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Inverse problem for the Rayleigh system with spectral data

We analyze an inverse problem associated with the time-harmonic Rayleigh system on a flat elastic half-space concerning the recovery of Lam\'{e} parameters in a slab beneath a traction-free surface. We employ the Markushevich substitution, while the data are captured in a Jost function, and point out parallels with a corresponding problem for the Schr\"{o}dinger equation. The Jost function can be identified with spectral data. We derive a Gel'fand-Levitan type equation and obtain uniqueness with two distinct frequencies.


I. INTRODUCTION
In the inverse problem, boundary spectral data directly encode the Weyl matrix of the transformed problem, and equivalently the Neumann-to-Dirichlet map associated with the original Rayleigh system. Boundary spectral data are, however, insufficient to guarantee unique recovery of both Lamé parameters in the slab. In fact, we require the Jost function as the spectral data.
We let x denote the coordinate tangential to the boundary with Fourier dual (wave vector) ξ . For fixed frequency, ω, we use the asymptotics as ω 2μ −1 0 − ξ 2 → ∞ of the Weyl matrix and the Jost function. We do require two distinct frequencies. Although the dependencies of Lamé parameters and required data are different from the inverse problem analyzed by Beals, Henkin and Novikova [11,12], various steps in our proofs follow the logic of their proofs. To keep the presentation self-contained, we will review certain aspects of the work of Beals, Henkin and Novikova.
The main results of this paper are the following: • We describe analytic properties of the Jost and Weyl solutions and functions for the Rayleigh Sturm-Liouville problem, in ξ , on the Riemann surface and study their asymptotic behaviors on the physical sheet.
• Using the Wronskian for the solutions of the adjoint Sturm-Liouville problems, we derive a formula representing the Weyl matrix in terms of the Jost function; we relate the Jost function to the boundary matrix for the original Rayleigh system.
• Following [8], we derive a Gel'fand-Levitan type equation relating the Weyl matrix to the apparent potential of the Rayleigh Sturm-Liouville equation, and establish the uniqueness of its solution.
• We show the unique recovery of the Lamé parameters in two steps: Determining the Markushevich substitution at the bottom of the slab and then recovering the potential as well as the Lamé parameters in the slab using two distinct frequencies.
The Rayleigh system has been considered for many decades in seismology, in particular with the aim to estimate Lamé parameters from the observation of Rayleigh waves at a few frequencies [13]. Empirically, seismologists have found that the eigenvalues corresponding with these waves as the data are insufficient and have considered additional types of data in the absence of a mathematical understanding of this inverse problem. The fixed-frequency Rayleigh system is an extraction from the time-harmonic elastic-wave system of equations. The inverse boundary value problem for time-harmonic elastic waves on a bounded, Lipschitz subdomain of R 3 has been studied before. Nakamura and Uhlmann [14,15] proved uniqueness assuming that the Lamé parameters are C ∞ and that the shear modulus is close to a positive constant. Eskin and Ralston [16] proved a related result. (In the context of the analyses of inverse boundary value problems for time-harmonic waves, we note that Complex Geometrical Optics solutions employed in these are multidimensional generalizations of Jost solutions.) Beretta et al. [17] proved uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition, with partial boundary data. Global uniqueness of the inverse problem in dimension three assuming general Lamé parameters remains an open problem. We note again that, here, the Lamé parameters only depend on the boundary normal coordinate.
We consider the case of an inhomogeneous isotropic elastic slab of thickness H bonded to a homogeneous isotropic elastic half space with Lamé parameters λ 0 and µ 0 . We assume that the layer's Lamé parameters, λ and µ, are three times continuously differentiable and smoothly matched to the half-space constants, λ 0 and µ 0 , respectively. In earlier papers [9], we used the notation λ I and µ I for λ 0 and µ 0 , respectively, and |Z I | = H.
Assumption II.1 can be weakened to letting λ ∈ C 1 (R + ). It reflects the strong ellipticity condition [18] as this appears in the existence and uniqueness of solutions of the boundary value problem for time-harmonic elastic waves. The parameters λ and µ will be further restricted through Assumption V.2 below.

PROBLEMS
We perform an analogue of the calibration transform on the Rayleigh system to obtain a matrixvalued (essentially non-diagonalizable) Sturm-Liouville problem. We follow [7].
Based on the Pekeris substitution [6], it was shown by Markushevich [3][4][5] that the boundary value problem (7)-(10) with χ 1 = χ 2 = 0 can be reduced to two matrix Sturm-Liouville problems with mutually transposed potentials and boundary conditions. Here, we briefly outline the transformations for arbitrary χ 1 and χ 2 . For conciseness of notation while suppressing the coordinate dual to ξ , in the remainder of the analysis, we use a ′ to denote differentiation with respect to x.
Let G be a 2 × 2-matrix solving the Cauchy problem, where I 2 is the unit matrix, and We have det G(x) = 1, cf. [3]. We adopt the notation of Markushevich [4], where g 0 stands for an arbitrary positive constant. It is convenient to put g 0 = µ 0 , which we do from now onwards.
By the substitution (x ∈ [0, ∞)) with signifying the inverse Markushevich transform, problem (7)-(10) reduces to the matrix Sturm-Liouville form Here, and Furthermore, Q is the matrix-valued potential given by with We note that the potential is not a symmetric matrix, that is, Q = Q T .
By the adjoint substitution problem (7)- (10) transforms to the matrix Sturm-Liouville form Here, is a 2 × 2-matrix with D(ξ ) and C(ξ ) being the matrices, In components, (11) has the form and introduce a class of potentials In view of Assumption II.1, our matrix-valued potential, Q, of Lamé type attains the form Q 0 on The Lamé parameters at x = 0 and x ≥ H, that is, λ (0), µ(0) as well as µ ′ (0), µ ′′ (0) and λ 0 and µ 0 are encoded in, and determine Θ (or θ 1 , θ 2 and θ 3 ) independently of Q. In this paper, we will not consider the problem of boundary determination.

IV. JOST AND WEYL SOLUTIONS, JOST FUNCTION AND WEYL MATRIX
We introduced V in (15) and obtain supplemented with (16), where Θ is given in (19).
First, we construct solutions to "background" equation, respectively. We make this precise below. We refer to q P and q S as quasi-momenta. We note that (40) are solutions to the Rayleigh system (7)-(8) with λ = λ 0 , µ = µ 0 constant for all x ≥ 0. We may construct similar solutions to the adjoint equation, We consider (15) and (25) on x ∈ (H, ∞), where the potential Q is given by formula (36) with the transformation matrix G determined by (33). Using that  , for x ∈ (H, ∞), implies that while (24), for x ∈ (H, ∞), implies that where w = (w 1 , w 2 ) T solves the Rayleigh system (7)- (8). For x ∈ (H, ∞) we have linearly independent solutions which correspond to the solutions of the original Rayleigh system (2)-(3) Using (33), substitution of (44) and (45) into (42) and (43), yields and In view of the presence of square roots, q p and q S , we introduce the cut complex plane, In Appendix A we introduce the corresponding Riemann surface and physical Riemann sheet, K + , by the condition Im q P (ξ ) > 0, Im q S (ξ ) > 0.
The Jost solutions, F ± P , F ± S , of (37) are determined by the conditions We define the matrix Jost solutions, F = F(x, ξ ) and F 0 = F 0 (x, ξ ) (emphasizing that, here, ξ is not the Fourier dual to x), as and the Jost function as (cf. (16)) where Θ(ξ ) is given in (19).
Remark IV.2. By (C.6), we have det F Θ (ξ ) = cξ ∆ R , where ∆ R is the Rayleigh determinant (see Remark IV.3) and c is a constant. The Rayleigh determinant originates from the reflection matrix that we will introduce later.
The Jost solutions, F a,± P , F a,± S , of (upon introducing V in (25)) are determined by the conditions We define the matrix Jost solutions, F a = F a (x, ξ ) and F a 0 = F a 0 (x, ξ ), as and the Jost function as (cf. (26)) Using (C.6)-(C.7), we find that We note that F a = (F ⋆ ) T , where F ⋆ denotes the solution to the adjoint problem according to [2].The Wronskian of the Jost solutions of the adjoint problems has the familiar property of being we obtain Lemma IV.1. Let F, F 0 , F a and F a 0 be given by (52) and (54), respectively. Then Now, we define the Weyl solution, Φ, as and the Weyl matrix, M, as This definition shows that M(ξ )F Θ (ξ ) = F(0, ξ ), whence M(ξ ) can be identified with the Robinto-Dirichlet map associated with the matrix Sturm-Liouville problem (15). Clearly, also Remark IV. 3. In a homogeneous half space, explicit calculations result in We have In a similar fashion, we introduce We have We make the following observation. Using (70) and (75), we evaluate the Wronskian, As, using the expressions for F 0 and F a 0 and independence of the Wronskian of x, we conclude that

Other solutions
Following [2], we introduce two other matrix-valued solutions, ϕ(x, ξ ), S(x, ξ ), of (15), that is, solutions to satisfying Hence, ϕ satisfies the Robin-type boundary condition Then the Weyl solution takes the form Furthermore, we introduce the Green's function, see Appendix B for explicit expressions for G. From the definition it follows that the Green's function is entire in ξ ∈ C (for any ω ∈ C fixed). The matrix Jost solution F(x, ξ ) then satisfies the Volterra-type integral equation In a similar fashion, we introduce two other matrix-valued solutions, ϕ a (x, ξ ), S a (x, ξ ), of (25), Hence, ϕ a satisfies the Robin-type boundary condition Then the Weyl solution takes the form Furthermore, we introduce the Green's function, The matrix Jost solution F a (x, ξ ) then satisfies the Volterra-type integral equation V. SPECTRAL DATA

Analytic continuation
We note that with an extension to the branch cuts, see Appendix A. We give conjugation properties of the matrix Jost solutions, in Proof. These properties are satisfied by the reference Jost solutions F 0 and F a 0 . Then we use the Volterra type integral equations (72), (77) and the properties of the kernels (as even functions in both q S and q P ) and identities (A.3).
The conjugation properties in Lemma V.1 also imply that and on K + .
We observe that M has a meromorphic continuation from the physical ("upper") sheet K + through the cuts to the unphysical ("lower") sheets and whole Riemann surface R (see Appendix A), which still satisfies this conjugation property.
We will simplify the analysis in the next subsection by introducing below, and note that this also defines the inverse,

A. Cauchy integral
From asymptotic expansion of the Weyl matrix which we develop in Lemma VI.1 below, specif- which implies that M has a finite number, N say, of poles ξ j ∈ K + . Here, N depends on ω. As q S (ξ j ) ∈ iR + , the poles are necessarily real. Moreover M has no other poles in K + . We order the set of poles of the Weyl matrix, Remark V.1. The poles of M are among the zeros of the determinant of the Jost function det F Θ or Rayleigh determinant on the physical sheet, K + , and correspond to the bound states or normal modes. Assumption V.1 is generically satisfied for all frequencies ω ∈ R + , that is, except possibly for a finite number of frequencies. In the seismology literature, the Rayleigh system is usually considered on a finite slab with traction-free boundary conditions when simplicity of the normal modes is well known [19]. In view of our outgoing radiation boundary condition at one end represented by (98) below, this result does not directly apply.
Proof. We fix a pole ζ j = ξ 2 j , use that (58) and (70)), and evaluate the residue As In a similar fashion, we get (75) and (67)). Integrating the derivative of the relevant Wronskian, we obtain Hence, yielding a representation of α j in terms of the Jost solutions.
We now prove (86). As the pole ζ j = ξ 2 j is simple, we also have At ξ = ξ j , we have Then using (57) at ξ = ξ j , we get and using (93) we obtain (86).
The jump across the branch cut is obtained through Using the definitions of the Jost functions, writing ξ * = ξ , we then get With Lemma IV.1, we obtain (87).
In (83) we distinguish, from a physics perspective, three contributions: from the evanescent modes, from the radiating modes, and Remark V.2. We note that through (86), (87) and using (55), α j and T can be expressed in terms of the Jost function only. Thus Lemma V.2 indicates that the Jost function encodes the boundary spectral data.

B. Even extension
In the original variable ξ , ξ 2 = ζ , (83) reads Using this representation, M(ξ ) has an artificial extension to as an even function which we will employ in the further analysis. We emphasize that this extension is fundamentally different from the above mentioned meromorphic continuation.
In the further analysis we invoke Assumption V.2. The parameter functions, λ and µ, are such that there is no pole of M(ξ ) with Im q S = 0 except, possibly, at ξ = ω √ µ 0 as a one-sided limit in K + .
C. Data for the original Rayleigh system

Weyl matrix
In Appendix C we develop a relation between the Neumann-to-Dirichlet map (ND) of the original Rayleigh system and the Weyl matrix induced by the Markushevich substitutions. From 13)) it follows that ND and M have the same poles and their jumps across branch cuts are explicitly related; this relationship depends on µ(0), µ ′ (0) and µ 0 .

Jost solution
In addition to the Weyl matrix, we need the Jost solution at x = 0 as the data. We let w = [w P w S ] denote the Jost solution of the Rayleigh system before the Markushevich transform, that is, both columns of w satisfy (7)-(8) and conditions where the "reference" Jost solution comprised of w P,0 + and w + S,0 is given in (44)-(45). The Jost solution can be excited by imposing the "outgoing radiation" conditions at the bottom of the slab supplemented with Dirichlet boundary value, at the bottom of the slab, x = H, and observed at x = 0, giving w(0, ξ ). Upon the Markushevich substitution, this yields F(0, ξ ).

Jost function
The Jost functions F Θ (ξ ) and F a Θ (ξ ) can be considered as alternative data. By (C.4)-(C.5) these are directly related to the boundary matrix of the original Rayleigh problem (cf. (C.3)) is the Jost solution discussed above. By (55), the Jost function determines the adjoint Jost function if µ(0), µ ′ (0) and µ 0 are known.

D. Unique recovery of a potential of Lamé type
In the following lemma, proposition and theorem we assume that H, λ 0 , µ 0 , µ(0) and µ ′ (0) are known. We introduce the expansion of the Jost solution at the boundary, In Theorem VI.1, we will construct explicit expressions for G 0 (0, ξ ) and G 1 (0, ξ ).
Furthermore, from a Lamé-type Q for any pair of frequencies, ω 1 = ω 2 ∈ R + , we recover λ and µ, which is proved in Subsection VII C.

A. Jost solutions
In this subsection, we establish fundamental properties of the Jost solutions. We refer to Appendix B for a representation of the Green's function G = G(x, y) introduced in (71).
Theorem VI.1. For any fixed x ≥ 0, the Jost solution, F, is analytic in ξ on K + , of exponential type, and satisfies As |ξ | → ∞, ξ ∈ K + , where and Analogous properties and an expansion can be obtained for F a .
Remark VI.1. The proof of Theorem VI.1 is based on an iteration of Volterra-type equation (72) (and (77)) following a standard argument [10]. We note that the very reason to perform the Markushevich transform was to re-write the Rayleigh problem in Schrödinger form and then as a Volterra-type integral equation with bounded kernel.
In the above theorem, G 0 contains a contribution O( 1 |ξ | ); this contribution is essential to ensure that G 0 (0, ξ ) is invertible while G 0 only depends on Q 0 . We write Furthermore, the Jost function, F Θ , and det F Θ are analytic in ξ where Im q P > 0, Im q S > 0 (ξ ∈ K + , see Appendix A) and continuous in ξ where Im q P ≥ 0, q P = 0, Im q S ≥ 0, q S = 0. We obtain Corollary VI.1. The Jost function admits the asymptotic expansion, as |ξ | → ∞, ξ ∈ K + , Corollary VI.2. Solutions S(x, ξ ) and ϕ(x, ξ ) (cf. (70) and above) are entire on C and even functions of ξ of exponential type. For the first and second derivatives, the following estimate holds true uniformly in x ≥ 0.

B. Weyl matrix
Next, we study the Weyl matrix introduced in (59), and derive its asymptotic expansion as |ξ | → ∞, ξ ∈ K + . It appears that it is more convenient to start the analysis with its inverse, where Θ is given in (19). The following result is an analogue of [8, Proposition 2].

VII. GEL'FAND-LEVITAN TYPE EQUATION AND PROOF OF THEOREM V.1
Using the results from the previous section, we obtain an asymptotic expansion for F(x, ξ )[F(0, ξ )] −1 in Lemma VII.1. The following asymptotic expansion holds true Proof. The statement follows from an explicit calculation using Theorem VI.1, that is, (101)- Corollary VII.1. The following asymptotic expansion holds true This corollary immediately implies that A. Recovery of V assuming that G H is known For the proof of Theorem V.1, we change variables through the transformation (Appendix A) The choice of sign is determined by letting ξ ∈ K + (where also Im q P (ξ ) > 0) correspond to k ∈ C + (Im k > 0). The inverse of the transformation is defined on C + and written as ξ (k). The branch cut in ξ ∈ K S corresponds with Im k = 0. We let where C + (Im k < 0). First, we give some basic asymptotic expansions. To next order, we have We introduce and two more solutions (cf. (70)) where we identify S(x, k), ϕ(x, k) and Θ(k) are entire and even in k as solutions and by the relevant boundary condi- Applying Lemma VI.1 and using (114)), we find that In the later analysis, for real-valued k, we will employ the notation Y (−ik) and E(−ik), substituting −ik for ξ (or ξ (k)). By abuse of notation, in this subsection, we will omit the in S, ϕ and Θ.
Clearly, the matrix functions Φ ± satisfy the boundary conditions (cf. (63)) Using Corollary VII.1 and (128), for real-valued k, we note that and the expansions in Corollary VII.1 imply Lemma VII.2. For k ∈ R, the following asymptotic expansion holds true where in which The lemma above implies that for k ∈ R, and we have We introduce (cf. (132)) for the later analysis.
The matrix function Ψ is meromorphic and defined on C through the above mentioned extension of Φ to branch cuts, which is important for the later contour integration in the proof of Proposition VII.2. However, as elucidated in Subsection V B, throughout we intrinsically use functions defined on the physical sheet only.
From (121) it follows that Ψ inherits its poles from M (see Subsection V A) Lemma VII.3. The matrix function Ψ has poles at ±k j with The residues are given by Proof. We have (cf. (125)) for Im k > 0, where S and ϕ are entire in k. As we obtain which, with (116), implies the statement. Here, we use that the poles of M are simple.

We define three new matrix functions in
Definition VII.2. We let j(k), e(x, k) andẽ(x, k) be given by Through the definition of M − , we obtain the extension of j(k) from C + to C − . It follows that that is, for Im k = 0, j(k) is even function j(−k) = j(k) with discontinuity on the real line for Using (116), we find that Consequently, j(k) = O 1 |k| 2 and j(k) − j(−k) = O 1 |k| 3 , k ∈ C + . The Weyl matrix directly determines j; this is, because the two leading terms in the asymptotic expansion of M determine E and Y 1 (cf. (114)-(117)).
Proposition VII.1. The function Ψ(x, k) has asymptotic expansion where D(x) is given in Lemma VII.1 and T 0 , T 1 are defined by (118). It admits the representation Moreover, the boundary values Ψ ± (x, k) = Ψ(x, k ± i0), Im k = 0, determine ϕ in (70) by Proof. Substituting (118)-(119) into Corollary VII.1 yields (143) for Im k < 0 and by taking a one-sided limit, we get (143) for Im k = 0. For real-valued k, Im k = 0, an explicit calculation gives Hence, cf. (134). This implies that the leading order in asymptotics (143) is the same for Ψ(x, k − i 0) and Ψ(x, k + i 0) + ikφ 0 (x, k)( j(k) − j(−k))E(−ik) in the full complex k-plane and also (as Ψ is bounded on C) that Ψ can be recovered from the Cauchy integral formula and its residues. Now, we prove representation (144). We let R > 0 and denote by γ ± R the closed half-circle contour in C ± with positive orientation containing all the poles ±k j inside. We denote by Γ ± R only the arc parts of these contours with negative orientation. We have Lemma VII.3 provides us with the expressions for the residues; we use that j(k ′ ) − j(−k ′ ) is entire for Im k = 0.
We substitute (147) in the integrand of the integral in the left-hand side and take the limit we obtain (144).
We factorize ϕ and ϕ, and introduce A and A according to and write We note that We introduce (cf. (132)) in Lemma VII.4. The functions A (x, k), A (x, k) and A j (x), j = 1, . . . , N, satisfy This equation holds in particular for k = k i , i = 1, . . . , N.
Proof. We take (144) as a point of departure. We first study the behavior of this equality, upon multiplication by ik T −1 0 , as k → ∞, in particular, the integral contribution (suppressing the factor 1 2π ) in the right-hand side, To establish uniform boundedness, it is sufficient to only consider the "leading" order of ϕ(x, k ′ ), that is, the second term in (131), or, in fact, the non-vanishing matrix element (suppressing the factor −2ϖ i), in the numerator of the integrand. Concerning the second term, we note that which holds true also for x = 0. Hence, We now let x > 0 and perform integration by parts, exploiting the exponent e 2ik ′ x , while analyzing the first term, We have This implies that Here, we used that the integrand is even in k ′ .
We use (145) and that ϕ is even in k to get Adding and subtracting terms, we rewrite this equality as 4ϕ(x, k) = 4T 0 e(x, k) − 2ikφ 0 (x, k)( j(k) − j(−k)) + I 1 (x, k) + I 2 (x, k) + I 3 (x, k) + I 4 (x, k) (157) where We note that I 1 and I 4 have poles, k j , in the complex k half plane C + (Im k > 0) and that I 2 and I 3 have poles, −k j , in the complex k half plane C − (Im k < 0). We introduce a positive half-circle (with radius R) contour γ + R in C + containing poles k j and k inside, and a negative half-circle (with radius R) contour γ − R in C − containing poles −k j and k inside. Then We observe that the contributions from the upper and lower semicircles in the limit vanish, that is, Hence, using that in the residues, the terms containing φ 0 (x, k ′ )( j(k ′ ) − j(−k ′ )) don't contribute, Using Lemma VII.3, the summations over the poles in (167) add up to Using (146), the integrals in (167) add up to We established boundedness of this integral in (156). Substituting (168) and (169) into (167) and the result into (158) implies the statements upon considering k ∈ R (and k = k j ).
With Proposition VII.1 and the proof of the previous lemma concerning the limit k → ∞ (Im k = 0), we obtain Lemma VII.5. The following holds true where D(x) is given in Lemma VII.1.
In the above, we note that T 1 depends on ω through θ 1 . This lemma provides an identity for D.
It is essential for this lemma that we analyzed the asymptotic expansions of F, M and Φ. The right-hand side of (170) motivates the introduction of (cf. (139)), the first term of which can be identified with a Fourier transform, and plays a key role in the proof of the next proposition. The right-hand of (170) is K(x, x).
Remark VII.1. From the analysis leading to (156) it follows that K(x, x) is bounded. In fact, the continuous differentiability of K(x, x) is directly related to the continuous differentiability of V through D(x).
We note that T 0 can be obtained from the asymptotic expansion of M (cf. (109) and (118)). Suppose that K(x, x) were known, then the potential, V , can be recovered upon differentiating (170): where D is given in Lemma VII.1 with The kernel, K(x, y), is determined by the boundary spectral data, which is the content of Proposition VII.2 (Gel'fand-Levitan). The kernel K(x, y) is the unique solution of the Gel'fand-Levitan type equation where E is given in (117), e(x, k j )C j e(y, k j ). and Proof. We distinguish two parts to complete the proof. Part I: construction of (172). We consider (171) and write A j (x)C j e ik j y E(∓ik j ).
We note that (cf. (149)) (176) and The absence of singularities means that A (x, k)e ±ikx has a bounded holomorphic extension to the half-plane ± Im k ≥ 0. Through exponential decay of e ik(y−x) , we find that if |y| > x then T ± (x, y) = 0 .
From the Fourier inversion formula, we obtain We substitute this expression in (151). Using that which shows that the summation over poles in (151) is cancelled, we obtain in which By straightforward calculations, we find that x −x K(x, y)B + (k, y) dy.
Substituting (180) and (181) into this equation and using that e(x, k) + e(−x, k) = 2 cos kx I 2 , we We multiply this equation by 2 j(k)e(y, k)k − 2πi ∑ N j=1 δ (k − k j )C j e(y, k j ) and obtain Dividing this equation by 2πi and integrating over k, leads to (172). More precisely, we first integrate over [−R, R] and establish that the integrals are uniformly bounded after which we interchange orders of integration and take the limit R → ∞.
the representation of g, in fact, can be simplified, cos(k j x)C j e(y, k j ).
Part II: (172) has a unique solution. We note that g(x, y) = 0 for |y| > x and that equation (172) is of Volterra type. We consider x as parameter and K(x, y) as unknown function. For unique solvability, we need to prove that, for some constant C > 0 (dependent on x), sup |y|≤x x −x |E(2δ (x + y ′ ))g(−y ′ , y)| dy ′ ≤ C.
Using special form of matrices g and E, it follows that (184) is satisfied. Then, using the Volterra property, it follows that the solution to the homogeneous problem is trivial and the solution to (172) can be constructed by iteration. This completes the proof of Proposition V.1.

B. Recovery of G H
Here, we prove that G H is determined by the two leading orders in asymptotic expansion of the Jost solution F at x = 0 as ξ → ∞, ξ ∈ K + . The asymptotic expansion of F(0, ξ ) is given by and det Q 2 = 1 µ 1 λ + 2µ , wherefrom λ and µ are recovered.
We observe that q P : ξ → ω 2 σ 0 − ξ 2 is a conformal mapping K P → K P . We obtain the Riemann surface R for both q P and q S by joining the separate Riemann surfaces for q P and q S so that q P and q S are single-valued holomorphic functions of ξ . We note that R is a four-fold cover of the plane. We identify the part of R where Im q P > 0, Im q S > 0 with the physical ("upper") sheet which coincides with K S,+ . Each sheet can be identified by the signs of Im q S and Im q P . We omit the subscript S in the notation and write K = K S and K + = K S,+ for the cut plane and the part of the cut plane corresponding to the physical sheet, respectively. We note that ξ has the meaning of Regge parameter.
In the main text, we introduce ζ = ξ 2 . We note that Im q S (ζ ) > 0, Im q P (ζ ) > 0 for ζ ∈ Π + , where corresponds to the physical sheet while (K + ) 2 = Π + . We introduce the notation We will use both parameters ξ (Jost solutions and Jost function) and ζ (Weyl solutions and Weyl matrix) and both cut planes K + and Π + , switching between them when it appears natural. By explicit construction, we obtain Lemma B.1. The following holds true G(x, y) = A(x) sin((x − y)q P ) q P + B(y) sin((x − y)q S ) q S + C cos((x − y)q S ) − cos((x − y)q P ) ω 2 , (B.1) Appendix C: Weyl matrix and Neumann-to-Dirichlet map of the Rayleigh system In this appendix, we study relationships between the orginal and transformed problems, that is, the Jost and Weyl solutions and the Jost function before the Markushevich transform. Consistent with the notation in (13) (cf. (6)) supplemented with boundary conditions (9)-(10), in the right-most equality, we reverted to the original notation (cf. (6)). Setting χ = B(w), (38) yields the expression for the Jost function, In a likewise manner, we obtain for the adjoint problem, We subject the Weyl solution to M −1 (cf. (14)), substitute (70)  . This equation provides a direct relationship between the Weyl matrix and the (observable) Neumann-to-Dirichlet map and, more specifically, between the associated spectral data as θ