The heterogeneous Helmholtz problem with spherical symmetry: Green’s operator and stability estimates

Abstract

We study wave propagation phenomena modelled in the frequency domain by the Helmholtz equation in heterogeneous media with focus on media with discontinuous, highly oscillating wave speed. We restrict to problems with spherical symmetry and will derive explicit representations of the Green’s operator and stability estimates which are explicit in the frequency and the wave speed.

Keywords

Helmholtz equation high frequency heterogeneous media stability estimates Bessel functions

1. Introduction

High-frequency scattering problems have many important applications which include, e.g., radar and sonar detection, medical and seismic imaging as well as applications in nano photonics and lasers. In physics, such problems are studied intensively in the context of wave scattering in disordered media and localisation of waves with the goal to design waves with prescribed intensity, interference, localized foci, parity-time symmetry, etc.; see, e.g., [2,13–16,21,24] for references to the theoretical and experimental physics literature.

In the frequency domain, these problems are often modelled by the Helmholtz equation with, possibly, large wavenumber. For heterogeneous media, the coefficients in these equation become variable and we focus here on the effects of variable wave speed. In general, wave propagation in heterogeneous media can exhibit interference phenomena in the form of, e.g., the localisation of waves or energies which grow exponentially with respect to the frequency. These events are rare in the set of all parameter configurations – however, their existence and their complicated behaviour make the analysis notoriously hard.

In this paper, we discuss the Helmholtz problem on a bounded, Lipschitz domain $Ω \subset R^{d}$ $\begin{matrix} (1.1) & \begin{matrix} - Δ u - {(\frac{ω}{c})}^{2} u = f in Ω, \\ B u = g on Γ : = \partial Ω, \end{matrix} \end{matrix}$ for given right-hand side f and boundary data g and for a suitable first order boundary differential operator $B$ which guarantees existence and uniqueness. We assume that the constant frequency satisfies $\begin{matrix} (1.2) & ω ⩾ ω_{0} > 0 \end{matrix}$ and that the variable wave speed $c \in L^{\infty} (Ω)$ is positive and bounded: $0 < c_{min} ⩽ c ⩽ c_{max}$ for some $c_{min}, c_{max} > 0$ .

We define the “energy space” by $H : = H^{1} (Ω)$ equipped with the norm $\begin{matrix} ‖ u ‖_{H} : = \sqrt{‖ \nabla u ‖^{2} + {‖ \frac{ω}{c} u ‖}^{2}}, \end{matrix}$ where $‖ \cdot ‖$ denotes the standard $L^{2}$ norm on the domain Ω induced by the $L^{2}$ -scalar product denoted by $(\cdot, \cdot)$ with the convention that complex conjugation is on the second argument. Also we write ${(\cdot, \cdot)}_{Γ} : = {(\cdot, \cdot)}_{L^{2} (Γ)}$ .

A stability estimate for the solution operator of the form (if available) $\begin{matrix} ‖ u ‖_{H} ⩽ C_{stab} (‖ f ‖ + ‖ g ‖_{H^{1 / 2} (Γ)}) \end{matrix}$ plays an important role in the design of numerical methods to approximate the solution of (1.1). In [10] the numerical discretization of heterogeneous Helmholtz equation is studied – in particular, a resolution condition is derived for abstract Galerkin discretization which explicitly depends on the stability constant $C_{stab}$ . Therefore the dependency of $C_{stab}$ on all the parameters given in the problem (e.g. frequency ω and wave speed c) is crucial for the design of a numerical discretization.

In this work, we will present a fully explicit analysis of the stability of the high frequency Helmholtz problem (1.1) in a setting with spherical symmetry.

Literature overview. First rigorous stability results for the heterogeneous Helmholtz equation go back to Aziz et al. [3], where the 1-dimensional problem with a wave speed $c \in C^{1} (Ω)$ is considered. For higher dimensions, similar results are proved in [20] and [10], for a wave speed that is assumed to be slowly varying. In [4] the full-space problem with one $C^{\infty}$ inclusion and a discontinuity of the wave speed c across the interface is discussed. It is shown that the stability constant in this setting cannot grow faster than exponential in the frequency ω. Transmission problems with one inclusion are also considered in [22] and [7] (for a convex, $C^{\infty}$ obstacle), where conditions on c are proposed such that the stability constant can grow super-algebraic in the frequency ω or such that the problem is stable independent of the frequency ω. A similar stability result is presented in [18] for a Lipschitz, star-shaped obstacle, together with a coefficient explicit estimate on the stability constant. If the obstacle is a ball, an analysis of the stability is carried out in [5] (for $d = 2$ ) and [6] (for $d = 3$ ), using estimates of Bessel and Hankel functions. The stability of the 1-dimensional Helmholtz problem with piecewise constant and possibly highly oscillating wave speed c is analysed in the thesis [8]. In the worst case, the estimate on the stability constant grows exponentially in the number of discontinuities of c. In [23] it is shown, for a one-dimensional model problem, that for a wave speed c which oscillates between two values, the stability constant can grow at most exponentially in the frequency ω and is independent of the number of discontinuities of c.

Our approach follows the same basic idea as in [23]: since the wave speed is assumed to be piecewise constant and spherically symmetric we employ a Fourier expansion in the spherical variables and end up with a radial transmission problem. We derive an explicit representation of the Green’s operator which is key for studying its stability.

Outline and main achievements of the paper. In this paper we investigate a heterogeneous Helmholtz problem in a spherical symmetric setting in general dimension $d \in N_{⩾ 1}$ . In order to focus on phenomena induced by the differential operator we have chosen $f = 0$ as the right-hand side and inhomogeneous Dirichlet-to-Neumann (DtN) boundary conditions. In Section 2 we introduce the model problem and the class of parameters under consideration.

We consider piecewise constant wave speed jumping at n radial points and the emphasis is that the number of jumps n may be arbitrary large. In Section 2, we employ a Fourier ansatz where the Fourier coefficients then only depend on the radial variable and satisfy an ordinary differential equation (ODE) of Bessel-type in each interval where the wave speed is constant. Interface conditions are imposed at the jump points and boundary conditions are derived for the ODE. The resulting system of equations can be represented as a linear system of dimension $2 n$ , whose solution is determined by the radial Green’s operator (i.e. the inverse matrix of the system).

In the literature, often restrictions are imposed on the wave speed c as, e.g., c belongs to $C^{0, 1} (Ω, R)$ ; the wave speed in radial coordinate satisfies a certain monotonicity behaviour; the number of jumps equals 1; the domain Ω is the full space and the coefficient c is periodic; the wave speed is given as a small fluctuation around the globally constant case; the Helmholtz equation is considered in a stochastic setting; or the problem is restricted to the one-dimensional case $d = 1$ . In contrast, the focus in this paper is on wavenumbers which do not satisfy such restrictions and to consider general dimensions d.

Our first main result is the derivation of a new representation of the radial Green’s operator for general dimension d and arbitrary number of jump points n. Since we restrict to a vanishing right-hand side $f = 0$ , only the last column of the Green’s operator is relevant. In Section 3 we introduce the representation of the last column in the radial Green’s operator $\begin{matrix} (1.3) & | {(M_{m}^{Green})}_{2 ℓ, 2 n} | = | \frac{Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{m, ℓ})}{β_{m, n}} |, | {(M_{m}^{Green})}_{2 ℓ - 1, 2 n} | = | \frac{β_{m, ℓ - 1}}{β_{m, n}} | . \end{matrix}$ Here, m denotes the Fourier mode, $c_{ℓ}$ are the constant values of the wave speed and $z_{ℓ} = ω x_{ℓ}$ , where $x_{ℓ}$ denotes the ℓ-th jump point in radial direction. The key is the sequence ${(β_{m, ℓ})}_{ℓ = 0}^{n} \subset C$ which satisfies a simple linear recursion (see Remark 3.2) and the analysis of the Green’s operator boils down to the investigation of this sequence. In [23] for the one-dimensional case $d = 1$ , the Green’s operator has also been expressed by a recursive sequence in the complex plane. However the recursion in [23] is more complicated via a (rational) Möbius transform instead of the linear recursion in this paper.

The proof of representation (1.3) is technical and shifted to Section 6. The representation suggests that the stability of the original problem depends on the maximal growth/decay of $| β_{m, ℓ} |$ with respect to ℓ and we state in Section 3.3 for the case $d = 3$ and $m = 0$ that the maximum/minimum can grow/decay exponentially with respect to the frequency ω, in particular we prove $\begin{matrix} | β_{0, ℓ} | ⩽ α^{ω}, | β_{0, ℓ} | ⩾ α^{- ω} for some α > 1, \end{matrix}$ which leads to a stability bound which is exponential with respect to ω. We present bounds for the energy norm $‖ \cdot ‖_{H}$ and also pointwise bounds. The proof of this main stability result is postponed to Section 5.

In Section 4, we characterise different parameter configurations which lead either to a localisation effect in the solution or to a globally stable solution. In the first example, we fix $n = 1$ and recall from the literature how the choice of the frequency ω and the Fourier mode m may lead to a wave localisation along the single jump interface of c (also known as “whispering gallery modes”, see also [6]). The other two examples are new and show that the localisation effect can also occur for $m = 0$ if the number n of jumps is “in resonance” with the frequency ω. In both examples, we consider the same wave speed c, they differ only on their choice of frequency ω. In the second example, we observe a localisation in the centre of the domain, leading to an exponential growth with respect to ω of the stability constant $C_{stab}$ as $n \to \infty$ . This underlines the sharpness of our main stability result. In the last example, although we also consider $n \to \infty$ , the stability constant $C_{stab}$ stays bounded independently of n and ω.

2. Helmholtz problem with spherical symmetry

In this section, we will specify the set of parameters (wave speed/frequency/ boundary conditions) and introduce the spherical symmetric setting which will be the basis of the Fourier expansion.

2.1. The Helmholtz problem with DtN boundary conditions

The Euclidean norm in $R^{d}$ is denoted by $| \cdot |$ . We fix the domain $Ω = B_{1}^{d} : = {x \in R^{d} ∣ | x | < 1}$ and denote by $\frac{\partial}{\partial n}$ the derivative in direction of the outward normal vector. We set $f = 0$ and the variable wave speed $c \in L^{\infty} (Ω)$ is assumed to be piecewise constant on annular regions. More precisely, we assume that there are given points $\begin{matrix} (2.1a) & 0 = x_{0} < x_{1} < \dots < x_{N} = 1, \end{matrix}$ corresponding to intervals $τ_{j} : = (x_{j - 1}, x_{j})$ of lengths $h_{j} : = x_{j} - x_{j - 1}$ and positive numbers $c_{j}$ such that $\begin{array}{l} (2.1b) & c (x) = c_{j} \forall x \in Ω with | x | \in τ_{j} \forall 1 ⩽ j ⩽ N, \\ (2.1c) & 0 < c_{min} ⩽ c ⩽ c_{max} < \infty \end{array}$ for some $c_{min}, c_{max} \in R_{> 0}$ . We consider the homogeneous Helmholtz problem $\begin{matrix} (2.2) & \begin{matrix} - Δ u - {(\frac{ω}{c})}^{2} u = 0 in Ω, \\ \frac{\partial u}{\partial n} - T_{\frac{ω}{c_{N}}} u = g on Γ \end{matrix} \end{matrix}$ with inhomogeneous Dirichlet-to-Neumann boundary conditions which are defined as follows. Let $Ω^{+} : = R^{d} ∖ \overline{Ω}$ . It can be shown that, for given $\tilde{g} \in H^{1 / 2} (Γ)$ and $κ \in R$ , the problem: $\begin{matrix} find w \in H_{loc}^{1} (Ω^{+}) such that \\ (- Δ - κ^{2}) w = 0 in Ω^{+}, \\ w = \tilde{g} on \partial Ω, \\ | ⟨ \frac{x}{| x |}, \nabla w (x) ⟩ - i κ w (x) | = o (| x |^{\frac{1 - d}{2}}), | x | \to \infty, \end{matrix}$ has a unique weak solution. The mapping $\tilde{g} \mapsto w$ is called the Steklov–Poincaré operator and denoted by $S_{P} : H^{1 / 2} (\partial Ω) \to H_{loc}^{1} (Ω^{+})$ . The Dirichlet-to-Neumann map is given by $T_{κ} : = γ_{1} S_{P} : H^{1 / 2} (\partial Ω) \to H^{- 1 / 2} (\partial Ω)$ , where $γ_{1} : = \partial / \partial n$ is the normal trace operator.

Remark 2.1.
The heterogeneous Helmholtz problem (2.2) is well-posed (cf. [1,10,12]).

In this paper, we discuss the stability of problem (2.2) for Ω being the unit ball centred at the origin and for a wave speed c that is piecewise constant on concentric layers. We will use an explicit representation of the Green’s operator to understand which parameter configurations are well-behaved (i.e. where the stability constant is bounded with respect to the frequency ω) and which configuration lead to a localisation effect (i.e. a stability constant that grows exponentially with respect to ω).
2.2. Helmholtz problem in spherical coordinates

For $d ⩾ 2$ , let $Y_{m, n}$ denote the eigenfunctions (spherical harmonics) of the (negative) Laplace–Beltrami operator $- Δ_{Γ}$ on $Γ : = \partial Ω$ (cf. [25, §22]) $\begin{matrix} (2.3) & - Δ_{Γ} Y_{m, n} = λ_{m} Y_{m, n}, m \in N_{0}, n \in ι_{m}, \end{matrix}$ for some finite index set $ι_{m}$ which corresponds to the multiplicity of the eigenvalue $λ_{m}$ . We assume that the eigenvalues $λ_{m}$ are numbered such that $\begin{matrix} 0 = λ_{0} < λ_{1} < \dots \end{matrix}$ and the eigenfunctions form a orthonormal basis of $L^{2} (Γ)$ . Explicitly it holds (cf. [25, §22]) $\begin{matrix} (2.4) & λ_{m} = m (m + d - 2), with multiplicity ♯ ι_{m} = \frac{2 m + d - 2}{m + d - 2} (\binom{m + d - 2}{d - 2}) . \end{matrix}$

We introduce spherical coordinates in $R^{d}$ by $x = r ξ$ , for $r : = | x |$ and $ξ : = x / r \in S_{d - 1}$ . This transformation is denoted by $x = ψ (r, ξ)$ and for a function w in Cartesian coordinates we write $\hat{w} = w \circ ψ$ . The Laplace operator in spherical coordinates is given by $\begin{matrix} (Δ v) \circ ψ = \frac{1}{r^{d - 1}} \partial_{r} (r^{d - 1} \partial_{r} \hat{v}) + \frac{1}{r^{2}} Δ_{Γ} \hat{v} . \end{matrix}$

Next, we transform the Helmholtz equation (2.2) to spherical coordinates and write $\hat{u} = u \circ ψ$ and $\hat{g} (ξ) = g \circ ψ (1, ξ)$ . We employ a Fourier expansion to the boundary data $\begin{matrix} \hat{g} (ξ) = \sum_{m \in N_{0}} \sum_{n \in ι_{m}} {\hat{g}}_{m, n} Y_{m, n} (ξ) \end{matrix}$ and a similar expansion for the solution $\begin{matrix} (2.5) & u \circ ψ (r, ξ) = \hat{u} (r, ξ) = \sum_{m \in N_{0}} \sum_{n \in ι_{m}} {\hat{u}}_{m, n} (r) Y_{m, n} (ξ) . \end{matrix}$ The ODE for the Fourier coefficients ${\hat{u}}_{m, n}$ (more precisely for the restrictions ${\hat{u}}_{m, n, j} : = {\hat{u}}_{m, n} |_{τ_{j}}$ ) reads for $r \in τ_{j}$ , $\begin{matrix} (2.6) & - \frac{1}{r^{d - 1}} \partial_{r} (r^{d - 1} {\hat{u}}_{m, n, j}^{'} (r)) + (\frac{m (m + d - 2)}{r^{2}} - {(\frac{ω}{c_{j}})}^{2}) {\hat{u}}_{m, n, j} (r) = 0 . \end{matrix}$ Moreover we have the conditions at each interface $\begin{matrix} (2.7) & {[{\hat{u}}_{m, n}]}_{x_{j}} = {[{\hat{u}}_{m, n}^{'}]}_{x_{j}} = 0, 1 ⩽ j ⩽ N - 1, \end{matrix}$ where ${[f]}_{x}$ denotes the jump of f at x.

Remark 2.2 (The case $d = 1$ ).

In one dimension, problem (2.2) can be written in a similar form. We set $\hat{u} (r) = u (| x |)$ and obtain for ${\hat{u}}_{j} : = {\hat{u}}_{|_{τ_{j}}}$ the ordinary differential equation $\begin{matrix} (2.8) & - {\hat{u}}_{j}^{″} - {(\frac{ω}{c_{j}})}^{2} {\hat{u}}_{j} = 0 in τ_{j} \end{matrix}$ with jump conditions $\begin{matrix} {[\hat{u}]}_{x_{j}} = {[{\hat{u}}^{'}]}_{x_{j}} = 0, 1 ⩽ j ⩽ N - 1 . \end{matrix}$ This corresponds to (2.6) for $d = 1$ and $m = n = 0$ as well as to (2.7). The function u then is given by $u (x) = \hat{u} (r) Y_{0, 0} (\frac{x}{| x |})$ , where for $d = 1$ we set $Y_{0, 0} (\pm 1) = 1$ .

We denote by

$J_{ν}$ : the Bessel function of first kind and order ν,

$j_{ν}$ : the spherical Bessel function of first kind and order ν,

$H_{ν}^{(1)}$ : the Hankel function of first kind and order ν,

$h_{ν}^{(1)}$ : the spherical Hankel function of first kind and order ν.

The fundamental system for the ODE (2.6) is generated, for $d ⩾ 2$ , by $\begin{matrix} f_{m, d, 1} (r) : = c_{d} \frac{H_{m + \frac{d}{2} - 1}^{(1)} (r)}{r^{\frac{d}{2} - 1}} and f_{m, d, 2} (r) : = c_{d} \frac{J_{m + \frac{d}{2} - 1} (r)}{r^{\frac{d}{2} - 1}} \end{matrix}$ and for the ODE (2.8) for $d = 1$ , by $\begin{matrix} f_{0, 1, 1} (r) : = c_{1} e^{i r} and f_{0, 1, 2} (r) : = c_{1} cos r, \end{matrix}$ for some normalization constants $c_{d} > 0$ . For $d = 1, 2, 3$ , we set $\begin{matrix} c_{1} : = 1, c_{2} : = 1, c_{3} : = \sqrt{π / 2} \end{matrix}$ so that $\begin{array}{l} f_{m, 1, 1} (r) : = e^{i r}, f_{m, 1, 2} (r) : = cos r, m = 0, \\ f_{m, 2, 1} (r) : = H_{m}^{(1)} (r), f_{m, 2, 2} (r) : = J_{m} (r), m \in N_{0}, \\ f_{m, 3, 1} (r) : = h_{m}^{(1)} (r), f_{m, 3, 2} (r) : = j_{m} (r), m \in N_{0} . \end{array}$

To include the case $d = 1$ in the notation, we set $N_{d} : = N_{0}$ for $d ⩾ 2$ and $N_{1} : = {0}$ . Note that $ι_{m}$ in (2.3) depends on the dimension d (cf. (2.4)) and for $d = 1$ we set $ι_{0} : = {0}$ . In this light, we write also short $N$ for $N_{d}$ . Also we set $S_{0} = {- 1, 1}$ and for $d ⩾ 2$ we denote by $S_{d - 1}$ the unit sphere in $R^{d}$ .

The DtN boundary conditions in (2.2) read (see, e.g., [17, (3.7), (3.10), (3.25)]): $\begin{matrix} {\hat{u}}_{m, n}^{'} (1) - \frac{ω}{c_{N}} \frac{f_{m, d, 1}^{'} (\frac{ω}{c_{N}})}{f_{m, d, 1} (\frac{ω}{c_{N}})} {\hat{u}}_{m, n} (1) = {\hat{g}}_{m, n} . \end{matrix}$ Next, we impose an appropriate condition at the origin which guarantees that the solution u in (2.5) is smooth at origin. We require for all $ξ \in S_{d - 1}$ $\begin{array}{l} lim_{r ↘ 0} u (r ξ) = lim_{r ↘ 0} u (- r ξ), \\ lim_{r ↘ 0} \frac{u (r ξ) - u (0)}{r} = lim_{r ↘ 0} \frac{u (0) - u (- r ξ)}{r} \end{array}$ which is equivalent to $\begin{array}{l} \sum_{m \in N} \sum_{n \in ι_{m}} {\hat{u}}_{m, n} (0) Y_{m, n} (ξ) = \sum_{m \in N} \sum_{n \in ι_{m}} {\hat{u}}_{m, n} (0) Y_{m, n} (- ξ), \\ \sum_{m \in N} \sum_{n \in ι_{m}} {\hat{u}}_{m, n}^{'} (0) Y_{m, n} (ξ) = - \sum_{m \in N} \sum_{n \in ι_{m}} {\hat{u}}_{m, n}^{'} (0) Y_{m, n} (- ξ) . \end{array}$ We use that the spherical harmonics satisfy by definition (cf. e.g., [9, p. 71]) $\begin{matrix} Y_{m, n} (ξ) = {(- 1)}^{m} Y_{m, n} (- ξ) . \end{matrix}$ Hence, $\begin{array}{l} {\hat{u}}_{m, n} (0) = {(- 1)}^{m} {\hat{u}}_{m, n} (0), \\ {\hat{u}}_{m, n}^{'} (0) = {(- 1)}^{m + 1} {\hat{u}}_{m, n}^{'} (0) . \end{array}$ This implies $\begin{array}{l} {\hat{u}}_{m, n} (0) = 0 for odd m, \\ {\hat{u}}_{m, n}^{'} (0) = 0 for even m \end{array}$ and these are the boundary conditions at $r = 0$ .

In summary, for the considered radial symmetric case we study the following system of ODEs: $\begin{matrix} (2.9) & - \frac{1}{r^{d - 1}} \partial_{r} (r^{d - 1} {\hat{u}}_{m, n, j}^{'} (r)) + (\frac{m (m + d - 2)}{r^{2}} - {(\frac{ω}{c_{j}})}^{2}) {\hat{u}}_{m, n, j} (r) = 0, \end{matrix}$ for $r \in τ_{j}$ , $1 ⩽ j ⩽ N$ , together with the interface conditions $\begin{matrix} (2.10) & {[{\hat{u}}_{m, n}]}_{x_{j}} = {[{\hat{u}}_{m, n}^{'}]}_{x_{j}} = 0, 1 ⩽ j ⩽ N - 1, \end{matrix}$ and boundary conditions $\begin{array}{l} {\hat{u}}_{m, n}^{'} (1) - \frac{ω}{c_{N}} \frac{f_{m, d, 1}^{'} (\frac{ω}{c_{N}})}{f_{m, d, 1} (\frac{ω}{c_{N}})} {\hat{u}}_{m, n} (1) = {\hat{g}}_{m, n}, \\ {\hat{u}}_{m, n} (0) = 0 for odd m, \\ {\hat{u}}_{m, n}^{'} (0) = 0 for even m . \end{array}$ For the solution of the homogeneous equation (2.9) we employ the ansatz for $m \in N$ and $n \in ι_{m}$ $\begin{matrix} (2.11) & {\hat{u}}_{m, n} |_{τ_{j}} = A_{(m, n), j} f_{m, d, 1} (\frac{ω}{c_{j}} \cdot) + B_{(m, n), j} f_{m, d, 2} (\frac{ω}{c_{j}} \cdot) . \end{matrix}$ The boundary conditions at the origin and the boundary condition at 1 imply1

¹
The Wronskian $W (φ_{1}, φ_{2})$ of two functions $φ_{1}$ and $φ_{2}$ is given by $\begin{matrix} W (φ_{1}, φ_{2}) (z) : = φ_{1} (z) φ_{2}^{'} (z) - φ_{1}^{'} (z) φ_{2} (z) . \end{matrix}$

\begin{array}{l} (2.12a) & A_{(m, n), 1} = 0, \\ (2.12b) & B_{(m, n), N} = \frac{f_{m, d, 1} (κ_{N, N})}{κ_{N, N} W (f_{m, d, 1}, f_{m, 2}) (κ_{N, N})} {\hat{g}}_{m, n}, \end{array}

with

\begin{matrix} z_{ℓ} : = ω x_{ℓ} and κ_{j, ℓ} : = z_{ℓ} / c_{j} . \end{matrix}

The interface conditions (2.10) can be rewritten as a system of linear equation given by $\begin{array}{l} A_{(m, n), ℓ} f_{m, d, 1} (κ_{ℓ, ℓ}) + B_{(m, n), ℓ} f_{m, d, 2} (κ_{ℓ, ℓ}) = A_{(m, n), ℓ + 1} f_{m, d, 1} (κ_{ℓ + 1, ℓ}) + B_{(m, n), ℓ + 1} f_{m, d, 2} (κ_{ℓ + 1, ℓ}), \\ \frac{A_{(m, n), ℓ}}{c_{ℓ}} f_{m, d, 1}^{'} (κ_{ℓ, ℓ}) + \frac{B_{(m, n), ℓ}}{c_{ℓ}} f_{m, d, 2}^{'} (κ_{ℓ, ℓ}) = \frac{A_{(m, n), ℓ + 1}}{c_{ℓ + 1}} f_{m, d, 1}^{'} (κ_{ℓ + 1, ℓ}) + \frac{B_{(m, n), ℓ + 1}}{c_{ℓ + 1}} f_{m, d, 2}^{'} (κ_{ℓ + 1, ℓ}), \end{array}$ for $1 ⩽ ℓ ⩽ N - 1$ , whose solution is given by $\begin{matrix} (2.13) & x_{m, n} = M_{m, n}^{Green} r_{m, n}, \end{matrix}$ where $\begin{matrix} x_{m, n} = {(B_{(m, n), 1}, A_{(m, n), 1}, B_{(m, n), 2}, \dots, A_{(m, n), n - 1}, B_{(m, n), N - 1}, A_{(m, n), N})}^{⊺}, \end{matrix}$ supplemented by $A_{(m, n), 1}$ , $B_{(m, n), N}$ as in (2.12) contains the coefficients in (2.11) and $M_{m, n}^{Green} \in C^{2 n \times 2 n}$ , $r_{m, n} \in C^{2 n}$ . The precise definition of the linear system corresponding to (2.13) is given in (3.2).

3. Representation of the Green’s operator and stability estimate

In this section, we introduce a representation of the radial Green’s operator and formulate the main stability estimate.

3.1. The radial Green’s operator

In this section, we rewrite the linear system (2.10) in a suitable way and find a representation of the entries of the Green’s operator $M_{m, n}^{Green}$ .

Notation 3.1.
In the following, we skip the indices n and d and write short $f_{m, ℓ}$ for $f_{m, d, ℓ}$ and $A_{m, j}$ for $A_{(m, n), j}$ and similar for other quantities. Using the definition of the Wronskian $W (\cdot, \cdot)$ , we define $\begin{array}{l} w_{m, j, k, ℓ}^{p, q} : = & W (f_{m, p} (\frac{\cdot}{c_{j}}), f_{m, q} (\frac{\cdot}{c_{k}})) (z_{ℓ}) = \frac{f_{m, p} (κ_{j, ℓ}) f_{m, q}^{'} (κ_{k, ℓ})}{c_{k}} - \frac{f_{m, p}^{'} (κ_{j, ℓ}) f_{m, q} (κ_{k, ℓ})}{c_{j}} . \end{array}$

The interface conditions (2.10) together with (2.12) give $\begin{matrix} (3.1) & [\begin{matrix} S_{m}^{(1)} & T_{m}^{(1)} & 0 & \dots & 0 \\ R_{m}^{(1)} & S_{m}^{(2)} & ⋱ & ⋱ & ⋮ \\ 0 & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & T_{m}^{(n - 1)} \\ 0 & \dots & 0 & R_{m}^{(n - 1)} & S_{m}^{(n)} \end{matrix}] (\begin{matrix} B_{1} \\ ⋮ \\ A_{n} \\ B_{n} \\ A_{n + 1} \end{matrix}) = C (\begin{matrix} 0 \\ ⋮ \\ 0 \\ f_{m, 2} (κ_{n + 1, n}) \\ \frac{f_{m, 2}^{'} (κ_{n + 1, n})}{c_{n + 1}} \end{matrix}) \end{matrix}$ with $\begin{matrix} n : = N - 1, C : = \frac{f_{m, 1} (κ_{n + 1, n + 1}) {\hat{g}}_{m}}{κ_{n + 1, n + 1} W (f_{m, 1}, f_{m, 2}) (κ_{n + 1, n + 1})} \end{matrix}$ and $\begin{array}{l} R_{m}^{(ℓ)} = [0 ∣ r_{m}^{(ℓ)}] = [\begin{matrix} 0 & f_{m, 1} (κ_{ℓ + 1, ℓ + 1}) \\ 0 & \frac{f_{m, 1}^{'} (κ_{ℓ + 1, ℓ + 1})}{c_{ℓ + 1}} \end{matrix}], 1 ⩽ ℓ ⩽ n - 1, \\ S_{m}^{(ℓ)} = [s_{m, 1}^{(ℓ)} ∣ s_{m, 2}^{(ℓ)}] = [\begin{matrix} f_{m, 2} (κ_{ℓ, ℓ}) & - f_{m, 1} (κ_{ℓ + 1, ℓ}) \\ \frac{f_{m, 2}^{'} (κ_{ℓ, ℓ})}{c_{ℓ}} & - \frac{f_{m, 1}^{'} (κ_{ℓ + 1, ℓ})}{c_{ℓ + 1}} \end{matrix}], 1 ⩽ ℓ ⩽ n, \\ T_{m}^{(ℓ)} = [t_{m}^{(ℓ)} ∣ 0] = [\begin{matrix} - f_{m, 2} (κ_{ℓ + 1, ℓ}) & 0 \\ - \frac{f_{m, 2}^{'} (κ_{ℓ + 1, ℓ})}{c_{ℓ + 1}} & 0 \end{matrix}], 1 ⩽ ℓ ⩽ n - 1 . \end{array}$

Next we transform this system to a tri-diagonal system and first introduce some notation. We define the block-diagonal matrix $\begin{matrix} D_{m} : = diag [D_{m}^{(ℓ)} : 1 ⩽ ℓ ⩽ n], \end{matrix}$ for2
²
We set ${(\begin{matrix} a \\ b \end{matrix})}^{⊥} : = (b, - a)$ .

$\begin{matrix} D_{m}^{(ℓ)} = \frac{1}{w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} [\begin{matrix} {(t_{m}^{ℓ})}^{⊥} \\ {(r_{m}^{ℓ - 1})}^{⊥} \end{matrix}], 1 ⩽ ℓ ⩽ n, \end{matrix}$ and multiply (3.1) from left with $D_{m}$ to obtain $\begin{matrix} (3.2) & {\hat{M}}_{m}^{(2 n)} (\begin{matrix} B_{1} \\ ⋮ \\ A_{N - 1} \\ B_{N - 1} \\ A_{N} \end{matrix}) = \frac{f_{m, 1} (κ_{N, N})}{ω w_{m, N, N, N}^{1, 2}} (\begin{matrix} 0 \\ ⋮ \\ 0 \\ {\hat{g}}_{m} \end{matrix}) \end{matrix}$ with $\begin{matrix} (3.3) & {\hat{M}}_{m}^{(2 n)} : = [\begin{matrix} {\hat{S}}_{m}^{(1)} & {\hat{T}}_{m}^{(1)} & 0 & \dots & 0 \\ {\hat{R}}_{m}^{(1)} & {\hat{S}}_{m}^{(2)} & ⋱ & ⋱ & ⋮ \\ 0 & ⋱ & ⋱ & 0 \\ ⋮ & ⋱ & {\hat{T}}_{m}^{(n - 1)} \\ 0 & \dots & 0 & {\hat{R}}_{m}^{(n - 1)} & {\hat{S}}_{m}^{(n)} \end{matrix}] \end{matrix}$ and $\begin{array}{l} {\hat{R}}_{m}^{(ℓ)} : = D_{m}^{(ℓ + 1)} R_{m}^{(ℓ)} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], {\hat{S}}_{m}^{(ℓ)} : = D_{m}^{(ℓ)} S_{m}^{(ℓ)} = \frac{1}{w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} [\begin{matrix} w_{m, ℓ + 1, ℓ, ℓ}^{2, 2} & w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2} \\ - w_{m, ℓ, ℓ, ℓ}^{1, 2} & w_{m, ℓ, ℓ + 1, ℓ}^{1, 1} \end{matrix}], \\ {\hat{T}}_{m}^{(ℓ)} : = D_{m}^{(ℓ)} T_{m}^{(ℓ)} = [\begin{matrix} 0 & 0 \\ - 1 & 0 \end{matrix}] . \end{array}$

For later use, we define the matrix $M^{(ℓ)}$ for odd ℓ by removing the last row and column in $M^{(ℓ + 1)}$ .

The Green’s operator is given by $M_{m}^{Green} : = {({\hat{M}}_{m}^{(2 n)})}^{- 1}$ and the right-hand side is defined by $r = (0, \dots, 0, \frac{f_{m, 1} (κ_{N, N})}{ω w_{m, N, N, N}^{1, 2}} g_{m})$ . Note that $\begin{matrix} W (f_{m, d, 1}, f_{m, d, 2}) (r) = \{\begin{matrix} - i & d = 1, \\ - i \frac{2}{π} c_{d}^{2} \frac{1}{r^{d - 1}} & d ⩾ 2, \end{matrix} \end{matrix}$ so that $\begin{matrix} (3.4) & w_{m, N, N, N}^{1, 2} = - \frac{i}{c_{N}} {(\frac{c_{N}}{ω})}^{d - 1} \times \{\begin{matrix} 1 & d = 1, \\ \frac{2}{π} c_{d}^{2} & d ⩾ 2 . \end{matrix} \end{matrix}$ We use Cramer’s rule to represent the entries of the Green’s operator and derive a recursive expression for the arising quotient of determinants (see Theorem 3.4). For $1 ⩽ ℓ ⩽ n = N - 1$ , we define the quantities $\begin{array}{l} (3.5a) & {\tilde{γ}}_{m, ℓ}^{+} : = \frac{f_{m, 1} (κ_{ℓ + 1, ℓ}) \overline{f_{m, 1}^{'} (κ_{ℓ, ℓ})}}{c_{ℓ}} - \frac{f_{m, 1}^{'} (κ_{ℓ + 1, ℓ}) \overline{f_{m, 1} (κ_{ℓ, ℓ})}}{c_{ℓ + 1}}, \\ (3.5b) & {\tilde{γ}}_{m, ℓ}^{-} : = \frac{f_{m, 1}^{'} (κ_{ℓ, ℓ}) f_{m, 1} (κ_{ℓ + 1, ℓ})}{c_{ℓ}} - \frac{f_{m, 1}^{'} (κ_{ℓ + 1, ℓ}) f_{m, 1} (κ_{ℓ, ℓ})}{c_{ℓ + 1}}, \\ (3.5c) & {\tilde{q}}_{m, ℓ} : = \frac{{\tilde{γ}}_{m, ℓ}^{-}}{{\tilde{γ}}_{m, ℓ}^{+}} . \end{array}$

We note that (3.5c) is well-defined since ${\tilde{γ}}_{m, ℓ}^{+} \neq 0$ (cf. proof of [11, Lemma 1]). We also remark that the fundamental solution $f_{m, 1} (x)$ scales with the prefactor $e^{i x}$ as $x \to \infty$ (cf. [19]). Therefore, it is natural to introduce $\begin{matrix} γ_{m, ℓ}^{\pm} : = i e^{i (\pm \frac{z_{ℓ}}{c_{ℓ}} - \frac{z_{ℓ}}{c_{ℓ + 1}})} {\tilde{γ}}_{m, ℓ}^{\pm}, q_{m, ℓ} : = \frac{γ_{m, ℓ}^{-}}{γ_{m, ℓ}^{+}} . \end{matrix}$ We define the sequence $β_{m, ℓ}$ for $0 ⩽ ℓ ⩽ n$ recursively by $\begin{matrix} (3.6) & \begin{matrix} β_{m, 0} : = 1, \\ β_{m, ℓ} : = \frac{γ_{m, ℓ}^{+}}{2 i w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2}} (e^{- i \frac{ω h_{ℓ}}{c_{ℓ}}} β_{m, ℓ - 1} + q_{m, ℓ} \overline{e^{- i ω \frac{h_{ℓ}}{c_{ℓ}}} β_{m, ℓ - 1}}) . \end{matrix} \end{matrix}$
Remark 3.2.
Recursion (3.6) can be written as a linear recursion for the real and imaginary part of $β_{m, ℓ} = β_{m, ℓ}^{R} + i β_{m, ℓ}^{I}$ . We set $\begin{array}{l} θ_{m, ℓ} : = \frac{γ_{m, ℓ}^{+}}{2 i w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2}} e^{- i \frac{ω h_{ℓ}}{c_{ℓ}}} = : θ_{m, ℓ}^{R} + i θ_{m, ℓ}^{I}, \\ ϕ_{m, ℓ} : = \frac{γ_{m, ℓ}^{+}}{2 i w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2}} q_{m, ℓ} e^{i ω \frac{h_{ℓ}}{c_{ℓ}}} = : ϕ_{m, ℓ}^{R} + i ϕ_{m, ℓ}^{I}, \end{array}$ and obtain $(β_{m, 0}^{R}, β_{m, 0}^{I}) = (1, 0)$ and $\begin{matrix} (\begin{matrix} β_{m, ℓ}^{R} \\ β_{m, ℓ}^{I} \end{matrix}) = [\begin{matrix} θ_{m, ℓ}^{R} + ϕ_{m, ℓ}^{R} & ϕ_{m, ℓ}^{I} - θ_{m, ℓ}^{I} \\ θ_{m, ℓ}^{I} + ϕ_{m, ℓ}^{I} & θ_{m, ℓ}^{R} - ϕ_{m, ℓ}^{R} \end{matrix}] (\begin{matrix} β_{m, ℓ - 1}^{R} \\ β_{m, ℓ - 1}^{I} \end{matrix}) . \end{matrix}$
Remark 3.3.
The Wronskian $w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2}$ is independent of m, but depends on the dimension d. Moreover it holds for $1 ⩽ ℓ ⩽ n$ $\begin{matrix} i w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2} = \frac{c_{ℓ + 1}^{d - 2}}{z_{ℓ}^{d - 1}} \times \{\begin{matrix} 1 & d = 1 \\ \frac{2}{π} c_{d}^{2} & d ⩾ 2 \end{matrix} \end{matrix}$ for all dimensions d. In particular, $i w_{m, ℓ + 1, ℓ + 1, ℓ}^{1, 2} \in R_{> 0}$ .
Theorem 3.4.
Let $β_{m, n, ℓ}$ be recursively defined as in ( 3.6 ). Then the entries of the Green’s operator are given by $\begin{matrix} (3.7) & \begin{matrix} {(M_{m}^{Green})}_{2 ℓ, 2 n} = - (\frac{Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{m, ℓ})}{e^{i \frac{z_{n}}{c_{n + 1}}} β_{m, n}}), {(M_{m}^{Green})}_{2 ℓ - 1, 2 n} = - (\frac{e^{i \frac{z_{ℓ - 1}}{c_{ℓ}}} β_{m, ℓ - 1}}{e^{i \frac{z_{n}}{c_{n + 1}}} β_{m, n}}), \end{matrix} \end{matrix}$ for $1 ⩽ ℓ ⩽ n$ .

The proof of Theorem 3.4 is postponed to Section 6.
3.2. Representation for the Fourier mode $m = 0$

In this section, we restrict to $d = 3$ and $m = 0$ . Then the variables in the definition of the sequence $β_{0, ℓ}$ simplify in the following way $\begin{array}{l} γ_{0, ℓ}^{+} = \frac{c_{ℓ + 1} + c_{ℓ}}{z_{ℓ}^{2}}, γ_{0, ℓ}^{-} = \frac{c_{ℓ + 1} - c_{ℓ}}{z_{ℓ}^{2}}, q_{ℓ} : = q_{0, ℓ} = \frac{c_{ℓ + 1} - c_{ℓ}}{c_{ℓ + 1} + c_{ℓ}}, \\ 2 i w_{m, k + 1, k + 1, k}^{1, 2} = \frac{2 c_{k + 1}}{z_{k}^{2}}, \frac{2 c_{k + 1}}{z_{k}^{2} γ_{0, k}^{+}} = \frac{2 c_{k + 1}}{c_{k} + c_{k + 1}} = 1 + q_{k} . \end{array}$ The recursion for $β_{0, ℓ}$ is then given by $\begin{matrix} (3.8) & \begin{matrix} β_{0, 0} : = 1, \\ β_{0, ℓ} : = \frac{1}{1 + q_{ℓ}} (e^{- i \frac{ω h_{ℓ}}{c_{ℓ}}} β_{0, ℓ - 1} + q_{ℓ} \overline{e^{- i ω \frac{h_{ℓ}}{c_{ℓ}}} β_{0, ℓ - 1}}) . \end{matrix} \end{matrix}$

3.3. Main stability theorem

First, we express the $‖ \cdot ‖_{H}$ -norm of the function u in (2.5) in spherical coordinates and recall well-known facts from spherical harmonics: The gradient in spherical coordinates is given by $\begin{matrix} (\nabla v) \circ ψ = \partial_{r} \hat{v} \vec{e_{r}} + \frac{1}{r} \nabla_{Γ} \hat{v}, \end{matrix}$ where $\nabla_{Γ}$ denotes the surface gradient and $\vec{e_{r}}$ the basis vector in r-direction. For the spherical harmonics $Y_{m, n}$ it holds $\begin{array}{l} {(Y_{m, n}, Y_{m^{'}, n^{'}})}_{Γ} = δ_{m, m^{'}} δ_{n, n^{'}}, \\ {(\nabla_{Γ} Y_{m, n}, \nabla_{Γ} Y_{m, n})}_{Γ} = λ_{m} δ_{m, m^{'}} δ_{n, n^{'}}, \\ {(\nabla_{Γ} Y_{m, n}, \vec{e_{r}})}_{Γ} = 0 . \end{array}$ Hence, the transformation rule of variables yields (with $g_{ν} (r) : = r^{(d - 1) / 2 - ν}$ and $I = (0, 1)$ ) $\begin{array}{l} ‖ u ‖_{H}^{2} & = ‖ \nabla u ‖^{2} + {‖ \frac{ω}{c} u ‖}^{2} \\ = \sum_{m \in N} \sum_{n \in ι_{m}} ({‖ {\hat{u}}_{m, n}^{'} g_{0} ‖}_{L^{2} (I)}^{2} ‖ Y_{m, n} ‖_{L^{2} (S_{d - 1})}^{2} + ‖ {\hat{u}}_{m, n} g_{1} ‖_{L^{2} (I)}^{2} ‖ \nabla_{Γ} Y_{m, n} ‖_{L^{2} (S_{d - 1})}^{2} \\ + {‖ \frac{ω}{c} {\hat{u}}_{m, n} g_{0} ‖}_{L^{2} (I)}^{2} ‖ Y_{m, n} ‖_{L^{2} (S_{d - 1})}^{2}) \\ = \sum_{m \in N} \sum_{n \in ι_{m}} ({‖ {\hat{u}}_{m, n}^{'} g_{0} ‖}_{L^{2} (I)}^{2} + {‖ \frac{ω}{c} {\hat{u}}_{m, n} g_{0} ‖}_{L^{2} (I)}^{2} + λ_{m} ‖ {\hat{u}}_{m, n} g_{1} ‖_{L^{2} (I)}^{2}) . \end{array}$ We split the integral over I into a sum of integrals over $τ_{j}$ and employ ansatz (2.11) to get the estimate $\begin{array}{l} ‖ u ‖_{H}^{2} & ⩽ 2 \sum_{m \in N} \sum_{n \in ι_{m}} \sum_{j = 1}^{N} ({(\frac{ω}{c_{j}} A_{(m, n), j})}^{2} {‖ f_{m, 1}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} + {(\frac{ω}{c_{j}} B_{(m, n), j})}^{2} {‖ f_{m, 2}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} \\ + {(\frac{ω}{c_{j}} A_{(m, n), j})}^{2} {‖ f_{m, 1} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} + {(\frac{ω}{c_{j}} B_{(m, n), j})}^{2} {‖ f_{m, 2} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} \\ + λ_{m} (A_{(m, n), j}^{2} {‖ f_{m, 1} (\frac{ω}{c_{j}} \cdot) g_{1} ‖}_{L^{2} (τ_{j})}^{2} + B_{(m, n), j}^{2} {‖ f_{m, 2} (\frac{ω}{c_{j}} \cdot) g_{1} ‖}_{L^{2} (τ_{j})}^{2})) . \end{array}$ If $m = 0$ , we have $λ_{m} = 0$ and the term $λ_{m} (\dots)$ vanishes.

We set $g_{m, n} = 0$ for all $m \in N$ and $n \in ι_{m}$ with $(m, n) \neq (0, 0)$ . Then, the solution u in the form (2.5) satisfies $u_{m, n} = 0$ for all $m \in N$ and $n \in ι_{m}$ with $(m, n) \neq (0, 0)$ . We set $A_{j} : = A_{(0, 0), j}$ and $B_{j} : = B_{(0, 0), j}$ and get $\begin{array}{l} (3.9) & ‖ u ‖_{H}^{2} & ⩽ 2 \sum_{j = 1}^{N} ({(\frac{ω}{c_{j}} A_{j})}^{2} {‖ f_{0, 1}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} + {(\frac{ω}{c_{j}} B_{j})}^{2} {‖ f_{0, 2}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} \\ (3.10) & + {(\frac{ω}{c_{j}} A_{j})}^{2} {‖ f_{0, 1} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} + {(\frac{ω}{c_{j}} B_{j})}^{2} {‖ f_{0, 2} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2}) . \end{array}$

We are also interested in pointwise estimates. For $r \in τ_{j}$ we have in the considered case (note that $\nabla_{Γ} Y_{0, 0} (ξ) = 0$ and $| Y_{0, 0} (ξ) | = | Γ |^{- 1 / 2}$ with the surface measure $| Γ |$ of $S_{d - 1}$ ) $\begin{array}{l} | \hat{u} (r, ξ) | ⩽ | Γ |^{- 1 / 2} (| A_{j} | | f_{0, 1} (\frac{ω}{c_{j}} r) | + | B_{j} | {‖ f_{0, 2} (\frac{ω}{c_{j}} \cdot) ‖}_{L^{\infty} (τ_{j})}), \\ | (\nabla u) \circ ψ (r, ξ) | ⩽ | Γ |^{- 1 / 2} (| A_{j} | | f_{0, 1}^{'} (\frac{ω}{c_{j}} r) | + | B_{j} | {‖ f_{0, 2}^{'} (\frac{ω}{c_{j}} \cdot) ‖}_{L^{\infty} (τ_{j})}) . \end{array}$

Remark 3.5.
The Hankel function $h_{0}^{(1)} (r)$ has a singularity of order $O (r^{- 1})$ and its derivative behaves as $O (r^{- 2})$ for $r \to 0$ (cf. [19]). Note, that $A_{1} = 0$ (by (2.12a)) and this singularity is not critical for the first interval $τ_{1}$ . A refined estimation of the coefficient $A_{ℓ}$ will be discussed in Section 5.2 in order to ensure that the “near-singularities” of the Hankel function on $τ_{2}$ , $τ_{3}$ ,…are damped by the smallness of the corresponding coefficients $A_{ℓ}$ .
Notation 3.6.
We write $A ≲ B$ if there exits a constant $C \in R_{> 0}$ such that $A ⩽ C B$ . We note that in the following estimates the constant C is independent of the boundary data ${\hat{g}}_{0}$ , the wave speed c, the frequency ω, and the number of jumps n but depends on $c_{min}$ and $c_{max}$ .

We present the main stability result of this paper. Theorem 3.7.
Let $d = 3$ and let u be the solution of ( 2.2 ) with boundary data given by $\hat{g} (ξ) = {\hat{g}}_{0} Y_{0, 0} (ξ)$ . Let the wave speed c be constant on concentric annular regions and oscillate between the two values $0 < c_{min} ⩽ c_{1}$ , $c_{2} ⩽ c_{max} < \infty$ , i.e. $\begin{matrix} c_{j} = \{\begin{matrix} c_{1} & if j is odd, \\ c_{2} & if j is even, \end{matrix} \forall 1 ⩽ j ⩽ n + 1 . \end{matrix}$ Then it holds $\begin{matrix} ‖ u ‖_{H} ≲ | {\hat{g}}_{0} | α^{ω} \end{matrix}$ for constants $α > 1$ which depends only on $c_{min}$ , $c_{max}$ . For $r \in (0, 1)$ , the pointwise estimates hold $\begin{array}{l} | \hat{u} (r, ξ) | ≲ α^{ω} | {\hat{g}}_{0} |, \\ | (\nabla u) \circ ψ (r, ξ) | ≲ r α^{ω} | {\hat{g}}_{0} | . \end{array}$
Remark 3.8.
We see that for the case $d = 3$ and $m = 0$ the function u is not only bounded with respect to the energy norm but also pointwise. However, the bound can grow exponentially with respect to the wavenumber.
Remark 3.9.
From Theorem 3.7 we know that the stability constant grows at most exponentially with respect to the frequency while the bound is independent of the number of jumps: $C_{stab} ⩽ C α^{ω}$ for some $C > 0$ and $0 < α < 1$ .

Such an explicit estimate for the continuous stability constant with respect to the coefficients in the problem is key for designing numerical discretization methods. Typcially, the well-posedness of the discretization can be guaranteed under a resolution condition (see, e.g., [10]) which depends on the stability constant. Future work will address the design of discretization methods for heterogeneous Helmholtz problem whose stability theory is based on the explicit knowledge of bounds of the continuous stability constant.

4. Sharpness of the stability theorem and examples of wave localisation

The representation of the entries of the Green’s operator in Theorem 3.4 allows us to construct examples with specific behaviours. In this chapter we discuss two localisation phenomena for $d = 3$ ; for wave localisations in one dimension we refer to [23].

4.1. Localisation at a single discontinuity (“whispering gallery modes”)

For only one discontinuity, the Green’s operator is given by a $2 \times 2$ -matrix and the solution of the system can be computed explicitly. Indeed one computes with the ansatz (2.11) for $d = 3$ $\begin{array}{l} A_{m, 1} = 0, A_{m, 2} = \frac{i ω}{c_{2}} h_{m}^{(1)} (\frac{ω}{c_{2}}) \frac{w_{m, 2, 1, 1}^{2, 2}}{w_{m, 2, 1, 1}^{1, 2}}, B_{m, 1} = \frac{1}{x_{1}^{2} ω} \frac{h_{m}^{(1)} (\frac{ω}{c_{2}})}{w_{m, 2, 1, 1}^{1, 2}}, B_{m, 2} = \frac{i ω}{c_{2}} h_{m}^{(1)} (\frac{ω}{c_{2}}) {\hat{g}}_{m} . \end{array}$ This is well-defined (cf. [11, Lem. 1]). For fixed $c_{2} > c_{1} > 0$ and $x_{1} \in (0, 1)$ , consider $w_{m, 2, 1, 1}^{1, 2} = W_{m} (ω)$ as a function, depending on a complex frequency ω. Let ${\tilde{ω}}_{m}^{0}$ be the smallest (in modulus) complex zero of $W_{m} (ω)$ , and choose the frequency $ω_{m} = Re ({\tilde{ω}}_{m}^{0})$ . Then it can be shown that the corresponding $B_{m, 1}$ has an super-algebraic growth in m (cf. [6]). This well-known phenomenon is referred to “whispering gallery modes” in the literature (see e.g. [6]). The wave localises along the interface of the jump of the wave speed.

4.2. Localisation for highly varying wave speed

In Section 3.2, we have seen that the representation of the Green’s operator is given by (3.7) with the simplified recursion for $β_{0, ℓ}$ as in (3.8). In this section, we describe parameter configurations where localisation of waves occurs.

Lemma 4.1.
Let $d = 3$ . Let $ω > 0$ and let the wave speed c be a piecewise constant function as described in ( 2.1 ) such that $\begin{matrix} (4.1) & e^{- i ω \frac{h_{ℓ}}{c_{ℓ}}} \in {i, - i} \forall 1 ⩽ ℓ ⩽ n + 1 with h_{ℓ} = x_{ℓ} - x_{ℓ - 1} . \end{matrix}$ Then, the last row of the radial Green’s operator satisfies $\begin{array}{l} (4.2) & | {(M_{0}^{Green})}_{2 ℓ - 1, 2 n} | = \prod_{k = ℓ}^{n} (\frac{1 + q_{k}}{1 - {(- 1)}^{k - 1} q_{k}}), \\ (4.3) & | {(M_{0}^{Green})}_{2 ℓ, 2 n} | = | Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} i^{ℓ}) | \prod_{k = ℓ}^{n} (\frac{1 + q_{k}}{1 - {(- 1)}^{k - 1} q_{k}}) \end{array}$ for $1 ⩽ ℓ ⩽ n$ with the relative jumps $q_{k}$ defined by $\begin{matrix} q_{k} : = \frac{c_{k + 1} - c_{k}}{c_{k + 1} + c_{k}} \in (- 1, 1) . \end{matrix}$
Proof.
The claim follows directly by using the representation (3.7) and that under the assumption (4.1) the recursion (3.8) for $| β_{0, ℓ} |$ simplifies to $\begin{matrix} | β_{0, ℓ} | = \prod_{k = 1}^{ℓ} (\frac{1 - {(- 1)}^{k - 1} q_{k}}{1 + q_{k}}), \forall 0 ⩽ ℓ ⩽ n . \end{matrix}$ □

If we assume that the jumps $q_{k}$ are such that $q_{ℓ} = {(- 1)}^{ℓ - 1} | q_{ℓ} |$ and ${min}_{1 ⩽ ℓ ⩽ n} | q_{ℓ} | ⩾ q_{min} > 0$ , then we can derive from (4.2) $\begin{matrix} (4.4) & \begin{matrix} | {(M_{0}^{Green})}_{2 ℓ - 1, 2 n} | & = \prod_{k = ℓ}^{n} (\frac{1 + {(- 1)}^{k - 1} | q_{k} |}{1 - | q_{k} |}) = \prod_{\begin{array}{c} k = ℓ \\ k odd \end{array}}^{n} (\frac{1 + | q_{k} |}{1 - | q_{k} |}) ⩾ {(\frac{1 + q_{min}}{1 - q_{min}})}^{(⌈ \frac{n}{2} ⌉ - ⌊ \frac{ℓ}{2} ⌋)}, \end{matrix} \end{matrix}$ i.e. the odd entries of the Green’s operator grow exponentially in the number of jumps. Numerical experiment show that in this case the solution u localizes in the center of the domain $Ω = B_{1}^{3}$ .
Definition 4.2 (Localisation interference).

Let $d = 3$ . For any number $n \in N$ of jumps, any set of radial jump points $\begin{matrix} (4.5) & 0 = x_{0} < x_{1} < \dots < x_{n + 1} = 1, \end{matrix}$ any piecewise constant wave speed, i.e. $\begin{array}{l} c (x) : = c_{ℓ} \forall x \in B_{1}^{d} with | x | \in (x_{ℓ - 1}, x_{ℓ}), 1 ⩽ ℓ ⩽ n + 1, \end{array}$ that is oscillatory with $q_{1} > 0$ , i.e. $\begin{matrix} q_{ℓ} = {(- 1)}^{ℓ - 1} | q_{ℓ} |, and min_{1 ⩽ ℓ ⩽ n} | q_{ℓ} | ⩾ q_{min} > 0 \end{matrix}$ and any frequency $ω > 0$ , we say that the wave speed c and the frequency ω are in localisation interference if $\begin{matrix} (4.6) & e^{- i ω \frac{h_{ℓ}}{c_{ℓ}}} \in {i, - i} with h_{ℓ} = x_{ℓ} - x_{ℓ - 1}, \forall 1 ⩽ ℓ ⩽ n + 1 . \end{matrix}$

Lemma 4.3.
Let $0 < q ⩽ \frac{c_{max} - c_{min}}{c_{max} + c_{min}}$ be fixed. For any $n \in N$ , there exists a frequency ω and a wave speed c such that they are in localisation interference and $n \sim ω$ . As a consequence, it holds for $1 ⩽ ℓ ⩽ n$ $\begin{matrix} | {(M_{0}^{Green})}_{2 ℓ - 1, 2 n} | = {(\frac{1 + q}{1 - q})}^{(⌈ \frac{n}{2} ⌉ - ⌊ \frac{ℓ}{2} ⌋)}, \end{matrix}$ i.e. the odd entries of the Green’s operator grow exponentially in the number of jumps $n \sim ω$ .
Proof.
Choose $0 < c_{1} < c_{2} < \infty$ such that $\frac{c_{2} - c_{1}}{c_{2} + c_{1}} = q$ . For any $n \in N$ fixed we set $\begin{array}{l} c_{j} = \{\begin{matrix} c_{1} & if j is odd, \\ c_{2} & if j is even, \end{matrix} \\ h_{ℓ} = \frac{π}{2 ω} c_{ℓ}, for 1 ⩽ ℓ ⩽ n + 1, \\ x_{ℓ} = \sum_{j = 1}^{ℓ} h_{ℓ}, 0 ⩽ ℓ ⩽ n + 1 \end{array}$ and define the wave speed c by $\begin{matrix} c (x) : = c_{ℓ} \forall x \in B_{1}^{d} with | x | \in (x_{ℓ - 1}, x_{ℓ}) . \end{matrix}$ In order to respect the condition $\sum_{j = 1}^{n + 1} h_{j} = 1$ , we define the frequency by $\begin{matrix} ω : = \frac{π}{2} \sum_{j = 1}^{n + 1} c_{j} \sim O (n) . \end{matrix}$ The representation of the odd entries of the Green’s operator follow from (4.4). □

We have seen that for any given $0 < c_{1} < c_{2}$ a set of jump points can be chosen along a frequency ω such that localisation appears. Figure 1 shows examples of the solution u of the Helmholtz equation with respect to the wave speed c and ω as constructed in Lemma 4.3 for different values of n.

Fig. 1.
Examples for a localisation with $m = 0$ (i.e. the solution is radial) and n jumps. The absolute value $| {\hat{u}}_{0, 0} (r) Y_{0, 0} (ξ) |$ on the disc $B_{1}^{3} \cap {x_{3} = 0}$ is shown. The maximal peaks grow exponentially in $n = O (ω)$ . For any $n \in N$ , the wave speed c and the frequency ω are chosen as constructed in Lemma 4.3 with $c_{1} = 1$ , $c_{2} = 3$ . For the boundary data we set ${\hat{g}}_{(0, 0)} = 1$ .
Remark 4.4.
A closer examination of the recursion (3.8) for $β_{0, ℓ}$ shows that there are other configurations of c and choices of ω that do not fit into Definition 4.2, but nevertheless the corresponding solution exhibits a localisation effect. In particular if $\begin{matrix} (4.7) & e^{- i ω \frac{h_{j}}{c_{j}}} \in {- 1, 1, i, - i}, \forall 1 ⩽ j ⩽ n, \end{matrix}$ the recursion simplifies and the representation of the last row of the Green’s operator differs only by some change of signs from (4.2). As an example, if the phase factors are such that $\begin{array}{l} e^{- i ω \frac{h_{1}}{c_{1}}} \in {1, - 1}, \\ e^{- i ω \frac{h_{ℓ}}{c_{ℓ}}} \in {i, - i} \forall 2 ⩽ ℓ ⩽ n + 1, \end{array}$ a lower estimate of the type (4.4) still holds, if the wave speed c satisfies $q_{ℓ} = {(- 1)}^{ℓ} | q_{ℓ} |$ (instead of $q_{ℓ} = {(- 1)}^{ℓ - 1} | q_{ℓ} |$ ). Other examples may be explored in a similar way.

From Theorem 3.7, we conclude that $‖ u ‖_{H}$ may grow exponentially in ω. Indeed for given n, the solution u of the problem (1.1) with wave speed c, frequency ω as constructed in Lemma 4.3 and boundary data $\hat{g} (ξ) = {\hat{g}}_{(0, 0)} Y_{(0, 0)} (ξ)$ satisfies $\begin{array}{l} ‖ u ‖_{H}^{2} & = {‖ {\hat{u}}_{0, 0}^{'} g_{0} ‖}_{L^{2} (I)}^{2} + {‖ \frac{ω}{c} {\hat{u}}_{0, 0} g_{0} ‖}_{L^{2} (I)}^{2} \\ ⩾ \frac{ω}{c_{1}} | B_{1} |^{2} {‖ sin (\frac{ω}{c_{1}} \cdot) ‖}_{L^{2} (τ_{1})}^{2} \\ = \frac{ω}{c_{1}} | B_{1} |^{2} (\frac{1}{2} x_{1} - \frac{x_{1}}{4} \frac{c_{1}}{ω} sin (2 x_{1} \frac{ω}{c_{1}})), \end{array}$ since $A_{1} = 0$ . The choice $x_{1} = h_{1} = \frac{π}{2} \frac{c_{1}}{ω}$ implies $sin (2 x_{1} \frac{ω}{c_{1}}) = 0$ . We recall the identities (2.13) together with (3.2), (4.2) and (8.1). Therefore, $\begin{array}{l} ‖ u ‖_{H} & ⩾ | B_{1} | \sqrt{\frac{π}{4}} = \sqrt{\frac{π}{4}} {(\frac{1 + q}{1 - q})}^{⌈ \frac{n}{2} ⌉} | h_{0}^{(1)} (\frac{ω}{c_{n + 1}}) \frac{ω}{c_{n + 1}} | | {\hat{g}}_{(0, 0)} | \\ = \sqrt{\frac{π}{4}} {(\frac{1 + q}{1 - q})}^{⌈ \frac{n}{2} ⌉} | {\hat{g}}_{(0, 0)} | \end{array}$ is a lower bound for the energy norm of the solution of problem (1.1). Combining this with the fact that $ω \sim n$ , implies that the estimate in Theorem 3.7 is sharp with respect to the frequency ω.
4.3. Globally stable solutions

Similarly to Sections 4.2 one can construct an example, where the entries of the Green’s operator are bounded independently of the number of jumps. Let c be oscillating between two values $0 < c_{1} < c_{2}$ as chosen in the proof of Lemma 4.3. For fixed $n \in N$ , we choose the jump locations $x_{j}$ and the frequency ω such that $\begin{matrix} (4.8) & e^{- i ω \frac{h_{j}}{c_{j}}} = - 1, \forall 1 ⩽ j ⩽ n . \end{matrix}$ In order to achieve (4.8) we set $\begin{matrix} ω = π \sum_{j = 1}^{n + 1} c_{j} \sim O (n), h_{j} = \frac{π}{ω} c_{j} . \end{matrix}$ Then the recursion simplifies to $\begin{matrix} β_{0, ℓ} = {(- 1)}^{ℓ}, \forall 1 ⩽ ℓ ⩽ n . \end{matrix}$ Finally, this leads to $\begin{matrix} | {(M_{0}^{Green})}_{2 ℓ - 1, 2 n} | = 1, \forall 1 ⩽ ℓ ⩽ n, \end{matrix}$ and $\begin{matrix} | {(M_{0}^{Green})}_{2 ℓ, 2 n} | ⩽ 1, \forall 1 ⩽ ℓ ⩽ n, \end{matrix}$ i.e. all entries of the Green’s operator are bounded in modulus from above by 1.

Fig. 2.

Examples with $m = 0$ (i.e. the solution is radial) and n jumps, where the solution behaves nicely. The absolute value $| {\hat{u}}_{0, 0} (r) Y_{0, 0} (ξ) |$ on the disc $B_{1}^{3} \cap {x_{3} = 0}$ is shown. For any $n \in N$ , the wave speed c and the frequency ω are chosen as constructed in Section 4.3 with $c_{1} = 1$ , $c_{2} = 3$ . For the boundary data we set ${\hat{g}}_{(0, 0)} = 1$ .

Figure 2 shows examples of solutions constructed in this way for different values of n.

Remark 4.5.

Although, we have seen in Section 4.2 that the stability estimate in Theorem 3.7 is sharp, these examples show that the estimate can be pessimistic for some configurations of c and ω. Moreover for fixed, large number of jumps n, a piecewise constant wave speed c along a specific frequency ω chosen such that the wave localizes is rare within the parameter space (1.2) together with (2.1) and can be interpreted as an interference phenomenon. As soon as such a critical choice is slightly perturbed (either by changing ω or c (and this includes changing a single jump point $x_{i}$ )) the wave becomes typically well-behaved. The examples of Section 4.2 and 4.3 have identical wave speed c and only differ by the frequency ω.

5. Stability for waves for the lowest Fourier mode

m = 0

In this section, we derive an estimate of the entries of the Green’s operator $M_{m}^{Green}$ for $m = 0$ , i.e. with no angular oscillations, and $d = 3$ . That is, a further investigation of the sequence $β_{0, ℓ}$ as defined in (3.8) is needed. In the first step, we will find an upper and lower bound on $| β_{0, ℓ} |$ which is independent of the position of the scaled jumps $z_{ℓ} = ω x_{ℓ}$ and their number n. In a second step, we derive a refined estimate of $β_{0, ℓ}$ , in particular of $Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{0, ℓ})$ (cf. (3.7)), for small values of $z_{ℓ}$ . Such a refined estimate of the odd entries in the Green’s operator, i.e. the coefficients belonging to the Hankel function $h_{0} (\frac{ω x}{c_{ℓ}})$ , is needed because $h_{0}^{'} (\frac{ω x}{c_{ℓ}})$ has a singularity of order $O (x^{- 2})$ at $x = 0$ (Lemma 8.1).

Throughout this section, we always assume that the wave speed c oscillates between two values, i.e. $\begin{matrix} (5.1) & q_{ℓ} = {(- 1)}^{ℓ + 1} q with - 1 < q = \frac{c_{2} - c_{1}}{c_{2} + c_{1}} < 1, \end{matrix}$ for some $c_{min} ⩽ c_{1}$ , $c_{2} ⩽ c_{max}$ . Also, we define $\begin{matrix} (5.2) & δ_{ℓ} : = \frac{ω h_{ℓ}}{c_{ℓ}} . \end{matrix}$

5.1. Maximal growth of the entries of the Green’s operator

The goal of this section is a lower and upper estimate on the sequence $β_{0, ℓ}$ as stated in Proposition 5.1.

Proposition 5.1.
It holds $\begin{array}{l} | β_{0, ℓ} | ⩽ α^{ω}, \\ | β_{0, ℓ} | ⩾ α^{- ω}, \end{array}$ for some $α = α (q) > 1$ for $q \in (- 1, 1)$ .
Proof.
From the recursion (3.8) and (5.1) we obtain $\begin{matrix} | β_{0, ℓ} |^{2} = \frac{1 + 2 q_{ℓ} Re (e^{- 2 i δ_{ℓ}} \frac{β_{0, ℓ - 1}^{2}}{| β_{0, ℓ - 1} |^{2}}) + q_{ℓ}^{2}}{{(1 + q_{ℓ})}^{2}} | β_{0, ℓ - 1} |^{2} . \end{matrix}$ This leads to $\begin{matrix} (5.3) & \frac{1 - | q_{ℓ} |}{1 + q_{ℓ}} | β_{0, ℓ - 1} | ⩽ | β_{0, ℓ} | ⩽ \frac{1 + | q_{ℓ} |}{1 + q_{ℓ}} | β_{0, ℓ - 1} | . \end{matrix}$ Next, we express $β_{0, ℓ}$ in terms of $β_{0, ℓ - 2}$ for $ℓ ⩾ 2$ : $\begin{matrix} (5.4) & β_{0, ℓ} : = \frac{e^{- i (δ_{ℓ - 1} + δ_{ℓ})}}{1 - q^{2}} ((1 - q^{2} e^{2 i δ_{ℓ}}) β_{0, ℓ - 2} + {(- 1)}^{ℓ} q (1 - e^{2 i δ_{ℓ}}) e^{2 i δ_{ℓ - 1}} \overline{β_{0, ℓ - 2}}) \end{matrix}$ and for the square of the modulus: $\begin{array}{l} | β_{0, ℓ} |^{2} = & \frac{| β_{0, ℓ - 2} |^{2}}{{(1 - q^{2})}^{2}} ((1 + 2 q^{2} + q^{4} - 4 q^{2} Re e^{2 i δ_{ℓ}}) \\ + 2 {(- 1)}^{ℓ} q Re ((1 - q^{2} e^{2 i δ_{ℓ}}) (1 - e^{- 2 i δ_{ℓ}}) e^{- 2 i δ_{ℓ - 1}} \frac{β_{0, ℓ - 2}^{2}}{| β_{0, ℓ - 2} |^{2}})) \\ = & \frac{| β_{0, ℓ - 2} |^{2}}{{(1 - q^{2})}^{2}} ((1 + 2 q^{2} + q^{4} - 4 q^{2} Re e^{2 i δ_{ℓ}}) \\ + 2 {(- 1)}^{ℓ} q ((1 + q^{2}) (1 - Re e^{2 i δ_{ℓ}})) Re (e^{- 2 i δ_{ℓ - 1}} \frac{β_{0, ℓ - 2}^{2}}{| β_{0, ℓ - 2} |^{2}}) \\ + 2 {(- 1)}^{ℓ} q (1 - q^{2}) (Im e^{2 i δ_{ℓ}}) Im (e^{- 2 i δ_{ℓ - 1}} \frac{β_{0, ℓ - 2}^{2}}{| β_{0, ℓ - 2} |^{2}})) . \end{array}$ Note that for $δ_{ℓ} ⩾ 0$ , it holds $\begin{array}{l} - Re e^{2 i δ_{ℓ}} = - cos (2 δ_{ℓ}) ⩽ - 1 + 2 min {δ_{ℓ}^{2}, 1}, \\ Im e^{2 i δ_{ℓ}} = sin (2 δ_{ℓ}) ⩽ min {2 δ_{ℓ}, 1} \end{array}$ so that $\begin{array}{l} | β_{0, ℓ} |^{2} ⩽ & \frac{| β_{0, ℓ - 2} |^{2}}{{(1 - q^{2})}^{2}} ({(1 - q^{2})}^{2} + 8 q^{2} min {δ_{ℓ}^{2}, 1} + 4 | q | (1 + q^{2}) min {δ_{ℓ}^{2}, 1} \\ + 2 | q | (1 - q^{2}) min {2 δ_{ℓ}, 1}) \\ ⩽ & | β_{0, ℓ - 2} |^{2} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{ℓ}, 1}) \end{array}$ for $C_{0} = 20$ . We consider the majorant $\begin{matrix} | β_{0, 2 ℓ} |^{2} ⩽ ρ_{ℓ}^{2} ({(δ_{2 k})}_{k = 1}^{ℓ}) : = \prod_{k = 1}^{ℓ} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 k}, 1}) . \end{matrix}$ For a fixed $0 < γ < 1$ , define the two disjoint subsets of indices $\begin{array}{l} I_{h} = {j \in 1, \dots, ℓ ∣ δ_{j} ⩽ γ} \\ I_{H} = {j \in 1, \dots, ℓ ∣ δ_{j} > γ} . \end{array}$ Then we can write $\begin{array}{l} \prod_{k = 1}^{ℓ} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 k}, 1}) = & \prod_{k \in I_{H}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 k}, 1}) \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) \end{array}$ and treat the two factors differently. The second factor can be treated with the techniques used in the proof of [23, Proposition 19]. That is, we maximize $\begin{matrix} (5.5) & \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) \end{matrix}$ under the restriction $\sum_{k \in I_{h}} δ_{2 k} ⩽ s$ . The restriction comes from the fact that $\begin{matrix} \sum_{k \in I_{h}} δ_{2 k} ⩽ \sum_{k = 1}^{2 ℓ} 2 ω \frac{h_{k}}{c_{k}} ⩽ 2 \frac{ω}{c_{min}} \sum_{k = 1}^{2 ℓ} h_{k} ⩽ 2 \frac{ω}{c_{min}} = : s . \end{matrix}$ The same analysis as in the proof of [23, Proposition 19] shows that there are three possible candidates for the maximizers of (5.5):
if $| I_{h} | ⩽ \frac{s}{γ}$ : Let $δ_{2 k} = γ$ for all $k \in I_{h}$ . Then we have $\begin{array}{l} \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) & = {(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} γ)}^{| I_{h} |} ⩽ {({(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} γ)}^{\frac{1}{γ}})}^{s} \\ ⩽ exp {(\frac{C_{0} | q |}{{(1 - q^{2})}^{2}})}^{s}, \end{array}$

if $| I_{h} | > \frac{s}{γ}$ : Let $\begin{array}{l} δ_{2 k_{j}} = γ for 1 ⩽ j ⩽ L - 1, \\ δ_{2 k_{L}} = s - (L - 1) γ, \\ δ_{2 k_{j}} = 0 for L + 1 ⩽ j ⩽ | I_{h} |, \end{array}$ where $L = min {ℓ \in N ∣ ℓ γ ⩾ s}$ and $I_{h} = {k_{1}, \dots, k_{| I_{h} |}}$ . Then we use the fact that $L ⩽ \frac{s}{γ} + 1$ and compute $\begin{array}{l} \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) & = {(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} γ)}^{L} ⩽ 2 {({(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} γ)}^{\frac{1}{γ}})}^{s} \\ ⩽ 2 exp {(\frac{C_{0} | q |}{{(1 - q^{2})}^{2}})}^{s}, \end{array}$

if $| I_{h} | > \frac{s}{γ}$ : Let $δ_{2 k} = \frac{s}{| I_{h} |}$ for all $k \in I_{h}$ . In this case $\begin{matrix} \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) ⩽ {(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} \frac{s}{| I_{h} |})}^{| I_{h} |} ⩽ exp {(\frac{C_{0} | q |}{{(1 - q^{2})}^{2}})}^{s} . \end{matrix}$

Since $s ⩽ 2 \frac{ω}{c_{min}}$ all choices lead to the estimate $\begin{matrix} \prod_{k \in I_{h}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) ⩽ {\tilde{α}}_{q}^{ω} \end{matrix}$ for some ${\tilde{α}}_{q} > 1$ depending on $C_{0}$ .

For the first factor, we note that $\begin{matrix} δ_{j} ⩾ γ ⟺ h_{j} ⩾ γ \frac{c_{j}}{ω} . \end{matrix}$ This means that $| I_{H} | ⩽ \frac{ω}{γ c_{min}}$ (with $| I_{H} |$ denoting the cardinality of $I_{H}$ ) and therefore $\begin{matrix} (5.6) & \prod_{k \in I_{H}} (1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 k}, 1}) ⩽ {(1 + \frac{C_{0} | q |}{{(1 - q^{2})}^{2}})}^{| I_{H} |} ⩽ {\tilde{α}}_{q}^{ω}, \end{matrix}$ for some ${\tilde{α}}_{q} > 1$ depending on $C_{0}$ . The lower bound can be treated similarly as the upper bound. First, we observe that $\begin{matrix} | β_{0, 2 ℓ} |^{2} ⩾ | β_{0, 2 ℓ - 2} |^{2} (1 - \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 ℓ}, 1}) . \end{matrix}$ From (5.3) we get by using (5.1) $\begin{matrix} | β_{0, 2 ℓ} | ⩾ \frac{1 - | q_{ℓ} |}{1 + q_{ℓ}} | β_{0, 2 ℓ - 1} | ⩾ \frac{1 - | q |}{1 + | q |} | β_{0, 2 ℓ - 2} | > 0 . \end{matrix}$ This leads to $\begin{matrix} | β_{0, 2 ℓ} |^{2} ⩾ | β_{0, 2 ℓ - 2} |^{2} max {(1 - \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} min {δ_{2 ℓ}, 1}), {(\frac{1 - | q |}{1 + | q |})}^{2}} . \end{matrix}$ We set $min {\frac{{(1 - q^{2})}^{2}}{C_{0} | q |}, 1} > γ > 0$ and again split $\begin{matrix} | β_{0, 2 ℓ} |^{2} ⩾ {(\frac{1 - | q |}{1 + | q |})}^{2 | I_{H} |} \prod_{k \in I_{h}} (1 - \frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k}) . \end{matrix}$ The estimate of the first factor follows by using the same arguments as for (5.6). For the second one, we use that $\frac{C_{0} | q |}{{(1 - q^{2})}^{2}} δ_{2 k} < 1$ for $k \in I_{h}$ and repeat the proof for the upper bound by the same symmetry and opposite monotonicity property. Finally, the estimate of $| β_{0, 2 ℓ - 1} |$ for odd indices follows easily by using (5.3). □
Remark 5.2.
We note that the constant $α_{q} > 1$ only depends on $| q | \in (0, 1)$ (however, in a continuous way). As $| q | \to 1$ (which implies $c_{max} \to \infty$ or $c_{min} \to 0$ ) the constant $α_{q}$ may tend to ∞. We emphasize that we do not need any periodicity on the wave speed c, and that the estimate is independent of the number of jumps n in c or the positions $x_{ℓ}$ of the jumps.

5.2. A refined estimation for the coefficients of the Hankel function close to the origin

In this chapter, we establish a refined estimate of $β_{0, ℓ}$ for small $\frac{ω x_{ℓ}}{c_{ℓ}}$ for reasons explained in Remark 3.5. In this section, we only consider indices ℓ such that $\begin{matrix} (5.7) & \frac{ω x_{ℓ}}{c_{min}} < \frac{1}{4 C_{1}} for C_{1} : = \frac{max {1, c_{max}}}{c_{min}^{2}} \end{matrix}$ (otherwise, the singular behaviour of the Hankel function at the origin is non-critical in the sense that all constants, in general, may depend on $c_{min}$ and $c_{max}$ ). Let L be the maximal index such that $\frac{ω x_{ℓ}}{c_{min}} < {(4 C_{1})}^{- 1}$ holds. In view of (3.7), our goal is to find an estimate of the term $\begin{matrix} Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{0, ℓ}) = sin (\frac{z_{ℓ}}{c_{ℓ + 1}}) Re (β_{0, ℓ}) + cos (\frac{z_{ℓ}}{c_{ℓ + 1}}) Im (β_{0, ℓ}) . \end{matrix}$ For $2 ⩽ ℓ ⩽ L$ we recall the recursion for $β_{0, ℓ}$ (cf. (5.4)): $\begin{array}{l} β_{0, ℓ} = & e^{- i \frac{ω h_{ℓ - 1}}{c_{ℓ - 1}}} (\frac{e^{- i \frac{ω h_{ℓ}}{c_{ℓ}}} - q^{2} e^{i \frac{ω h_{ℓ}}{c_{ℓ}}}}{1 - q^{2}}) β_{0, ℓ - 2} - {(- 1)}^{ℓ} \frac{2 i q}{1 - q^{2}} e^{i \frac{ω h_{ℓ - 1}}{c_{ℓ - 1}}} sin (\frac{ω h_{ℓ}}{c_{ℓ}}) \overline{β_{0, ℓ - 2}} \end{array}$ and split it into the real part $β_{0, ℓ}^{R} : = Re (β_{0, ℓ})$ and imaginary part $β_{0, ℓ}^{I} : = Im (β_{0, ℓ})$ of $β_{0, ℓ}$ . This leads to $\begin{matrix} (\begin{matrix} β_{0, ℓ}^{R} \\ β_{0, ℓ}^{I} \end{matrix}) = P_{ℓ} (\begin{matrix} β_{0, ℓ - 2}^{R} \\ β_{0, ℓ - 2}^{I} \end{matrix}), P_{ℓ} : = [\begin{matrix} 1 + s_{ℓ} (μ_{ℓ}) & t_{ℓ} (μ_{ℓ}) \\ - t_{ℓ} (μ_{ℓ}^{- 1}) & 1 + s_{ℓ} (μ_{ℓ}^{- 1}) \end{matrix}] \end{matrix}$ with $δ_{ℓ}$ as in (5.2) and $\begin{array}{l} μ : = \frac{1 - q}{1 + q} = \frac{c_{1}}{c_{2}}, μ_{ℓ} : = \{\begin{matrix} μ ℓ & is even, \\ μ^{- 1} & ℓ is odd, \end{matrix} \\ s_{ℓ} (κ) : = cos (δ_{ℓ - 1}) cos (δ_{ℓ}) - 1 - κ sin (δ_{ℓ - 1}) sin (δ_{ℓ}), \\ t_{ℓ} (κ) : = sin (δ_{ℓ - 1}) cos (δ_{ℓ}) + κ cos (δ_{ℓ - 1}) sin (δ_{ℓ}) . \end{array}$ Since $β_{0, 0} = 1$ , the recursion gives for $β_{0, 2 ℓ}$ for even indices $\begin{matrix} (5.8) & (\begin{matrix} β_{0, 2 ℓ}^{R} \\ β_{0, 2 ℓ}^{I} \end{matrix}) = P_{2 ℓ} \cdot \dots \cdot P_{4} \cdot P_{2} (\begin{matrix} 1 \\ 0 \end{matrix}) . \end{matrix}$ The values of $β_{0, k}$ for odd indices k are given by $\begin{matrix} (5.9) & (\begin{matrix} β_{0, 2 ℓ + 1}^{R} \\ β_{0, 2 ℓ + 1}^{I} \end{matrix}) = [\begin{matrix} cos (δ_{2 ℓ + 1}) & sin (δ_{2 ℓ + 1}) \\ - μ sin (δ_{2 ℓ + 1}) & μ cos (δ_{2 ℓ + 1}) \end{matrix}] (\begin{matrix} β_{0, 2 ℓ}^{R} \\ β_{0, 2 ℓ}^{I} \end{matrix}) . \end{matrix}$ At this point, we only discuss the recursion for the even values $2 ℓ$ of $β_{0, \cdot}$ . The result for the odd values will be proved in Lemma 5.8. The proof of the following Lemma is straightforward by induction.

Lemma 5.3.
It holds $\begin{matrix} (5.10) & P_{2 ℓ} \cdot \dots \cdot P_{2} = [\begin{matrix} 1 & L_{2 ℓ} (μ) \\ - L_{2 ℓ} (μ^{- 1}) & 1 \end{matrix}] + [\begin{matrix} S_{2 ℓ} (μ) & T_{2 ℓ} (μ) \\ - T_{2 ℓ} (μ^{- 1}) & S_{2 ℓ} (μ^{- 1}) \end{matrix}], \end{matrix}$ with $\begin{matrix} L_{2 ℓ} (κ) : = \sum_{j = 1}^{ℓ} t_{2 j} (κ) . \end{matrix}$ $S_{2 ℓ} (κ)$ , $T_{2 ℓ} (κ)$ are given by the recursion $\begin{matrix} (5.11) & S_{2} (κ) : = s_{2} (κ), T_{2} (κ) : = 0, \end{matrix}$ and $\begin{array}{l} S_{2 ℓ + 2} (κ) = (1 + s_{2 ℓ + 2} (κ)) S_{2 ℓ} (κ) + s_{2 ℓ + 2} (κ) - t_{2 ℓ + 2} (κ) (L_{2 ℓ} (κ^{- 1}) + T_{2 ℓ} (κ^{- 1})), \\ T_{2 ℓ + 2} (κ) = (1 + s_{2 ℓ + 2} (κ)) T_{2 ℓ} (κ) + s_{2 ℓ + 2} (κ) L_{2 ℓ} (κ) + t_{2 ℓ + 2} (κ) S_{2 ℓ} (κ^{- 1}) . \end{array}$

We first consider the first term on the right hand side of (5.10). The following lemma will be useful.
Lemma 5.4.
For a constant $C > 0$ only depending on $c_{min}$ , $c_{max}$ the following holds for $1 < 2 ℓ ⩽ L$ $\begin{matrix} L_{2 ℓ} (μ^{- 1}) = \frac{z_{2 ℓ}}{c_{2 ℓ + 1}} + R_{2 ℓ}^{3}, | R_{2 ℓ}^{3} | ⩽ C z_{2 ℓ}^{3} . \end{matrix}$
Proof.
This follows directly from the definitions of $t_{ℓ}$ and $L_{2 ℓ}$ and using a Taylor expansion with respect to $δ_{1}, \dots, δ_{2 ℓ}$ . □

In view of $β_{0, 0} = 1$ , i.e., $(β_{0, 0}^{R}, β_{0, 0}^{I}) = (1, 0)$ we observe that the first term in the product of $P_{2 k}$ (cf. (5.10)) applied to the initial coefficient $β_{0, 0}$ yields $\begin{matrix} (5.12) & [\begin{matrix} 1 & L_{2 ℓ} (μ) \\ - L_{2 ℓ} (μ^{- 1}) & 1 \end{matrix}] (\begin{matrix} 1 \\ 0 \end{matrix}) = (\begin{matrix} 1 \\ - \frac{z_{2 ℓ}}{c_{2 ℓ + 1}} - R_{2 ℓ}^{3} \end{matrix}) . \end{matrix}$

Next, we focus on the second term of (5.10). The goal is to show that the matrix entries are of higher order than the first term with respect to $z_{ℓ}$ . We start with an estimate of $s_{ℓ}$ , $t_{ℓ}$ and $L_{2 ℓ}$ .
Lemma 5.5.
The following estimates hold $\begin{array}{l} | s_{ℓ} (κ) | ⩽ R_{ℓ}^{2} : = C_{1} (δ_{ℓ - 1}^{2} + δ_{ℓ}^{2}), \\ | t_{ℓ} (κ) | ⩽ R_{ℓ}^{1} : = C_{1} (δ_{ℓ - 1} + δ_{ℓ}), \\ | L_{2 ℓ} (κ) | ⩽ Z_{2 ℓ} : = C_{1} z_{2 ℓ}, \end{array}$ for $κ \in {μ, μ^{- 1}}$ and the constant $C_{1} = \frac{max {1, c_{max}}}{c_{min}^{2}}$ as in ( 5.7 ). In particular, condition ( 5.7 ) implies $\begin{matrix} (5.13) & 0 < R_{ℓ}^{1} ⩽ 1 / 2 . \end{matrix}$
Proof.
This is a simple consequence of the definitions of $s_{ℓ}$ , $t_{ℓ}$ and $L_{2 ℓ}$ . □

The estimates in Lemma 5.5 can be used to define the majorant of $S_{2 ℓ} (κ)$ , $T_{2 ℓ} (κ)$ by $\begin{array}{l} v_{ℓ} : = {({\overset{ˇ}{S}}_{2 ℓ} (κ), {\overset{ˇ}{T}}_{2 ℓ} (κ))}^{⊺}, \\ {\overset{ˇ}{S}}_{2 ℓ} (κ) : = max {| S_{2 ℓ} (κ) |, | S_{2 ℓ} (κ^{- 1}) |}, \\ {\overset{ˇ}{T}}_{2 ℓ} (κ) : = max {| T_{2 ℓ} (κ) |, | T_{2 ℓ} (κ^{- 1}) |} \end{array}$ via the affine recursion $\begin{matrix} v_{ℓ + 1} = Q_{ℓ} v_{ℓ} + R_{ℓ} \end{matrix}$ for $\begin{matrix} Q_{ℓ} : = [\begin{matrix} 1 + R_{2 ℓ + 2}^{2} & R_{2 ℓ + 2}^{1} \\ R_{2 ℓ + 2}^{1} & 1 + R_{2 ℓ + 2}^{2} \end{matrix}] and R_{ℓ} : = [\begin{matrix} R_{2 ℓ + 2}^{2} + R_{2 ℓ + 2}^{1} Z_{2 ℓ} \\ R_{2 ℓ + 2}^{2} Z_{2 ℓ} \end{matrix}], \end{matrix}$ where $R_{2 ℓ + 2}^{1}$ , $R_{2 ℓ + 2}^{2}$ , $Z_{2 ℓ}$ are defined in Lemma 5.5. The recursion can be resolved and we get $\begin{matrix} (5.14) & v_{ℓ + 1} = P_{0, ℓ} v_{1} + \sum_{j = 1}^{ℓ} P_{j, ℓ} R_{j}, with P_{j, ℓ} : = Q_{ℓ} \cdot \dots \cdot Q_{j + 1} . \end{matrix}$ A simple algebraic calculus (cf. Lemma 7.1) yields $\begin{matrix} P_{j, ℓ} = [\begin{matrix} p_{j, ℓ}^{(1)} & p_{j, ℓ}^{(2)} \\ p_{j, ℓ}^{(2)} & p_{j, ℓ}^{(1)} \end{matrix}], \end{matrix}$ with $\begin{array}{l} p_{j, ℓ}^{(1)} = \frac{1}{2} (\prod_{k = j + 1}^{ℓ} (1 + R_{2 k + 2}^{2} + R_{2 k + 2}^{1}) + \prod_{k = j + 1}^{ℓ} (1 + R_{2 k + 2}^{2} - R_{2 k + 2}^{1})), \\ p_{j, ℓ}^{(2)} = \frac{1}{2} (\prod_{k = j + 1}^{ℓ} (1 + R_{2 k + 2}^{2} + R_{2 k + 2}^{1}) - \prod_{k = j + 1}^{ℓ} (1 + R_{2 k + 2}^{2} - R_{2 k + 2}^{1})) . \end{array}$ We may apply Lemma 7.2 with $a_{k} = 1 + R_{2 k + 2}^{2}$ and $b_{k} = R_{2 k + 2}^{1}$ by taking into account (5.13). This leads to $\begin{array}{l} | p_{j, ℓ}^{(1)} | ⩽ exp (\sum_{k = j + 1}^{ℓ} R_{2 k + 2}^{2}) cosh (\sum_{ℓ = j + 1}^{k} R_{2 k + 2}^{1}), \\ | p_{j, ℓ}^{(2)} | ⩽ exp (\sum_{k = j + 1}^{ℓ} R_{2 k + 2}^{2}) sinh (\sum_{k = j + 1}^{ℓ} R_{2 k + 2}^{1}) . \end{array}$
Lemma 5.6.
It holds $\begin{matrix} (5.15) & \begin{matrix} | p_{j, ℓ}^{(1)} | ⩽ 1 + C_{2} {(z_{2 ℓ + 2} - z_{2 j + 2})}^{2}, \\ | p_{j, ℓ}^{(2)} | ⩽ C_{2} (z_{2 ℓ + 2} - z_{2 j + 2}), \end{matrix} \end{matrix}$ for some constant $C_{2} > 0$ depending only on $c_{min}$ , $c_{max}$ .
Proof.
By Lemma 5.5, we have $\begin{array}{l} \sum_{k = j + 1}^{ℓ} R_{2 k + 2}^{1} ⩽ C_{1} \sum_{k = j + 1}^{ℓ} (δ_{2 k + 1} + δ_{2 k + 2}) ⩽ \frac{C_{1}}{c_{min}} (z_{2 ℓ + 2} - z_{2 j + 2}), \\ \sum_{k = j + 1}^{ℓ} R_{2 k + 2}^{2} ⩽ C_{1} {(\sum_{k = j + 1}^{ℓ} (δ_{2 k + 1} + δ_{2 k + 2}))}^{2} ⩽ \frac{C_{1}}{c_{min}^{2}} {(z_{2 ℓ + 2} - z_{2 j + 2})}^{2}, \end{array}$ which leads to the assertion. □
Lemma 5.7.
The sum in ( 5.14 ) can be estimated componentwise by $\begin{array}{l} \sum_{j = 1}^{ℓ} | {(P_{j, ℓ} R_{j})}_{1} | ⩽ C_{3} z_{2 ℓ + 2}^{2}, \\ \sum_{j = 1}^{ℓ} | {(P_{j, ℓ} R_{j})}_{2} | ⩽ C_{3} z_{2 ℓ + 2}^{3}, \end{array}$ for a constant $C_{3} > 0$ depending only on $c_{min}$ , $c_{max}$ .
Proof.
For the components of $R_{j}$ we get from Lemma 5.5 $\begin{array}{l} | R_{2 j + 2}^{2} + R_{2 j + 2}^{1} Z_{2 j} | ⩽ C_{1} (δ_{2 j + 1}^{2} + δ_{2 j + 2}^{2}) + C_{1} (δ_{2 j + 1} + δ_{2 j + 2}) Z_{2 j}, \\ | R_{2 j + 2}^{2} Z_{2 j} | ⩽ C_{1} (δ_{2 j + 1}^{2} + δ_{2 j + 2}^{2}) Z_{2 j} . \end{array}$ We use (5.15) to get $\begin{array}{l} | {(P_{j, ℓ} R_{j})}_{1} | ⩽ C (1 + Z_{2 ℓ + 2}^{2}) ((δ_{2 j + 1} + δ_{2 j + 2}) Z_{2 ℓ + 2} + (δ_{2 j + 1}^{2} + δ_{2 j + 2}^{2})), \\ | {(P_{j, ℓ} R_{j})}_{2} | ⩽ C Z_{2 ℓ + 2} (Z_{2 ℓ + 2} (δ_{2 j + 1} + δ_{2 j + 2}) + (1 + Z_{2 ℓ + 2}^{2}) (δ_{2 j + 1}^{2} + δ_{2 j + 2}^{2})) \end{array}$ for a constant $C > 1$ depending only on $c_{min}$ , $c_{max}$ . A summation over j leads to the assertion. □

Finally, we can combine Lemma 5.7 and the fact that $\begin{matrix} {\overset{ˇ}{S}}_{2} (μ) = max {| S_{2} (μ) |, | S_{2} (μ^{- 1}) |} ⩽ C_{1} (δ_{1}^{2} + δ_{2}^{2}) ⩽ {\overset{ˇ}{C}}_{1} z_{2 ℓ}^{2} \end{matrix}$ for all $1 ⩽ ℓ ⩽ L$ to estimate ${\overset{ˇ}{S}}_{2 ℓ}$ and ${\overset{ˇ}{T}}_{2 ℓ}$ given by the recursion (5.14) for $κ \in {μ, μ^{- 1}}$ by $\begin{matrix} (5.16) & \begin{matrix} | S_{2 ℓ} (κ) | ⩽ {\overset{ˇ}{S}}_{2 ℓ} (μ) ⩽ C z_{2 ℓ}^{2}, \\ | T_{2 ℓ} (κ) | ⩽ {\overset{ˇ}{T}}_{2 ℓ} (μ) ⩽ C z_{2 ℓ}^{3} . \end{matrix} \end{matrix}$ We summarize our findings in the next lemma, including the estimate of $β_{0, ℓ}$ for odd indices ℓ. Lemma 5.8.
It holds that $\begin{matrix} Re β_{0, ℓ} = 1 + s_{ℓ}, Im β_{0, ℓ} = - \frac{z_{ℓ}}{c_{ℓ + 1}} - t_{ℓ}, \end{matrix}$ for some $s_{ℓ}$ , $t_{ℓ} \in R$ , bounded by $\begin{matrix} (5.17) & | s_{ℓ} | ⩽ C {(\frac{ω x_{ℓ}}{c_{ℓ}})}^{2}, | t_{ℓ} | ⩽ C {(\frac{ω x_{ℓ}}{c_{ℓ}})}^{3}, \end{matrix}$ for a constant $C > 0$ only depending on $c_{min}$ , $c_{max}$ .
Proof.
For the even entries the claim follows from the straightforward combination of (5.8), (5.10), (5.12), (5.16). For the odd entries, we first use (5.9) and compute $\begin{array}{l} Re (β_{0, 2 ℓ + 1}) & = cos (δ_{2 ℓ + 1}) Re (β_{0, 2 ℓ}) + sin (δ_{2 ℓ + 1}) Im (β_{0, 2 ℓ}) = 1 + s_{2 ℓ + 1}, \end{array}$ for some $s_{2 ℓ + 1}$ with $| s_{2 ℓ + 1} | ⩽ C z_{2 ℓ + 1}^{2}$ by a Taylor argument. On the other hand we also know $\begin{matrix} Re (β_{0, 2 ℓ}) = 1 + s_{2 ℓ}, Im (β_{0, 2 ℓ}) = - \frac{z_{2 ℓ}}{c_{2 ℓ + 1}} - t_{2 ℓ}, \end{matrix}$ for some $s_{2 ℓ}$ , $t_{2 ℓ} \in R$ , bounded by $\begin{matrix} | s_{2 ℓ} | ⩽ C {(\frac{ω x_{2 ℓ}}{c_{2 ℓ}})}^{2}, | t_{2 ℓ} | ⩽ C {(\frac{ω x_{2 ℓ}}{c_{2 ℓ}})}^{3} . \end{matrix}$ For the imaginary part, we employ again (5.9) and (5.17) for (even) $2 ℓ$ to get $\begin{array}{l} Im β_{0, 2 ℓ + 1} & = - μ sin (δ_{2 ℓ + 1}) β_{0, 2 ℓ}^{R} + μ cos (δ_{2 ℓ + 1}) β_{0, 2 ℓ}^{I} \\ = - μ sin (δ_{2 ℓ + 1}) (1 + s_{2 ℓ}) + μ cos (δ_{2 ℓ + 1}) (- \frac{z_{2 ℓ}}{c_{2 ℓ + 1}} - t_{2 ℓ}) \\ = - \frac{z_{2 ℓ + 1}}{c_{2 ℓ}} + μ \frac{{\tilde{δ}}_{2 ℓ + 1}^{3}}{6} - μ δ_{2 ℓ + 1} s_{2 ℓ} + μ \frac{{(δ_{2 ℓ + 1}^{'})}^{2}}{2} \frac{z_{2 ℓ}}{c_{2 ℓ + 1}} + μ cos (δ_{2 ℓ + 1}) (- t_{2 ℓ}), \end{array}$ for some $0 ⩽ {\tilde{δ}}_{2 ℓ + 1}$ , $δ_{2 ℓ + 1}^{'} ⩽ δ_{2 ℓ + 1}$ . By using (5.17) (for (even) $2 ℓ$ ) the assertion follows. □
Proposition 5.9.
For all $ℓ ⩽ L$ it holds $\begin{matrix} Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{0, ℓ}) ⩽ C z_{ℓ}^{3} . \end{matrix}$ The constant C depends only on $c_{min}$ , $c_{max}$ but is independent of L.
Proof.
Using Lemma 5.8, we compute $\begin{array}{l} Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{0, ℓ}) & = cos (\frac{z_{ℓ}}{c_{ℓ + 1}}) Im (β_{0, ℓ}) + sin (\frac{z_{ℓ}}{c_{ℓ + 1}}) Re (β_{0, ℓ}) \\ = - cos (\frac{z_{ℓ}}{c_{ℓ + 1}}) (\frac{z_{ℓ}}{c_{ℓ + 1}} + t_{ℓ}) + sin (\frac{z_{ℓ}}{c_{ℓ + 1}}) (1 + s_{ℓ}) \\ = - \frac{1}{6} {(\frac{z_{ℓ}^{'}}{c_{ℓ + 1}})}^{3} - t_{ℓ} - \frac{1}{2} {(\frac{{\tilde{z}}_{ℓ}}{c_{ℓ + 1}})}^{2} (\frac{z_{ℓ}}{c_{ℓ + 1}} + t_{ℓ}) + s_{ℓ} sin (\frac{z_{ℓ}}{c_{ℓ + 1}}) \end{array}$ for some $0 ⩽ {\tilde{z}}_{ℓ}$ , $z_{ℓ}^{'} ⩽ z_{ℓ}$ . Using the estimates (5.17) yields the assertion. □

5.3. Proof of Theorem 3.7

From Proposition 5.1, Proposition 5.9, the representation (2.13), definition (3.2) and the results of Theorem 3.4, we conclude $\begin{array}{l} | A_{ℓ + 1} | = | {\hat{g}}_{0} | \frac{ω}{c_{N}} | h_{0}^{(1)} (\frac{ω}{c_{N}}) | | \frac{Im (e^{i \frac{z_{ℓ}}{c_{ℓ + 1}}} β_{0, ℓ})}{e^{i \frac{z_{n}}{c_{n + 1}}} β_{0, n}} | ≲ | {\hat{g}}_{0} | α^{ω} min {z_{ℓ}^{3}, 1}, \\ | B_{ℓ} | = | {\hat{g}}_{0} | \frac{ω}{c_{N}} | h_{0}^{(1)} (\frac{ω}{c_{N}}) | | \frac{e^{i \frac{z_{ℓ - 1}}{c_{ℓ}}} β_{0, ℓ - 1}}{e^{i \frac{z_{n}}{c_{n + 1}}} β_{0, n}} | ≲ | {\hat{g}}_{0} | α^{ω}, \end{array}$ for all $1 ⩽ ℓ ⩽ n$ and some $α > 1$ . We note that we also used (8.1), to estimate the term $\frac{ω}{c_{N}} | h_{0}^{(1)} (\frac{ω}{c_{N}}) |$ . Moreover we recall (2.12a), (2.12b) that $\begin{matrix} | A_{1} | = 0 = z_{0}, | B_{N} | = | \frac{ω}{c_{N}} h_{0}^{(1)} (\frac{ω}{c_{N}}) | | {\hat{g}}_{0} | = | {\hat{g}}_{0} | . \end{matrix}$ Now we apply these estimates of $A_{ℓ}$ , $B_{ℓ}$ to (3.10) and (3.9). The combination of this with estimates of Hankel and Bessel functions in Lemma 8.2 yields $\begin{array}{l} ‖ u ‖_{H}^{2} & ⩽ 2 max_{2 ⩽ j ⩽ N} A_{j}^{2} (2 + {(\frac{c_{j}}{z_{j - 1}})}^{2}) + 2 max_{1 ⩽ j ⩽ N} B_{j}^{2} (\frac{16 z_{j}^{4}}{{(2 c_{j}^{2} + z_{j}^{2})}^{2}} + {(\frac{2 z_{j}}{c_{j} + z_{j}})}^{2}) \\ ≲ | {\hat{g}}_{0} |^{2} α^{ω} {max_{2 ⩽ j ⩽ N} (min {z_{j}^{6}, 1} (2 + {(\frac{c_{j}}{z_{j - 1}})}^{2})) + max_{1 ⩽ j ⩽ N} (\frac{16 z_{j}^{4}}{{(2 c_{j}^{2} + z_{j}^{2})}^{2}} + {(\frac{2 z_{j}}{c_{j} + z_{j}})}^{2})} \\ ≲ | {\hat{g}}_{0} |^{2} α^{ω} . \end{array}$

Next we will prove the pointwise estimates in Theorem 3.7. For $r \in τ_{j}$ , $2 ⩽ j ⩽ N$ , we obtain $\begin{array}{l} | \hat{u} (r, ξ) | & ≲ | {\hat{g}}_{0} | α^{ω} (min {z_{j - 1}^{3}, 1} \frac{c_{j}}{z_{j - 1}} + \frac{2}{1 + \frac{z_{j - 1}}{c_{j}}}) \\ ≲ \frac{α^{ω}}{1 + z_{j - 1}} | {\hat{g}}_{0} | ⩽ α^{ω} | {\hat{g}}_{0} | \end{array}$ and in a similar fashion $\begin{matrix} | (\nabla u) \circ ψ (r, ξ) | ≲ | {\hat{g}}_{0} | α^{ω} (min {z_{j - 1}^{3}, 1} (\frac{c_{j}}{ω r} (1 + \frac{c_{j}}{ω r})) + \frac{ω r}{c_{j}}) ≲ r {\tilde{α}}^{ω} | {\hat{g}}_{0} | . \end{matrix}$ For the first interval $τ_{1}$ we use that $A_{1} = 0$ and obtain the same estimates.

6. Proof of the representation of the Green’s operator (Theorem 3.4)

We denote by $M_{m}^{(2 n, i, j)}$ the matrix which arises by removing the i-th row and the j-th column of $M_{m}^{(2 n)}$ . According to Cramer’s rule, we have $\begin{matrix} {({\hat{M}}_{m}^{(2 n)})}_{i, j}^{- 1} = {(- 1)}^{i + j} \frac{det {\hat{M}}_{m}^{(2 n, j, i)}}{det {\hat{M}}_{m}^{(2 n)}} . \end{matrix}$ From (3.3) and the well-known recursion formula for determinants of tri-diagonal matrices we get $\begin{matrix} (6.1) & det {\hat{M}}_{m}^{(2 n, 2 n, i)} = (\prod_{ℓ = ⌊ \frac{i + 1}{2} ⌋}^{n - 1} {({\hat{T}}_{m}^{(ℓ)})}_{2, 1}) (\prod_{ℓ = ⌊ \frac{i + 2}{2} ⌋}^{n} {({\hat{S}}_{m}^{(ℓ)})}_{1, 2}) det {\hat{M}}_{m}^{(i - 1)} . \end{matrix}$ Note that $\begin{matrix} det {\hat{S}}_{m}^{(ℓ)} = \frac{w_{m, ℓ, ℓ + 1, ℓ}^{2, 1}}{w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} \end{matrix}$ which is well-defined (cf. proof of [11, Lemma 1]). Next, we express the determinant $det {\hat{M}}_{m}^{(2 n)}$ in a recursive way. For $n \in N_{⩾ 1}$ and $q \in {1, 2}$ , let (with Kronecker’s delta $δ_{i, j}$ ) $\begin{matrix} W_{m, ℓ, q} : = \{\begin{matrix} δ_{1, q} & for ℓ = 0, \\ det [\begin{matrix} W_{m, ℓ - 1, 1} & w_{m, ℓ, ℓ + 1, ℓ}^{1, q} \\ W_{m, ℓ - 1, 2} & w_{m, ℓ, ℓ + 1, ℓ}^{2, q} \end{matrix}] & for ℓ ⩾ 1 . \end{matrix} \end{matrix}$

Lemma 6.1.
Let $n ⩾ 1$ . We have $\begin{matrix} det {\hat{M}}_{m}^{(2 n)} = \frac{W_{m, n, 1}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} \end{matrix}$ and $\begin{matrix} det {\hat{M}}_{m}^{(2 n - 1)} = \frac{- W_{m, n, 2}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} . \end{matrix}$
Proof.
By induction: The case $n = 1$ can be easily checked for both cases. Then, we have $\begin{array}{l} det {\hat{M}}_{m}^{(2 (n + 1))} & = det {\hat{S}}_{m}^{(n + 1)} det {\hat{M}}_{m}^{(2 n)} + {({\hat{S}}_{m}^{(n + 1)})}_{2, 2} det {\hat{M}}_{m}^{(2 n - 1)} \\ = \frac{w_{m, n + 1, n + 2, n + 1}^{2, 1}}{w_{m, n + 2, n + 1, n + 1}^{2, 1}} \frac{W_{m, n, 1}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} - \frac{w_{m, n + 1, n + 2, n + 1}^{1, 1}}{w_{m, n + 2, n + 1, n + 1}^{2, 1}} \frac{W_{m, n, 2}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} \\ = \frac{W_{m, n + 1, 1}}{\prod_{ℓ = 1}^{n + 1} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} \end{array}$ and, in turn, $\begin{array}{l} det {\hat{M}}_{m}^{(2 n + 1)} & = {({\hat{S}}_{m}^{(n + 1)})}_{1, 1} det {\hat{M}}_{m}^{(2 n)} + det {\hat{M}}_{m}^{(2 n - 1)} \\ = \frac{w_{m, n + 2, n + 1, n + 1}^{2, 2}}{w_{m, n + 2, n + 1, n + 1}^{2, 1}} \frac{W_{m, n, 1}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} - \frac{W_{m, n, 2}}{\prod_{ℓ = 1}^{n} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} \\ = \frac{- W_{m, n + 1, 2}}{\prod_{ℓ = 1}^{n + 1} w_{m, ℓ + 1, ℓ, ℓ}^{2, 1}} . \end{array}$ □

In the next step we will derive a representation of $W_{m, n, 1}$ . We define the sequence ${({\tilde{β}}_{m, ℓ})}_{ℓ = 0}^{n}$ by $\begin{matrix} (6.2) & \begin{matrix} {\tilde{β}}_{m, 0} : = 1, \\ {\tilde{β}}_{m, ℓ} : = \frac{{\tilde{γ}}_{m, ℓ}^{+}}{2} ({\tilde{β}}_{m, ℓ - 1} - {(- 1)}^{ℓ} {\tilde{q}}_{m, ℓ} \overline{{\tilde{β}}_{m, ℓ - 1}}) \end{matrix} \end{matrix}$ using the definition in (3.5) for ${\tilde{γ}}_{m, ℓ}^{+}$ and ${\tilde{q}}_{m, ℓ}$ .
Lemma 6.2.
For $n \in N$ it holds $\begin{matrix} W_{m, n, 1} = {(- 1)}^{n} {\tilde{β}}_{m, n} and W_{m, n, 2} : = - \frac{\overline{{\tilde{β}}_{m, n}} - {(- 1)}^{n} {\tilde{β}}_{m, n}}{2} . \end{matrix}$
Proof.
By using the definition of $w_{m, n, n + 1, n}^{2, 1}$ and $f_{m, 2} = \frac{1}{2} (f_{m, 1} + \overline{f_{m, 1}})$ it is easy to verify $\begin{array}{l} w_{m, ℓ, ℓ + 1, ℓ}^{1, 1} = - {\tilde{γ}}_{m, ℓ}^{-}, w_{m, ℓ, ℓ + 1, ℓ}^{1, 2} = - \frac{1}{2} (\overline{{\tilde{γ}}_{m, ℓ}^{+}} + {\tilde{γ}}_{m, ℓ}^{-}), \\ w_{m, ℓ, ℓ + 1, ℓ}^{2, 1} = - \frac{1}{2} ({\tilde{γ}}_{m, ℓ}^{+} + {\tilde{γ}}_{m, ℓ}^{-}), w_{m, ℓ, ℓ + 1, ℓ}^{2, 2} = - \frac{1}{4} ({\tilde{γ}}_{m, ℓ}^{+} + {\tilde{γ}}_{m, ℓ}^{-} + \overline{{\tilde{γ}}_{m, ℓ}^{+}} + \overline{{\tilde{γ}}_{m, ℓ}^{-}}) . \end{array}$ Now we start to prove the statement by induction. For $n = 1$ we have $\begin{array}{l} \begin{matrix} W_{m, 1, 1} & = w_{m, 1, 2, 1}^{2, 1} = - \frac{1}{2} ({\tilde{γ}}_{m, 1}^{+} + {\tilde{γ}}_{m, 1}^{-}) = - \frac{{\tilde{γ}}_{m, 1}^{+}}{2} (1 + {\tilde{q}}_{m, 1}) = - {\tilde{β}}_{m, 1}, \end{matrix} \\ \begin{matrix} W_{m, 1, 2} & = w_{m, 1, 2, 1}^{2, 2} = - \frac{1}{4} ({\tilde{γ}}_{m, 1}^{+} + {\tilde{γ}}_{m, 1}^{-} + \overline{{\tilde{γ}}_{m, 1}^{+}} + \overline{{\tilde{γ}}_{m, 1}^{-}}) = - \frac{1}{2} (\frac{{\tilde{γ}}_{m, 1}^{+}}{2} (1 + {\tilde{q}}_{m, 1}) + \frac{\overline{{\tilde{γ}}_{m, 1}^{+}}}{2} (1 + \overline{{\tilde{q}}_{m, 1}})) \\ = - \frac{\overline{{\tilde{β}}_{m, 1}} + {\tilde{β}}_{m, 1}}{2} . \end{matrix} \end{array}$ Assume the statement is true for $n - 1$ . We have from the definition of $W_{m, n, 1}$ $\begin{array}{l} W_{m, n, 1} & = (W_{m, n - 1, 1} w_{m, n, n + 1, n}^{2, 1} - W_{m, n - 1, 2} w_{m, n, n + 1, n}^{1, 1}) \\ = \frac{1}{2} ({(- 1)}^{n} {\tilde{β}}_{m, n - 1} ({\tilde{γ}}_{m, n}^{+} + {\tilde{γ}}_{m, n}^{-}) - \overline{{\tilde{β}}_{m, n - 1}} {\tilde{γ}}_{m, n}^{-} - {(- 1)}^{n} {\tilde{β}}_{m, n - 1} {\tilde{γ}}_{m, n}^{-}) \\ = {(- 1)}^{n} {\tilde{β}}_{m, n} . \end{array}$ For $W_{m, n, 2}$ , we compute $\begin{array}{l} W_{m, n, 2} & = (W_{m, n - 1, 1} w_{m, n, n + 1, n}^{2, 2} - W_{m, n - 1, 2} w_{m, n, n + 1, n}^{1, 2}) \\ = \frac{1}{4} ({(- 1)}^{n} {\tilde{β}}_{m, n - 1} ({\tilde{γ}}_{m, n}^{+} + \overline{{\tilde{γ}}_{m, n}^{-}}) - \overline{{\tilde{β}}_{m, n - 1}} (\overline{{\tilde{γ}}_{m, n}^{+}} + {\tilde{γ}}_{m, n}^{-})) \\ = - \frac{1}{2} (\overline{{\tilde{β}}_{m, n}} - {(- 1)}^{n} {\tilde{β}}_{m, n}) . \end{array}$ □

The combination of representation in (6.1) and applying Lemma 6.1 and 6.2 yields for odd $i = 2 ℓ - 1$ , $1 ⩽ ℓ ⩽ n$ $\begin{matrix} (6.3) & \begin{matrix} {({\hat{M}}_{m}^{(2 n)})}_{2 ℓ - 1, 2 n}^{- 1} & = - \frac{det {\hat{M}}_{m}^{(2 n, 2 n, 2 ℓ - 1)}}{det {\hat{M}}_{m}^{(2 n)}} = {(- 1)}^{n - ℓ} \frac{(\prod_{k = ℓ}^{n} {\hat{S}}_{1, 2}^{(k)}) det {\hat{M}}_{m}^{(2 (ℓ - 1))}}{det {\hat{M}}_{m}^{(2 n)}} \\ = - (\prod_{k = ℓ}^{n} w_{m, k + 1, k + 1, k}^{1, 2}) \frac{{\tilde{β}}_{m, ℓ - 1}}{{\tilde{β}}_{m, n}} . \end{matrix} \end{matrix}$ For $i = 2 ℓ$ even, $1 ⩽ ℓ ⩽ n$ , we compute $\begin{array}{l} {({\hat{M}}_{m}^{(2 n)})}_{2 ℓ, 2 n}^{- 1} & = \frac{det {\hat{M}}_{m}^{(2 n, 2 n, 2 ℓ)}}{det {\hat{M}}_{m}^{(2 n)}} = - {(- 1)}^{n - ℓ} (\prod_{k = ℓ + 1}^{n} w_{m, k + 1, k + 1, k}^{1, 2}) \frac{W_{m, ℓ, 2}}{W_{m, n, 1}} \\ (6.4) & = \frac{{(- 1)}^{ℓ} \overline{{\tilde{β}}_{m, ℓ}} - {\tilde{β}}_{m, ℓ}}{2 {\tilde{β}}_{m, n}} \prod_{k = ℓ + 1}^{n} w_{m, k + 1, k + 1, k}^{1, 2} . \end{array}$ Finally, we introduce $\begin{matrix} β_{m, ℓ} = \frac{e^{- i \frac{z_{ℓ}}{c_{ℓ + 1}}}}{\prod_{k = 1}^{ℓ} w_{m, k + 1, k + 1, k}^{1, 2}} {\tilde{β}}_{m, ℓ}, 0 ⩽ ℓ ⩽ n . \end{matrix}$

Recalling Remark 3.3, one can easily check that the recursion for $β_{m, ℓ}$ given in (3.8) follows from the recursion of ${\tilde{β}}_{m, ℓ}$ defined in (6.2). The representations (3.7) are a direct consequence of the definition of $β_{m, ℓ}$ and equations (6.3) and (6.4).
7. Some basic facts from linear algebra

Lemma 7.1.
Let $\begin{matrix} A_{ℓ} : = [\begin{matrix} a_{ℓ} & b_{ℓ} \\ b_{ℓ} & a_{ℓ} \end{matrix}] . \end{matrix}$ Then $\begin{matrix} A_{k} \dots A_{1} = \frac{1}{2} [\begin{matrix} \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) + \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) & \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) - \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) \\ \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) - \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) & \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) + \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) \end{matrix}] . \end{matrix}$

The proof of this lemma follows in a straightforward way by induction and is skipped. To estimate the matrix product in Lemma 7.1 for positive coefficients we need the following lemma. Lemma 7.2.
For $0 < b_{ℓ} < a_{ℓ}$ , one has $\begin{array}{l} \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) + \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) ⩽ 2 (\prod_{ℓ = 1}^{k} a_{ℓ}) cosh (\sum_{ℓ = 1}^{k} \frac{b_{ℓ}}{a_{ℓ}}), \\ \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) - \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) ⩽ 2 (\prod_{ℓ = 1}^{k} a_{ℓ}) sinh (\sum_{ℓ = 1}^{k} \frac{b_{ℓ}}{a_{ℓ}}) . \end{array}$
Proof.
We compute $\begin{matrix} \prod_{ℓ = 1}^{k} (a_{ℓ} + b_{ℓ}) \pm \prod_{ℓ = 1}^{k} (a_{ℓ} - b_{ℓ}) = (\prod_{ℓ = 1}^{k} a_{ℓ}) (\prod_{ℓ = 1}^{k} (1 + \frac{b_{ℓ}}{a_{ℓ}}) \pm \prod_{ℓ = 1}^{k} (1 - \frac{b_{ℓ}}{a_{ℓ}})) \end{matrix}$ and recall that for $0 < γ_{ℓ} \in R$ $\begin{array}{l} \prod_{ℓ = 1}^{k} (1 + γ_{ℓ}) + \prod_{ℓ = 1}^{k} (1 - γ_{ℓ}) = 2 \sum_{ℓ = 0}^{⌊ \frac{k}{2} ⌋} \sum_{\begin{array}{c} α \in {0, 1}^{k}, \\ | α | = 2 ℓ \end{array}} \prod_{i = 1}^{k} γ_{i}^{α_{i}}, \\ \prod_{ℓ = 1}^{k} (1 + γ_{ℓ}) - \prod_{ℓ = 1}^{k} (1 - γ_{ℓ}) = 2 \sum_{ℓ = 0}^{⌊ \frac{k - 1}{2} ⌋} \sum_{\begin{array}{c} α \in {0, 1}^{k}, \\ | α | = 2 ℓ + 1 \end{array}} \prod_{i = 1}^{k} γ_{i}^{α_{i}} . \end{array}$ Finally, we note that for any $n \in N$ $\begin{matrix} {(\sum_{i = 1}^{k} γ_{i})}^{n} = \sum_{\begin{array}{c} α \in N^{k} \\ | α | = n \end{array}} \frac{n!}{α_{1}! \dots α_{k}!} \prod_{i = 1}^{k} γ_{i}^{α_{i}} ⩾ n! \sum_{\begin{array}{c} α \in {0, 1}^{k} \\ | α | = n \end{array}} \prod_{i = 1}^{k} γ_{i}^{α_{i}} \end{matrix}$ and $\begin{array}{l} cosh (\sum_{i = 1}^{k} γ_{i}) = \sum_{ℓ = 0}^{\infty} \frac{{(\sum_{i = 1}^{k} γ_{i})}^{2 ℓ}}{(2 ℓ)!} ⩾ \sum_{n = 0}^{n} \frac{{(\sum_{i = 1}^{k} γ_{i})}^{2 n}}{(2 n)!}, \\ sinh (\sum_{i = 1}^{k} γ_{i}) = \sum_{ℓ = 0}^{\infty} \frac{{(\sum_{i = 1}^{k} γ_{i})}^{2 ℓ + 1}}{(2 ℓ + 1)!} ⩾ \sum_{ℓ = 0}^{n} \frac{{(\sum_{i = 1}^{k} γ_{i})}^{2 ℓ + 1}}{(2 ℓ + 1)!} . \end{array}$ □

8. Some facts about Hankel and Bessel functions

In this section, we state some properties of spherical Hankel and Bessel functions.

Lemma 8.1.

It holds $\begin{matrix} h_{0}^{(1)} (x) = \frac{1}{x} (sin x + i cos x) and j_{0} (x) = \frac{sin x}{x} . \end{matrix}$ In particular it holds for $x > 0$ $\begin{matrix} (8.1) & | x h_{0} (x) | = 1 and | x j_{0} (x) | ⩽ | sin x | ⩽ \frac{2 | x |}{1 + | x |} \end{matrix}$ and $\begin{matrix} (8.2) & {| {(h_{0}^{(1)})}^{'} (x) |}^{2} = \frac{1}{x^{2}} + \frac{1}{x^{4}} and {| j_{0}^{'} (x) |}^{2} = {(\frac{cos x}{x} - \frac{sin x}{x^{2}})}^{2} . \end{matrix}$

Lemma 8.2.

For $d = 3$ and $m = 0$ the fundamental system satisfies the integral estimates $\begin{array}{l} {‖ h_{0}^{(1)} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} ⩽ {(\frac{c_{j}}{ω})}^{2} h_{j}, \\ {‖ {(h_{0}^{(1)})}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} ⩽ (1 + {(\frac{c_{j}}{z_{j - 1}})}^{2}) {(\frac{c_{j}}{ω})}^{2} h_{j}, \\ {‖ j_{0} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} ⩽ {(\frac{2 z_{j}}{c_{j} + z_{j}})}^{2} {(\frac{c_{j}}{ω})}^{2} h_{j}, \\ {‖ j_{0}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} ⩽ \frac{16 z_{j}^{4}}{{(2 c_{j}^{2} + z_{j}^{2})}^{2}} {(\frac{c_{j}}{ω})}^{2} h_{j}, \end{array}$ as well as the pointwise estimates for $r \in τ_{j}$ (with $y = ω r / c_{j}$ ) $\begin{array}{l} | h_{0}^{(1)} (y) | ⩽ \frac{1}{y}, | {(h_{0}^{(1)})}^{'} (y) | ⩽ \frac{1}{y} (1 + \frac{1}{y}), \\ | j_{0} (y) | ⩽ \frac{2}{1 + y}, | j_{0}^{'} (y) | ⩽ \frac{4 y}{2 + y^{2}} . \end{array}$

Proof.

By using Lemma 8.1, we have $\begin{array}{l} {‖ h_{0}^{(1)} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} = \int_{x_{j - 1}}^{x_{j}} {(\frac{ω}{c_{j}} r)}^{- 2} r^{2} d r = {(\frac{c_{j}}{ω})}^{2} h_{j}, \\ {‖ {(h_{0}^{(1)})}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} = \int_{x_{j - 1}}^{x_{j}} {| h_{0}^{'} (\frac{ω}{c_{j}} r) |}^{2} r^{2} d r ⩽ {(\frac{c_{j}}{ω})}^{2} (1 + {(\frac{c_{j}}{z_{j - 1}})}^{2}) h_{j} . \end{array}$ For the spherical Bessel functions we get $\begin{array}{l} \begin{matrix} {‖ j_{0} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} & = \int_{x_{j - 1}}^{x_{j}} {(\frac{sin (\frac{ω}{c_{j}} r)}{(\frac{ω}{c_{j}} r)})}^{2} r^{2} d r = {(\frac{c_{j}}{ω})}^{3} \int_{z_{j - 1} / c_{j}}^{z_{j} / c_{j}} {(sin y)}^{2} d y \\ ⩽ 4 {(\frac{c_{j}}{ω})}^{3} \int_{z_{j - 1} / c_{j}}^{z_{j} / c_{j}} \frac{y^{2}}{{(1 + y)}^{2}} d y ⩽ {(2 \frac{c_{j}}{ω} \frac{z_{j}}{c_{j} + z_{j}})}^{2} h_{j}, \end{matrix} \\ {‖ j_{0}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} = {(\frac{c_{j}}{ω})}^{3} \int_{z_{j - 1} / c_{j}}^{z_{j} / c_{j}} {(cos x - \frac{sin x}{x})}^{2} d x . \end{array}$ We set $κ (x) : = cos x - \frac{sin x}{x}$ and a straightforward calculation leads to $| κ^{″} (x) | ⩽ 2$ . This together with the trivial estimate $| κ (x) | ⩽ 2$ leads to $\begin{matrix} | κ (x) | ⩽ min {x^{2}, 2} ⩽ \frac{4 x^{2}}{x^{2} + 2} . \end{matrix}$ Hence, $\begin{matrix} {‖ j_{0}^{'} (\frac{ω}{c_{j}} \cdot) g_{0} ‖}_{L^{2} (τ_{j})}^{2} ⩽ \frac{16 z_{j}^{4}}{{(2 c_{j}^{2} + z_{j}^{2})}^{2}} {(\frac{c_{j}}{ω})}^{2} h_{j} . \end{matrix}$ The first three pointwise estimates follow from (8.1) and (8.2). For the last one, we estimate $\begin{matrix} | j_{0}^{'} (y) | = \frac{1}{y} | κ (y) | ⩽ \frac{4 y}{2 + y^{2}} . \end{matrix}$ □

Footnotes

Acknowledgement

The authors gratefully acknowledge support by the Swiss National Science Foundation under Grant No. 172803.

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