Local inverse scattering at fixed energy in spherically symmetric asymptotically hyperbolic manifolds
aa r X i v : . [ m a t h - ph ] J a n Local inverse scattering at fixed energy in spherically symmetricasymptotically hyperbolic manifolds
Thierry Daud´e ∗ , Damien Gobin and Fran¸cois Nicoleau † Abstract
In this paper, we adapt the well-known local uniqueness results of Borg-Marchenko type in theinverse problems for one dimensional Schr¨odinger equation to prove local uniqueness results in thesetting of inverse metric problems. More specifically, we consider a class of spherically symmetricmanifolds having two asymptotically hyperbolic ends and study the scattering properties of masslessDirac waves evolving on such manifolds. Using the spherical symmetry of the model, the stationaryscattering is encoded by a countable family of one-dimensional Dirac equations. This allows usto define the corresponding transmission coefficients T ( λ, n ) and reflection coefficients L ( λ, n ) and R ( λ, n ) of a Dirac wave having a fixed energy λ and angular momentum n . For instance, the reflectioncoefficients L ( λ, n ) correspond to the scattering experiment in which a wave is sent from the left endin the remote past and measured in the same left end in the future. The main result of this paper isan inverse uniqueness result local in nature. Namely, we prove that for a fixed λ = 0, the knowledgeof the reflection coefficients L ( λ, n ) (resp. R ( λ, n )) - up to a precise error term of the form O ( e − nB )with B > B of the error term. The crucial ingredients in theproof of this result are the Complex Angular Momentum method as well as some useful uniquenessresults for Laplace transforms. Keywords . Inverse Scattering, Black Holes, Dirac Equation. . Primaries 81U40, 35P25; Secondary 58J50.
The aim of this short paper is to extend the local inverse uniqueness results of Borg-Marchenko type forone dimensional Schr¨odinger equation obtained first in [23], and improved in [2, 11, 24], to the setting ofinverse metric problems, that is inverse problems on three or four dimensional curved manifolds whoseunknown - the object we wish to determine by observing waves at infinity - is the (Riemanniann orLorentzian) metric itself. We shall consider for the moment a very specific and simple class of 3D-Riemanniann manifolds that we name Spherically Symmetric Asymptotically Hyperbolic Manifolds, inshort SSAHM. Precisely, these are described by the setΣ = R x × S θ,ϕ , ∗ D´epartement de math´ematiques, Universit´e de Cergy-Pontoise, UMR CNRS 8088, 2 Av. Adolphe Chauvin, 95302Cergy-Pontoise cedex. Email: [email protected]. Research supported by the French National Research ProjectAARG, No. ANR-12-BS01-012-01 † D´epartement de Math´ematiques, Universit´e de Nantes, 2, rue de la Houssini`ere, BP 92208, 44322 Nantes Cedex 03.Emails: [email protected], [email protected]. Research supported by the French National Re-search Project NOSEVOL, No. ANR- 2011 BS0101901 σ = dx + a − ( x ) dω , where dω = (cid:0) dθ + sin θ dϕ (cid:1) is the euclidean metric on the 2D-sphere S . The assumptions on thefunction a ( x ) - that determines completely the metric - are: a ∈ C ( R ) , a > , (1.1)and ∃ a ± > , κ + < , κ − > , a ( x ) = a ± e κ ± x + O ( e κ ± x ) , x → ±∞ ,a ′ ( x ) = a ± κ ± e κ ± x + O ( e κ ± x ) , x → ±∞ . (1.2)Under these assumptions, (Σ , σ ) is clearly a spherically symmetric Riemanniann manifold with twoasymptotically hyperbolic ends { x = ±∞} . Note indeed that the metric σ is asymptotically a small perturbation of the ”hyperbolic like” metrics σ ± = dx + e − κ ± x dω ± , x → ±∞ , where dω ± = 1 / ( a ± ) dω are fixed metrics on S . From this, we see easily that the sectional curvature of σ tends to the constant negative values − ( κ ± ) on the corresponding ends { x = ±∞} . Hence the name”asymptotically hyperbolic” for this kind of geometry. Note in passing that we allow κ ± to take differentvalues leading to different sectional curvatures in the two ends. We emphasize at last that such SSAHMare very particular cases (because of our assumption of spherical symmetry) of the much broader class ofasymptotically hyperbolic manifolds for instance described in [13, 14, 21] (to cite only a few papers thatdeal with inverse problems).On the manifold (Σ , σ ), we are interested in studying how (scalar, electromagnetic, Dirac, . . . ) wavesevolve, scatter at late times and ultimately, in trying to answer the question: can we determine the metricby observing these waves at the infinities of the manifold (in our model, the two ends { x = ±∞} ). Fordefiniteness, we shall consider in this paper how massless Dirac waves propagate and scatter towards thetwo asymptotically hyperbolic ends. Note that the same results should hold with the Dirac equationreplaced by the wave equation. Precisely, let us consider the massless Dirac equation i∂ t ψ = D σ ψ, (1.3)where D σ denotes a representation of the Dirac operator on (Σ , σ ) and the 2-spinor solution ψ belongsto L (Σ; C ). It will be shown in Section 2 that we have a very simple connection between D σ and thefunction a ( x ) appearing in the metric σ , precisely D σ = Γ D x + a ( x ) D S , (1.4)where D S denotes the intrinsic Dirac operator on S , represented here by the expression D S = Γ (cid:18) D θ + i cot θ (cid:19) + Γ θ D ϕ , (1.5)with D x = − i∂ x , D θ = − i∂ θ , D ϕ = − i∂ ϕ and where the 2 ×
2- Dirac matrices Γ , Γ , Γ satisfy theusual anti-commutation relations Γ i Γ j + Γ j Γ i = 2 δ ij . (1.6)Due to the spherical symmetry of the problem and the existence of generalized spherical harmonics { Y kl } that ”diagonalize” D S , we can decompose the energy Hilbert space H = L (Σ; C ) onto a Hilbert2um of partial Hilbert spaces H kl with the property that the H kl ’s are let invariant through the actionof the Dirac operator (1.4). More precisely, if we introduce the set of indices I = { k ∈ / Z , l ∈ / N , | k | ≤ l } , we have H = ⊕ kl H kl , H kl = L ( R ; C ) ⊗ Y kl , and D klσ := D σ |H kl = Γ D x − ( l + 1 / a ( x )Γ . Note that the partial Dirac operator D klσ ’s only depend on the angular momentum l + 1 / ∈ N ∗ . Forsimplicity, we shall denote l + by n (hence the new parameter n runs over the integers N ∗ ) and also D nσ = Γ D x − na ( x )Γ , (1.7)for the partial Dirac operators on each generalized spherical harmonic Y kl .We are thus led to consider the restriction of the Dirac equation (1.3) to each partial Hilbert space H kl separatly and study the properties of the family of 1D Dirac Hamiltonians D nσ , n ∈ N ∗ in order toobtain spectral, direct and inverse scattering results for the complete Dirac Hamiltonian D σ . This hasbeen done in [9] in a very similar context (see also [1, 7, 8]). Let us summarize here these results. Werefer to Section 3 for more explanations.First, the Dirac Hamiltonian D σ is selfadjoint on the Hilbert space H = L (Σ; C ) and has absolutelycontinuous spectrum. In particular, the pure point spectrum of D σ is empty. As a consequence, theenergy of massless Dirac fields cannot remain trapped on any compact subsets of Σ, i.e. for all compactsubset K ⊂ R , lim t →±∞ k K ( x ) e − it D σ ψ k = 0 . In other words, the massless Dirac fields scatter towards the asymptotic ends { x = ±∞} of the manifoldΣ at late times.Second, a complete direct scattering theory can be established for D σ on (Σ , σ ). For all energy λ ∈ R ,we denote the scattering matrix at energy λ by S ( λ ). It is a unitary operator on L ( S ; C ) and thus hasthe structure of an operator valued 2 × i.e. S ( λ ) = (cid:20) T L ( λ ) R ( λ ) L ( λ ) T R ( λ ) (cid:21) , (1.8)where T L , T R are the transmission operators and R, L the reflection operators. The formers measure thepart of a signal having energy λ transmitted from an end to the other end in a scattering process whereasthe latters measure the part of a signal of energy λ reflected from an end to itself ( { x = −∞} for L and { x = + ∞} for R ).Due to the spherical symmetry of the model, the scattering matrix lets invariant all the partial Hilbertspaces H kl and can be thus decomposed into a Hilbert sum of unitary operators acting C . We write asa shorthand S ( λ ) = X k,l ∈ I S kl ( λ ) , S kl ( λ ) := S ( λ ) |H kl . Since the 1D Dirac operator (1.7) only depends on n = l + 1 / ∈ N ∗ , the partial scattering matrices S kl ( λ ) also only depend on n . We shall thus use the notation S ( λ, n ) = (cid:20) T ( λ, n ) R ( λ, n ) L ( λ, n ) T ( λ, n ) (cid:21) . (1.9) Note that the models studied in [9] (see also [7, 8]) come from General Relativity. More precisely, the direct and inversescattering of Dirac waves propagating in the exterior region of a Reissner-Nordstr¨om-de Sitter black holes were studiedtherein. It turns out that these ”relativistic” models and the one presented in this paper are equivalent. This was brieflymentioned in [9] and made rigorous in the next Section 2. n ∈ N ∗ , we emphasize that the partial scattering matrices S ( λ, n ) are unitary matrices thatencode the stationary scattering at a fixed energy λ on a given generalized spherical harmonics H kl with n = l + 1 /
2. As above, the transmission coefficients T ( λ, n ) correspond to the transmitted part of a signal(from one end to the other) whereas the left L ( λ, n ) and right R ( λ, n ) reflection coefficients correspondto the reflected part of a signal in a given end.In [9], we addressed the question whether it was possible to determine uniquely the metric fromthe knowledge of the reflection coefficients L ( λ, n ) or R ( λ, n ). Using essentially the Complex AngularMomentum method (see [20] for the first appearance of this method and [19] for an application toSchr¨odinger inverse scattering), we were able to answer positively to the question with some interestingimprovements in the hypotheses. Precisely, we state here the inverse scattering uniqueness result provedin [9]. Theorem 1.1.
Let
Σ = R × S be a SSAHM equipped with the Riemanniann metric σ = dx + a − ( x ) dω , where the function a ( x ) satisfies the assumptions (1.1) - (1.2). Let D σ = Γ D x + a ( x ) D S be an expressionof the massless Dirac operator associated to (Σ , σ ) . To the evolution equation i∂ t ψ = D σ ψ with ψ ∈ H = L (Σ; C ) , we associate the countable family of partial waves scattering matrices S ( λ, n ) for λ ∈ R and n ∈ N ∗ as above. Consider also a subset L of N ∗ that satisfies a M¨untz condition X n ∈L n = ∞ . Then the knowledge of either R ( λ, n ) or L ( λ, n ) for a fixed λ = 0 and for all n ∈ L determines uniquelythe function a ( x ) (and thus the metric σ ) up to a discrete set of translations. Remark 1.2.
First, we emphasize that the above result is not true if λ = 0 , (see Remark 3.7, [9]).Secondly, in [9], Theorem 1.1, it is claimed that the knowledge of the transmission coefficients T ( λ, n ) fora fixed λ = 0 and for all n ∈ L also determines uniquely the function a ( x ) up to a translation. The crucialingredient of the proof can be found in the Proposition 3.13 of [9] which states that if T ( λ, n ) = ˜ T ( λ, n ) for all n ∈ L , then the corresponding reflection coefficients L ( λ, n ) and ˜ L ( λ, n ) (resp. R ( λ, n ) and ˜ R ( λ, n ) ) coincide up to a multiplicative constant. The proof of this result given in [9] is unfortunatelyincomplete and therefore, this last point is not so clear and could even be false. However, in this paper,in Proposition B.5, Addendum B, we prove that the knowledge of the transmission coefficients T ( λ, n ) forall n ∈ L together with the knowledge of the reflection coefficients L ( λ, k ) for a finite number of integer k , (and a technical assumption on the sectional curvatures), uniquely determines the function a ( x ) up toa translation. The question whether these last hypotheseses are necessary remains open. For more general Asymptotically Hyperbolic Manifolds (AHM in short) with no particular symmetry,difficult direct and inverse scattering results for scalar waves have been proved by Joshi, S´a Barreto in[14], by S´a Barreto in [21] and by Isozaki, Kurylev in [13]. In [14] for instance, it is shown that theasymptotics of the metric of an AHM are uniquely determined (up to certain diffeomorphisms) by thescattering matrix S ( λ ) at a fixed energy λ off a discrete subset of R . In [21], it is proved that the metricof an AHM is uniquely determined (up to certain diffeomorphisms) by the scattering matrix S ( λ ) forevery λ ∈ R off an exceptional subset. Similar results are obtained recently in [13] for even more generalclasses of AHM. At last, we also mention [4] where related inverse problems - inverse resonance problems- are studied in certain subclasses of AHM.The new inverse scattering results of this paper are local in nature, in the same spirit as [2, 11, 23, 24].Instead of assuming the full knowledge of one of the reflection operators, we instead assume the knowledge4f one of these operators up to some precise error remainder (see below). Using the particular analyticproperties of the scattering coefficients L ( λ, z ) and R ( λ, z ) with respect to the complex angular momentum z and some well known uniqueness properties of the Laplace transform (see [12, 23]), we are able to provethe following improvement of our previous result. Theorem 1.3.
Let (Σ , σ ) and (Σ , ˜ σ ) two a priori different SSAHM. We denote by a ( x ) and ˜ a ( x ) the tworadial functions defining the metrics σ and ˜ σ . We define A = Z R a ( x ) dx, ˜ A = Z R ˜ a ( x ) dx, as well as the diffeomorphisms g : R −→ (0 , A ) ,x −→ g ( x ) = R x −∞ a ( s ) ds, , ˜ g : R −→ (0 , ˜ A ) ,x −→ ˜ g ( x ) = R x −∞ ˜ a ( s ) ds, . We also denote by h = g − : (0 , A ) −→ R , ˜ h = ˜ g − : (0 , ˜ A ) −→ R their inverse diffeomorphisms. Asabove, we define S ( λ, n ) and ˜ S ( λ, n ) the corresponding partial scattering matrices. Let λ = 0 be a fixedenergy and < B < min ( A, ˜ A ) . Then the following assertions are equivalent: ( i ) L ( λ, n ) = ˜ L ( λ, n ) + O (cid:0) e − nB (cid:1) , n → + ∞ . (1.10)( ii ) ∃ k ∈ Z , a ( x ) = ˜ a ( x + kπλ ) , ∀ x ≤ h ( B ) = ˜ h ( B ) − kπλ . Symmetrically, the following assertions are also equivalent: ( iii ) R ( λ, n ) = ˜ R ( λ, n ) + O (cid:0) e − nB (cid:1) , n → + ∞ . (1.11)( iv ) ∃ k ∈ Z , a ( x ) = ˜ a ( x + kπλ ) , ∀ x ≥ h ( A − B ) = ˜ h ( ˜ A − B ) − kπλ . The above result asserts that the partial knowledge of the reflection coefficients in the sense of (1.10)or (1.11) allows to determine uniquely the metric σ in the neighbourhoods of the two ends { x = ±∞} .The size of these neighbourhoods depend on the magnitude of the error terms in (1.10) - (1.11). Of course, h ( B ) (resp. h ( A − B )) depends on the metric a ( x ) which is a priori unknown. But, it is not difficult toprove using (1.2) that h ( B ) ∼ κ − log B when B →
0. In the same way, h ( A − B ) ∼ − κ + log( A − B )when B → A . We also emphasize that the ”surface gravities” κ ± can be explicitly recover from theasymptotics of L ( λ, n ) or R ( λ, n ), n → + ∞ , (see [9], Theorem 4.22).As a direct consequence, we obtain immediately the following global uniqueness result for the metricof a SSAHM. This result slightly improves our earlier version obtained in [9] and stated in Theorem 1.1. Corollary 1.4.
Assume that for a given C ≥ min( A, ˜ A ) and λ = 0 a fixed energy, one of the followingassertions holds : ( i ) L ( λ, n ) = ˜ L ( λ, n ) + O (cid:0) e − nC (cid:1) , n → + ∞ . ( ii ) R ( λ, n ) = ˜ R ( λ, n ) + O (cid:0) e − nC (cid:1) , n → + ∞ . Then, there exists k ∈ Z , a ( x ) = ˜ a ( x + kπλ ) , ∀ x ∈ R . roof. Let us treat for instance the case ( i ) and assume that A ≤ ˜ A . From our hypothesis, for all B < A ,we have L ( λ, n ) = ˜ L ( λ, n ) + O (cid:0) e − nB (cid:1) . Hence Theorem 1.3 implies that a ( x ) = ˜ a ( x + kπλ ) , ∀ x ≤ h ( B ). The result follows letting B tend to A and using that lim X → A h ( X ) = + ∞ . Note that we also obtain A = ˜ A .Let us give here a possible interpretation of the above local uniqueness result Theorem 1.3. Considerfor instance the reflection coefficients L ( λ, n ) and recall that it encodes the following scattering experi-ment: a wave having energy λ is sent from the end { x = −∞} in the past and evolves on the SSAHM.Then L ( λ, n ) measures the part of this wave that is reflected to the same end { x = −∞} in the farfuture. Now our result asserts that if we know L ( λ, n ) up to a precise error term of the form O (cid:0) e − nB (cid:1) ,then the metric is uniquely determined in a neighbourhood of { x = −∞} , the size of the neighbourhooddepending only on the constant B defining the error term. We infer thus that, under our assumption, thewave sent from { x = −∞} hasn’t the time to travel through the whole manifold before being measuredback in the end { x = −∞} . This explains heuristically why the partial knowledge of L ( λ, n ), in theprecise sense given by our assumption, is not enough to determine the full metric.At last, when using the transmission coefficients as the starting point of our inverse problem, we geta result different in nature than the one obtained with the reflection coefficients. Precisely, we obtaina global uniqueness result. Moreover, as we have said before, we have to assume that the reflectioncoefficients L ( λ, n ) are equal for a finite number of integer n and we make a technical assumption on thesectional curvatures κ ± . The question whether these last hypotheses are necessary remains open. Theorem 1.5.
Assume that κ + + 1 κ − < , κ + + 1˜ κ − < , and that for a fixed energy λ = 0 and for some B > max( A, ˜ A ) , T ( λ, n ) = ˜ T ( λ, n ) + O (cid:0) e − nB (cid:1) , n → + ∞ . (1.12) Assume also that L ( λ, n ) = ˜ L ( λ, n ) , (1.13) for a finite but large enough number of indices n ∈ N . Then there exists a constant σ ∈ R such that ˜ a ( x ) = a ( x + σ ) . In consequence, the two SSAHM (Σ , g ) and ( ˜Σ , ˜ g ) coincide up to isometries. Let us make a few comments on this result. First, we provide an heuristic reason why we don’t havea local uniqueness result when we assume the knowledge of the transmission coefficients up to a preciseerror. The transmission coefficients - by definition - measure the part of a wave transmitted from oneend, say { x = −∞} , to the other end { x = + ∞} . In our case where the SSAHM has only two ends, thetransmitted wave has thus the time to propagate into the whole manifold. It is then natural that thetransmission coefficients encode all the information of the SSAHM.Second, the asymptotics of the transmission coefficients when n tends to infinity are computed in [9].Precisely, we have | T ( λ, n ) |∼ C e − nA , | ˜ T ( λ, n ) |∼ ˜ C e − n ˜ A . B in Theorem 1.5 cannot be weaker that B > max( A, ˜ A ). Note then thatthe assertion (i) implies immediately A = ˜ A from the above asymptotics. We mention that a globaluniqueness inverse result in the case where max( A, ˜ A ) < B ≤ max( A, ˜ A ) is still an open question.This paper is organised as follows. In Section 2, we recall how to compute the Dirac equation ona curved manifold and apply this formalism to obtain a representation of a massless Dirac operator ona SSAHM that is suitable for us. In Section 3, we recall the main results from [9] where a completedescription of the stationary scattering corresponding to massless Dirac fields evolving in a SSAHM wasobtained. In Section 4, we prove our main results, Theorem 1.3, Theorem 1.5. Eventually, we includethe last Section 5 in which an application of our local inverse uniqueness results on SSAHM is given inthe context of black hole spacetimes, precisely on Reissner-Nordstr¨om-de-Sitter black holes. Since the Dirac equation is by essence a relativistic equation, we prefer to work directly on the fourdimensional Lorentzian manifold (
M, τ ) defined by M = R t × Σ , and equipped with the metric τ = dt − σ, where (Σ , σ ) is the SSAHM we aim to study. Below we recall how to compute the massless Diracequation on such a 4D curved background and obtain a representation of it that we put under the genericHamiltonian form i∂ t ψ = D τ ψ. We shall see in Remark 2.1 that this procedure leads to an equivalent form of the massless Dirac equationthan the one i∂ t ψ = D σ ψ, we would have obtained working on the 3D-Riemanniann manifold (Σ , σ ). In other words, the Diracoperators D τ and D σ that we obtain by these two formalisms are shown to be unitarily equivalent. Weprefer to work with the relativistic point of view nevertheless since we are also interested in applicationsof our inverse results to spacetimes coming from General Relativity, namely black hole spacetimes. Wepostpone this parenthesis till Section 5. Dcurved space-time
To calculate the massless Dirac equation in a 4D curved spacetime M equipped with a Lorentzian metric τ of signature (1 , − , − , − { e A } A =0 , , , a given local Lorentz frame, i.e. a set of vector fields satisfying τ ( e A , e B ) = η AB where η AB = diag(1 , − , − , −
1) is the flat (Lorentz)metric. We also denote by { e A } A =0 , , , the set of dual 1-forms of the frame { e A } . Latin letters A,B willdenote in what follows local Lorentz frame indices, while Greek letters µ, ν run over four-dimensionalspace-time coordinate indices. The massless Dirac equation takes then the generic form D φ = γ A ( ∂ A + Γ A ) φ = 0 . (2.1)7ere, the γ A ’s are the gamma Dirac matrices satisfying the anticommutation relations { γ A , γ B } = γ A γ B + γ B γ A = 2 η AB . (2.2)The differential operators ∂ A ’s are given by ∂ A = e µA ∂ µ in terms of the local differential operators andthe Γ A ’s are the components of the spinor connection Γ = Γ A e A = Γ µ dx µ in the local Lorentz frame.In order to derive the latter, we first compute the spin-connection 1-form ω AB = ω ABµ dx µ = f ABC e C thanks to Cartan’s first structural equation and the skew-symmetric condition de A + ω AB ∧ e B = 0 , ω AB = η AC ω CB = − ω BA . (2.3)Note here that we use the flat metric η AB or its inverse η AB to raise or lower Latin indices. We also useEinstein summation convention. With this definition, the spinor connection Γ is then defined asΓ = 18 [ γ A , γ B ] ω AB = 14 γ A γ B ω AB = 14 γ A γ B f ABC e C . (2.4) We now apply this formalism to calculate the massless Dirac equation on the 4D- Lorentzian manifold(
M, τ ) given by M = R t × R x × S θ,ϕ , and τ = dt − dx − a − ( x ) (cid:0) dθ + sin θ dϕ (cid:1) , where a ∈ C ( R ) , a > . The spherical symmetry of the metric leads to the natural choice of local Lorentz frame e = ∂ t , e = ∂ x , e = a ( x ) ∂ θ , e = a ( x )sin θ ∂ ϕ . (2.5)The dual 1-forms are then given by e = dt, e = dx, e = a − ( x ) dθ, e = sin θa ( x ) dϕ. (2.6)The exterior derivatives of the e A ’s are readily computed de = 0 , de = 0 , de = a ′ ( x ) a ( x ) e ∧ e , de = a ′ ( x ) a ( x ) e ∧ e − a ( x ) cot θ e ∧ e . Using (2.3), we then easily get ω = ω = ω = 0 , ω = a ′ ( x ) a ( x ) e , ω = a ′ ( x ) a ( x ) e , ω = − a ( x ) cot θ e , or equivalently ω = ω = ω = 0 , ω = − a ′ ( x ) a ( x ) e , ω = − a ′ ( x ) a ( x ) e , ω = a ( x ) cot θ e . γ A γ B ω AB = (cid:18) − a ′ ( x )2 a ( x ) γ γ (cid:19) e + (cid:18) − a ′ ( x )2 a ( x ) γ γ + a ( x ) cot θ γ γ (cid:19) e = Γ A e A . (2.7)The massless Dirac equation γ A ( ∂ A + Γ A ) φ = 0 on ( M, τ ) thus takes the form (cid:20) γ ∂ t + γ ∂ x + γ (cid:18) a ( x ) ∂ θ − a ′ ( x )2 a ( x ) γ γ (cid:19) + γ (cid:18) a ( x )sin θ ∂ ϕ − a ′ ( x )2 a ( x ) γ γ + a ( x ) cot θ γ γ (cid:19)(cid:21) φ = 0 , or using (2.2) (cid:20) γ ∂ t + γ ∂ x + a ( x ) (cid:18)(cid:18) ∂ θ − cot θ (cid:19) γ + 1sin θ ∂ ϕ γ (cid:19) − a ′ ( x ) a ( x ) γ (cid:21) φ = 0 . We can get rid of some potentials by considering the weighted spinor ψ = a − ( x ) φ. (2.8)Then ψ satifies the equation (cid:20) γ ∂ t + γ ∂ x + a ( x ) (cid:18)(cid:18) ∂ θ − cot θ (cid:19) γ + 1sin θ ∂ ϕ γ (cid:19)(cid:21) ψ = 0 . We finally put this equation under Hamiltonian form. The spinor ψ thus satisfies i∂ t ψ = D ψ, where the Dirac operator D is given by D = γ γ D x + a ( x ) (cid:20)(cid:18) D θ + i cot θ (cid:19) γ γ + 1sin θ D ϕ γ γ (cid:21) , and D x = − i∂ x , D θ = − i∂ θ , D ϕ = − i∂ ϕ . Let us introduce some notations. We denoteΓ = γ γ , Γ = γ γ , Γ = γ γ . From (2.2), it is clear that the Dirac matrices Γ , Γ , Γ satisfy the usual anticommutation relations { Γ i , Γ j } = 2 δ ij , ∀ i, j = 1 , , . (2.9)We choose the following representation for these Dirac matricesΓ = (cid:18) − (cid:19) , Γ = (cid:18) (cid:19) , Γ = (cid:18) i − i (cid:19) . (2.10)We also denote D S = Γ (cid:18) D θ + i cot θ (cid:19) + Γ θ D ϕ , (2.11)which turns out to be an expression of the intrinsic Dirac operator on S . With all these notations, themassless Dirac equation on ( M, τ ) takes its final Hamiltonian form i∂ t ψ = D ψ, D = Γ D x + a ( x ) D S . (2.12)9 emark 2.1. Consider the Riemanniann manifold
Σ = R x × S θ,ϕ equipped with the metric σ = dx + a − ( x ) dω where dω denotes the euclidean metric on S . Using the same Cartan’s orthonormal frameformalism as described above, we could associate to (Σ , σ ) a Dirac operator D σ and consider the associatedDirac equation i∂ t ψ = D σ ψ, (2.13) with ψ given by (2.8).The Dirac equation (2.13) is the one we obtain if we adopt the Schr¨odinger viewpoint, namely if weconsider the evolution of Dirac fields on the fixed D-Riemanniann manifold (Σ , σ ) . On the other hand,the Dirac equation (2.12) is the one we obtain if we adopt the relativistic viewpoint, that is the naturalDirac equation associated to the D-Lorentzian manifold ( M = R t × Σ , τ = dt − σ ) . It turns out thatthe two points of view are equivalent in the sense that the corresponding Dirac operators D and D σ areunitarily equivalent. This can be seen by a direct calculation.To each point of (Σ , σ ) , we associate the orthonormal local frame e = ∂ x , e = a ( x ) ∂ θ , e = a ( x )sin θ ∂ ϕ . Note that the vector fields { e A } A =1 , , satisfy σ ( e A , e B ) = δ AB where δ AB = diag (1 , , is the flat(Riemanniann) D metric. Now following the same procedure as above (still introducing the spinorweight (2.8)), we obtain the following Dirac equation i∂ t ψ = D ψ, D = γ D x + a ( x ) (cid:20) γ (cid:18) D θ + i cot θ (cid:19) + γ θ D ϕ (cid:21) , where the gamma Dirac matrices γ , γ , γ satisfy the anticommutation formulae (2.9). Hence we con-clude that the Dirac equations (2.12) and (2.13) only differ by a choice of equivalent representationof the gamma Dirac matrices satisfying (2.9). But it is well known that such two different choices ofrepresentation lead to unitarily equivalent Dirac operators (see [25]). In this section, we recall the construction of the stationary representation of the scattering matrix S ( λ, n )for a fixed energy λ ∈ R and all angular momentum n ∈ N , (we refer to [1] and [9] for details). Letus consider first the stationary solutions of equation (2.12) restricted to each spin weighted sphericalharmonic, i.e. the solutions of [Γ D x − na ( x )Γ ] ψ = λψ, ∀ n ∈ N ∗ . (3.1)For λ ∈ R , we define the Jost solution from the left F L ( x, λ, n ) and the Jost solution from the right F R ( x, λ, n ) as the 2 × F L ( x, λ, n ) = e i Γ λx ( I + o (1)) , x → + ∞ , (3.2) F R ( x, λ, n ) = e i Γ λx ( I + o (1)) , x → −∞ . (3.3)From (3.1), (3.2) and (3.3), it is easy to see that such solutions (if there exist) must satisfy the integralequations F L ( x, λ, n ) = e i Γ λx − in Γ Z + ∞ x e − i Γ λ ( y − x ) a ( y )Γ F L ( y, λ, n ) dy, (3.4)10 R ( x, λ, n ) = e i Γ λx + in Γ Z x −∞ e − i Γ λ ( y − x ) a ( y )Γ F R ( y, λ, n ) dy. (3.5)Since the potential a belongs to L ( R ), it follows that the integral equations (3.4) and (3.5) are uniquelysolvable by iteration and that k F L ( x, λ, n ) k ≤ e n R + ∞ x a ( s ) ds , k F R ( x, λ, n ) k ≤ e n R x −∞ a ( s ) ds . Since the Jost solutions are fundamental matrices of (3.1), there exists a 2 × A L ( λ, n ) suchthat F L ( x, λ, n ) = F R ( x, λ, n ) A L ( λ, n ). From (3.3) and (3.4), we get the following expression for A L ( λ, n ) A L ( λ, n ) = I − in Γ Z R e − i Γ λy a ( y )Γ F L ( y, λ, n ) dy. (3.6)Moreover, the matrix A L ( λ, n ) satisfies the following equality (see [1], Proposition 2.2) A ∗ L ( λ, n )Γ A L ( λ, n ) = Γ , ∀ λ ∈ R , n ∈ N . (3.7)Using the notation A L ( λ, n ) = (cid:20) a L ( λ, n ) a L ( λ, n ) a L ( λ, n ) a L ( λ, n ) (cid:21) , (3.8)the equality (3.7) can be written in components as | a L ( λ, n ) | − | a L ( λ, n ) | = 1 , | a L ( λ, n ) | − | a L ( λ, n ) | = 1 ,a L ( λ, n ) a L ( λ, n ) − a L ( λ, n ) a L ( λ, n ) = 0 . (3.9)The matrices A L ( λ, n ) encode all the scattering information of equation (3.1). In particular, it is shownin [1] that the scattering matrix S ( λ, n ) has the representation S ( λ, n ) = (cid:20) T ( λ, n ) R ( λ, n ) L ( λ, n ) T ( λ, n ) (cid:21) , (3.10)where T ( λ, n ) = 1 a L ( λ, n ) , R ( λ, n ) = − a L ( λ, n ) a L ( λ, n ) , L ( λ, n ) = a L ( λ, n ) a L ( λ, n ) . (3.11) Remark 3.1.
It follows from (3.4) that if we define the new potential ˜ a ( x ) = a ( x + c ) , the associatedJost solutions satisfy ˜ F L ( x, λ, n ) = F L ( x + c, λ, n ) e − i Γ λc . (3.12) Hence, it follows from (3.6) that (with obvious notations) ˜ A L ( λ, n ) = e i Γ λc A L ( λ, n ) e − i Γ λc , (3.13) and so, using (3.10) and (3.11), we conclude that ˜ S ( λ, n ) = e i Γ λc S ( λ, n ) e − i Γ λc , (3.14) or in components (cid:20) ˜ T ( λ, n ) ˜ R ( λ, n )˜ L ( λ, n ) ˜ T ( λ, n ) (cid:21) = (cid:20) T ( λ, n ) e iλc R ( λ, n ) e − iλc L ( λ, n ) T ( λ, n ) (cid:21) . (3.15) Hence the transmission coefficients T ( λ, n ) are invariant under any radial translations of the potential a , whereas the reflection coefficients L ( λ, n ) and R ( λ, n ) are invariant under the discrete set of radialtranslations ˜ a ( x ) = a ( x + kπλ ) for k ∈ Z and λ = 0 . n ∈ N to take complex values z and study the analytic properties of the above scattering data with respect to z ∈ C .Precisely, it was shown in [9] that we can define for z ∈ C , the Jost solutions F L ( x, λ, z ) and F R ( x, λ, z )which are the unique solutions of the stationary equation[Γ D x − za ( x )Γ ] ψ = λψ, ∀ z ∈ C . (3.16)with the asymptotics (3.2) and (3.3). Similarly, we can define the matrix A L ( λ, z ) for all z ∈ C . Allthese matrix-functions are analytic in the complex variable z ∈ C . Moreover, they satisfy the followingproperties: Lemma 3.2. (i) Set A = Z R a ( x ) dx . Then | a L ( λ, z ) | , | a L ( λ, z ) | ≤ cosh( A | z | ) , ∀ z ∈ C , (3.17) | a L ( λ, z ) | , | a L ( λ, z ) | ≤ sinh( A | z | ) , ∀ z ∈ C . (3.18) (ii) The functions a L ( λ, z ) and a L ( λ, z ) are entire and even in z whereas the functions a L ( λ, z ) and a L ( λ, z ) are entire and odd in z . Moreover they satisfy the symmetries a L ( λ, z ) = a L ( λ, ¯ z ) , ∀ z ∈ C , (3.19) a L ( λ, z ) = a L ( λ, ¯ z ) , ∀ z ∈ C . (3.20) (iii) The following relations hold for all z ∈ C a L ( λ, z ) a L ( λ, ¯ z ) − a L ( λ, z ) a L ( λ, ¯ z ) = 1 , (3.21) a L ( λ, z ) a L ( λ, ¯ z ) − a L ( λ, z ) a L ( λ, ¯ z ) = 1 . (3.22)At this stage, we have proved that the components of the matrix A L ( λ, z ) are entire functions ofexponential type in the variable z . Precisely, from (3.17) and (3.18), we have | a Lj ( λ, z ) | ≤ e A | z | , ∀ z ∈ C , j = 1 , .., . (3.23)Using the relations (3.21), (3.22) and the parity properties of the a Lj ( λ, z ), we can improve these estimates(see Lemma 3.4. in [9]). Lemma 3.3.
Let λ ∈ R be fixed. Then for all z ∈ C | a Lj ( λ, z ) | ≤ e A | Re ( z ) | , j = 1 , .., . (3.24)It follows from Lemma 3.3 that the functions z → a Lj ( λ, z ) belong to the Nevanlinna class in theright half-plane (see for instance [22] for a definition). We emphasize that this property is the key pointto prove Theorem 1.1 (see [9]).Similarly, if we use the notation F L ( x, λ, z ) = (cid:20) f L ( x, λ, z ) f L ( x, λ, z ) f L ( x, λ, z ) f L ( x, λ, z ) (cid:21) , F R ( x, λ, z ) = (cid:20) f R ( x, λ, z ) f R ( x, λ, z ) f R ( x, λ, z ) f R ( x, λ, z ) (cid:21) , we have the corresponding estimates for the Jost functions f Lj ( x, λ, z ) and f Rj ( x, λ, z ) for j = 1 , . . . , emma 3.4. For all j = 1 , .., and for all x ∈ R , | f Lj ( x, λ, z ) | ≤ C e | Re ( z ) | R ∞ x a ( s ) ds , (3.25) | f Rj ( x, λ, z ) | ≤ C e | Re ( z ) | R x −∞ a ( s ) ds . (3.26)Finally, we shall need later the asymptotic expansion of the scattering data when the angular mo-mentum z → + ∞ , z real. The main tool to obtain these asymptotics easily is a simple change of variable X = g ( x ), called the Liouville transformation which we precise here. Let us define X = g ( x ) = Z x −∞ a ( t ) dt. (3.27)Clearly, since a > g : R → ]0 , A [ is a C -diffeomorphism where A = Z R a ( t ) dt. (3.28)In what follows, we denote by h = g − the inverse diffeomorphism of g and we use the notation f ′ ( X ) = ∂f∂X ( X ). We also define for j = 1 , ...,
4, and for X ∈ ]0 , A [, f j ( X, λ, z ) = f Lj ( h ( X ) , λ, z ) , (3.29) g j ( X, λ, z ) = f Rj ( h ( X ) , λ, z ) . (3.30)Observe at last that in the variable X , Lemma 3.4 can be written as ∀ z > , | f j ( X, λ, z ) | ≤ C e z ( A − X ) , | g j ( X, λ, z ) | ≤ C e zX . (3.31)The interest in introducing the variable X is that the components f j ( X, λ, z ) and g j ( X, λ, z ) of theJost solutions satisfy now singular Sturm-Liouville differential equations in the variable X , in which thecomplex angular momentum z plays the role of the spectral parameter. More precisely, we have thefollowing lemma. Lemma 3.5.
1. For j = 1 , , f j ( X, λ, z ) and g j ( X, λ, z ) satisfy on ]0 , A [ the Sturm-Liouville equation y ′′ + q ( X ) y = z y. (3.32)
2. For j = 3 , , f j ( X, λ, z ) and g j ( X, λ, z ) satisfy on ]0 , A [ the Sturm-Liouville equation y ′′ + q ( X ) y = z y, (3.33) where the potential q ( X ) = λ h ′ ( X ) − iλh ′′ ( X ) = λ a ( x ) + iλ a ′ ( x ) a ( x ) , has the asymptotics q ( X ) − ω − X = O (1) , X → , with ω − = λ κ − + i λκ − , (3.34) q ( X ) − ω + ( A − X ) = O (1) , X → A , with ω + = λ κ + i λκ + . (3.35)13 short glance at Lemma 3.5 suggests that the Jost functions f j and g j can be constructed as smallperturbations of usual modified Bessel functions I ν ( z ( A − X )) and I µ ( zX ) for suitable µ, ν . This was donein details in [9]. As a consequence of this construction and using the well known asymptotic expansionof the modified Bessel functions, the large z asymptotics of the scattering data a Lj ( λ, z ) were calculatedin [9]. More precisely, if we set ν + = 12 − i λκ + , µ − = 12 + i λκ − , (3.36)the following asymptotics hold. Theorem 3.6.
1. For X ∈ ]0 , A [ fixed and z ∈ S θ where S θ = { z ∈ C , | arg( z ) | ≤ θ } for a given < θ < π , we havefor the Jost solutions f ( X ) and g ( X ) f ( X, λ, z ) = 2 − ν + √ π ( − κ + a + ) iλκ + Γ(1 − ν + ) z − iλκ + e z ( A − X ) (cid:16) O ( 1 z ) (cid:17) . (3.37) g ( X, λ, z ) = i − µ − √ π ( κ − a − ) − iλκ − Γ(1 − µ − ) z iλκ − e zX (cid:16) O ( 1 z ) (cid:17) . (3.38)
2. For the scattering data a L ( λ, z ) and a L ( λ, z ) , we have a L ( λ, z ) = 12 π (cid:18) − κ + a + (cid:19) iλκ + (cid:18) κ − a − (cid:19) − iλκ − Γ (cid:18) − iλκ − (cid:19) Γ (cid:18)
12 + iλκ + (cid:19) × (cid:16) z (cid:17) iλ ( κ − − κ + ) e zA (cid:18) O ( 1 z ) (cid:19) , (3.39) a L ( λ, z ) = i π (cid:18) − κ + a + (cid:19) iλκ + (cid:18) κ − a − (cid:19) iλκ − Γ (cid:18)
12 + iλκ − (cid:19) Γ (cid:18)
12 + iλκ + (cid:19) × (cid:16) z (cid:17) − iλ ( κ + + κ − ) e zA (cid:18) O ( 1 z ) (cid:19) .
3. For the scattering coefficients T ( λ, z ) and L ( λ, z ) , we have T ( λ, z ) = 2 π (cid:16) − a + κ + (cid:17) iλκ + (cid:16) a − κ − (cid:17) − iλκ − Γ (cid:16) − iλκ − (cid:17) Γ (cid:16) + iλκ + (cid:17) (cid:16) z (cid:17) iλ ( κ + − κ − ) e − zA (cid:18) O ( 1 z ) (cid:19) ,L ( λ, z ) = i (cid:18) κ − a − (cid:19) iλκ − Γ (cid:16) + iλκ − (cid:17) Γ (cid:16) − iλκ − (cid:17) (cid:16) z (cid:17) − iλκ − (cid:18) O ( 1 z ) (cid:19) . (3.40) ( i ) ⇒ ( ii ) . Assume that L ( λ, n ) = ˜ L ( λ, n ) + O (cid:0) e − nB (cid:1) , n → + ∞ . Our first step is to extend these asymptotics(which are true for n integer → + ∞ ) to the case of z → + ∞ ( z real and positive). To do this, we shalluse some well-known uniqueness results for Laplace transforms obtained in [12, 23].14e begin with an elementary result for functions of the complex variable belonging to the Hardyclass. We recall (see for instance [16], Lecture 19) that the Hardy class H is the set of analytic functions F in the right half-plane Ω = { z ∈ C , Re z > } , satisfying the conditionsup x> Z R | F ( x + iy ) | dy < ∞ , (4.1)and equipped with the norm || F || = (cid:18) sup x> Z R | F ( x + iy ) | dy (cid:19) . (4.2)The Paley-Wiener Theorem asserts that a function F ( z ) belongs to the Hardy space H if and only ifthere exists a function f ∈ L (0 , + ∞ ) such that F ( z ) = 1 √ π Z + ∞ e − tz f ( t ) dt , ∀ z ∈ Ω . (4.3)Moreover, we have || F || = || f || L (0 , ∞ ) . (4.4)Let us also recall a uniqueness result for Laplace transforms given in [12], Prop. 2.4., (see also [23]for a continuous version): Proposition 4.1.
Let f ∈ L (0 , a ) . If for all ǫ > , Z a e − nt f ( t ) dt = O ( e − an (1 − ǫ ) ) , n → + ∞ , then f = 0 a.e. We now put together all the previous results and prove the following Proposition.
Proposition 4.2.
Let F be a function in the Hardy class H . Assume that for some B > , we have F ( n ) = O (cid:0) e − Bn (cid:1) , n → + ∞ , ( n integer). Then, | F ( z ) | ≤ || F ||√ πRez e − BRez , ∀ z ∈ Ω . (4.5) Proof.
For n ∈ N , the Paley-Wiener theorem and the Cauchy-Schwarz inequality imply Z B e − nt f ( t ) dt = √ π F ( n ) − Z + ∞ B e − nt f ( t ) dt = √ π F ( n ) + O (cid:0) e − nB (cid:1) = O (cid:0) e − nB (cid:1) . So, Proposition 4.1 entails that f = 0 a.e in (0 , B ). Using (4.3) again and (4.4), we obtain at once(4.5).Let us give a direct consequence (which could be certainly improved) of the previous result to ourinverse problem. 15 roposition 4.3. Assume that the reflection coefficients L ( λ, n ) and ˜ L ( λ, n ) satisfy for some < B < min( A, ˜ A ) , L ( λ, n ) = ˜ L ( λ, n ) + O ( e − nB ) , n → + ∞ , n integer . Then L ( λ, z ) = ˜ L ( λ, z ) + O ( √ z e − zB ) , z → + ∞ , z real . (4.6) Proof.
First, let us recall that L ( λ, n ) = a L ( λ, n ) a L ( λ, n ) , ˜ L ( λ, n ) = ˜ a L ( λ, n )˜ a L ( λ, n ) . (4.7)Using Lemma 3.3, we obtain immediately a L ( λ, n )˜ a L ( λ, n ) − ˜ a L ( λ, n ) a L ( λ, n ) = O ( e n ( A + ˜ A − B ) ) . (4.8)For z ∈ Ω, we set F ( z ) = a L ( λ, z )˜ a L ( λ, z ) − ˜ a L ( λ, z ) a L ( λ, z ) z + 1 e − z ( A + ˜ A ) . (4.9)Clearly, F is holomorphic in Ω, and by Lemma 3.3, we have | F ( z ) | ≤ | z + 1 | . (4.10)It follows that F ∈ H and by (4.8), we have F ( n ) = O ( e − nB ). Using Proposition 4.2, we see that F ( z ) = O ( z − e − zB ), z → + ∞ . For z >
0, we write L ( λ, z ) − ˜ L ( λ, z ) = ( z + 1) e z ( A + ˜ A ) a L ( λ, z ) ˜ a L ( λ, z ) F ( z ) . (4.11)We conclude the proof using (3.39).This concludes the first step of the proof of Theorem 1.3, ( i ) ⇒ ( ii ). The second step of the proofconsists in an adaption of the strategy used to prove the local Borg-Marchenko Theorem in [23, 2] forone-dimensional Schr¨odinger operators to our setting of Dirac operators on a SSAHM. This strategy isrelatively close to the proof of Theorem 1.1 given in [9], itself inspired by [10]. Let us introduce for X ∈ ]0 , B [ the matrix P ( X, λ, z ) = (cid:18) P ( X, λ, z ) P ( X, λ, z ) P ( X, λ, z ) P ( X, λ, z ) (cid:19) , defined by P ( X, λ, z ) ˜ F R (˜ h ( X ) , λ, z ) = F R ( h ( X ) , λ, z ) , (4.12)where F R = ( f Rk ) and ˜ F R = ( ˜ f Rk ) are the Jost solutions from the right associated with a ( x ) and ˜ a ( x ).To simplify the notations, for k = 1 , ...,
4, we set as previously: f k ( X, λ, z ) = f Lk ( h ( X ) , λ, z ) , ˜ f k ( X, λ, z ) = ˜ f Lk (˜ h ( X ) , λ, z ) ,g k ( X, λ, z ) = f Rk ( h ( X ) , λ, z ) , ˜ g k ( X, λ, z ) = ˜ f Rk (˜ h ( X ) , λ, z ) . Using that det F R = 1 and det ˜ F R = 1, we obtain the following equalities : (cid:26) P ( X, λ, z ) = g ˜ g − g ˜ g ,P ( X, λ, z ) = − g ˜ g + g ˜ g . (4.13)16t follows from (4.13) and the analytical properties of the Jost functions that, for j = 1 ,
2, the applications z → P j ( X, λ, z ) are analytic on C and of exponential type. Moreover, by Lemma 3.4, these applicationsare bounded on the imaginary axis i R .We shall now prove that the applications z → P j ( X, λ, z ) are also bounded on the real axis. Todo this, we first perform some elementary algebraic transformations on P j ( X, λ, z ). Since F L ( x, λ, z ) = F R ( x, λ, z ) A L ( λ, z ), we easily get for z > g = f a L − L ( λ, z ) g , (4.14)˜ g = ˜ f ˜ a L − ˜ L ( λ, z )˜ g . (4.15)Thus, P ( X, λ, z ) = ( ˜ L ( λ, z ) − L ( λ, z )) g ˜ g + f ˜ g a L − ˜ f g ˜ a L ! . (4.16)Using (3.31) and (3.39), it is easy to see that the function z → f ˜ g a L − ˜ f g ˜ a L ! is bounded on R + for allfixed X ∈ ]0 , A [. Moreover, (3.31) and Proposition 4.3 imply | ( ˜ L ( λ, z ) − L ( λ, z )) g ˜ g | ≤ C √ ze − z ( B − X ) , (4.17)and thus, this term remains bounded when z → + ∞ for all X ∈ ]0 , B [. Summarizing, for all fixed X ∈ ]0 , B [, the function z → P ( X, λ, z ) is bounded on R + .Similarly, we have P ( X, λ, z ) = ( ˜ L ( λ, z ) − L ( λ, z )) g ˜ g + ˜ f g ˜ a L − f ˜ g a L ! , (4.18)and using the same arguments as above, we obtain that, for all fixed X ∈ ]0 , B [, z → P ( X, λ, z ) isbounded on R + .Clearly, these last results remain true on R by an elementary parity argument. Finally, applying thePhragmen-Lindel¨of’s Theorem ([3], Thm 1.4.2.) on each quadrant of the complex plane, we deduce that z → P j ( X, λ, z ) is bounded on C . By Liouville’s Theorem, and a standard continuity argument in thevariable X , we have thus obtained P j ( X, λ, z ) = P j ( X, λ, , ∀ z ∈ C , ∀ X ∈ ]0 , B ] . (4.19)Now, we return to the definition of P j ( X, λ, z ) for z = 0. We observe first that F R ( x, λ,
0) = e iλ Γ x and similarly ˜ F R ( x, λ,
0) = e iλ Γ x . This is immediate from the definition of the Jost functions. Thus wededuce from (4.12) that P ( X, λ,
0) = e iλ ( h ( X ) − ˜ h ( X )) Γ . (4.20)Then, putting (4.20) and (4.19) into (4.12) we get ( ˜ g ( X, λ, z ) = e iλ (˜ h ( X ) − h ( X )) g ( X, λ, z ) , ˜ g ( X, λ, z ) = e iλ (˜ h ( X ) − h ( X )) g ( X, λ, z ) . (4.21)17y Lemma 4.2 in [9], the Wronskians W ( g , g ) = W (˜ g , ˜ g ) = iz . Then, a straightforward calculationgives e iλ (˜ h ( X ) − h ( X )) = 1 . (4.22)Thus, by a standard continuity argument, there exists k ∈ Z such that˜ h ( X ) = h ( X ) + kπλ , ∀ X ∈ ]0 , B ] . (4.23)Note that, for the particular choice X = B , we obtain ˜ h ( B ) = h ( B ) + kπλ . Let us differentiate (4.23)with respect to X . We obtain easily 1 a (˜ h ( X )) = 1 a ( h ( X )) , (4.24)and using again (4.23), we have a ( x ) = ˜ a ( x + kπλ ) , ∀ x ∈ ] − ∞ , h ( B )] . (4.25)Thus, we have proved the first part of Theorem 1.3. ✷ ( ii ) ⇒ ( i ) . Let us assume there exists k ∈ Z such that a ( x ) = ˜ a ( x + kπλ ) , ∀ x ≤ h ( B ). It follows immediatelyfrom the definition of the diffeomorphisms h and ˜ h , that ˜ h ( B ) = h ( B ) + kπλ . Moreover, if we set ˘ a ( x ) =˜ a ( x + kπλ ) , ∀ x ∈ R , and using (3.15), we see that (with obvious notation),˘ L ( λ, n ) = e − iλ kπλ ˜ L ( λ, n ) = ˜ L ( λ, n ) . (4.26)Thus, it remains to prove the implication ( ii ) ⇒ ( i ) in the case k = 0. Now, let us begin with an obviouslemma (whose proof is omitted) : Lemma 4.4.
Assume that a ( x ) = ˜ a ( x ) , ∀ x ≤ h ( B ) = ˜ h ( B ) . Then, a − = ˜ a − , κ − = ˜ κ − . (4.27) and g j ( X, λ, z ) = ˜ g j ( X, λ, z ) , ∀ X ≤ B, ∀ j = 1 , . . . . (4.28)Using again the relation F L ( x, λ, z ) = F R ( x, λ, z ) A L ( λ, z ) and (4 . z > X ≤ B , f a L = g + L ( λ, z ) g , ˜ f ˜ a L = g + ˜ L ( λ, z ) g . (4.29)For z > g = 0. So, for such z , we can write L ( λ, z ) − ˜ L ( λ, z ) = 1 g f a L − ˜ f ˜ a L ! . (4.30)Now, using Theorem 3.6, we obtain easily: L ( λ, z ) − ˜ L ( λ, z ) = O (cid:0) e − zX (cid:1) , ∀ X ∈ ]0 , B ] , (4.31)and taking X = B , the proof is complete. ✷ .3 Proof of Theorem 1.3, ( iii ) ⇔ ( iv ) . The local uniqueness result for the reflection coefficient R ( λ, n ) is actually a by-product of the previousone using the following trick. If we set a ⋆ ( x ) = a ( − x ), a straightforward calculation using (3.4) - (3.5)shows that the associated Jost solutions satisfy (cid:26) F ⋆R ( x, λ, n ) = F L ( − x, − λ, − n ) ,F ⋆L ( x, λ, n ) = F R ( − x, − λ, − n ) . (4.32)It follows immediately that A ⋆L ( λ, n ) = A − L ( − λ, − n ) which implies the equality R ⋆ ( λ, n ) = − L ( − λ, n ).Thus, it suffices to use the previous result for the reflection coefficients L , with λ replaced by − λ , toprove the equivalence ( iii ) ⇔ ( iv ) of Theorem 1.3. ✷ Assume that T ( λ, n ) = ˜ T ( λ, n ) + O (cid:0) e − nB (cid:1) , (4.33)with B > max(
A, A ′ ). Using the asymptotics in Theorem 3.6, we obtain A = ˜ A and a L ( λ, n ) − ˜ a L ( λ, n ) = O (cid:16) e − n ( B − A ) (cid:17) . (4.34)Now, we set for z ∈ Ω, F ( z ) = a L ( λ, z ) − ˜ a L ( λ, z ) z + 1 e − zA . (4.35)As previously, we see that F belongs to the Hardy space H and F ( n ) = O (cid:0) e − (2 B − A ) n (cid:1) . By Proposition4.2, we have | F ( z ) | ≤ || F ||√ πRez e − (2 B − A ) Rez , ∀ z ∈ Ω . (4.36)It follows that there exists C > z > | a L ( λ, z ) − ˜ a L ( λ, z ) | ≤ C √ z e − z ( B − A ) . (4.37)Thus, f ( z ) := a L ( λ, z ) − ˜ a L ( λ, z ) is bounded on R + . Moreover, this function is of exponential type, andbounded on i R . The Phragmen - Lindel¨of Theorem implies that f is bounded on Ω and consequently, isalso bounded on C using parity arguments. Hence Liouville’s Theorem entails that f ( z ) = f (0) = 0. Weconclude the proof using Proposition B.5. ✷ . In this Section, we adapt the previous local inverse uniqueness results to the setting of general relativityand more precisely to Reissner-Nordstr¨om-de-Sitter black holes. We emphasize that the link betweensuch black holes and SSAHM was already given in [9]. Considering the scattering of massless Dirac fieldsevolving in the outer region of a RN-dS black holes, we shall prove that the partial knowledge of thecorresponding reflection coefficients in the sense of (1.10) - (1.11) not only determines the metric of suchblack holes in the neighbourhood of the event and cosmological horizons (see below for the definition),but in fact determines the whole metric. This is due to the fact that the metric of RN-dS black holesonly depend on 3 parameters - their mass, electric charge and positive cosmogical constant - parametersthat can be deduced from the explicit form of the metric in the neighbourhoods of the horizons.19 .1 Reissner-Nordst¨om-de-Sitter black holes
Refering to Wald [26] for more general details on black hole spacetimes, we summarize here the essentialfeatures of Reissner-Nordstr¨om-de-Sitter (RN-dS) black holes given in [8, 9]. First, RN-dS are spheri-cally symmetric electrically charged exact solutions of the Einstein-Maxwell equations. In Schwarzschildcoordinates, the exterior region of a RN-dS black hole is described by the four-dimensional manifold M = R t × ] r − , r + [ r × S θ,ϕ equipped with the Lorentzian metric τ = F ( r ) dt − F ( r ) − dr − r (cid:0) dθ + sin θ dϕ (cid:1) , (5.1)where F ( r ) = 1 − Mr + Q r − Λ3 r . (5.2)The constants M > Q ∈ R appearing in (5.2) are interpreted as the mass and the electric charge of theblack hole and Λ > F ( r )has three simple positive roots 0 < r c < r − < r + and a negative one r n <
0. This is always achieved ifwe suppose for instance that Q < M and that Λ M be small enough (see [15]). The sphere { r = r c } is called the Cauchy horizon whereas the spheres { r = r − } and { r = r + } are the event and cosmologicalhorizons respectively. These horizons which appear as singularities of the metric (5.1) are in fact merecoordinates singularities. This means that using appropriate coordinates system, these horizons can beunderstood as regular null hypersurfaces that can be crossed one way but would require speeds greaterthan that of light to be crossed the other way: hence their names horizons.In what follows, we shall only consider the exterior region of the black hole, that is the region { r −
Using the notations of the Theorem 1.3, the following assertions are equivalents : ( i ) L ( λ, n ) = ˜ L ( λ, n ) + O (cid:0) e − nB (cid:1) . ( ii ) R ( λ, n ) = ˜ R ( λ, n ) + O (cid:0) e − nB (cid:1) . ( iii ) M = ˜ M , Q = ˜ Q and Λ = ˜Λ . ( iv ) ∃ k ∈ Z , a ( x ) = ˜ a ( x + kπλ ) , ∀ x ∈ R . Proof.
We first use Theorem 1.3 to obtain the equality of the potential a and ˜ a on a half-line ] − ∞ , b ](respectively [ b , + ∞ [), b < b ∈ R . Then, a line by line inspection of the proof given in [9] p. 43-44shows that this information is enough to prove the uniqueness of the mass M , the square of the charge Q and the cosmological constant Λ of the black hole. Finally, since the parameters of the black holedetermine uniquely the metric, we obtain the equality of the potentials a and ˜ a on R (up to a discreteset of translations as stated in (iv)). 21 Other formulation of the main inverse uniqueness results
In this Section, we formulate our main Theorems 1.3 in a more global way, avoiding the use of a decom-position onto generalized spherical harmonics. More precisely, we replace the main assumptions (1.10)and (1.11) by L ( S )-operator norms conditions on the global reflection coefficients. We recall first thedefinition and essential properties of these operators. Proposition A.1.
For all ( n, k ) ∈ I where I = { n ∈ N ∗ , k ∈ / Z , | k | ≤ n − } , we use the notation Y kn = ( Y kn , Y kn ) for the corresponding generalized spherical harmonics. Then,1) The families { Y kn } ( n,k ) ∈ I and { Y kn } ( n,k ) ∈ I form Hilbert bases of l = L ( S ; C ) ; precisely for all ψ ∈ l ,we can decompose ψ as ψ = X n,k ∈ I ψ jkn Y jkn , j = 1 , , with k ψ k = 12 X n,k ∈ I | ψ jkn | .
2) Let λ ∈ R be a fixed energy. Then, the transmission operators T L ( λ ) and T R ( λ ) are defined as operatorsfrom l to l as follows. For all ψ = P n,k ∈ I ψ jkn Y jkn ∈ l T L ( λ ) ψ = T L ( λ ) X n,k ∈ I ψ kn Y kn ( λ ) = X n,k ∈ I (cid:0) T ( λ, n ) ψ kn (cid:1) Y kn ( λ ) , (A.1) and T R ( λ ) ψ = T R ( λ ) X n,k ∈ I ψ kn Y kn ( λ ) = X n,k ∈ I (cid:0) T ( λ, n ) ψ kn (cid:1) Y kn ( λ ) , (A.2) where T ( λ, n ) are the transmission coefficients defined in (3.10) - (3.11). In short, we write T L ( λ ) Y kn = T ( λ, n ) Y kn , T R ( λ ) Y kn = T ( λ, n ) Y kn , ∀ n, k ∈ I, (A.3) and thus the operators T L ( λ ) (resp. T R ( λ ) ) are diagonalizable on the Hilbert basis of eigenfunctions ( Y kn ) k,n ∈ I (resp. ( Y kn ) k,n ∈ I ) associated to the eigenvalues T ( λ, n ) (in both cases).3) Let λ ∈ R be a fixed energy. Then, the reflection operators L ( λ ) and R ( λ ) are defined as operatorsfrom l to l as follows. For all ψ = P n,k ∈ I ψ jkn Y jkn ∈ l R ( λ ) ψ = R ( λ ) X n,k ∈ I ψ kn Y kn ( λ ) = X n,k ∈ I (cid:0) R ( λ, n ) ψ kn (cid:1) Y kn ( λ ) , (A.4) and L ( λ ) ψ = L ( λ ) X n,k ∈ I ψ kn Y kn ( λ ) = X n,k ∈ I (cid:0) L ( λ, n ) ψ kn (cid:1) Y kn ( λ ) , (A.5) where R ( λ, n ) and L ( λ, n ) are defined in (3.10) - (3.11). In short, we write R ( λ ) Y kn = R ( λ, n ) Y kn , L ( λ ) Y kn = L ( λ, n ) Y kn , ∀ n, k ∈ I. (A.6)22t is immediate from the above definitions to express the L ( S )-operator norms of the transmission op-erators T L ( λ ) , T R ( λ ) and of the reflection operators L ( λ ) , R ( λ ) in terms of the coefficients T ( λ, n ) , L ( λ, n )and R ( λ, n ). For a fixed λ ∈ R , we have k T L ( λ ) k = k T R ( λ ) k = k T ( λ, n ) k ∞ , k L ( λ ) k = k L ( λ, n ) k ∞ , k R ( λ ) k = k R ( λ, n ) k ∞ . (A.7)To reformulate the assumptions (1.10) and (1.11) by L ( S )-operator norms conditions, we observethat the selfadjoint operator | D S | acts as multiplication by n on each generalized spherical harmonics Y kn in the Hilbert decomposition L ( S , C ) = ⊕ n,k ∈ I C ⊗ Y kn . We still denote by | D S | the restrictionof this operator to l = L ( S , C ) and thus, | D S | acts as multiplication by n on each generalized sphericalharmonics Y kn or Y kn in the two Hilbert decompositions l = ⊕ n,k ∈ I C ⊗ Y jkn , j = 1 , λ ∈ R and 0 < B < min( A, ˜ A ). Then, using (A.7), the assumptions (1.10) and (1.11) for thereflection coefficients can be written as(1 . ⇐⇒ (cid:13)(cid:13)(cid:13) e B | D S | (cid:16) L ( λ ) − ˜ L ( λ ) (cid:17)(cid:13)(cid:13)(cid:13) B ( l ) = O (1) , (A.8)and (1 . ⇐⇒ (cid:13)(cid:13)(cid:13) e B | D S | (cid:16) R ( λ ) − ˜ R ( λ ) (cid:17)(cid:13)(cid:13)(cid:13) B ( l ) = O (1) . (A.9)Similarly, let B > max( A, ˜ A ). Then we get for the assumption (1.12) on the transmission coefficients theequivalences (1 . ⇐⇒ (cid:13)(cid:13)(cid:13) e B | D S | (cid:16) T L ( λ ) − ˜ T L ( λ ) (cid:17)(cid:13)(cid:13)(cid:13) B ( l ) = O (1) , (A.10) ⇐⇒ (cid:13)(cid:13)(cid:13) e B | D S | (cid:16) T R ( λ ) − ˜ T R ( λ ) (cid:17)(cid:13)(cid:13)(cid:13) B ( l ) = O (1) . B Addendum on the inverse scattering problem from the trans-mission coefficients T ( λ, n ) In [9], Theorem 1.1, it is claimed that the knowledge of the transmission coefficients T ( λ, n ) for a fixed λ = 0 and for all n ∈ L where L is a subset of N satisfying the M¨untz condition X n ∈L n = + ∞ , also determines uniquely the function a ( x ) up to a translation. The crucial ingredient of the proof canbe found in the Proposition 3.13 of [9] which states that”If T ( λ, n ) = ˜ T ( λ, n ) for all n ∈ L , then the corresponding reflection coefficients L ( λ, n ) and ˜ L ( λ, n )(resp. R ( λ, n ) and ˜ R ( λ, n )) coincide up to a multiplicative constant”.The proof of this result given in [9] is unfortunately incomplete. In fact, this last point is not so clearand could even be false. We shall try in this Appendix to give some insights of what happens when wetry to determine the metric from the transmission coefficient T ( λ, n ).We first give a correct version of the above result that is weaker than the Proposition 3.13. given in[9]. 23 roposition B.1. Let (Σ , g ) ans ( ˜Σ , ˜ g ) be two SSAHM whose metrics depend on the functions a ( x ) and ˜ a ( x ) satisfying the assumptions (1.1) - (1.2). For a fixed energy λ = 0 , consider the correspondingcountable family of transmission coefficients T ( λ, n ) and ˜ T ( λ, n ) for all n ∈ N ∗ . Consider also a subset L of N ∗ that satisfies a M¨untz condition X n ∈L n = ∞ . Assume that T ( λ, n ) = ˜ T ( λ, n ) , ∀ n ∈ L . Then T ( λ, z ) = ˜ T ( λ, z ) , ∀ z ∈ C . Assume moreover that κ + + κ − < . • If κ + + κ − < , there exists a rational function g ( z ) such that L ( λ, z ) = g ( z ) ˜ L ( λ, z ) , ∀ z ∈ C . • If κ + + κ − > and (cid:16) ˜ a − a + (cid:17) iλκ + = (cid:16) ˜ a + a − (cid:17) iλκ − , there exists a rational function h ( z ) such that L ( λ, z ) = h ( z ) ˜ R ( λ, z ) , ∀ z ∈ C . Proof.
By definition of the transmission coefficients and using Corollary 3.9. and Theorem 3.10. in [9],our assumption implies that T ( λ, z ) = ˜ T ( λ, z ) , ∀ z ∈ C , or equivalently a L ( λ, z ) = ˜ a L ( λ, z ) , ∀ z ∈ C . (B.1)Now, we set f ( z ) = a L ( λ, z ) z . Using that a L ( λ,
0) = 0, we see that f ( z ) is an even entire functionof order 1 thanks to Lemma 3.2. Thus, we can write f ( z ) = g ( z ) where g is an entire function of order . Using Hadamard’s factorization Theorem, we obtain the following expression for ff ( z ) = G z m ∞ Y n =1 (cid:16) − z z n (cid:17) , (B.2)where 2 m is the multiplicity of 0, G is a constant and the z n are the zeros of f belonging to C + = { z ∈ C , ℑ ( z ) > , or ℑ ( z ) = 0 , ℜ ( z ) > } counted according to their multiplicity. From (B.1) and Lemma3.2, (iii), we have f ( z ) f (¯ z ) = ˜ f ( z ) ˜ f (¯ z ) , where ˜ f ( z ) = ˜ a L ( λ, z ) z . Thus we get | G | z m ∞ Y n =1 (cid:16) − z z n (cid:17)(cid:16) − z ¯ z n (cid:17) = | ˜ G | z m ∞ Y n =1 (cid:16) − z ˜ z n (cid:17)(cid:16) − z ¯˜ z n (cid:17) . It follows that | G | = | ˜ G | , m = ˜ m and z n = ± ˜ z n or z n = ± ˜ z n , ∀ n ∈ N ∗ . (B.3)24 emark B.2.
1) The equation (B.3) is where we made an error in [9], Proposition 3.13. since weasserted that z n = ˜ z n , ∀ n ∈ N ∗ .
2) If ℑ ( z n ) > , then we must have z n = ˜ z n or z n = − ˜ z n , ∀ n ∈ N ∗ . (B.4) Hence, the zeros z n and ˜ z n with positive imaginary parts coincide up to ”-” complex conjugation .On the other hand, if ℑ ( z n ) = 0 and ℜ ( z n ) > , then z n = ˜ z n , or z n = ˜ z n , (B.5) holds. In some cases, we can prove that the large zeros z n and ˜ z n coincide using the asymptotics of a L ( λ, z )for large z in the complex plane. We shall use Lemma B.3.
For | z | large in the complex plane, we have a L ( λ, z ) = π (cid:16) − κ + a + (cid:17) iλκ + (cid:16) κ − a − (cid:17) − iλκ − Γ(1 − ν + )Γ(1 − µ − ) (cid:0) z (cid:1) iλ (cid:16) κ − − κ + (cid:17) × (cid:18) e zA + e − zA e − sg(Im( z )) πλ (cid:16) κ + − κ − (cid:17) (cid:19) (cid:0) O ( z ) (cid:1) ,a L ( λ, z ) = i π (cid:16) − κ + a + (cid:17) iλκ + (cid:16) κ − a − (cid:17) iλκ − Γ(1 − ν + )Γ(1 − ν − ) (cid:0) z (cid:1) − iλ (cid:16) κ − + κ + (cid:17) × (cid:18) e zA + e − zA e sg(Im( z )) iπ (cid:16) iλ (cid:16) κ + + κ − (cid:17)(cid:17) (cid:19) (cid:0) O ( z ) (cid:1) . Proof.
We refer to [6, 10] where similar asymptotics have been obtained.Using Rouche’s Theorem and a standard argument (see [10]), we obtain from Lemma B.3 the followingasymptotics for the large zeros z n with positive imaginary part. Corollary B.4.
There exists p ∈ Z such that for large n , we have z n = i πA ( n + p ) − λπ A (cid:18) κ + + 1 κ − (cid:19) + O ( 1 n ) . We conclude from Corollary B.4 and the previous Remark that there exists N ∈ N large enough suchthat for all n > N , we have z n = ˜ z n or z n = − ˜ z n . Assume from now on that κ + + κ − <
0. We conclude that the large zeros of a L ( λ, z ) with positiveimaginary part are located in the quadrant I = { z ∈ C , ℜ ( z ) > , ℑ ( z ) > } . By parity, the zeros withnegative imaginary part are located in the quadrant III = { z ∈ C , ℜ ( z ) < , ℑ ( z ) < } .Since the ˜ z n ’s with positive imaginary part also satisfy the asymptotics in Corollary B.4, we get thefollowing dichotomy. 25 If κ + + κ − <
0, then the zeros ˜ z n ’s with positive imaginary part are located in the quadrant I .Hence, using (B.4), we have the following. There exists a N ∈ N such that z n = ˜ z n , ∀ n > N. (B.6)Using (B.1), Lemma B.3 and Corollary B.4, we get in this case1˜ κ − − κ + = 1 κ − + 1 κ + , κ − + 1˜ κ + = 1 κ − + 1 κ + , which gives ˜ κ − = κ − , ˜ κ + = κ + . Also, we use (B.2) and (B.6) to obtain f ( z ) = G ˜ G N Y n =1 (cid:16) − z z n (cid:17)(cid:16) − z ˜ z n (cid:17) ˜ f ( z ) . (B.7)Finally, denote by E N = { n ∈ { , . . . , N } , z n = ˜ z n , and z n = − ˜ z n } . Then we obtain from (B.7) f ( z ) = G ˜ G Y n ∈ E N (cid:16) − z ˜ z n (cid:17)(cid:16) − z ˜ z n (cid:17) ˜ f ( z ) . (B.8)Denoting by g ( z ) the rational function g ( z ) = G ˜ G Q n ∈ E N (cid:16) − z zn (cid:17)(cid:16) − z zn (cid:17) , we finally get a L ( λ, z ) = g ( z ) ˜ a L ( λ, z ) , and thus L ( λ, z ) = g ( z ) ˜ L ( λ, z ) . • If κ + + κ − >
0, then the zeros ˜ z n ’s with positive imaginary part are located in the quadrant II = { z ∈ C , ℜ ( z ) < , ℑ ( z ) > } . Hence, using (B.4), we have the following. There exists a N ∈ N such that z n = − ˜ z n , ∀ n > N. (B.9)Using (B.1), Lemma B.3 and Corollary B.4, we get in this case1˜ κ − − κ + = 1 κ − + 1 κ + , κ − + 1˜ κ + = − κ − − κ + , which gives ˜ κ − = − κ + , ˜ κ + = − κ − . Using again (B.1) and the asymptotics of a L ( λ, z ) from Lemma B.3, we get the necessary condition (cid:18) ˜ a − a + (cid:19) iλκ + = (cid:18) ˜ a + a − (cid:19) iλκ − . f ( z ) = G ˜ G N Y n =1 (cid:16) − z z n (cid:17)(cid:16) − z ˜ z n (cid:17) ˜ f (¯ z ) . (B.10)Finally, denote by F N = { n ∈ { , . . . , N } , z n = ˜ z n , and z n = − ˜ z n } . Then we obtain from (B.7) f ( z ) = G ˜ G Y n ∈ F N (cid:16) − z ˜ z n (cid:17)(cid:16) − z ˜ z n (cid:17) ˜ f (¯ z ) . (B.11)Denoting by h ( z ) the rational function h ( z ) = G ˜ G Y n ∈ F N (cid:16) − z ˜ z n (cid:17)(cid:16) − z ˜ z n (cid:17) , we finally get a L ( λ, z ) = h ( z ) ˜ a L ( λ, ¯ z ) = h ( z ) ˜ a L ( λ, z ) , and thus L ( λ, z ) = − h ( z ) ˜ R ( λ, z ) . Both above cases prove the results stated in the Proposition.Even in the case when the reflection coefficients L ( λ, z ) and ˜ L ( λ, z ) (resp. R ( λ, z ) and ˜ R ( λ, z )) coincideup to a rational function in the z variable, we cannot conclude from this fact the result stated in [9], thatis the uniqueness of the function a ( x ) and ˜ a ( x ) up to a translation. This question remains thus open andwe conjecture that this is false. We refer to the last Section of [6] for more details about this point in asimilar and more general model.What we can prove however is the following weaker statement. Proposition B.5.
Assume that κ + + 1 κ − < , κ + + 1˜ κ − < . Let L be a subset of N such that X n ∈L n = ∞ . Assume that T ( λ, n ) = ˜ T ( λ, n ) , ∀ n ∈ L . Assume also that L ( λ, k ) = ˜ L ( λ, k ) , (B.12) for a finite but large enough number of indices k ∈ N . Then there exists a constant σ ∈ R such that ˜ a ( x ) = a ( x + σ ) . In consequence, the two SSAHM (Σ , g ) ans ( ˜Σ , ˜ g ) coincide up to isometries.Proof. From Proposition B.1, we know that there exists a rational function g ( z ) such that L ( λ, z ) = g ( z ) ˜ L ( λ, z ) . From (B.12), we infer that g ( z ) = 1 for all z ∈ C and thus L ( λ, z ) = ˜ L ( λ, z ) . Hence the Proposition is proved using Theorem 1.1. in [9].27 eferences [1] Aktosun T., Klaus M., van der Mee C.,
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