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\title[Series solutions of integral equations in scattering]{On the series solutions of integral equations in scattering}

\alttitle{Sur les solutions en série des équations intégrales en diffusion}


\author{\firstname{Faouzi} \lastname{Triki}\IsCorresp}

\address{Laboratoire Jean Kuntzmann, Université Grenoble-Alpes, Grenoble, France}


\email{Faouzi.Triki@univ-grenoble-alpes.fr}


\author{\firstname{Mirza} \lastname{Karamehmedović}}

\address{Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark}


\email{mika@dtu.dk}


\keywords{\kwd{Helmholtz equation} \kwd{Born series} \kwd{scattering}}

\altkeywords{\kwd{Équation de Helmholtz} \kwd{série de Born} \kwd{diffusion}}

\subjclass{35R30, 34L25, 78A46}




\begin{abstract} We study the validity of the Neumann or Born series approach in solving the Helmholtz equation 
and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The 
obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. 
Our approach makes use of reduction space techniques proposed by Suzuki~\tralicstex{[21]}{\cite{Suzuki-1976}}. Furthermore 
we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results. 

\end{abstract}


\begin{altabstract} Nous étudions la validité de l'approche série de Neumann ou de Born pour résoudre l'équation de Helmholtz, et pour l'identification de coefficients dans des problèmes inverses de diffusion. Plus précisément, nous obtenons une condition nécessaire et suffisante sous laquelle la série converge fortement. Cette condition est beaucoup plus faible que celle utilisée traditionnellement. Nous examinons également le taux de convergence de la série. Notre approche utilise des techniques d'espace de réduction proposées par Suzuki~\tralicstex{[21]}{\cite{Suzuki-1976}}. De plus, nous proposons une méthode d'interpolation qui permet l'utilisation de la série de Neumann dans tous les cas. Enfin, nous fournissons plusieurs tests numériques avec différentes fonctions de milieu et valeurs de fréquence pour valider nos résultats théoriques.
\end{altabstract}


\begin{document}



\maketitle

\section{Introduction and main results}\label{sec:intro}
Let $d=1,2,\dots$, fix a positive $k_0$ and $\hat{k}\in S^{d-1}$, let $q\in L^{\infty}(\R^d)$ be compactly supported with $q(x)>-1$, and consider the following Helmholtz equation in $\R^d$ with the Sommerfeld radiation condition:
\begin{equation}\label{eqn:HS}
\begin{cases}
(\Delta+k_0^2(1+q(x)))u=-k_0^2q(x)\expe^{\irm k_0\hat{k}.x}&\text{in}\,\,\,\R^d,\\
\lim_{|x|\to \infty}|x|^{(d-1)/2}\left(\partial_{|x|}-\irm k_0\right)u=0&\text{uniformly in}\,\,\,x/|x|\in S^{d-1}.
\end{cases}
\end{equation}
The system~\eqref{eqn:HS} is satisfied by the scattered field arising from the interaction of the incident uniform plane wave $\expe^{\irm k_0\hat k.x}$ with the medium $q(x)$. Convolving the PDE in~\eqref{eqn:HS} with the outgoing fundamental solution\footnote{Here $H_0^{(1)}$ the Hankel function of the first kind and order zero.} of the Helmholtz operator $\Delta+k_0^2$ in $\R^d$,
\[
\Phi_d(x)=\begin{cases}(-2\pi|x|)^{(1-d)/2}(\irm /2k_0)\partial_{|x|}^{(d-1)/2}\expe^{\irm k_0|x|},&x\in\R^d\setminus\{0\},\,\,\,d\,\,\,\text{odd},\\(-2\pi|x|)^{(2-d)/2}(\irm /4)\partial_{|x|}^{(d-2)/2}H_0^{(1)}(k_0|x|),&x\in\boldsymbol{R}^d\setminus\{0\},\,\,\,d\,\,\,\,\text{even},\end{cases}
\]
%\Mirza{The 1D fundamental solution at least should be multiplied by $-1$ to get $-\delta$.}
%\Mirza{Check the other cases.}
and integrating by parts, we get the Lippmann--Schwinger equation
\begin{equation}\label{eqn:LSE}
(I-V_q(k_0))u=V_{q}(k_0)\expe^{\irm k_0\hat{k}(\,\cdot\,)}\quad\text{in}\,\,\,\R^d,
\end{equation}
where 
\[
V_q(k_0)u(x)=k_0^2\int_{y\in\supp{q}}\Phi_d(x-y)q(y)u(y)\,\dd y
\]
exists as an improper integral for each $x\in\R^d$. It is well-known that the integral equation~\eqref{eqn:LSE} is equivalent with~\eqref{eqn:HS}, and that it suffices to solve~\eqref{eqn:LSE} in, say, a 
bounded open ball $B\subset\R^d$ that includes $\supp{q}$, followed by the continuous extension $u(x)=V_q(k_0)[u(\,\cdot\,)+\exp(\irm k_0\hat{k}(\,\cdot\,))](x)$ for $x\in\R^d\setminus B$. The mapping $V_q(k_0):L^2(B)\to L^2(B)$ is compact, and we shall in the following consider only the restriction of the Lippmann--Schwinger equation in~\eqref{eqn:LSE} to $B$. The objective of the paper is to study the successive approximations for solving the integral equation~\eqref{eqn:LSE}:
\begin{equation}\label{iterative scheme}
u_0 = V_q(k_0)\expe^{\irm k_0\hat{k}(\,\cdot\,)}; \; u_{n+1} = u_0+ V_q(k_0)u_n, \;\; n\in \N. 
\end{equation}
 The computational advantage of this iterative method is that it does not need to solve the partial differential equation~\eqref{eqn:HS} in the whole space and deal with the radiation conditions. Instead, one can 
 obtain a good approximation $u_n$ of the solution $u$ by applying successively the integral operator $V_q(k_0)$ if the
 sequence converges. %\\

 On the other hand the strong convergence of the sequence $(u_n)_{n\in \N}$ to the solution $u$ of 
the integral equation~\eqref{eqn:LSE} is equivalent to the convergence 
of the Neumann series: 
\begin{equation} \label{Neumann series} \lim_{n\to \infty} u_n =\sum_{j=0}^\infty 
V_q(k_0)^{j+1}\expe^{(\,\cdot\,)k_0\hat{k}(\,\cdot\,)} =(I-V_q(k_0))^{-1}V_q(k_0)\expe^{(\,\cdot\,)k_0\hat{k}(\,\cdot\,)}.
\end{equation}

In inverse scattering problems the Neumann series approach known more under the name of {\it Born approximation } was initially employed to study the quantum mechanical inverse backscattering problem in one dimension (see for instance~\cite{MS3} and references therein). The principal advantage 
of using this technique in inverse medium problem is that it
requires solving a linear equation instead
of an oscillatory nonlinear one~\cite{BT1, BT2}. It has also been applied to various other inverse problems, including optical and electrical impedance tomographies, acoustic and electromagnetic parameters identification~\cite{BCM, Kleinman1990, AGKLS, MS, MS2, PMCS, MaS, KMS, KMS2, BG, AMS}. However, it is important to note, that the strategies considered in these works are based on purely formal analysis or assume strong conditions on the targeted physical parameters. %\\

It is well known that a sufficient condition for the convergence of the Neumann series~\eqref{Neumann series} is that the spectral radius $\spr(V_q(k_0))$ of the compact operator $V_q(k_0)$ is strictly less than one, that is $\spr(V_q(k_0))<1$. But this latter condition while it is optimal for the expansion of the operator $(I-V_q(k_0))^{-1}$ in $L^2(B)$, it is obviously too restrictive for the convergence of~\eqref{Neumann series}. Then is it possible to derive a necessary and sufficient condition for the convergence of only~\eqref{Neumann series}? On the other hand the strong convergence
\begin{equation} \label{optimal condition}
V_q(k_0)^{j}\expe^{\irm k_0\hat{k}(\,\cdot\,)} \to 0, \; \; j \to \infty,
\end{equation}
in $L^2(B)$, is evidently a necessary condition for the convergence of the series 
\eqref{Neumann series}. Suzuki in his seminal work~\cite{Suzuki-1976} wondered if this condition is also sufficient. Surprisingly, it turns out that this condition 
also guarantees the convergence of the series. The main idea of the proof is to derive a minimal invariant space
where the expansion of the restriction of $(I-V_q(k_0))^{-1}$ to that space is equivalent to the convergence of the series~\eqref{Neumann series}. 

Let
\begin{equation} \label{ reduced space}
L^2_{k_0,\hat k}(B) = \Span\left(V_q(k_0)\expe^{\irm k_0\hat{k}(\,\cdot\,)}, 
V_q(k_0)^{2}\expe^{\irm k_0\hat{k}(\,\cdot\,)} , \ldots, V_q(k_0)^{j+1}\expe^{\irm k_0\hat{k}(\,\cdot\,)} , \ldots \right).
\end{equation}
By construction $L^2_{k_0,\hat k}(B)$ is invariant by $V_q(k_0)$. Denote $\tilde V_q(k_0)$ the restriction of 
$V_q(k_0)$ to $L^2_{k_0,\hat k}(B)$. 
Suzuki showed that condition~\eqref{optimal condition} indeed implies $\spr(\tilde V_q(k_0))<1$, and hence ensures the convergence of the
Neumann series to 
the unique solution.

 
\begin{proposition}
\label{main result}
The convergence of the Neumann series~\eqref{Neumann series} is equivalent to
the condition~\eqref{optimal condition}. 
\end{proposition}
 
\begin{remark}\label{rem:1}
Since $V_q(k_0)$ is a compact operator the strong convergence~\eqref{optimal condition} can be replaced by a weak convergence of a subsequence.
Notice that $L^2_{k_0,\hat k}(B)$ can also be generated by finite sums of the sequence 

\[
L^2_{k_0,\hat k}(B) = \Span\left( \sum_{j=0}^J V_q(k_0)^{j+1}\expe^{\irm k_0\hat{k}(\,\cdot\,)};\; \; J\in \N \right).
\]
Recall that the traditional condition to ensure the convergence of the Neumann series is~\cite{BT2}
\begin{equation} \label{norm condition}
 \|V_q(k_0)\|\leq C_{k_0,q} = \left(\int_{B}\int_{B}|k_0^2\Phi_d(x-y)q(x)|^2\, \dd x\, \dd y\right)^{1/2}<1.
\end{equation}
This condition occurs in the situation for weak scattering, and is not
valid for high wave number $k_0$, or large magnitude of the medium function
$q$. But since $\expe^{\irm k_0\hat{k}(\,\cdot\,)}$ is sparse 
we expect that $L^2_{k_0,\hat k}(B)$ has a lower dimensionality than the whole space 
$L^2(B)$, and consequently the convergence of the Neumann series~\eqref{Neumann series} may happen beyond the conventional 
limitation~\eqref{norm condition}. In other words $\spr(\tilde V_q(k_0))<1$ can be satisfied by a larger class of wave numbers and 
medium functions not necessary within the weak scattering regime. This was also observed in many numerical experiments in the past, has
fueled many discussions and was the origin of several investigations~\cite{MS, MS2, PMCS, MaS, KMS, KMS2, AMS}. This pattern
is clearly confirmed by many numerical examples in Section~\ref{Sect4}.

\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{main2}
 Assume that the condition~\eqref{optimal condition} is satisfied, that is 
\[
 \lim_{n\to \infty}\left\|V_q(k_0)^{n}\expe^{\irm k_0\hat{k}(\,\cdot\,)}\right\|=0.
\] 
 Then there exists
 a constant $C>0$ independent of $n$ such that the following
 error estimate 
 \begin{equation}
 \|u-u_n\| \leq C \left\|V_q(k_0)^{n}\expe^{\irm k_0\hat{k}(\,\cdot\,)}\right\|,
 \end{equation}
 holds for all $n \in \N.$
 
\end{theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The rest of the paper is organized as follows. In Section~\ref{section 2},
we provide the proofs for Proposition~\ref{main result}, and Theorem~\ref{main2}. Section~\ref{Sect3} is devoted to the construction of a preconditioner for the integral equation~\eqref{eqn:LSE}. Precisely, we propose an interpolation method that allows the use of the Neumann series independently of the fact that the condition~\eqref{optimal condition} is fulfilled or not. We present then several numerical experiments to show the effectiveness of the derived theoretical results
in Section~\ref{Sect4}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the main results} \label{section 2}
In this section we shall prove the main results of the paper. To ease the notation we 
set 
\[
A = V_q(k_0);\;\; \psi = V_q(k_0)\expe^{\irm k_0\hat{k}(\,\cdot\,)}; \;\; \h =L^2(B); \;\;\h_0=
 L^2_{k_0,\hat k}(B).
\]

\begin{proof} [Proof of Proposition~\ref{main result}] If the series 
\begin{equation} \label{abstract Neumann series}
\sum_{j=0}^\infty A^j\Psi = (I-A)^{-1}\Psi,
\end{equation}
is convergent then obviously 
we will have $A^{j}\Psi\to 0$ strongly in $\h$.
Now assume that 
 $A^{j}\Psi$ converges strongly to zero in $\h$,
and focus on the 
nontrivial opposite direction. %\\

The main observation
of Suzuki is that the convergence of the series~\eqref{abstract Neumann series}
depends more on the specific local behavior of the operator $A$ relative to the 
given vector $\psi$ rather than its global properties on the whole space $\h$
which requires that $\spr(A) <1$. Indeed 
consider the Hilbert subspace $\h_0 \subset \h$ space generated by the vectors $A^j\Psi, \, j\in \N $, that is 
\begin{equation} 
\h_0 = \Span\left(\Psi, A\Psi, \ldots, A^j\Psi, \ldots \right).
\end{equation}
Clearly $\h_0$ is invariant by $A$, and since $\Psi$ lies in $\h_0$ to prove
that the series~\eqref{abstract Neumann series} strongly in $\h_0$
it is sufficient to show that $A_0$ the restriction of $A$ to $\h_0$ verifies
$\spr(A_0)<1$. Remark that since $\h_0 \subset \h$ we have $\spr(A_0) \leq \spr(A)$. %\\

Let $\M$ be the linear 
manifold formed by the vectors $v \in \h_0$ satisfying $A^jv$ tends strongly to zero as $j\to \infty$. 
We first remark that the fact $j\to A^{j}\Psi$ tends strongly to zero, $\M$ contains all the vectors $A^j\Psi, \, j\in \N $, and consequently is dense in $\h_0$. %\\

Let $\sigma(A_0)$ denotes the spectrum of $A_0$, and set $\sigma_-(A_0) =
\{\lambda \in \Sigma(A_0); \; |\lambda| <1 \}$ and $\sigma_+(A_0) =
\{\lambda \in \Sigma(A_0); \; |\lambda| \geq 1 \}$. Since $A_0$ is compact 
$\sigma_+(A_0)$ is finite, in addition there exists a rectifiable Jordan curve $\mathcal C_+$ in the resolvent set surrounding $\sigma_+(A_0)$
and does not contain other eigenvalues of $\sigma(A_0)$. Similarly there exists a rectifiable Jordan curve $\mathcal C_-$ in the resolvent set surrounding only $\sigma_-(A_0) $. 
Then following~\cite{Kato}, the spectral projections 
\begin{equation}
P_\pm = \frac{1}{2\irm \pi}\int_{\mathcal C_\pm} (\lambda I- A_0)^{-1} \dd\lambda,
\end{equation}
verify the following identities
\begin{equation} \label{identities}
 P_- +P_+ = I; \; P_-P_+= P_+P_-= 0; \; P_\pm A = A P_\pm. 
\end{equation}


Recalling that $\spr(A_0) = \sup_{\lambda \in \sigma(A_0)}{|\lambda|}.$ Since $\sigma(A_0)$ is a sequence of complex
values that may converge to zero, proving that $\spr(A_0)<1$ is equivalent to show that $\sigma_+(A_0)$ is an empty set. Let now 
$v \in P_+ \h_0$. By the density of the set $\M$ in $\h_0$, there 
exists a sequence $(v_n)_{n\in \N_0} \in \M$ that converges 
strongly to $v$. $\N_0 $ here is the set $\N\setminus\{0\}$. Denote $v_{n, \pm} = P_\pm v_n$. Therefore 
$v_n = v_{n, +}+v_{n, -}$. Remarking that $ A v_{n, -}= A P_- v_n=
P_-Av_n$ converges strongly to $P_- v =0$ leads to $v_{n, -} \in \M$. 
Hence $ v_{n, +} = v_n - v_{n, -}$ lies in fact in $\M \cap P_+ \h_0$. This shows that $\M \cap P_+ \h_0$ is indeed dense in $ P_+ \h_0$. But since $\sigma_+(A_0)$ is finite $ P_+ \h_0$ is finite 
dimensional space and consequently $\M \cap P_+ \h_0= P_+ \h_0$. 
This is clear not correct if $P_+ \h_0$ is not trivial (take any
eigenvector of $A_0$ associated to $\lambda \in \sigma_+(A_0)$, 
it obviously does not belong to $\M$). Then $\sigma_+(A_0)$ is an empty set, and finally $\spr(A_0)<1$, which achieves the proof. 




\end{proof}
\begin{proof} [Proof of Theorem~\ref{main2}] 
Since $\psi \in \M$ we deduce from Proposition~\ref{main result} that the Neumann series~\eqref{abstract Neumann series} is convergent.
On the other hand we deduce from~\eqref{identities} $A^j \psi = A^j P_-\psi = A_0^j \psi$. Therefore 
\begin{equation} \label{bound}
\left\|\sum_{j=0}^\infty A^j\Psi\right\| = \left\|\sum_{j=0}^\infty A_0^j\Psi\right\| = \left\|(I-A_0)^{-1}\Psi\right\| \leq \left\|(I-A_0)^{-1}\right\|\|\Psi\|.
\end{equation}
Let $n\in \N$ be fixed. The fact that 
$\psi \in \M$ implies that $A^{n+1}\psi \in \M$. Applying inequality~\eqref{bound} to the 
vector $A^{n+1}\psi$ leads to 
\begin{equation}
\left\|\sum_{j=n+1}^\infty A^j\Psi\right\| = \left\|\sum_{j=0}^\infty A^j A^{n+1}\Psi\right\| \leq \left\|(I-A_0)^{-1}\right\|\left\|A^{n+1}\Psi\right\| \leq
C \left\|A^{n}\Psi\right\|,
\end{equation}
with $C = \|(I-A_0)^{-1}\| \|A_0\|$, which finishes the proof of the Theorem.
\end{proof}

\begin{remark} The proofs stay valid for any general compact operator $A$ and even if $\h$ is a Banach space. 
In the particular case where $A$ is in addition normal, that is $AA^* = A^*A$, the obtained results are straightforward. Indeed
if $\sigma(A)=\{\lambda_k; k\in \N_0\}$ the eigenvalues of $A$, and $P_k$ is the orthogonal spectral projection 
associated to $\lambda_k$, we have 
\[A = \sum_{k=1}^\infty \lambda_k P_k,\]
and it is clear that the condition $\psi \in \M$ is equivalent to
\[
\psi = \sum_{|\lambda_k|<1} P_k \psi. 
\]
Therefore
\[
\left\|\sum_{j=n+1}^\infty A^j\Psi \right\|^2 = \left \| \sum_{|\lambda_k|<1} \frac{\lambda_k^{n+1}}{1-\lambda_k} P_k \Psi \right \|^2 
= \sum_{|\lambda_k|<1} \frac{\lambda_k^{2(n+1)}}{(1-\lambda_k)^2} \|P_k\Psi\|^2
\leq \frac{r_0^{2}}{(1-r_0)^2}\|A^{n}\Psi\|^2,
\]
where $r_0 = \max_{|\lambda_k|<1}|\lambda_k| = \|A_0\| =\spr(A_0).$ One can verify that $
C = \|(I-A_0)^{-1}\|\|A_0\| = \frac{r_0}{1-r_0}.$
Finally it is easy to find examples of $A$ such that 
the inequalities
\[
\spr(A_0) \ll 1 \ll \spr(A) =\|A\|,
\]
hold, and where the benefit of considering the reduced space $\h_0$ is indeed remarkable.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preconditioning}\label{Sect3}
By ``preconditioning'' we here mean the transformation of the original Lippmann--Schwinger equation $(I-V_q(k_0))u=\psi$ to an integral equation solvable by a convergent Neumann series regardless of the value of $\|V_q(k_0)\|_{L^2(B)\to L^2(B)}$ and of whether or not the sequence $(\|V_q(k_0)^j\psi\|_{L^2})_{j\in\N}$ converges to zero. See~\cite{Engl-Nashed-1981,Kleinman1988,Kleinman1988-2, Kleinman1990,Osnabrugge2016} for related approaches. Throughout this section we assume the problem dimension $d\in\{1,2,3\}$.
\begin{lemma}\label{lemma:precon}
 If $q(x)\ge0$, $q\not\equiv0$, then there is a complex constant $\gamma$ such that the solution of the equation $(I-V_q(k_0))u=\psi$ in $L^2(B)$ is expressible in terms of the convergent Neumann series
 \[
 u=\sum_{j=0}^{\infty}M^j\gamma\psi,
 \]
 where $M=(1-\gamma)I+\gamma V_q(k_0)$.
\end{lemma}
\begin{proof}
 Let $V_q(k_0)\varphi=\lambda\varphi$ in $B$ with nonzero $\lambda$.
 % and with $\varphi$ not identically zero outside any bounded ball in $\R^d$. 
 Then
\begin{equation}
\begin{cases}
\varphi''+k_0^2(1+q(x)/\lambda)\varphi=0,& x\in{}]-R,R[,\\
-\varphi'(-R)=\irm k_0\varphi(-R),\\
\varphi'(R)=\irm k_0\varphi(R)
\end{cases}
\end{equation}
for $d=1$, and
\begin{equation*}
\begin{cases}
(\Delta+k_0^2(1+q(x)/\lambda))\varphi=0&\text{in}\,\,\,\R^d,\\
\lim_{|x|\to \infty}|x|^{(d-1)/2}\left(\partial_{|x|}-\irm k_0\right)\varphi=0&\text{uniformly in}\,\,\,x/|x|\in S^{d-1},
\end{cases}
\end{equation*}
for $d\in\{2,3\}$. Thus, for sufficiently large $R>0$ we have
\begin{align}\label{eqn:former}
0&=\int_{|x|<R}(\bar \varphi\Delta \varphi+k_0^2|\varphi|^2+k_0^2\lambda^{-1}q(x)|\varphi|^2)\nonumber\\&=\int_{|x|<R}(k_0^2|\varphi|^2-|\nabla \varphi|^2)+\int_{|x|=R}\bar{\varphi}\partial_r\varphi+k_0^2\lambda^{-1}\int_{x\in\supp q}q(x)|\varphi|^2,
\end{align}
as well as $(\Delta+k_0^2)\varphi=0$ in $\{|x|>R\}$. In the case $d=1$ we readily find that
\[
\Im\int_{|x|=R}\bar{\varphi}\partial_r\varphi=k_0(|\varphi(-R)|^2+|\varphi(R)|^2)>0,
\]
while in the case $d\in\{2,3\}$ we can follow the argument in, e.g.,~\cite[Theorem 2.13, p. 38]{CK4} to find
\[
\Im\int_{|x|=R}\bar{\varphi}\partial_r\varphi>0.
\]
Hence 
\[
\int_{x\in\supp q}q(x)|\varphi|^2\, \dd x >0, 
\]
and this in conjunction with~\eqref{eqn:former} gives
\[
\Im(\lambda^{-1})=-\frac{\Im\int_{|x|=R}\bar{\varphi}\partial_r\varphi}{k_0^2\int_{x\in\supp q}q(x)|\varphi|^2}<0,
\]
so $\Im\lambda>0$ and finally $\Re(\expe^{\irm \alpha}(1-\lambda))>0$ if $\alpha\in\mathopen]-\pi/2,\pi/2[$ satisfies
\begin{equation}\label{eqn:condition}
\tan\alpha>\max_{\lambda'\in\sigma(V_q(k_0))}\frac{\Re\lambda'-1}{\Im\lambda'},
\end{equation}
where $\sigma(V_q(k_0))$ is the spectrum of $V_q(k_0)$. The existence of the maximum in~\eqref{eqn:condition} follows from the fact that the eigenvalues of the compact operator $V_q(k_0):L^2(B)\to L^2(B)$ reside in the closed ball $\{\lambda'\in\boldsymbol{C},\,\,\,|\lambda'|\le\|V_q(k_0)\|_{L^2(B)\to L^2(B)}\}$ and can accumulate only at zero. These facts also imply that
there exists $\varepsilon>0$ such that $|\gamma(1-\lambda')-1|<1$ for all $\lambda'\in\sigma(V_q(k_0))$, where $\gamma=\varepsilon \expe^{\irm \alpha}$. It remains to notice that 
each eigenvalue $\mu$ of $M$ is of the form $\mu=1-\gamma+\gamma\lambda'$, where $\lambda'$ is some eigenvalue of $V_q(k_0)$, and finally that the equation $(I-M)u=\gamma\psi$ is equivalent with the equation $(I-V_q(k_0))u=\psi$.
\end{proof}

We can in fact be more specific in a special case in dimension one. Let $L$ be a positive constant, set $B=\mathopen]0,L[$, and let $q(x)\equiv q_0={\rm const.}>0$ for $x\in\bar{B}$.
\begin{lemma}\label{lemma:precon2}
If $\varepsilon'>k_0Lq_0/2$,
\[
\alpha=\arctan\frac{1+\frac{k_0Lq_0}{2}\varepsilon'}{\varepsilon'-\frac{k_0Lq_0}{2}},
\qquad
0<\varepsilon<\frac{1}{{1+k_0 Lq_0/2}}\frac{\lvert\tan(2\max\{\alpha,\arctan\varepsilon'\})|}{\tan(\max\{\alpha,\arctan\varepsilon'\})},
\]
and $\gamma=\varepsilon \expe^{\irm \alpha}$,
then each eigenvalue $\mu$ of $M=(1-\gamma)I+\gamma V_q(k_0)$ satisfies $|\mu|<1$.
\end{lemma}
\begin{proof}
Assume $\lambda\in\boldsymbol{C}\setminus\{0\}$ and $\varphi\in L^2(]0,L[)$, $\varphi\not\equiv0$, satisfy 
\begin{equation}\label{eqn:sys}
V_q(k_0)\varphi(x)=\lambda\varphi(x),\quad x\in(0,L).
\end{equation}
Integration by parts readily shows the equivalence of the Lippmann--Schwinger equation~\eqref{eqn:sys} with the Helmholtz system
\begin{equation}\label{eqn:hs}
\begin{cases}
\varphi''(x)+k_0^2s^2\varphi(x)=0,&x\in{}]0,L[,\\
-\varphi'(0)=\irm k_0\varphi(0),\\
\varphi'(L)=\irm k_0\varphi(L),
\end{cases}
\end{equation}
where $s=\left(1+q_0/\lambda\right)^{1/2}$.
The eigenvectors of the Laplacian on $]0, L[$ are generally of the form
\begin{equation}\label{eqn:form}
\varphi(x)=A\exp(\irm k_0sx)+B\exp(-\irm k_0sx),
\end{equation}
and we then readily find that~\eqref{eqn:hs} is equivalent with~\eqref{eqn:form} together with \begin{equation}\label{eqn:equiv}
s\neq1,\quad B=A(s+1)/(s-1),\quad \expe^{2\irm k_0 Ls}=(s+1)^2/(s-1)^2.
\end{equation}
Next define the constant $T(k_0 L)>0$ by $T(k_0 L)\sinh(k_0 LT(k_0 L))=1$. Using the last condition in~\eqref{eqn:equiv}, % the fact that
we find that, necessarily,
\[
s\in S_{\pm}(k_0 L)=\left\{\frac{-\cosh k_0 Lt\pm\sqrt{1-t^2\sinh^2k_0 Lt}}{\sinh k_0 Lt}+it,\,\,t\in(0,T(k_0 L)]\right\},
\]
which in turn implies
\[
\lambda\in\Lambda(q_0,k_0 L)=\left\{\frac{q_0}{s^2-1},\,\,s\in S_-(k_0 L)\cup S_+(k_0 L)\right\}.
\]
As an example, Figure~\ref{fig:example_spectrum} shows the set $\Lambda(q_0,k_0L)$, as well as the numerically computed spectrum, for two sets of parameter values for $k_0$, $L$, and $q_0$.

\begin{figure}[htb]
 \centering
 \includegraphics[scale=0.3]{Figures/spectrum_1}
 \includegraphics[scale=0.3]{Figures/spectrum_2}
 \caption{The theoretically predicted set $\Lambda(q_0,k_0L)$ that includes the eigenvalues of $V_q(k_0)$, plotted against the numerically computed eigenvalues. The parameter values are: top, $k_0=1$, $L=1$, and $q_0=5$; bottom, $k_0=50$, $L=1$, and $q_0=1$. The reference circle is centered at the origin and has radius one.}
 \label{fig:example_spectrum}
\end{figure}

Since $\Im\lambda>0$ and since the eigenvalues of $V_q(k_0)$ accumulate precisely at zero, we have
\[
\lim_{t\to 0}\frac{\Re\omega(t)-1}{\Im\omega(t)}=-\infty
\]
for
\[
\omega(t)=\frac{q_0}{s(t)^2-1},\quad t\in(0,T(k_0 L)];
\]
here $s(t)$ is given by the above definition of $S_{\pm}(k_0L)$. Furthermore, if $\Re\omega(t)<1$ then $(\Re\omega(t)-1)/\Im\omega(t)<0$, while $\Re\omega(t)\ge1$ implies
\[
\frac{1-t^2\sinh^2k_0 Lt+\cosh(k_0 Lt)\sqrt{1-t^2\sinh^2k_0 Lt}}{\sinh^2k_0 Lt}\le\frac{q_0}{2};
\]
the latter can be seen by rewriting $\Re\omega-1\ge0$ as
\[
((\Re s)^2-(\Im s)^2-1)((\Re s)^2-(\Im s)^2-1-q_0)+4(\Re s)^2(\Im s)^2\le0,
\]
and noting that $(\Re s)^2-(\Im s)^2-1>0$ and $(\Re s)^2(\Im s)^2>0$. Now since also
\begin{multline*}
\frac{1-t^2\sinh(k_0Lt)^2-\cosh(k_0Lt)\sqrt{1-t^2\sinh(k_0Lt)^2}}{t\sinh(k_0Lt)(\cosh(k_0Lt)-\sqrt{1-t^2\sinh(k_0Lt)^2})}
\\
\le\frac{1-t^2\sinh(k_0Lt)^2+\cosh(k_0Lt)\sqrt{1-t^2\sinh(k_0Lt)^2}}{t\sinh(k_0Lt)(\cosh(k_0Lt)+\sqrt{1-t^2\sinh(k_0Lt)^2})},
\end{multline*}
we have
\[
\frac{\Re\omega(t)-1}{\Im\omega(t)}<\frac{\Re\omega(t)}{\Im\omega(t)}<\frac{1-t^2\sinh^2k_0 Lt+\cosh(k_0 Lt)\sqrt{1-t^2\sinh^2k_0 Lt}}{t\sinh(k_0 Lt)(\cosh k_0Lt+\sqrt{1-t^2\sinh^2k_0 Lt})},
\]
so $\Re\omega(t)\ge1$ implies
\begin{align*}
\frac{\Re\omega(t)-1}{\Im\omega(t)}&<\frac{q_0}{2}\frac{\sinh k_0 Lt}{t(\cosh k_0 Lt+\sqrt{1-t^2\sinh^2k_0 Lt})}\le\frac{q_0}{2}\frac{\tanh k_0 Lt}{t}\\&\le\frac{q_0}{2}\lim_{\tau\searrow0}\frac{\tanh k_0 L\tau}{\tau}=\frac{q_0k_0 L}{2},
\end{align*}
that is, we get a similar estimate on $(\Re\omega(t)-1)/\Im\omega(t)$ as we do on $\|V_q(k_0)\|$. Next, define
\begin{equation}\label{eqn:ang1}
\xi_+=\arctan\sup_{\omega\in\Lambda(q_0,k_0 L)}\frac{\Im\left(\expe^{\irm \alpha}(1-\omega)\right)}{\Re\left(\expe^{\irm \alpha}(1-\omega)\right)}
\end{equation}
and
\begin{equation}\label{eqn:ang2}
\xi_-=\arctan\inf_{\omega\in\Lambda(q_0,k_0 L)}\frac{\Im\left(\expe^{\irm \alpha}(1-\omega)\right)}{\Re\left(\expe^{\irm \alpha}(1-\omega)\right)}.
\end{equation}
With $\alpha\in(0,\pi/2)$, $\varepsilon>0$, and $\gamma=\varepsilon \expe^{\irm {\alpha}}$, we have $\arg(\gamma(1-\omega))\in[\xi_-,\xi_+]$ for all $\omega\in\Lambda(q_0,k_0 L)$, and if
\begin{equation}\label{eqn:epsilon}
\varepsilon<\frac{1}{{1+k_0 Lq_0/2}}\frac{\lvert\tan(2\max\{|\xi_+|,|\xi_-|\})|}{\tan(\max\{|\xi_+|,|\xi_-|\})}
\end{equation}
then $|\gamma(1-\omega)-1|<1$ for all $\omega\in\Lambda(q_0,k_0 L)$, and specifically $|\gamma(1-\lambda')-1|<1$ for all eigenvalues $\lambda'$ of $V_q(k_0)$; the condition~\eqref{eqn:epsilon} can be deduced by requiring $\Im(\gamma(1-\omega))<|\tan(\pi-2\arg(\gamma(1-\omega)))|$ and using $|1-\omega|\le1+\|V_q(k_0)\|$. Note that we must choose $\alpha<\pi/2$ rather than $\alpha=\pi/2$ since
\[
\frac{\Im(\expe^{\irm \alpha}(1-\omega(t)))}{\Re(\expe^{\irm \alpha}(1-\omega(t)))}\underset{t\to 0}{\to}\tan\alpha,
\]
which for $\alpha=\pi/2$ forces $\xi_+=\pi/2$ and thus $\varepsilon<0$. Note furthermore that we can bound~\eqref{eqn:ang1}{\rm --}\eqref{eqn:ang2} analytically for
\[
\alpha=\arctan\frac{1+\frac{k_0Lq_0}{2}\varepsilon'}{\varepsilon'-\frac{k_0Lq_0}{2}},
\]
with $\varepsilon'>k_0Lq_0/2$, since
\[
-\cot\alpha\le\frac{\Im\left(\expe^{\irm \alpha}(1-\omega)\right)}{\Re\left(\expe^{\irm \alpha}(1-\omega)\right)}\le\tan\alpha,\quad\omega\in\Lambda(q_0,k_0 L),\quad\Re\omega\le1,
\]
since furthermore (recall that $(\Re\omega-1)/\Im\omega<k_0Lq_0/2<\tan\alpha$)
\[
%-\frac{1}{\varepsilon'\cos^2\alpha}
-\frac{\frac{k_0Lq_0}{2}\tan\alpha+1}{\tan\alpha-\frac{k_0Lq_0}{2}}\le\frac{\Im\left(\expe^{\irm \alpha}(1-\omega)\right)}{\Re\left(\expe^{\irm \alpha}(1-\omega)\right)}<0,\quad\omega\in\Lambda(q_0,k_0 L),\quad\Re\omega>1,
\] 
and finally since
\[
-\cot\alpha>-\frac{\frac{k_0Lq_0}{2}\tan\alpha+1}{\tan\alpha-\frac{k_0Lq_0}{2}}=-\varepsilon'.\qedhere
\]
\end{proof}

\begin{remark}\label{rem:drawback}
A drawback of preconditioning is that, with increasing $k_0Lq_0$, the acceptable values of $\alpha$ and of $\arctan\varepsilon'$ tend to $\pi/2$, and $\varepsilon$ therefore tends to zero. Thus, while the Neumann series remains convergent, the equation
\[
(I-M)u=\gamma\psi
\]
may be said, especially in a numerical context, to lose information about the operator $V_q(k_0)$ and about the original inhomogeneity $\psi$, as both are multiplied with $\gamma=\varepsilon \expe^{\irm\alpha}$ there.
\end{remark}
\begin{remark}\label{rem:better}
Instead of using the bound on $\varepsilon$ stated in Lemma~\ref{lemma:precon2}, we can estimate $\xi_+$ and $\xi_-$ from~\eqref{eqn:ang1}{\rm --}\eqref{eqn:ang2} numerically and arrive at a larger sufficiently small $\varepsilon$ using~\eqref{eqn:epsilon}.
\end{remark}
We show numerical examples of the use of preconditioning in Section~\ref{sec:preconnum}.


\section{Numerical examples} \label{sec:preconnum}\label{Sect4}
We here present several numerical examples in dimension one. Fix a positive wavenumber $k_0$, obstacle size $L>0$, medium function $q\in L^2(]0,L[)$, $q(x)>-1$,
and consider the following system for the scattered wave $u(x)$ corresponding to the left excitation $\exp(\irm k_0x)$ in dimension one:
\begin{equation}\label{eqn:psi1}
\begin{cases}
\psi^{\prime \prime}(x)+k_0^2(1+q(x))\psi(x) = -k_0^2q(x)\exp(\irm k_0x),& x\in{}]0,L[,\\
-\psi^\prime(0)=\irm k_0\psi(0),
\\
\psi^\prime(L)=\irm k_0\psi(L).
\end{cases}
\end{equation}
The function $G(x,y)=(i/2k_0)\exp(\irm k_0|x-y|)$, $x,y\in[0,L]$, is the free-space Green's function associated with the boundary problem~\eqref{eqn:psi1}, since
\begin{equation*}
\begin{cases}
(\partial_y^2+k_0^2)G(x,y)=-\delta(x-y),& x,y\in{}]0,L[,
\\
-\partial_yG(x,0)=\irm k_0G(x,0),&x\in{}]0,L[,
\\
\partial_yG(x,L)=\irm k_0G(x,L),& x\in[0,L].
\end{cases}
\end{equation*}
Multiplying the differential equation in~\eqref{eqn:psi1} with $G(x,y)$ 
and integrating by parts, we get the Lippmann--Schwinger equation
\begin{equation}
(I-V_q(k_0))\psi(x)=V_q(k_0)\exp(\irm k_0(\,\cdot\,))(x),\quad x\in{}]0,L[\,,
\end{equation}
where
\[
V_q(k_0)u(x)=\frac{\irm k_0}{2}\int_{y=0}^L\expe^{\irm k_0|x-y|}q(y)u(y)\, \dd y,\quad x\in{}]0,L[\,.
\]
The operator $V_q(k_0):L^2(]0,L[)\to L^2(]0,L[)$ is compact, with norm satisfying
\[
\|V_q(k_0)\|^2\le\int_{x=0}^L\int_{y=0}^L\left|\frac{\irm k_0}{2}\expe^{\irm k_0|x-y|}q(y)\right|^2\, \dd y\, \dd x=\frac{k_0^2L\|q\|^2_2}{4}.
\]

Now let $q_0 \in \mathopen]-1, +\infty[$, and assume that $q|_{]0, L[}\equiv q_0$. One can easily prove 
\[
\mathfrak H_0= \Span\left(V_{q}(k_0)^j \expe^{\irm k_0(\,\cdot\,)}; 
\; j\in \mathbb N\right) = \Span \left(V_{1}(k_0)^j\expe^{\irm k_0(\,\cdot\,)}; \; j\in \mathbb N\right).
\]
On the other hand, since $V_q(k_0) = q_0 V_1(k_0)$, we have
\[\spr\left((V_{q})_0(k_0)\right) = 
|q_0|\spr\left((V_1)_0(k_0)\right),\]
where $ (V_{q})_0(k_0)$ and $(V_{1})_0(k_0)$ are the restrictions of $ V_{q}(k_0)$ and $V_{1}(k_0)$, respectively, to the 
space $\mathfrak H_0$. Following the proof of Proposition~\ref{main result}, the Born series 
\[
\sum_{j=1}^\infty \left(V_q(k_0)\right)^j \expe^{\irm k_0(\,\cdot\,)} 
\]
is convergent if and only if $\spr\left((V_{q})_0(k_0)\right)<1$, that is, if and only if
\[
|q_0|<\frac{1}{\spr\left((V_1)_0(k_0)\right)}.
\]
To illustrate this, we show in Figure~\ref{fig:konverg} two cases of repeated application of $V_q(k_0)$ on the original right-hand side $V_q(k_0)\expe^{\irm k_0(\,\cdot\,)}$.

\begin{figure}[htb]
 \centering
 \includegraphics[scale=0.58]{Figures/koverg_1_new.png}
 
 \vspace*{2mm}
 \includegraphics[scale=0.58]{Figures/koverg_2_new.png}
 \caption{Top: $k_0=1$, $L=1$, $q_0=1$. Bottom: $k_0=50$, $L=1$, $q_0=1$.}
 \label{fig:konverg}
\end{figure}

In the case where Spr$((V_q)_0(k_0))<1$, the series $(V_q(k_0)^J\expe^{\irm k_0(\,\cdot\,)})$ converges strongly to zero, while $(I-V_q(k_0))\sum_{j=1}^JV_q(k_0)^j\expe^{\irm k_0(\,\cdot\,)}$ converges strongly to $V_q(k_0)\expe^{\irm k_0(\,\cdot\,)}$, The second of the two convergence processes plateaus for large values of $J$ due to the effect of numerical errors. In contrast, neither sequence converges in the case where Spr$((V_q)_0(k_0))\ge1$.


\begin{figure}[p]
 \centering
 \includegraphics[scale=0.27]{Figures/spectrum_1.png}
 \includegraphics[scale=0.27]{Figures/spectrum_1_MU.png}

 \includegraphics[scale=0.27]{Figures/spectrum_1_PROV.png}

 
 a) $k_0$=1, $L=1$, $q_0=5$, $\spr(V_q(k_0))=2.42$, $\varepsilon'=5.2$, $\alpha=1.3803$, $\varepsilon=0.02$, $T=0.9320$.
 
 \includegraphics[scale=0.27]{Figures/spectrum_2.png}
 \includegraphics[scale=0.27]{Figures/spectrum_2_MU.png}

 \includegraphics[scale=0.27]{Figures/spectrum_2_PROV.png}

 b) $k_0=50$, $L=1$, $q_0=1$, $\spr(V_q(k_0))=7.21$, $\varepsilon'=50$, $\alpha=1.5508$, $\varepsilon=3\cdot10^{-5}$, $T=0.0677$.
 
 \caption{The original theoretically predicted and numerically computed spectra of $V_q(k_0)$; the transformed curve $\{z=1-\gamma(1-\omega),\,\,\,\omega\in\Lambda(q_0,k_0 L)\}\subset\boldsymbol{C}$ is included in the unit open disk centered at 1.}
 \label{fig:orig_transf_1}
\end{figure}

Finally, we illustrate Lemma~\ref{lemma:precon2} numerically in Figures~\ref{fig:orig_transf_1} and~\ref{fig:precon_works}, picking parameter values that violate the strong (norm) condition~\eqref{norm condition}: the computed values of $\spr(V_q(k_0))$ %$\|V_q(k_0)\|$ 
in the examples in Figures~\ref{fig:orig_transf_1} and~\ref{fig:precon_works} are 2.42 and 7.21. In spite of this, we indeed do produce the operator $M=(1-\gamma)I+\gamma V_q(k_0)$ with $\|M\|<1$, and we specifically show in Figure~\ref{fig:precon_works} that we get a convergent Neumann series solution for the case $k_0=50$, $L=1$, $q_0=1$, %($\|V_q(k_0)\|=7.1791$)
$\spr(V_q(k_0))=7.21$, albeit the convergence is rather slow.

\begin{figure}[htb]
 \centering
 \includegraphics[scale=0.4]{Figures/precon_works}
 \caption{Preconditioning of the equation $(I-V_q(k_0))u=V_q(k_0)\expe^{\irm k_0(\,\cdot\,)}$ results in a convergent Neumann series solution. The parameters here are as in Figure~\ref{fig:konverg} (bottom) and Figure~\ref{fig:orig_transf_1} b), and specifically $\spr(V_q(k_0))=7.21$.}
 \label{fig:precon_works}
\end{figure}
% \subsection{The Born series is not always convergent} 

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no affiliations other than their research organizations.



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\end{document}

