Elastic scattering by finitely many point-like obstacles

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Elastic scattering by finitely many point-like obstacles

G. Hu, M. Sini

RICAM-Report 2014-30

Elastic scattering by finitely many point-like obstacles

Guanghui Hu^1,a) and Mourad Sini^2,b)

1)Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany

2)RICAM, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria

(Dated: 18 March 2013)

This paper is concerned with the time-harmonic elastic scattering by a finite number N of point-like obstacles in Rⁿ (n = 2,3). We analyze the N-point interactions model in elasticity and derive the associated Green’s tensor (integral kernel) in terms of the point positions and the scattering coefficients attached to them, following the approach in quantum mechanics for modeling N-particle interactions. In particular, explicit expressions are given for the scattered near and far fields corresponding to elastic plane waves or point-source incidences. As a result, we rigorously justify the Foldy method for modeling the multiple scattering by finitely many point-like obstacles for the Lam´e model. The arguments are based on the Fourier analysis and the Weinstein-Aronszajn inversion formula of the resolvent for the finite rank perturbations of closed operators in Hilbert spaces.

a)[email protected]; http://www.wias-berlin.de/people/hu/

b)Corresponding address: [email protected]; http://www.ricam.oeaw.ac.at/people/page/sini/

I. INTRODUCTION

We consider the time-harmonic elastic scattering by N point-like scatterers located at y^(j), j = 1,2,· · · , N inRⁿ(n = 2,3). We set Y :={y^(j) : j = 1,2,· · · , N}. Physically, such point-like obstacles are related to highly concentrated inhomogeneous elastic medium with sufficiently small diameters compared to the wave-length of incidence. Define the Navier operator

H_ωu:= (−∆^∗−ω²)u, ∆^∗ :=µ∆ + (λ+µ) grad div (1) where λ, µ are the Lam´e constants of the background homogeneous medium, and ω > 0 denotes the angular frequency. Denote by U^tol = U^I +U^S the sum of the incident field U^I and the scattered field U^S. The N-point interactions mathematical model we wish to analyze is the following: find the total elastic displacement U^tol such that

H_ω(U^tol) =

∑N j=1

a_jδ(x−y^(j))U^tol inRⁿ\Y, (2)

rlim→∞r^(n−1)/2(∂U_p

∂r −ik_pU_p) = 0, lim

r→∞r^(n−1)/2(∂U_s

∂r −ik_sU_s) = 0, r=|x|, (3) where the last two limits are uniform in all directions ˆx := x/r ∈ S := {|xˆ| = 1}. Here k_p :=ω/√

λ+ 2µ, k_s :=ω/√µare the compressional and shear wavenumbers, and Up :=−k_p⁻²grad divU^S, Us =−k⁻_s²curlcurlU^S

denote the longitudinal and transversal parts of the scattered field, respectively. The con- ditions in (3) are referred to as the Kupradze radiation condition in elasticity¹⁰.

The equation (2) formally describes the elastic scattering by N obstacles with densities concentrated on the point y^(j). This concentration is modeled by the Dirac impulses δ(· − y^(j)). In (2) the numbera_j ∈Cis the coupling constant (scattering strength) attached to the j-th scatterer, which can be viewed as the limit of the density coefficients for approximating the idealizedδ-functions in (2).

Let us describe the Foldy method, see Refs.^4,11 for more details in the acoustic case, to solve the problem (2)-(3). Let Γ_ω(x, z) be the fundamental tensor of the Lam´e model. Using (2) and (3), we obtain formally the following representation

U^tol(x) =U^I(x) +

∑N j=1

a_jΓ_ω(x, y^(j))U^tol(y^(j)), x̸=y^(j), j = 1,2, . . . , N. (4)

There is no easy way to calculate the values of U^tol(y^(j)), j = 1,2, . . . , N, and we cannot evaluate (4). There are several approximations to handle this point. We can cite the Born, Foldy and also the intermediate levels of approximations, see Refs.^3,11 for more details about these approximations. Here, we only discuss the Foldy method. Following this method, proposed in⁴ to model the multiple interactions occurring in the acoustic scattering, see also¹¹ for more details, the total fieldU^tol(x) has the form

U^tol(x) = U^I(x) +

∑N j=1

a_jΓ_ω(x, y^(j))U_j(y^(j)), (5) where the approximating terms U_j(y^(j))’s can be calculated from the Foldy linear algebraic system given by

U_j(y^(j)) = U^I(y^(j)) +

∑N m=1m̸=j

a_mΓ_ω(y^(j), y^(m))U_m(y^(m)), ∀ j = 1, . . . , N. (6)

This last system is invertible except for some particular distributions of the points y^(j)’s, see Ref.³ for a discussion about this issue. Hence the systems (5)-(6) provides us with a close form of the solution to the scattering by N-point scatterers. As it can be seen, the system (6) is obtained from (5) by taking the limits of x to the points y^(m)’s and removing the singular part.

Our objective is to rigorously justify and give sense to this method in the framework of elastic propagation. To do it, we follow the approaches, presented in Ref.¹, known in quantum mechanics for describing the interaction of N-particles. As pointed out in Ref.¹ for quantum mechanical systems, the Dirac potentials on the right hand side of (2) cannot be regarded as an operator or quadratic form perturbation of the Laplacian operator in Rⁿ. This is also our main difficulty to deal with the scattering problem in elasticity. One way to solve this problem is to employ the self-adjoint extensions of symmetric operators and the Krein’s inversion formula of the resolvents; see e.g. Ref.¹ (Part I) and Ref.⁵ for the basic mathematical framework in quantum mechanics. An alternative approach is the renormalization techniques, see Ref.¹ (Part 2), based on introducing appropriate coupling constants which vanish in a suitable way in the process of approximation such that the resolvent of the model makes sense. Precisely, replacing the scattering coefficients a_j by parameter dependent coefficients a_j(ϵ), ϵ∈R+, decaying in a suitable way when ϵ→0, and the Fourier transform of the delta distribution by its truncated part, up to ¹_ϵ, one obtains

a parameter family of self-adjoint operators, with ϵ as a parameter, in the Fourier variable.

These operators are finite-rank perturbations of the multiplication operator (which is the Fourier transform of the Laplacian). Based on the Weinstein-Aronszajn inversion formula, one shows that the resolvent of this family of operators converges, asϵ→0, to the resolvent of a closed and self-adjoint operator. This last operator is taken to be the Fourier transform of the operator modeling the finitely many pointlike obstacles scattering problem.

The purpose of this paper is to develop the counterpart in elasticity for the model (2), following this renormalization procedure. As a result, we show that the Foldy system (5)-(6) is indeed a natural model to describe the scattering by N-point scatterers provided that we take the coefficients aj of the form (cj−κ)⁻¹ with cj being real valued and

κ:=







−_4π¹ [

λ+3µ

µ(λ+2µ)C+_µ(λ+2µ)^λ+µ −¹₂(^ln_µ^µ+ ^ln(λ+2µ)_λ+2µ ) ]

inR²,

iω_{12πµ(λ+2µ)}^2λ+5µ inR³.

(7)

The constantC in (7) denotes Euler’s constant. Let us finally mention that the system (5)- (6) is used in Refs.^3,6as a model for the detection of point-like obstacles from the longitudinal or the transversal parts of the far field pattern.

The rest of the paper is organized as follows. In Section II we present a detailed investi- gation of theN point interactions in elasticity inR². Section II A gives the Green’s tensors for the Navier and Lam´e equations, in the absence of the obstacles, and the limit of their difference as the argument tends to origin. Such a limit will be used in Section II B for deriving the Green’s tensor (integral kernel) of the model in the presence of the obstacles.

An immediate consequence of this tensor is the explicit far field pattern for plane wave incidence in terms of point positions and the associated scattering coefficients; see Section II B. Finally, in Section III we extend the main Theorem II.6 in two-dimensions to the case of three-dimensions.

II. ELASTIC SCATTERING BY POINT-LIKE OBSTACLES IN R²

Throughout the paper the notation (·)^⊤ means the transpose of a vector or a matrix, e_j, j = 1,2,· · · , N denote the Cartesian unit vectors in Rⁿ, and the notation I stands for the n×n identity matrix in Rⁿ. We first review some basic properties of the fundamental solutions to the Navier and Lam´e equations inR².

A. Fundamental solutions

We begin with the Green’s tensor for the operator H_ω, given by Γ_ω(x, y) := i

4µH₀⁽¹⁾(k_s|x−y|)I+ i

4ω²grad_xgrad^⊤_x[H₀⁽¹⁾(k_s|x−y|)−H₀⁽¹⁾(k_p|x−y|)] (8) forx, y ∈R², x̸=y, whereH₀⁽¹⁾(t) denotes the Hankel function of the first kind and of order zero. Foru= (u1, u2)^⊤ and ω = 0, we have the Lam´e operator

H₀u:=−∆^∗u=−



(λ+ 2µ)∂₁²u1+µ∂₂²u1+ (λ+µ)∂1∂2u2

(µ∂₁²u₂+ (λ+ 2µ)∂²₂u₂+ (λ+µ)∂₁∂₂u₁



, ∂_j :=∂x_j, j = 1,2.

Define the Fourier transform F :L²(R²)² →L²(R²)² by (Ff)(ξ) = ˆf(ξ) := 1

2π lim

R→∞

∫

|x|≤R

f(x)e^−ix·ξdx, ξ = (ξ1, ξ2)^⊤. Its inverse transform is given by

(F⁻¹g)(x) := 1 2π lim

R→∞

∫

|ξ|≤R

g(ξ)e^ix·ξdξ.

With simple calculations, we obtain (FH0)u=



(λ+ 2µ)ξ₁²+µξ₂² (λ+µ)ξ₁ξ₂ (λ+µ)ξ₁ξ₂ µξ₁²+ (λ+ 2µ)ξ₂²







uˆ₁ ˆ u₂



=:M0(ξ)ˆu

where ˆu:=Fu. Moreover, we have the non-vanishing determinant of M₀: det(M₀) = (λ+ 2µ)µ|ξ|⁴ ̸= 0, if |ξ| ̸= 0, implying that M0 is invertible, with its inverseM₀⁻¹ given by

M₀⁻¹(ξ) = 1 (λ+ 2µ)µ|ξ|⁴



µξ₁²+ (λ+ 2µ)ξ₂² −(λ+µ)ξ₁ξ₂

−(λ+µ)ξ₁ξ₂ (λ+ 2µ)ξ₁²+µξ₂²





= 1

µ|ξ|²I− λ+µ

µ(λ+ 2µ)|ξ|⁴Ξ(ξ) (9)

for |ξ| ̸= 0, where

Ξ(ξ) :=ξ^⊤ξ =



 ξ²₁ ξ₁ξ₂ ξ1ξ2 ξ₂²



 for ξ= (ξ₁, ξ₂)∈R².

SettingM_ω =M₀−ω²I, we then analogously have M_ω⁻¹(ξ) = 1

µ|ξ|²−ω² I− λ+µ

(µ|ξ|²−ω²) [(λ+ 2µ)|ξ|²−ω²]Ξ(ξ). (10)

Let Γ₀(x, y) = Γ₀(|x−y|) be the Green’s tensor to the operator H₀, i.e., the Kelvin matrix of fundamental solutions to the Lam´e system, given by (see Ref.⁷ (Chapter 2.2))

Γ₀(x,0) = 1 4π

[

− 3µ+λ

µ(2µ+λ)ln|x|I+ µ+λ

µ(2µ+λ)|x|²Ξ(x) ]

. (11)

Then, there holds 1

2π(F⁻¹M₀⁻¹)(x) = Γ₀(x,0), 1

2π(F⁻¹M_ω⁻¹)(x) = Γ_ω(x,0), |x| ̸= 0, x∈R².

The following lemma gives the entries of the matrix Γ_ω−Γ₀ taking the value at the origin.

Lemma II.1. There holds the limit

|xlim|→0[Γ_ω(x,0)−Γ₀(x,0)] =ηI, where

η :=− 1 4π

[ λ+ 3µ

µ(λ+ 2µ)(lnω

2 +C−iπ

2 ) + λ+µ

µ(λ+ 2µ)− 1 2(lnµ

µ + ln(λ+ 2µ) λ+ 2µ )

]

, (12) with C = 0.57721· · · being Euler’s constant.

Proof. Recall Ref.⁹ that Γ_ω can be decomposed into Γ_ω(x,0) = 1

π ln(|x|) ˜Γ₁(x) + ˜Γ₂(x), (13) with the martices ˜Γ_j taking the form

Γ˜₁(x) := ˜Ψ₁(|x|)I+ ˜Ψ₂(|x|) 1

|x|²Ξ(x), Γ˜₂(x) := χ₁(|x|)I+χ₂(|x|) 1

|x|²Ξ(x), (14) where χ_j(τ) (j = 1,2) are C^∞ functions on R⁺ and

Ψ˜₁(τ) =− 1

2µJ₀(k_sτ) + 1

2ω²τ [k_sJ₁(k_sτ)−k_pJ₁(k_pτ)], Ψ˜₂(τ) = 1

2ω² [

k²_sJ₀(k_sτ)− 2k_s

τ J₁(k_sτ)−k_p²J₀(k_pτ) + 2k_p

τ J₁(k_pτ) ]

.

Here J_n denotes the Bessel function of order n. Moreover, making use of the asymptotic behavior

J₀(t) = 1− 1

4t²+ 1

64t⁴+O(t⁶), J₁(t) = 1 2t− 1

16t³+O(t⁵), t→0⁺, we get (see also Ref.⁹)

Ψ˜₁(τ) =−η₁+η₂ τ²+O(τ⁴), Ψ˜₂(τ) = η₃ τ²+O(τ⁴), (15) χ₁(τ) =η+O(τ²), χ₂(τ) =η₄1

π +O(τ²)

as τ →0, where

η:=−_4πω¹2

[

k²_sln^k₂^s +k_p²ln^k₂^p +^k

2 s−k²p

2 + (C−^iπ₂)(k_s²+k_p²) ]

, η1 := _4ω¹2(k_s²+k²_p), η2 := _32ω¹2(3k_s⁴+k_p⁴), η3 := _16ω¹2(k⁴_p−k⁴_s), η4 := ^k

2 s−k²_p 4ω² ,

with Euler’s constantC = 0.57721· · ·. Note that the coefficientsη, η₁, η₄ can be respectively rewritten as (12) and

η₁ = λ+ 3µ

4µ(λ+ 2µ), η₄ = λ+µ 4µ(λ+ 2µ)

in terms of the Lam´e constants λ and µ. Insertion of (14) and (15) into (13) yields the asymptotic behavior

Γ_ω(x,0) = 1

πln|x|[−η₁+o(1)]I+ηI+ η₄

π|x|²Ξ(x) +o(1) as|x| →0, which together with (11) proves Lemma II.1.

B. Solvability of elastic scattering by N point-like obstacles

Consider a new operator H u= [H₀−

∑N j=1

a_jδ(x−y^(j))]u, y^(j)= (y₁^(j), y₂^(j))^⊤∈R².

The objective of this section is to give a mathematically rigorous meaning of this operator and describe the scattered field corresponding to incident plane waves or point-sources. As mentioned in the introduction, our arguments are in the lines of the approach known in quantum mechanics for describing the point interactions of N particles; see Ref.¹.

To start, we set

He :=FHF⁻¹ =FH₀F⁻¹−

∑N j=1

a_jF[δ(x−y^(j))]F⁻¹.

Forf = (f₁, f₂)^⊤∈L²(R²)², we have (FH₀F⁻¹) ˆf = (FH₀)f =M₀fˆ, and formally (Fδ(x−y^(j))F⁻¹f)(ξ) = (ˆ Fδ(x−y^(j))f)(ξ)

= (2π)⁻¹f(y^(j))e^−iy(j)·ξ

= (2π)⁻¹e^−iy(j)·ξ ( 1

2π

∫

R²

f(ξ)eˆ ^iy(j)·ξdξ )

=

⟨f , φˆ ¹_y(j)

⟩

φ¹_y(j)(ξ) +

⟨f , φˆ ²_y(j)

⟩

φ²_y(j)(ξ),

where φⁱ_y(j)(ξ) := ϕ_y(j)(ξ) (e_i)^⊤ fori= 1,2, withϕ_y(j)(ξ) := (2π)⁻¹e^−iy(j)·ξ. Here we used the inner product

⟨f, g⟩:=

∫

R²

f(ξ)·g(ξ)dξ, forf, g ∈L²(R²)². Therefore, formally we have

Hfe = (FH₀F⁻¹)f =M₀(ξ)f −

∑N j=1

{⟨

a_jf, φ¹_y(j)

⟩

φ¹_y(j)(ξ) +

⟨

a_jf, φ²_y(j)

⟩

φ²_y(j)(ξ) }

.

Our aim is to prove the existence of the resolvent of He and to deduce an explicit expression of its Green’s tensor. To make the computations rigorous, we introduce the cut-off function

χ_ϵ(ξ) =





1, if ϵ≤ |ξ| ≤1/ϵ,

0, if |ξ|< ϵ or |ξ|>1/ϵ,

for some 0< ϵ < 1, and define the operator

He^ϵf :=M₀(ξ)f −

∑N j=1

∑2 i=1

⟨

a_j(ϵ)f, φ^ϵ,i_y(j)

⟩

φ^ϵ,i_y(j)(ξ), φ^ϵ,i_y(j)(ξ) :=χ_ϵ(ξ)φⁱ_y(j)(ξ). (16) We will choose the coupling constants a_j(ϵ) in a suitable way such that the resolvent of He^ϵ has a reasonable limit as ϵ tends to zero. Let us first recall the Weinstein-Aronszajn determinant formula from Ref.¹ (Lemma B.5), which is our main tool for analyzing the resolvent of He^ϵ.

Lemma II.2. Let H be a (complex) separable Hilbert space with a scalar product ⟨

·,·⟩ . Let A be a closed operator in H and Φ_j, Ψ_j ∈ H, j = 1, ..., m. Then

( A+

∑m j=1

⟨·,Φ_j⟩ Ψ_j−z

)₋1

=(

A−z)₋1

−

∑m j=1

[Π(z)]₋1 j,j^′

⟨·, [(

A−z)₋1

]^∗Φ_j′⟩(

A−z)₋1

Ψ_j (17) for z in the resolvent set of A such that det[

Π(z)]

̸

= 0, with the entries of Π(z) given by [Π(z)]

j,j^′ :=δ_j,j′+⟨(

A−z)₋1

Ψ_j′,Φ_j⟩

. (18)

Note that in Lemma II.2, the notation [Π(z)]₋₁

j,j^′ denotes the (j, j^′)-th entry of the matrix [Π(z)]₋1

, and [ ]^∗ stands for the adjoint operator of [ ]. To apply Lemma II.2, we take H :=L²(R²)², A := M0, m := 2N and Φj := Φ^ϵ_j, Ψj = −˜ajΦ^ϵ_j for j = 1,· · · ,2N, with ˜aj

and Φ^ϵ_j defined as follows:

˜

a_j(ϵ) = a_l(ϵ) if j ∈ {2l−1,2l}, Φ^ϵ_j :=





φ^ϵ,1_y(l) ifj = 2l−1, φ^ϵ,2_y(l) ifj = 2l,

for some l∈ {1,2,· · · , N}. The multiplication operator A is closed with a dense domain D(A) := {

ˆ

u∈L²(R²)², M₀uˆ∈L²(R²)²}

in L²(R²)² hence He^ϵ, ϵ >0, is also closed with the same domain. Moreover, we set z :=ω² for ω ∈ C such that Imω > 0. For such complex-valued number ω, one can observe that det(M₀−ω²I)̸= 0 so that (M₀−ω²I)⁻¹ always exists. Further, it holds that

[(M₀−ω²I)⁻¹]^∗ = [(M₀−ω²I)⁻¹]^T = (M₀−ω²I)⁻¹, (19) where [ ]^T denotes the conjugate transpose of a matrix, and ω denotes the conjugate of ω.

Simple calculations show that

(A−z)₋1

Ψ_j =−˜a_j(M₀−ω²I)⁻¹Φ^ϵ_j and

δ_j,j′ +⟨(

A−z)₋1

Ψ_j′,Φ_j⟩

= ˜a_j[˜a⁻_j¹δ_j,j′ −⟨

(M₀−ω²I)⁻¹Φ^ϵ_j′,Φ^ϵ_j′

⟩].

Therefore, by Lemma II.2 we arrive at an explicit expression of the inverse ofHe^ϵ−ω², given by

(He^ϵ−ω²)⁻¹f = (M0−ω²I)⁻¹f +

∑2N j,j^′=1

[Πϵ(ω)]⁻_j,j¹′

⟨

f, χϵF₋^(j_ω^′)

⟩

χϵF_ω^(j), Imω > 0, (20) with

Π_ϵ(ω) :=[

˜

a⁻_j¹δ_j,j′ −⟨

(M₀−ω²I)⁻¹Φ^ϵ_j′, Φ^ϵ_j⟩]2N

j,j^′=1, χ_ϵF_ω^(j):= (M₀−ω²I)⁻¹Φ^ϵ_j, (21) provided that Imω >0 and det[Π_ϵ(ω)]̸= 0.

In order to obtain (He−ω²)⁻¹, we need to remove the cut-off function in (20) by evaluating the limits of Π_ϵ(ω) and

⟨

f, χ_ϵF₋^(j_ω^′)

⟩

χ_ϵFω^(j′) as ϵ → 0. This will be done in the subsequent Lemmas II.3 and II.5.

Lemma II.3. The coefficients ˜a_j(ϵ) can be chosen in such a way that the limit Π_B,Y(ω) :=

lim_ϵ→0Π_ϵ(ω) exists and takes the form

Π_B,Y(ω) =















(b₁−η)I −Γ_ω(y⁽¹⁾−y⁽²⁾) · · · −Γ_ω(y⁽¹⁾−y^(N))

−Γ_ω(y⁽²⁾−y⁽¹⁾) (b₂−η)I · · · −Γ_ω(y⁽²⁾−y^(N))

. . . .

. . . .

. . . .

−Γω(y^(N)−y⁽¹⁾) −Γω(y^(N)−y⁽²⁾) ... (bN −η)I















, (22)

where η is given in (12) and B := (b₁,· · · , b_N)is an arbitrary vector in C^1×N. If in addition we choose

b_l:=c_l− λ+ 3µ

4πµ(λ+ 2µ)(lnω 2 − iπ

2), l = 1,2,· · · , N, (23) with c_l∈R arbitrary, then we have

(Π_B,Y(ω))^∗ = Π_B,Y(−ω).¯ (24)

Remark II.4. In this paper, the number η is referred to as the normalizing constant and b_l ∈C is viewed as the scattering coefficient attached to the l-th scatterer. The coefficient b_l characterizes the scattering density concentrated at y^(l). The relation between the scattering coefficient bl and the scattering strength al will be given in Remark II.7.

Proof. The proof will be carried out in the following three cases of j, j^′ ∈ {1,· · · ,2N}. Case 1: |j^′−j|= 1, and j, j^′ ∈ {2l−1,2l} for somel ∈ {1,· · · , N}.

We have j^′−j = 1 if j is an odd number, and j−j^′ = 1 if j is an even number. Assume firstly that j = 2l −1, j^′ = 2l for some l = 1,· · · , N. Then, we have Φ^ϵ_j = χϵφ_y^(j)(1,0)^⊤, Φ^ϵ_j′ =χ_ϵφ_y(j)(0,1)^⊤. Hence

⟨(M₀−ω²I)⁻¹Φ^ϵ_j,Φ^ϵ_j′

⟩=⟨

(M₀−ω²I)⁻¹χ_ϵ(1,0)^⊤, χ_ϵ(0,1)^⊤⟩ since φ_y^(l)(ξ)φ_y^(l)(ξ) = 1.Consequently, it holds that

⟨(M₀−ω²I)⁻¹Φ^ϵ_j,Φ^ϵ_j′

⟩=

∫

ϵ<|ξ|<1/ϵ

(M₀−ω²I)⁻¹(1,0)^⊤·(0,1)^⊤dξ = 0

because the scalar function (M₀−ω²I)(ξ)⁻¹(1,0)^⊤·(0,1)^⊤ is odd in bothξ₁ and ξ₂; see (10).

By symmetry, we have also ⟨

(M₀−ω²I)⁻¹Φ^ϵ_j′,Φ^ϵ_j⟩

= 0.

Case 2: j =j^′ ∈ {2l−1,2l}for some l ∈ {1,2,· · · , N}. In this case, we set

˜

a⁻_j¹(ϵ) := 1 4π²

∫

ϵ<|ξ|<1/ϵ

M0(ξ)⁻¹Φ^ϵ_j·Φ^ϵ_jdξ+bl, bl ∈C. Hence, if j = 2l−1 is an odd number, then by (9) we have

˜

a⁻_j¹(ϵ) = 1 4π²

∫

ϵ<|ξ|<1/ϵ

M₀(ξ)⁻¹(1,0)^⊤·(1,0)^⊤dξ+b_l

= 1 4π²

∫

ϵ<|ξ|<1/ϵ

µξ₁²+ (λ+ 2µ)ξ₂²

µ(λ+ 2µ)|ξ|⁴ dξ+bl

=− (λ+ 3µ)

2π(λ+ 2µ)µlnϵ+b_l. (25)

Moreover, by the choice of ˜a_j(ϵ), limϵ→0

[˜a⁻_j¹(ϵ)−⟨

(M₀−ω²I)⁻¹Φ^ϵ_j, Φ^ϵ_j⟩]

= lim

ϵ→0

1 4π²

∫

ϵ<|ξ|<1/ϵ

[M0(ξ)⁻¹−(M0(ξ)−ω²I)⁻¹]

(1,0)^⊤·(1,0)^⊤dξ+bl

= 1 4π²

∫

R²

[M₀(ξ)⁻¹−(M₀(ξ)−ω²I)⁻¹]

(1,0)^⊤·(1,0)^⊤dξ+b_l (26) From the definition of the inverse Fourier transformation, we have

|xlim|→0[ Γ0(x,0)−Γω(x,0)] = 1 4π² lim

|x|→0

∫

R²

[M0(ξ)⁻¹−(M0(ξ)−ω²I)⁻¹] e^iξ·xdξ

= 1 4π²

∫

R²

[M0(ξ)⁻¹−(M0(ξ)−ω²I)⁻¹] dξ, where the last step follows from the uniform convergence

[M₀(ξ)⁻¹−(M₀(ξ)−ω²I)⁻¹]

m,n(1−e^iξ·x)→0, as|x| →0, m, n= 1,2,

inξ∈R², which can be easily proved using the expressions ofM₀(ξ)⁻¹ and (M₀(ξ)−ω²I)⁻¹ given in (9) and (10). Therefore, the first term on the right hand side of (26) is just the (1,1)-th entry of the matrix Γ₀(x,0)−Γ_ω(x,0) taking the value at|x|= 0. Recalling Lemma II.1, we obtain

limϵ→0

[˜a⁻_j¹(ϵ)−⟨

(M₀−ω²I)⁻¹Φ^ϵ_j, Φ^ϵ_j⟩]

=−η+b_l, where η is given in (12).

Analogously, if j = 2l for some l = 1,· · · , N, then ˜a⁻_j¹ takes the same form as in (25) and

limϵ→0

[a˜⁻_j¹(ϵ)−⟨

(M₀−ω²I)⁻¹Φ^ϵ_j, Φ^ϵ_j⟩]

= lim

ϵ→0

1 4π²

∫

ϵ<|ξ|<1/ϵ

[M₀(ξ)⁻¹−(M₀(ξ)−ω²I)]₋1

dξ(0,1)^⊤·(0,1)^⊤+b_l

= [Γ₀(x,0)−Γ_ω(x,0)]_|x|=0 (0,1)^⊤·(0,1)^⊤+b_l

=−η+b_l.

To sum up Cases 1 and 2, we deduce that the 2 × 2 diagonal blocks of the matrix Π_B,Y := lim_ϵ→0Π_ϵ(ω) are given by the 2×2 matrices



−η+b_l 0 0 −η+b_l



, l= 1,2,· · · , N.

Case 3: j ∈ {2l−1,2l},j^′ ∈ {2l^′−1,2l^′}for somel, l^′ ∈ {1,· · · , N}such that|l−l^′| ≥1, i.e. the element [Π_B,Y]_j,j′ lies in the off diagonal-by-2×2-blocks of Π_B,Y.

Without loss of generality we assume j = 2l−1, j^′ = 2l^′ −1. Then,

Φ^ϵ_j =χϵφ¹_y(l) =χϵ(1,0)^⊤ϕ_y^(l), Φ^ϵ_j+1 =χϵφ²_y(l) =χϵ(0,1)^⊤ϕ_y^(l), Φ^ϵ_j′ =χ_ϵφ¹_y(l′) =χ_ϵ(1,0)^⊤ϕ_y(l′), Φ^ϵ_j′+1 =χ_ϵφ²_y(l′) =χ_ϵ(0,1)^⊤ϕ_y(l′), Define the 2×2 matrix Υ_l:= (Φ^ϵ_j,Φ^ϵ_j+1) = χ_ϵϕ_y(l)I. A short computation shows

⟨(M₀−ω²I)⁻¹(ξ)Υ_l(ξ), Υ_l′(ξ)⟩

=

∫

ϵ<|ξ|<1/ϵ

(M₀−ω²I)⁻¹(ξ)ϕ_y(l)(ξ)ϕ_y(l′)(ξ)dξ

= 1

4π²

∫

ϵ<|ξ|<1/ϵ

(M0−ω²I)⁻¹(ξ) exp[i(y^(l′)−y^(l))·ξ]dξ

→[

Γ_ω(y^(l′)−y^(l)) ]

asϵ→0,

where the last step follows from the inverse Fourier transformation.

Finally, combining Cases 1-3 gives the matrix (22).

In addition, if we choose the vectorB in the form (23) withc_l ∈R,l= 1, ..., N, then from the explicit forms of Π_B,Y(ω) in (22) andη in (II.1), we obtain (Π_B,Y(ω))^∗ = Π_B,Y(−ω).

We next prove the convergence of the operator K_j,j^ϵ _′ : L²(R²)² →L²(R²)² defined by K_j,j^ϵ ′(f) :=

⟨

f, χ_ϵF₋^(j_ω^′)

⟩

χ_ϵF_ω^(j), f ∈L²(R²)².

To be consistent with the definitions of Φ^ϵ_j and χ_ϵFω^(j), we introduce the functions Φ_j(ξ) :=





(2π)⁻¹e^−iξ·y(l)(1,0)^⊤ if j = 2l−1, (2π)⁻¹e^−iξ·y(l)(0,1)^⊤ if j = 2l,

, F_ω^(j) := (M₀−ω²I)⁻¹Φ_j(ξ), (27) With these notations, we define the matrix

Θ_l,ω(ξ) := (F_ω^(2l−1), F_ω^(2l)) = (M₀−ω²I)⁻¹exp(−iξ·y^(l))(2π)⁻¹ ∈C^2×2, Imω >0 (28) for l= 1, ..., N.

Lemma II.5. Suppose that Imw > 0. Then the operator K_j,j^ϵ ′ converges to K_j,j′ in the operator norm, where the operator K_j,j′ :L²(R²)² →L²(R²)² is defined by

K_j,j′(f) :=

⟨

f, F₋^(j_ω^′)

⟩ F_ω^(j).

Proof. It is easy to see (F⁻¹Θ_l,ω)(x) = 1

4π²

∫

R²

(M₀−ω²I)⁻¹(ξ) exp(iξ·(x−y^(l)))dξ= Γ_ω(x−y^(l)). (29) By the definition of Γ_ω and the asymptotic behavior of Hankel functions for a large complex argument, it follows that bothFω^(j)andF₋^(j)_ω belong toL²(R²)² for everyj provided Im (ω)>

0. Then obviouslyχ_ϵF_−ω^(j′)−F_−ω^(j′) and χ_ϵF_−ω^(j)−F_−ω^(j) tend to zero inL²(R²)² whenϵ tends to 0. We write

⟨

f, χ_ϵF₋^(j_ω^′)

⟩

χ_ϵF_ω^(j)−⟨

f, F₋^(j_ω^′)

⟩ F_ω^(j)

=

⟨ f,

(

χ_ϵF₋^(j_ω^′)−F₋^(j_ω^′) )⟩ (

χ_ϵF_ω^(j)−F_ω^(j) )−⟨

f, F₋^(j_ω^′)

⟩ (

F_ω^(j)−χ_ϵF_ω^(j) ) +

⟨ f,

(

χϵF₋^(j_ω^′)−F₋^(j_ω^′) )⟩

F_ω^(j),

which combined with the Cauchy-Schwartz inequality implies the convergence sup

f∈L²(R²)²

∥⟨

f, χ_ϵF₋^(j_ω^′)

⟩

χ_ϵFω^(j)−⟨

f, F₋^(j_ω^′)

⟩

Fω^(j)∥L²(R²)²

∥f∥L²(R²)²

→0, ϵ→0.

This proves the convergence ||K_j,j′ −K_j,j^ϵ ′||L²(R²)²→L²(R²)² →0 as ϵ→0.

Combining Lemmas II.3 and II.5, we obtain the convergence in the operator norm of (He^ϵ−ω²)⁻¹ to

L(ω) ˆf := (M₀−ω²I)⁻¹fˆ+

∑2N j,j^′=1

[

Π_B,Y(ω) ]₋1

j,j^′

⟨f , Fˆ ₋^(j_ω^′)

⟩

F_ω^(j), ∀fˆ∈L²(R²)², (30)

for all Imw > 0 such that det[ΠB,Y(ω)] ̸= 0. Recall again that [ΠB,Y(ω)]⁻_j,j¹′ stands for the (j, j^′)-th entry of the matrix [Π_B,Y(ω)]⁻¹. The main theorem of this paper is stated as follows.

Theorem II.6. Suppose that the operator He^ϵ is given by (16), with a_j(ϵ) =

[

− (λ+ 3µ)

2π(λ+ 2µ)µlnϵ+b_j ]₋1

, b_j ∈C, j ∈ {1,2,· · · , N}.

Write B = (b1,· · · , bN) satisfying the condition (23), and let ΠB,Y, Fω^(j) be defined by (22), (27) respectively. Then

(i): The operatorHe^ϵ converges in norm resolvent sense to a closed and self-adjoint operator

∆ˆ_B,Y as ϵ → 0, where the resolvent of ∆ˆ_B,Y is given by (30). That is, for Imω > 0