγ. ThenU isasymplecticmanifold. The symplectic map κ :U U is defined as the first return map: γ → κ (m)=exp(T(m)H )(m) U, exp(tH )(m) / U, 0<t<T(m), γ p p ∈ ∈ where,strictlyspeaking,thedefinitionmayrequirereplacingU byalargerneighbourhoodasthe range of κ . γ The relation between the spectrum and closed orbits is provided by trace formulæ: the Gutzwiller and Balian-Bloch trace formulæ in physics and their mathematical precursor, the Selberg trace formula, and successors, the Duistermaat-Guillemin and the Guillemin-Melrose formulæ(see[17]andreferencesgiventhere). Gutzwiller’sformulastatesthatifdet(I Pk)=0, for 0=kT suppg, then for g,χ ∞(R), χ 1 near 0, g 0 near 0, − γ 6 6 γ ∈ ∈Cc ≡ ≡ (1.3) tr gˆ P(h) χ(P(h)) ∞ eikSγ/h+iνγ,kπ2 Tγ ∞ a (g)hj, a =g(kT ). (cid:18) h (cid:19) ∼Xγ 06=kX=−∞ |det(Pγk−I)|21 Xj=0 j,γ,k 0,γ,k γ whereT istheperiodofγ,andtheγ summationisoverallsimpleclosedorbitsofH onp−1(0). γ p The Maslov index, ν , may in principle depend on the number of times of “going around” the γ,k orbit. The coefficients a are distributions on R: ∞(R) g a . j,γ,k Cc ∋ 7→ j,γ,k Itisnowstandardthatunderthenon-degeneracyandsimplicityassumptions,det(I P )=0, γ − 6 Tγ = Tγ′ γ = γ′, we can recover the actions, Sγ, periods, Tγ, Maslov indices, νγ,k, and ⇒ det(I Pk), for all k. Under stronger non-degeneracy conditions (see (1.4) and (2.9) below), | − γ | a theorem of Fried [5] (see Proposition 3.2 below) then shows that the eigenvalues of the linear Poincarémap, P , can be recovered. γ Thehigherordercoefficientsa containfurtherinformationandtheyformafamilyofsemi- j,γ,k classical spectral invariants. It turns out that under a stronger version of the non-degeneracy hypothesis, we can recover from them the full infinitesimal information about the non-linear Poincaré map κ . For simplicity we state it here in the case of elliptic orbits only, which is the γ semi-classical analogue of Guillemin’s result [7] – see Sect.5 for the general statement. Thus suppose that the eigenvalues of dκ (w ), w = U γ, are given by exp( iθ ), j = γ 0 0 j ∩ ± 1, ,n, θ (0,π). We strengthen the non-degeneracy assumption to independence from 2π j ··· ∈ over rationals: n (1.4) k θ 2πZ, k Z = k =0, j =1, ,n. j j j j ∈ ∈ ⇒ ··· j=1 X The now classicalnormalformtheoremofBirkhoff, LewisandSternberg saysthatthere exist symplecticcoordinates(x,ξ) R2n,centeredatw ,suchthatinaneighbourhoodofw ,wehave 0 0 ∈ κ =κ +κ♭, κ♭(x,ξ)= ((x2+ξ2)∞), γ 0 O ∂p κ (ι,ω)= ι,ω , ι=(ι , ,ι ), ω =(ω , ,ω ), 0 1 n 1 n − ∂ι ··· ··· (cid:18) (cid:19) (1.5) 1 ι = (x2+ξ2), ω =arg(x +iξ ), j 2 j j j j j n p(ι)=p(0) θ ι + (ι2), j j − O j=1 X see [7,Theorem2.1]. The Taylorseriesof the twist mapκ is calledthe Birkhoff normal form of 0 κ. It invariantly describes the infinitesimal properties of the flow near γ. Since we chose θ ’s in j a unique way, 0<θ <π, the Birkhoff normal form is unique. j Specialized to this case our general result Theorem 4 can be stated as NORMAL FORMS IN INVERSE PROBLEMS 3 Theorem 1. Suppose that P(h) = h2∆+V(x) E is a semi-classical Schro¨dinger operator − − satisfying (1.2), and that γ is an elliptic closed orbit of H on p−1(0), with eigenvalues of P , p γ e±iθj, satisfying (1.4). Then the leading term and the coefficients aj,γ,k in (1.3), determine the Birkhoff normal form of κ . γ As stated above, our motivation for the recovery of the classical Birkhoff normal forms came from an attempt to understand the results of Guillemin [7] and Zelditch [19],[20],[21], in the contextoftherecenttraceformulaofSjo¨strand-Zworski[17],andofthe (quantum)Birkhoffnor- malformforsemi-classicalFourierintegraloperatorofIantchenko[11]andIantchenko-Sjo¨strand [12] (see also [16]). Guillemin and Zelditch considered the high energy (or ∞ singularities) C case, corresponding to h2∆ 1 semi-classically. Guillemin’s starting point was his earlier work g − with Françoise[4], andanobservationofZelditch onthe non-commutativeresidues of(classical) Fourierintegraloperator[6]. Zelditch’sapproachalsousedthenon-commutativeresiduebutwas more concrete and computational. Inadditionto recoveringthe classicalBirkhoffnormalform,[7],[19], [20],[21]wereconcerned withrecoveringthefullquantumBirkhoffnormalformoftheoperatorneartheclosedtrajectory. Without giving a definition of the Birkhoff normal form this result follows from Theorem 2. Suppose that the assumptions of Theorem 1 are satisfied for potentials V and V, with orbits γ and γ˜, respectively. If, in the notation of (1.3), S = S˜ , T = T , and det(I γ γ˜ γ γ˜ | − Pk) = det(I Pk), ν =ν˜ , for all k, then e γ | | − γ˜ | γ,k γ˜,k e There exists a unitary h-Fourier integral operator F, e microlocally defined in a neighbourhood of γ, j,k, a =a˜  . ∀ j,γ,k j,γ˜,k ⇐⇒  msuiccrholtohcaatlly(−tho2i∆nfi+niVte)Ford=erFa(l−onhg2∆γ.+V), ThegeneralresultfollowsfromProposition4.1andTheeorem4below,andisgiveninCorollary 5.2. The definitions of h-Fourier integral operators, and of agreement “to infinite order” are recalled in Sect. 2. Ourapproachisastraightforwardapplicationof (2.3)and(2.13)quotedbelowfromourearlier papers, and of a result of Fried [5] on the recovery of the eigenvalues of P (see Proposition 3.2 γ below). The traceformula showsthat the tracesofpowersof the quantum monodromyoperator of [17] are spectral invariants. The quantum monodromy operator is a semi-classical operator whichquantizesthe Poincarémapforagivenorbitγ,andonceitisputintothe Birkhoffnormal formusing(2.13)thetracesofitspowersareeasytocompute–seeProposition3.1. Therecovery of the classical Birkhoff normal form is then very clear – see Theorem 3. One of the most striking applications of this approach to inverse problems is the result of Zelditch[21]ontherecoveryofbi-axialanalyticplanardomainsfromthespectrumofthe(Dirich- let or Neumann) Laplacian. By applying our general methods we avoid any specific new work involvingtheboundaryvalueproblem–seeTheorem5andCorollary5.4. Thisandotherinverse results are presented in Sect.5. What may seem like an excessive generalityin the trace formulæ of [17] (see (2.3) below) and Theorem 4 is motivated by their potential use for the recovery of classical dynamics of effective Hamiltonians. One of the most impressive inverse procedures is the application of the Onsager rules for determining Fermi surfaces – see for instance [1] for an introductory physics discussion. They can be interpreted in terms of a trace formula involving an effective Hamiltonian coming from the Peierls substitution – see [9]. The question which will be investigatedelsewhere is: can a more detailed information about the Fermi surface be obtained fromhigher order terms in the expansions of magnetic susceptibilities of metals. 4 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI Throughout the paper we will write ∞ u(x,ǫ)= u (x)ǫj, j j=0 X to denote asymptotic expansions as ǫ 0. When no confusion is likely to arise all equalities → are meant modulo (ǫ∞) where ǫ is the relevant small parameter (mostly h, the mathematical “Planck constant”)O. We denote by N the set of natural numbers, 0,1,2, . ··· Acknowledgements. The third author would like to thank the National Science Foundation forpartialsupportundergrantDMS-9970614. HewouldalsoliketothankBillMillerandHoward Taylor for references to the chemistry literature. 2. Preliminaries In this section we will recall the general trace formula of [17] and the classical and quantum Birkhoff normal forms of [12]. We consider X which is either a compact ∞ manifold of dimension n+1 or Rn+1. C We introduce the usual class of semi-classical symbols on X: Sm,k(T∗X)= a ∞(T∗X (0,1]): ∂α∂βa(x,ξ;h) C h−m ξ k−|β| , { ∈C × | x ξ |≤ α,β h i } andthe correspondingclassofpseudodifferentialoperators,Ψm,k(X), withthe quantizationand h symbol maps: Opw : Sm,k(T∗X) Ψm,k(X) h −→ h σ : Ψm,k(X) Sm,k(T∗X)/Sm−1,k−1(T∗X), h h −→ with both maps surjective, and the usual properties σ (A B)=σ (A)σ (B), h h h ◦ 0 Ψm−1,k−1(X)֒ Ψm,k(X) σh Sm,k(T∗X)/Sm−1,k−1(T∗X) 0, → → → → a short exact sequence, and σ Opw :Sm,k(T∗X) Sm,k(T∗X)/Sm−1,k−1(T∗X), h◦ h −→ the natural projection map. Let P(z) ∞(I ;Ψ0,k(X)), I = ( a,a) R, be a family of self-adjoint, principal type ∈ C z h − ⊂ operators, such that Σ = m : σ(P(z))=0 T∗X is compact. We assume that z { }⊂ σ(∂ P(z)) C <0, near Σ , σ(P(z)) C ξ k, z z (2.1) ≤− | |≥ h i for ξ C when X is a compact manifold, and for (x,ξ) C if X =Rn+1. | |≥ | |≥ We also assume that for z near 0, the Hamilton vector field, H , p(z)=σ(P(z)), has a simple p(z) closed orbit γ(z) Σ with period T(z), and that γ(z) has a neighbourhood Ω such that z ⊂ (2.2) m Ω and exptH (m)=m, p(z,m)=0, 0< t T(z)N +ǫ, z I, = m γ(z), p(z) ∈ | |≤ ∈ ⇒ ∈ where T(z) is the period of γ(z), assumed to depend smoothly on z. We also write p(z,m) for σ(P(z))(m). Let A Ψ0,0(X) be a microlocal cut-off to a sufficiently small neighbourhood of ∈ h γ(0). NORMAL FORMS IN INVERSE PROBLEMS 5 ThenifP(z)isanalmostanalyticextensionofP(z),z R,χ ∞(I),χ˜ ∞(C),itsalmost ∈ ∈Cc ∈Cc analytic extension, f ∞(R), and suppfˆ ( N(C ǫ)+C,N(C ǫ) C) 0 , we have, p p ∈C ⊂ − − − − \{ } 1 tr f(z/h)∂¯ χ˜(z)∂ P(z)P(z)−1 A (dz)= z z π L Z (2.3) N−1 (cid:2) (cid:3) 1 d tr f(z/h)M(z,h)k M(z,h)χ(z)dz+ (h∞), −2πi R dz O −16=k=−N−1 Z X where M(z,h) is the quantum monodromy operator. The constant C > 0, in the condition on p fˆdepends on p(z). Thequantummonodromy operator,M(z,h)isdefinedasfollows. Forapointonγ,m γ,we 0 ∈ can define the microlocalkernel of P(z) at m , to be the set of families u(h), such that u(h) are 0 microlocally defined near m and P(z)u(h)= (h∞) near m . We denote it by ker (P(z)). 0 O 0 m0(z) Any solutioncanbe continuedmicrolocallyalong γ(z)and we denote the correspondingforward and backward continuation by I (z) – see [17, 4] for precise definitions. Now, let m = m be ± 1 0 § 6 another point on γ(z). We then define I (z) (z)=I (z), near m , − + 1 (2.4) M (z) : ker (P(z)) ker (P(z)). M m0(z) −→ m0(z) The operator P(z) is assumed to be self-adjoint with respect to some inner product , , h• •i and we define the quantum flux norm on ker (P(z)) as follows: let χ be a microlocal cut-off m0(z) function supported near γ and equal to one near the part of γ between m and m . We denote 0 1 by [P(z),χ] the part of the commutator supported in m , and put + 0 def u,v = [(h/i)P(z),χ] u,v , u,v ker (P(z)). h iQF h + i ∈ m0(z) It is easy to check that this norm is independent of the choice of χ – see [17, Lemma 4.4]. This independence leads to the unitarity of (z): M (z)u, (z)u = u,u , u ker (P(z)). hM M iQF h iQF ∈ m0(z) Forpracticalreasonsweidentifyker (P(z))with ′(Rn),microlocallynear(0,0),andchoose m0(z) D the identification so that the corresponding monodromy map is unitary (microlocally near (0,0) where(0,0)correspondstotheclosedorbitintersectingatransversalidentifiedwithT∗Rn). This gives M(z,h) : ′(Rn) ′(Rn), D −→ D microlocally defined near (0,0) and M(z,h)isasemi-classicalFourierintegraloperatorwhichquantizesthePoincarémap of γ(z). This, and (2.3), (2.6) are the only properties of M(z,h) which will be used in this paper. Basicideas become clear whenone considersthe simplest example P(z)=hD z, onS1. In x − that case M(z,h)=exp(2πiz/h) – see [17, 2] for a careful presentation. § We will now assume that the symbol of P(z) has an asymptotic expansion in powers of h, near Σ : 0 (2.5) ∞ (P(z))(x,ξ;h)= hj(p (z))(x,ξ), (x,ξ) U, U a precompact neighbourhood of Σ . j 0 ∈ j=0 X 6 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI Under this assumption we have a simple extension of [17, Proposition 7.5] (2.6) trM(z,h)k−1hD M(z,h)=eikI(z)eiνk(z)π2I′(z) +eikI(z) ∞ a hj, z dκk 1 21 j,k | γ(z)− | Xj=1 whereI(z)istheclassicalaction,ν theMaslovindexofthekthiterateofγ(z)(itisindependent k of z for z near 0). We start by recalling the general statement of the classical Birkhoff normal form – see [12, Theorem 1.3, Proposition 4.3] and references there. Let (W,ω) be a symplectic manifold of dimension 2n, and let κ : W W , κ(w )=w , 0 0 −→ be a symplectic transformation, κ∗ω = ω, fixing w W. In (1.5), the twist map, κ , is the 0 0 ∈ Birkhoff normal form of κ, (2.7) κ =expH , 0 p where the Hamiltonian p is in the Birkhoff normal form. The general case presented in [12] is formulated using (2.7): suppose that the eigenvalues of dκ(w ) are given by 0 λ ,λ−1, j =1, ,n j j ··· λ =expµ , µ =α +iβ , α >0, 0<β / 2πZ, 1 j n , j j j j j j j ch ∈ ≤ ≤ (2.8) λ =expµ , µ =µ , 1 j n , j+nch j+nch j+nch j ≤ ≤ ch λ =expµ , µ >0, 2n +1 j 2n +n , j j j ch ch rh ≤ ≤ λ =expµ , µ i(R 2πZ), n +2n +1 j n. j j j rh ch ∈ \ ≤ ≤ Here n +n +2n = n, and e stands for elliptic, rh for real hyperbolic, and ch, for complex e rh ch hyperbolic. We replace the condition in the purely elliptic case (1.4) by the more general condition n (2.9) k µ 2πiZ, k Z = k =0. j j j j ∈ ∈ ⇒ j=1 X Thenthegeneralizationof (1.5)saysthatthereexistsymplecticcoordinates(x,ξ) Rn centered ∈ at (0,0) (2.10) κ=expH +κ♭, κ♭(x,ξ)= ((x2+ξ2)∞), p O p=p +r, r(x,ξ)= ((x2+ξ2)2), 0 O nch p (x,ξ)= α (x ξ +x ξ ) β (x ξ x ξ ) 0 j 2j−1 2j−1 2j 2j j 2j−1 2j 2j 2j−1 − − j=1 X 2nch+nrh n µ + µ x ξ + j(x2+ξ2) j j j 2i j j j=2Xnch+1 j=2ncXh+nrh+1 r(x,ξ)=R(ι), R(ι)= r ια, α |αX|≥2 (x ξ +x ξ +i(x ξ x ξ ))/2 1 j n 2j−1 2j−1 2j 2j 2j−1 2j 2j 2j−1 ch − ≤ ≤ ιj = ¯xιj′ξ 2j′n=j+−1nchj, nc2hn+1+≤nj ≤2nch  (xj2j+ξ2)/(2i) 2nch+n≤+≤1 jch n.rh j j ch rh ≤ ≤  NORMAL FORMS IN INVERSE PROBLEMS 7 Observe that p = ι,µ . By the Birkhoff normal form of κ we mean p and the asymptotic 0 0 h i expansion of r. WenowrecallthequantumBirkhoffnormalforms[11]and[12,Theorem5.1,(4.11)]. Westart with some generalfacts about h-Fourierintegraloperators. Let W above be a neighbourhoodof w T∗X, where X is a ∞ manifold. To κ : W W we can associate a class of operators, 0 ∈ C → microlocally defined near w (see [17, 3] for the precise definition of this notion): 0 § Im(X,X,κ′), h the semi-classical Fourier integral operators quantizing κ. One way to define them is by consid- ering oscillatory integrals: we identify w with (0,0) T∗Rn so that 0 ∈ U Im(X,X,κ′) Uu(x)=h−m−n+2N eiφ(x,y,θ)/ha(x,y,θ;h)u(y)dydθ, ∈ h ⇐⇒ Z Z (2.11) φ ∞(Rn Rn RN), graph(κ)=π(C ), φ ∈C × × C = (x,y,θ) : φ′(x,y,θ)=0 (x,y,θ) π (x,φ′;y,φ′), φ { θ }∋ 7−→ x y where φ is assumed to be a non-degenerate phase function in the sense of Ho¨rmander (see [10, Def. 21.2.5] for a related discussion), that is φ is smooth and real, C is a smooth manifold of φ dimension 2n, and π has an injective differential. The amplitude a is assumed to be supported near (0,0,0), and to be a classical symbol in the sense of (2.5). Beforestatingthemainresultof[12]wehavetointroducethenotionofequivalenceoffamilies of operators used in that paper [12, 2, 5]: let U(z) I0(X,X,κ′) and U(z) I0(X,X,κ˜′) be § § ∈ z ∈ z two families of operators. Then e (2.12) U U ≡ if κ0(w0)= κ˜0(w0)= w0, the two families of transeformations, κz and κ˜z, agree to infinite order at w and z = 0, and the terms in the asymptotic expansions (in powers of h) of φ , φ˜ , a , a˜ 0 z z z z (with amplitudes as in (2.11)) agree to infinite order at (0,0,0) and z =0. Suppose that U(z) quantizes κ and κ satisfies the assumptions of (2.10). Also assume that z 0 U(z) is elliptic. Then U(z) V(z)−1eiP(z,h)/hV(z), ≡ V(z) I0(X,X, ′) is a family of elliptic Fourier integral operators near ((0,0),w ) ∈ Cz 0 P(z,h)=p (z)+hp (z)+h2p (z)+ , 0 1 2 (2.13) ··· p (z)=p (z;I , ,I ), I =ιw(x,hD ), j j 1 ··· j j j x n p (z)= µ (z)I +R(z;I , ,I ), R(z;ι)= (ι2). 0 j j 1 n ··· O j=1 X When U(z) is microlocally unitary near w , P(z,h), V(z) can be chosen to be microlocally self- 0 adjoint and unitary, respectively, at ((0,0),w ). We will call the expansion of P(z,h) at z = 0, 0 ι=0, the infinitesimal Quantum Birkhoff Normal Form of U(z) at z =0. We recall that a Fourier integral operator is elliptic if it is associated to a canonical transformation, that is above is the graph of a canonical transformation, ((0,0),w ) , and that its 0 C ∈C symboliselliptic,thatistheleadingtermintheasymptoticexpansionoftheamplitudein(2.11) does not vanish. 8 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI 3. Recovering Birkhoff normal forms from traces From [17, Lemma 3.2, Proposition 7.3] we know that if U(z) I0(X,X,κ(z)′), is a family ∈ h of h-Fourier integral operators, and that the canonical relations κ(z)k, k = 0, have only one 6 non-degenerate fixed point then (3.1) tr U(z)k =eikS(z)/h a (z)+a (z)h+ +a (z)hl+ (hl+1) . 0k 1k lk ··· O In this section we will prove the fol(cid:0)lowing (cid:1) Theorem 3. Let U(z) I0(X,X,κ(z)′) be a family of h-Fourier integral operator associated to ∈ h smooth locally defined canonical transformations, κ(z) : T∗X T∗X. Suppose that κ has a 0 → unique fixed point, w , and that the eigenvalues of dκ (w ) given in (2.8) satisfy (2.9). Then 0 0 0 the Taylor series of the coefficients a (z) in (3.1) determine the infinitesimal quantum Birkhoff lk normal form of U(z), P(z,h) in (2.13). In particular, the Birkhoff normal form of κ at z =0, 0 p(z), in (2.10), is determined. Our proof of Theorem 3 is based on Proposition 3.1. Suppose that (3.2) G(ι;h)= ι,µ +F(ι,h), F(ι,h)=F (ι)+hF (ι)+ , eµj =1, F (ι)= (ι2), 0 1 0 h i ··· 6 O where we consider F(ι,h) as a formal power series in h, and each F (ι), a formal power series j in ι, and where we supressed the dependence on µ in higher order terms. If I (h)=ιw(x,hD ), then j j x n 1 (3.3) tr e−iG(I(h);h)/h =e−iF(ih∂µ;h)/h .  2sinh(µ /2) j j=1 Y   Proof. When G(ι;h)= ι,µ this follows from computations in each of the cases given in (2.10): h i tr e−ihI(h),µi/h =tr e−ihI(1),µi, where we can separate the trace into separate blocks: elliptic blocks: • ∞ tre−µ(Dx2+x2)/2 = e−µ(n+12) = 2sin1hµ , k=0 2 X where the formal computation (µ iR ) can be justified by condsidering the limit ǫ 0, + ∈ → with µ replaced by µ+ǫ. real hyperbolic blocks: • tre−iµ(xDx+Dxx)/2 =tre−µ(x∂x+21) =tr u e−µ/2u(e−µ ) , 7→ • (cid:16) (cid:17) Formally, 1 tre−µ(x∂x+12) =e−µ/2 δ0((e−µ−1)x)dx= 2sinhµ , Z 2 andthiscanbejustifiedbyapproximatingδ byasequenceofsmoothcompactlysupported 0 functions. NORMAL FORMS IN INVERSE PROBLEMS 9 complex hyperbolic blocks: herewehavetoworkintwodimensions,µ =α+iβ,µ =µ¯ = 1 2 1 • α iβ, and the corresponding ι and ι (see the last two lines in (2.10)) are ι = z ζ, 1 2 1 − · ι =w ω, where z =(x ix )/√2, w =z¯and ζ =(ξ +iξ )/√2, ω =(ξ iξ )/√2, the 2 1 2 1 2 1 2 corresp·onding dual coordi−nates, ω =ζ¯. With I =ιw(x,D ), j =1,2, we ha−ve j j x tre−iµ1I1−iµ2I2 =tre−µ1(z∂z+12)−µ¯1(z¯∂z¯+21) =tr u( ,¯) e−µ1/2−µ¯1/2u(e−µ1 ,e−µ¯1¯) . • • 7→ • • The last operator can be written as a contour(cid:16)integral, (cid:17) e−µ1/2−µ¯1/2u(e−µ1,e−µ¯1z¯)= 1 ei((e−µ1z−z˜)ζ˜+(e−µ¯1z¯−w˜)ω˜)u(z˜,w˜)dz˜dw˜dζ˜dω˜, (2π)2 ZZZZ with the integration contour given by w˜ =¯z˜, ω˜ =¯ζ˜. Formal stationary phase argument applied when taking the trace gives 2 1 . 2sinhµj j=1 2 Y Once we understand the contribution of the leading term the following formal argument essentially gives (3.3). Since I(h) is unitarily equivalent to hI(1) we see that tr e−iG(I(h);h)/h =tr e−iG(hI(1);h)/h. On the other hand, I(1)=∂ I(1),µ , and hence µ h i e−iG(hI(1);h)/h =e−iF(hI(1);h)/he−ihI(1),µi =e−iF(ih∂µ;h)/he−ihI(1),µi. By moving differentiation outside of the trace we obtain the expansion (3.3). We also need a result of Fried [5] used in the same context in [7]: Proposition 3.2. If the assumption (2.9) holds then the set n ak sinh(kµ /2), k N, 0 j ∈ j=1 Y determines expµ ’s and a uniquely. j 0 Proposition 3.2 and an argument based on Kronecker’s Theorem give the following inverse result crucial in the proof of Theorem 3 (see [7, 8], [20, 6]) for different original approaches): § § Lemma 3.3. Suppose that µ satisfy (2.9). Then for any polynomial, p(ξ)= a ξα, the j |α|≤M α coefficients a can be recovered from the asymptotics of α P n 1 (3.4) p(ik−1∂ ) , k . µ 2sinh(kµ /2) −→∞ j j=1 Y Proof. UsingtheargumentoftheproofofProposition3.1wecanrewrite(3.4)withe =(1,...,1): 0 n 1 p(ik−1∂ ) =p(ik−1∂ ) e−hm+e0/2,kµi µ µ 2sinh(kµ /2) (3.5) jY=1 j mX∈Nn = p((m+e /2)/i)e−hm+e0/2,kµi. 0 m∈Nn X To justify this expression we will consider it as a distribution obtained as a boundary value of a holomorphic function. We introduce new complex variables z Ω1 = D(0,1)ne and w Ω2 = ∈ ∈ 10 A. IANTCHENKO, J. SJO¨STRAND, AND M. ZWORSKI (D(0,1) 0)nch inapolydiscandapuncturedpolydiscrespectively: attheboundary,zj =eiµj/2, \ and wj =eiβj+ne+nrh/2. We then write m=(m′ ,m′′ ,m ,m ) Nn, e =(ech,ech,erh,ee), ch ch rh e ∈ 0 0 0 0 0 µ=(µ ,µ¯ ,µ ,µ ), µ =α+iβ, µ Rnrh, µ (iR)ne, ch ch rh e ch rh ∈ + e ∈ m+e /2,µ = n +ech,α + n +erh/2,µ +i l +ee/2,µ /i +i l ,β , h 0 i h 2 0 i h 1 0 rhi h1 e i h2 i n =m′ +m′′ , l =m′ m′′ , n =m , l =m . 2 ch ch 2 ch− ch 1 rh 1 e Here m Nn is arbitrary, and ∈ n Nnrh, l Nne, n Nnch, l Znch, 1 1 2 2 ∈ ∈ ∈ ∈ with the constraint that n l 2± 2 Nnch, 2 ∈ which implies that l n , component wise. We then put 2 2 | |≤ q(n,l)=p((m+e /2)/i), z =e−µe Cne, w =e−iβ/2 Cnch, 0 ∈ ∈ (component-wise in the natural sense). In this notation (3.5) becomes (3.6) e−k(hn1+er0h/2,µrhi+hn2+ec0h/2,αi)Fn(zk,wk), n∈NXnrh+nch where Fn(z,w)= q(n,l)z2l1+ee0w2l2, l1∈NneX, l2∈Znch n22±l2∈Nnch (z,w) Ω1 Ω2 Cne Cnch. Here we note that for any fixed n, q(n,l) is non-zero for finitely ∈ × ⊂ × many l ’s and hence F (z,w) is holomorphic in the punctured polydisc. Our task to recover 2 n the coefficients q(n,l) from asymptotics as k 0. That amounts to recovering the holomorphic → functions F . We observe that (3.5) shows that the distributional boundary values F ’s are n n smooth away from z4 =1. j An application of Kronecker’s theorem1 shows that for any (x,y) (∂D(0,1))ne (∂D(0,1))nch, ∈ × there exist a sequence kr such that (zkr,wkr) (x,y). →∞ → From (2.9) we deduce that nrh+nch k Reµ =0 = k =0. j j j ⇒ i=1 X Hence by choosing sequences of k , we can determine the coefficients, F (x,y) in the first n → ∞ sum (3.6) (the terms have different rates of exponential decay). From that we determine the holomorphic functions F , and the coefficients, q(n,l). n Proof of Theorem 3. Applying[12,Theorem3.2]and[12,Theorem4.4]weseethat,upto (h∞) O errors,we can write tr U(z)k =tr e−kiG(I(h),z;h)/h, 1which says that if α ∈Zn satisfies hα,ki∈ 2πZ, k ∈ Zn, ⇒ k = 0, then for any y ∈Rn and ǫ > 0, there existsN ∈NandK∈Zn,suchthat|y−Nα−2πK|<ǫ.Itsuseisjustifiedby(2.9).