remarkRemark exampleExample hypothesisHypothesis claimClaim Stability for a
multi-frequency inverse random source problemT. Wang, X. Xu and Y. Zhao

# Stability for a multi-frequency inverse random source problem

Tianjiao Wang School of Mathematical Sciences, Zhejiang University, Hangzhou,
Zhejiang, China, 310058<EMAIL_ADDRESS>Xiang Xu School of
Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China,310058.
XX’s work was supported in part by National Key Research and Development
Program of China (2023YFA1009100) and National Natural Science Foundation of
China (12071430), Key Laboratory of Collaborative Sensing and Autonomous
Unmanned Systems of Zhejiang Province<EMAIL_ADDRESS>Yue Zhao  School of
Mathematics and Statistics, and Key Lab NAA–MOE, Central China Normal
University, Wuhan 430079, China<EMAIL_ADDRESS>
###### Abstract

We present stability estimates for the inverse source problem of the
stochastic Helmholtz equation in two and three dimensions by either near-field
or far-field data. The random source is assumed to be a microlocally isotropic
generalized Gaussian random function. For the direct problem, by exploring the
regularity of the Green function, we demonstrate that the direct problem
admits a unique bounded solution with an explicit integral representation,
which enhances the existing regularity result. For the case using near-field
data, the analysis of the inverse problem employs microlocal analysis to
achieve an estimate for the Fourier transform of the micro-correlation
strength by the near-field correlation data and a high-frequency tail. The
stability follows by showing the analyticity of the data and applying a novel
analytic continuation principle. The stability estimate by far-field data is
derived by investigating the correlation of the far-field data. The stability
estimate consists of the Lipschitz type data discrepancy and the logarithmic
stability. The latter decreases as the upper bound of the frequency increases,
which exhibits the phenomenon of increasing stability.

###### keywords:

increasing stability, inverse source problem, generalized Gaussian random
function.

35Q74, 35R30, 78A46

##  1 Introduction

Inverse source scattering problem is concerned with recovering the unknown
source from near-field or far-field data away from its support. Such problem
has generated tremendous interest due to its wide applications in scientific
and engineering fields such as seismology, telecommunications, medical
imaging, antenna synthesis, radar technology, and magnetoencephalography [1,
3, 8, 15]. However, the non-uniqueness of the inverse source problem at a
single frequency, caused by the existence of non-radiating sources, poses a
challenge [9, 12]. Consequently, additional information is required for a
unique determination of the source. To resolve this issue, the use of multi-
frequency data has been realized to be an effective approach to regain the
uniqueness and achieve enhanced stability [7, 8, 11, 24].

In many applications, the source is often considered as a random field due to
uncertainties in the surrounding environment or random measurement noise [2].
The presence of randomness introduces additional challenges compared to
deterministic source scattering. Specifically, the regularities of wave fields
tend to be lower, and the measurements become statistical data. Inverse source
problems driven by Wiener process have been extensively investigated [4, 5, 6,
19, 23]. In recent studies, uniqueness of inverse source problems have been
studied in [17, 18, 21] by assuming the source to be a generalized Gaussian
random field. The covariance of such random field is a classical pseudo-
differential operator with a principal symbol taking the form
$h(x)|\xi|^{-m}$, where $h$ is called micro-correlation strength. This model
encompasses various important stochastic processes, including white noise,
fractional Brownian motion, and Markov fields [16]. Compared with the many
uniqueness results of the inverse random source problems, the stability has
been much less studied. To the best of our knowledge, the only existing
stability results were obtained in [20, 22] driven by Wiener process. The
corresponding stability for the generalized Gaussian random field remains
unsolved.

In numerical experiments in [4, 19], it has been observed that the ill-
posedness of the inverse random source problem can be overcome by using multi-
frequency data which yields increasing stability, i.e., as the frequency
increases the inverse problem becomes more stable. The goal of this work is to
mathematical verify the increasing stability with a generalized Gaussian
random field. Specifically, consider the stochastic Helmholtz equation

(1.1) $\Delta u+k^{2}u=f,\quad\mathrm{in}\quad\mathbb{R}^{d},\,d=2,3,$

where $k>0$ is the wavenumber. The source function $f$ is a microlocally
isotropic general Gaussian random field of order $m$ in a bounded Lipschitz
domain $D\subset\mathbb{R}^{d}$ (see Section 2 for a detailed definition). The
wave field $u$ is required to satisfy the Sommerfeld radiation condition

(1.2) $\lim_{r\to\infty}r^{\frac{d-1}{2}}(\partial_{r}u-iku)=0,\quad r=|x|.$

Let $B_{R}=\\{x:x\in\mathbb{R}^{d},\,|x|\leq R\\}$ with boundary $\partial
B_{R}$ and assume that $D\subset\subset B_{R}$. The inverse problem is to
determine the principle symbol of $f$ from either the near-field data on
$\partial B_{R}$ or far-field data.

Compared with the deterministic case, new challenges arise due to the
roughness and randomness of the source. The well-posedness of the direct
problem (1.1)–(1.2) has been discussed in [18]. However, it is not clear if
the solution can be represented by the convolution of the Green function and
the source as in the classical setting. By studying the regularity of the
Green function, we establish an explicit integral representation of the
solution which holds pointwisely. As a consequence, we obtain an enhanced
regularity result. The analysis of the inverse problem employs microlocal
analysis to achieve an estimate for the Fourier transform of the micro-
correlation strength by the near-field correlation data and a high-frequency
tail. Next, we show that the correlation data is analytic and derive an upper
bound with respect to complex wave number. The stability estimate follows by
an application of a novel analytic continuation developed in [25]. For the
case of far-field data, the stability estimate can be derived by investigating
the correlation of the far-field data. The stability has a unified form which
consists of the Lipschitz data discrepancy and a logarithmic stability, where
the latter decreases as the frequency increases.

The rest of this paper is organized as follows. Section 2 is devoted to the
definition and some properties of the microlocally isotropic Gaussian random
function. In Section 3, we derive a regularity result for the Green function
of the Helmholtz equation which leads to an explicit integral representation
of the direct problem. The main increasing stability results are presented in
Section 4 and Section 5 for near-field data and far-field data, respectively.
A conclusion is given in Section 6. Throughout this paper, $a\lesssim b$
stands for $a\leq Cb$, where $C>0$ is a generic constant whose special value
is not required but should be clear from the context.

##  2 Preliminaries

In this section, we state the properties of microlocally isotropic generalized
Gaussian random functions. Let $(\Omega,\mathcal{A},\mathbb{P})$ be a complete
probability space. Denote the test function space by $\mathcal{D}$, which
consists of smooth functions with compact supports in $\mathbb{R}^{d}(d=2,3)$.
Then the dual space of $\mathcal{D}$ is denoted as $\mathcal{D}^{\prime}$. A
scalar field $f$ is said to be a real-valued generalized Gaussian random
function if $f\,:\,\Omega\to\mathcal{D}^{\prime}$ is a distribution such that
for each $\omega\in\Omega$ the path $f[\cdot](\omega)\in\mathcal{D}^{\prime}$
is a linear functional on $\mathcal{D}$, and $\omega\mapsto\langle
f(\omega),\phi\rangle$ is a real-valued Gaussian random variable for all
$\phi\in\mathcal{D}$. The expectation of $f$ is a generalized function defined
by

$\mathbb{E}f\,:\,\phi\mapsto\mathbb{E}\langle
f,\phi\rangle,\quad\phi\in\mathcal{D},$

and the covariance is a bilinear form given by

$\mathrm{Cov}f\,:\,(\phi_{1},\phi_{2})\mapsto\mathrm{Cov}(\langle
f,\phi_{1}\rangle,\langle
f,\phi_{2}\rangle),\quad\phi_{1},\phi_{2}\in\mathcal{D}.$

Then define the covariance operator
$C_{f}\,:\,\mathcal{D}\to\mathcal{D}^{\prime}$ by

$\langle C_{f}\phi_{1},\phi_{2}\rangle=\mathrm{Cov}(\langle
f,\phi_{1}\rangle,\langle
f,\phi_{2}\rangle),\quad\phi_{1},\phi_{2}\in\mathcal{D},$

which is associated with a Schwartz kernel denoted by $K_{f}(x,y)$ as follows

$\langle
C_{f}\phi_{1},\phi_{2}\rangle=\int_{\mathbb{R^{d}}}\int_{\mathbb{R^{d}}}K_{f}(x,y)\phi_{1}(x)\phi_{2}(y)\,\mathrm{d}x\mathrm{d}y,\mbox{
and }K_{f}(x,y)=\mathbb{E}[(f(x)-\mathbb{E}f(x))(f(y)-\mathbb{E}f(y))].$

In this paper, the random source is assumed to be characterized by a special
class of generalized Gaussian random functions as follows.

######  Definition 2.1.

A generalized Gaussian random function $f$ with zero expectation is called
microlocally isotropic of order $m\in\mathbb{R}$ in the domain
$D\subset\mathbb{R}^{d}$, if $\text{supp}\,f\subset\subset D$ for almost
surely $\omega\in\Omega$ and its covariance operator $C_{f}$ is a classical
pseudo-differential operator with the principal symbol
$h(x)\big{|}\xi\big{|}^{-m}$ where $0\leq h\in C^{\infty}_{0}(\mathbb{R}^{d})$
and $\text{supp}\,h\subset\subset D$.

The smooth function $h$ is called the micro-correlation strength of the random
function $f$. Let $c(x,\xi)$ be the symbol of $C_{f}$. The operator can be
represented by

$C_{f}(\phi)(x)=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{ix\cdot\xi}c(x,\xi)\hat{\phi}(\xi)\,\mathrm{d}\xi,\quad\phi\in\mathcal{D},$

where $\hat{\phi}(\xi)$ stands for the Fourier transform of $\phi$ defined by
$\hat{\phi}(\xi)=\mathcal{F}\phi(\xi):=\int_{\mathbb{R}^{d}}e^{-ix\cdot\xi}\phi(x)\,\mathrm{d}x.$
The kernel $K_{f}$ can be represented as an oscillatory integral of the form

(2.1)
$K_{f}(x,y)=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{i(x-y)\cdot\xi}c(x,\xi)\,\mathrm{d}\xi.$

We introduce the following regularity result in [21] for microlocally
isotropic Gaussian random functions.

######  Lemma 2.1.

Let $f$ be a microlocally isotropic Gaussian random function of order $m$.
Then $f\in W^{\frac{m-d}{2}-\epsilon,p}(D)$ almost surely for all $\epsilon>0$
and $1<p<\infty$.

By Sobolev embedding theorem, Lemma 2.1 gives the following corollary.

######  Corollary 2.1.

Let $f$ be a microlocally isotropic Gaussian random function of order $m$.
Suppose $D$ is a Lipschitz domain such that $\text{ supp}f\subset D$. For any
$a\in\mathbb{N}$ satisfying $d+2a<m\leq d+2(a+1)$, we have $f\in
C^{a,\gamma}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\bar{D})}$
almost surely with $\gamma\in(0,\frac{m-d}{2}-a)$.

##  3 Direct problem

In this section, we investigate the direct scattering problem. The Green
function of the Helmholtz equation $\Phi(x,y)$ has the explicit form

(3.3)
$\displaystyle\Phi(x,y)=\left\\{\begin{array}[]{cc}\frac{i}{4}H^{(1)}_{0}(k|x-y|),&d=2,\\\
\frac{e^{ik|x-y|}}{4\pi|x-y|},&d=3,\end{array}\right.$

where $H^{(1)}_{0}$ is the Hankel function of the first kind with order zero.
According to Corollary 2.1, for $m>d$, the source $f\in
C^{0,\gamma}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(\bar{D})}\subset
L^{2}(D)$ with some $\gamma>0$ almost surely. Hence, the scattering problem
(1.1)-(1.2) is classical which admits a unique solution $u\in
H^{2}_{loc}(\mathbb{R}^{d})$ with the following explicit integral form

(3.4) $\displaystyle u=-\int_{\mathbb{R}^{d}}\Phi(x,y)f(y)\,\mathrm{d}x.$

However, for $m\leq d$, Lemma 2.1 shows that such source is too rough to exist
pointwisely which should be taken as distribution. In this case, the following
theorem proved in [18] gives the well-posedness and regularity of the direct
problem.

######  Theorem 3.1.

Let $f$ be a microlocally isotropic Gaussian random function of order
$m\in(d-4,d]$. The problem (1.1)-(1.2) is well-posed and the unique solution
can be represented by (3.3) in sense of distributions, which satisfies $u\in
W^{\alpha,q}_{loc}(\mathbb{R}^{3})$ almost surely for any $q>1$ and

$0<\alpha<\min\left\\{2-\frac{d-m}{2},2-\frac{d-m}{2}+d\left(\frac{1}{q}-\frac{1}{2}\right)\right\\}.$

For $m>d$, by classical acoustic wave scattering theory we know that the
solution of the direct problem admits a pointwise explicit integral
representation (3.4). However, for $m\leq d$, we only know that (3.4) holds in
sense of distributions. In the rest of this section, we show that (3.4) holds
pointwisely, which implies $u\in L^{\infty}(\mathbb{R}^{d})$.

The following lemma is useful in the subsequent analysis.

######  Lemma 3.1.

Supposing that $p_{1},p_{2}\in(1,d)$ and $p_{1}+p_{2}>d$, we have that the
inequalities

$\displaystyle\int_{D}\frac{1}{|y-x|^{p_{1}}|y-z|^{p_{2}}}\,\mathrm{d}y\lesssim\frac{1}{|x-z|^{p_{1}+p_{2}-d}}$

and

$\displaystyle\int_{D}\frac{|\log|z-y||}{|y-x|^{p_{1}}|y-z|^{p_{2}}}\,\mathrm{d}y\lesssim\frac{1+|\log|x-z||}{|x-z|^{p_{1}+p_{2}-d}}$

hold for $x,z\in\mathbb{R}^{d}$.

###### Proof.

Without loss of gengerality, assume $z=0$. Letting $y^{\prime}=\frac{y}{|x|}$
we obtain

$\displaystyle\int_{D}\frac{1}{|y-x|^{p_{1}}|y|^{p_{2}}}\,\mathrm{d}y\leq\int_{\mathbb{R}^{d}}\frac{|x|^{d-p_{1}-p_{2}}}{|y^{\prime}-\hat{x}|^{p_{1}}|y^{\prime}|^{p_{2}}}\,\mathrm{d}y^{\prime}\lesssim\frac{1}{|x|^{p_{1}+p_{2}-d}},$

which proves the first inequality. The second inequality can be proven
similarly.

######  Lemma 3.2.

For any $x\in\mathbb{R}^{d}$ we have $\Phi(x,\cdot)\in
W^{1+\mu,p}_{loc}(\mathbb{R}^{d})$ with $\mu\in(0,1)$,
$1<p<\frac{d}{\mu+d-1}$.

###### Proof.

Consider the Slobodeckij semi-norm

$|u|^{p}_{W^{\mu,p}(D)}:=\int_{D}\int_{D}\frac{|u(x)-u(y)|^{p}}{|x-y|^{p\mu+d}}\,\mathrm{d}x\,\mathrm{d}y.$

The norm of $W^{1+\mu,p}(D)$ can be expressed by

$\|u\|^{p}_{W^{1+\mu,p}(D)}:=\|u\|^{p}_{W^{1,p}(D)}+\sum_{i=1}^{d}|\partial_{i}u|^{p}_{W^{\mu,p}(D)}.$

Without loss of generality, we assume $x\in D\subset\subset\mathbb{R}^{d}$.
Firstly we consider the case when $d=3$. Notice

$\Phi(x,y)=\frac{e^{ik|x-y|}}{4\pi|x-y|},\quad\partial_{i}\Phi(x,y)=e^{ik|x-y|}\frac{ik(y_{i}-x_{i})|x-y|-(y_{i}-x_{i})}{4\pi|x-y|^{3}},$

which gives $\Phi(x,\cdot)\in W^{1,p}(D)$ with $1<p<\frac{3}{2}$. Therefore,
we only need to show

$\displaystyle|\partial_{j}\Phi(x,y)|^{p}_{W^{\mu,p}(D)}<\infty,\quad
j=1,2...d,$

for $0<\mu<1$ and $1<p<\frac{3}{2+\mu}$. Clearly, we only need to consider the
term which has higher singularity. Hence, it suffices to show

$e^{ik|x-y|}\frac{y_{j}-x_{j}}{|x-y|^{3}}\in W^{\mu,p}(D)$

for $0<\mu<1$ and $1<p<\frac{3}{2+\mu}$. Using the fractional Leibniz rule in
[14] gives

(3.5)
$\displaystyle\left|e^{ik|x-y|}\frac{y_{j}-x_{j}}{|x-y|^{3}}\right|_{W^{\mu,p}(D)}$
$\displaystyle\lesssim\|e^{ik|x-y|}\|_{L^{\infty}(D)}\left|\frac{y_{j}-x_{j}}{|x-y|^{3}}\right|_{W^{\mu,p}(D)}+|e^{ik|x-y|}|_{W^{\mu,\infty(D)}}\left\|\frac{y_{j}-x_{j}}{|x-y|^{3}}\right\|_{L^{p}(D)}.$

It is straightforward to verify $e^{ik|x-y|}\in W^{1,\infty}(D)$, which
implies

(3.6)
$\|e^{ik|x-y|}\|_{L^{\infty}(D)}<\infty,\quad|e^{ik|x-y|}|_{W^{\mu,\infty(D)}}<\infty.$

Combining (3.5)–(3.6), we only need to prove

$\frac{x_{i}-y_{i}}{|x-y|^{3}}\in W^{\mu,p}(D)$

for $0<\mu<1$ and $1<p<\frac{3}{2+\mu}$. It is easy to verify

$\frac{x_{i}-y_{i}}{|x-y|}\in W^{1,p^{\prime}}(D),$

where $1<p^{\prime}<3$. Applying Sobolev embedding theorem yields

(3.7) $\frac{x_{i}-y_{i}}{|x-y|}\in W^{\mu,q}(D),$

where $0<\mu<1$ and $q<\frac{3}{\mu}$. Next we will show that

(3.8) $\frac{1}{|x-y|^{2}}\in W^{\mu,p}(D)$

for $0<\mu<1$ and $1<p<\frac{3}{2+\mu}$. To this end, direct calculations
imply

$\displaystyle\int_{D}\int_{D}\frac{\left||x-y|^{-2}-|x-z|^{-2}\right|^{p}}{|y-z|^{p\mu+3}}\,\mathrm{d}y\mathrm{d}z\lesssim\int_{D}\int_{D}\frac{(|x-z|^{p}+|x-y|^{p})|z-y|^{p}}{|y-z|^{p\mu+3}|x-z|^{2p}|x-y|^{2p}}\,\mathrm{d}y\mathrm{d}z$
(3.9) $\displaystyle=$
$\displaystyle\int_{D}\int_{D}|y-z|^{-(p(\mu-1)+3)}|x-y|^{-p}|x-z|^{-2p}\,\mathrm{d}y\mathrm{d}z+\int_{D}\int_{D}|y-z|^{-(p(\mu-1)+3)}|x-z|^{-p}|x-y|^{-2p}\,\mathrm{d}y\mathrm{d}z.$

Applying Lemma 3.1 gives

(3.10)
$\displaystyle\int_{D}\int_{D}|y-z|^{-(p(\mu-1)+3)}|x-y|^{-p}|x-z|^{-2p}\,\mathrm{d}y\mathrm{d}z\lesssim\int_{D}\frac{1}{|x-z|^{2p+\mu
p}}<\infty,$

when $0<\mu<1$ and $p<\frac{3}{2+\mu}$. Inserting (3.10) into (3.9) gives
(3.8).

Applying fractional Leibniz rule yields

(3.11)
$\displaystyle\left|\frac{x_{i}-y_{i}}{|x-y|^{3}}\right|_{W^{\mu,p}(D)}\lesssim\left\|\frac{x_{i}-y_{i}}{|x-y|}\right\|_{L^{\infty}(D)}\left|\frac{1}{|x-y|^{2}}\right|_{W^{\mu,p}(D)}+\left|\frac{x_{i}-y_{i}}{|x-y|}\right|_{W^{\mu,q}(D)}\left|\frac{1}{|x-y|^{2}}\right|_{L^{q^{\prime}}(D)},$

where $\frac{1}{q^{\prime}}+\frac{1}{q}=\frac{1}{p}$. We have known that

(3.12) $\left|\frac{1}{|x-y|^{2}}\right|_{W^{\mu,p}(D)}<\infty,$

for $0<\mu<1$, $1<p<\frac{3}{2+\mu}$ and

(3.13) $\left|\frac{x_{i}-y_{i}}{|x-y|}\right|_{W^{\mu,q}(D)}<\infty$

for $0<\mu<1$, $1<q<\frac{3}{\mu}$. Obviously, it can be verified

(3.14) $\left|\frac{1}{|x-y|^{2}}\right|_{L^{q^{\prime}}(D)}<\infty$

with $1<q^{\prime}<\frac{3}{2}$. Hence, we can choose $1<q<\frac{3}{\mu}$ and
$1<q^{\prime}<\frac{3}{2}$ which gives

$p=\frac{1}{q^{-1}+q^{\prime-1}}<\frac{3}{2+\mu}.$

Then by combining (3.11)–(3.14) we arrive at

$\displaystyle\left|\frac{x_{i}-y_{i}}{|x-y|^{3}}\right|_{W^{\mu,p}(D)}<\infty,$

which completes the proof when $d=3$.

When $d=2$, the discussion is analogous as $d=3$ by noticing the asymptotic
relations

$H^{(1)}_{0}(t)=\frac{2i}{\pi}\log{\frac{t}{2}}+O(1)$

and

$H^{(1)}_{0}(t)=-\frac{2i}{\pi t}+O(t)$

when $t\to 0$. We omit it for brevity.

Combining Lemma 2.1 and 3.2, we have the following regularity result.

######  Theorem 3.2.

Let $f$ be a microlocally isotropic Gaussian random function of order
$m\in(d-4,d]$. The solution to the direct scattering problem admits the
following representation which holds pointwisely

(3.15) $u(x)=-\int_{\mathbb{R}^{d}}\Phi(x,y)f(y)\,\mathrm{d}x.$

Moreover, $u\in L^{\infty}(\mathbb{R}^{d})$ almost surely.

###### Proof.

For any fixed $x\in\mathbb{R}^{d}$, combining Lemma 2.1 and 3.2 gives that
(3.15) is well-defined, i.e.

(3.16)
$\displaystyle|u(x)|\leq\|\Phi(x,\cdot)\|_{W^{1+\mu,p}(D)}\|f\|_{W^{-(1+\mu),p^{\prime}}(D)}<\infty,$

with

$0<\mu<1,\quad 1<p<\frac{d}{\mu+d-1}\quad\text{\rm
and}\quad\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

Now we show $u\in L^{\infty}(\mathbb{R}^{d})$. Recalling (3.3), there exists a
sufficient large $R_{0}$ such that $D\subset B_{R_{0}}$ and

$|\Phi(x,y)|\leq C,\quad|\nabla_{y}\Phi(x,y)|\leq C\quad\text{\rm
and}\quad|\nabla_{y}^{2}\Phi(x,y)|\leq C,\quad x\in\mathbb{R}^{d}\backslash
B_{R_{0}},\,y\in D,$

where $C$ is a constant independent of $x$ and $y$. Hence for any
$x\in\mathbb{R}^{d}\backslash B_{R_{0}}$, there holds

(3.17)
$\displaystyle\|\Phi(x,\cdot)\|_{W^{1+\mu,p}(D)}\leq\|\Phi(x,\cdot)\|_{W^{2,p}(D)}\leq
C,$

where $C$ is a constant independent of $x$. For any $x\in B_{R_{0}}$, we have

(3.18)
$\displaystyle\|\Phi(x,\cdot)\|_{W^{1+\mu,p}(D)}\leq\|\Phi(x,\cdot)\|_{W^{1+\mu,p}(B_{R_{0}})}\leq\|\Phi(x,\cdot)\|_{W^{1+\mu,p}(B_{2R_{0})}}=\|\Phi(0,\cdot)\|_{W^{1+\mu,p}(B_{2R_{0}})}.$

Combining (3.16)–(3.18) we complete the proof.

It can be verified that using the regularity result in Theorem 4.1 and Sobolev
embedding theorem one can only have $u\in L^{\infty}(\mathbb{R}^{d})$ when
$m>2(d-2)$. Hence, the above theorem enhances the regularity result in Theorem
4.1.

##  4 Inverse problem using near-field data

In this section, we derive the stability estimate by near-field data. Denote
$a(x,\xi)=c(x,\xi)-h(x)|\xi|^{-m}$. Assume that $f$ satisfies the following
assumption.

_Assumption (A). The random source $f$ is a real-valued microlocally isotropic
Gaussian random function of order $m>d-1$. The covariance operator has the
symbol $c(x,\xi)$ with the principal symbol $h(x)|\xi|^{-m}$ satisfying (i)
$|h(x)|\leq M$ for $x\in D$; (ii) $|c(x,\xi)|\leq M(1+|\xi|)^{-m}$ for
$\xi\in\mathbb{R}^{d}$ and $x\in D$; (iii) $|a(x,\xi)|\leq M|\xi|^{-(m+1)}$
for $|\xi|\geq 1$ and $x\in D$_. Here $M>0$ stands for a constant.

We first give a bound with a high frequency tail for the Fourier transform of
$h$ in terms of the source function. Recalling the definitions in Section 2,
the covariance of $\hat{f}$ can be expressed by

$\displaystyle\mathbb{E}(\hat{f}(\xi)\hat{f}(\eta))$
$\displaystyle=\mathbb{E}\left(\int_{B_{R}}\int_{B_{R}}f(x)f(y)e^{-i\xi\cdot
x}e^{-i\eta\cdot
y}\,\mathrm{d}x\mathrm{d}y\right)=\int_{B_{R}}\int_{B_{R}}K_{f}(x,y)e^{-i\xi\cdot
x}e^{-i\eta\cdot y}\,\mathrm{d}x\mathrm{d}y$
$\displaystyle=(2\pi)^{-d}\int_{B_{R}}\int_{B_{R}}\int_{\mathbb{R}^{d}}e^{-i(\xi\cdot
x+\eta\cdot
y)}e^{i(x-y)\cdot\zeta}c(x,\zeta)\,\mathrm{d}\zeta\mathrm{d}x\mathrm{d}y$
$\displaystyle=\int_{B_{R}}\int_{\mathbb{R}^{d}}e^{ix\cdot(\zeta-\xi)}c(x,\zeta)\delta(\zeta+\eta)\,\mathrm{d}\zeta\mathrm{d}x=\int_{B_{R}}e^{-ix\cdot(\eta+\xi)}c(x,-\eta)\,\mathrm{d}x$
$\displaystyle=\int_{B_{R}}e^{-ix\cdot(\eta+\xi)}h(x)|\eta|^{-m}\,\mathrm{d}x+MR^{d}O(|\eta|^{-(m+1)})$
$\displaystyle=|\eta|^{-m}\hat{h}(\xi+\eta)+MR^{d}O(|\eta|^{-(m+1)}).$

Therefore, we obtain

$\hat{h}(\xi+\eta)=|\eta|^{m}\mathbb{E}(\hat{f}(\xi)\hat{f}(\eta))+MR^{d}O\left(\frac{1}{|\eta|}\right).$

Taking $\xi=k\theta_{1}$ and $\eta=k\theta_{2}$ with
$\theta_{1},\,\theta_{2}\in\mathbb{S}^{d-1}$, one arrives at

(4.1)
$\hat{h}(k(\theta_{1}+\theta_{2}))=|k|^{m}\mathbb{E}(\hat{f}(k\theta_{1})\hat{f}(k\theta_{2}))+MR^{d}O\left(\frac{1}{|k|}\right).$

Next, we bound the term $\mathbb{E}(\hat{f}(k\theta_{1})\hat{f}(k\theta_{2}))$
by the correlation data on $\partial B_{R}$. Multiplying the governing
equation (1.1) by the plane wave $e^{-ik\theta_{j}\cdot x}$ and integrating by
parts yield

(4.2) $\displaystyle\int_{B_{R}}fe^{-ik\theta_{j}\cdot
x}\,\mathrm{d}x=\int_{\partial B_{R}}\partial_{\nu}u(x)e^{-ik\theta_{j}\cdot
x}-\partial_{\nu}(e^{-ik\theta_{j}\cdot x})u(x)\,\mathrm{d}x.$

Taking expectation gives that

$\displaystyle\mathbb{E}(\hat{f}(k\theta_{1})\hat{f}(k\theta_{2}))$
$\displaystyle=\int_{\partial B_{R}}\int_{\partial
B_{R}}\mathbb{E}(\partial_{\nu}u(x)\partial_{\nu}u(y))e^{-ik(\theta_{1}\cdot
x+\theta_{2}\cdot y)}\,\mathrm{d}s(x)\mathrm{d}s(y)$
$\displaystyle\quad-\int_{\partial B_{R}}\int_{\partial
B_{R}}\mathbb{E}(\partial_{\nu}u(x)u(y))e^{-ik(\theta_{1}\cdot
x+\theta_{2}\cdot
y)}(-ik\theta_{2}\cdot\hat{y})\,\mathrm{d}s(x)\mathrm{d}s(y)$
$\displaystyle\quad-\int_{\partial B_{R}}\int_{\partial
B_{R}}\mathbb{E}(\partial_{\nu}u(y)u(x))e^{-ik(\theta_{1}\cdot
x+\theta_{2}\cdot
y)}(-ik\theta_{1}\cdot\hat{x})\,\mathrm{d}s(x)\mathrm{d}s(y)$
$\displaystyle\quad-\int_{\partial B_{R}}\int_{\partial
B_{R}}\mathbb{E}(u(x)u(y))e^{-ik(\theta_{1}\cdot x+\theta_{2}\cdot
y)}k^{2}(\theta_{2}\cdot\hat{y})(\theta_{1}\cdot\hat{x})\,\mathrm{d}s(x)\mathrm{d}s(y)$
(4.3) $\displaystyle\quad:=\sum_{j=1}^{4}I_{j}(k,\theta_{1},\theta_{2}),$

where $\hat{x}=x/|x|$. By inserting (4.3) into (4.1) and noticing

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\{\theta:\theta=\theta_{1}+\theta_{2},\theta_{1},\theta_{2}\in\mathbb{S}^{d-1}\\}=\\{\theta\in\mathbb{R}^{d}:|\theta|\leq
2\\},}$

we derive the inequality

(4.4)
$\displaystyle|\hat{h}(\xi)|^{2}\lesssim\sup_{\theta_{1},\theta_{2}\in\mathbb{S}^{d-1}}k^{2m}|\sum_{j=1}^{4}I_{j}(k,\theta_{1},\theta_{2})|^{2}+\frac{M^{2}R^{2d}}{k^{2}},$

which holds for all $|\xi|\leq 2k$. For convenience, denote

$\epsilon^{2}(k,\theta_{1},\theta_{2}):=\sum_{j=1}^{4}|I_{j}(k,\theta_{1},\theta_{2})|^{2}$

with $k>0$ and $\theta_{1},\theta_{2}\in\mathbb{S}^{d-1}$. From the equation
(1.1) and the Sommerfeld radiation condition (1.2), it can be verified that

$u(x;-k)=\overline{u(x;k)},\quad k\in\mathbb{R},$

where the notation $u(x;k)$ is used to exhibit the dependence of the solution
on the wavenumber $k$. Therefore, the definition of
$\epsilon^{2}(k,\theta_{1},\theta_{2})$ can be extended to $\mathbb{C}$ as

$\epsilon^{2}(k,\theta_{1},\theta_{2}):={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\sum_{j=1}^{4}I_{j}(k,\theta_{1},\theta_{2})I_{j}(-k,\theta_{1},\theta_{2}).}$

Consider the multi-frequency data characterized as

$\epsilon^{2}=\sup_{k\in[0,K],\theta_{1},\theta_{2}\in\mathbb{S}^{d-1}}\epsilon^{2}(k,\theta_{1},\theta_{2}),$

where $K>0$ is the upper bound of the frequency. Denote a sectorial domain by
$\mathcal{R}=\\{z\in\mathbb{C}:|\arg z|<\pi/4\\}$. In what follows, we show
that $\epsilon^{2}(k,\theta_{1},\theta_{2})$ is analytic and has an upper
bound for $k\in\mathcal{R}$. We only consider the term $I_{1}$ since the
discussions for $I_{2},I_{3},I_{4}$ are similar. Recalling (3.15), we deduce

$\displaystyle I_{1}(k,\theta_{1},\theta_{2})$ $\displaystyle=\int_{\partial
B_{R}}\int_{\partial B_{R}}\mathbb{E}(u(x)u(y))e^{-ik(\theta_{1}\cdot
x+\theta_{2}\cdot y)}\,\mathrm{d}s(x)\mathrm{d}s(y)$
$\displaystyle=\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}J(x,y,\tau,t;k)\mathbb{E}(f(\tau)f(t))\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)$
$\displaystyle=\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}J(x,y,\tau,t;k)K_{f}(\tau,t)\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y),$

where we denote

$J(x,y,\tau,t;k):=\partial_{\nu(x)}\Phi(x,\tau)\partial_{\nu(y)}\Phi(y,t)e^{-ik(\theta_{1}\cdot
x+\theta_{2}\cdot y)}.$

Obviously, $J$ is analytic with respective to $k\in\mathcal{R}$. In order to
show $I_{1}$ is also analytic, we shall verify the following estimates

(4.5) $\displaystyle\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}|K_{f}(\tau,t)J(x,y,\tau,t;k)|\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)<\infty$
(4.6) $\displaystyle\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}|K_{f}(\tau,t)\partial_{k}J(x,y,\tau,t;k)|\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)<\infty$

which hold uniformly with respect to $k\in\mathcal{R}$. As a consequence, the
derivative can be taken under the integral. We only prove (4.5), since (4.6)
can be proven similarly. It is necessary to discuss the singularity of
$K_{f}(\tau,t)$. Inspired by [10], we show in the following lemma that the
kernel $K_{f}(x,y)$ can be represented as the sum of a singular part and a
continuous remainder which is bounded under _assumption (A)_.

######  Lemma 4.1.

Let the random function $f$ satisfy assumption (A). The covariance function
$K_{f}$ has the following form:

(i) If $a<m-(d-1)\leq a+1$ with $a=1,2,3...$,

$K_{f}(x,y)=ch(x)|x-y|^{m-d}+F_{m}(x,y)$

where $c$ is a constant dependent on $m,d$ and $F_{m}(x,y)\in
C^{a,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ with $\alpha\in(0,m-d-a+1)$.

(ii) If $d-1<m<d$,

$K_{f}(x,y)=ch(x)|x-y|^{m-d}+F_{m}(x,y)$

where $c$ is a constant dependent on $m,d$ and $F_{m}(x,y)\in
C^{0,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ with $\alpha\in(0,m-d+1)$.

(iii) If $m=d$,

$K_{f}(x,y)=ch(x)\log{|x-y|}+F_{m}(x,y)$

where $c$ is a constant dependent on $d$ and $F_{m}(x,y)\in
C^{0,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ with $\alpha\in(0,1)$.

For all of the above three cases, we have

$\|F_{m}\|_{L^{\infty}(D\times D)}\lesssim M.$

###### Proof.

We first consider the case $a<m-(d-1)\leq a+1$. Choose a radially symmetric
cut-off function $\chi(\xi)\in C^{\infty}_{0}(\mathbb{R}^{d})$ such that
$\chi(\xi)=1$ when $|\xi|\leq 1$. Recalling (2.1) , we deduce

$\displaystyle\langle K(x,x-\cdot),\phi\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{F}^{-1}(\chi
c(x,\cdot))(y)\phi(y)\,\mathrm{d}y+\int_{\mathbb{R}^{d}}\mathcal{F}^{-1}((1-\chi)a(x,\cdot))(y)\phi(y)\,\mathrm{d}y$
(4.7)
$\displaystyle\quad+\int_{\mathbb{R}^{d}}(1-\chi(\xi))h(x)|\xi|^{-m}(\mathcal{F}^{-1}\phi)(\xi)\,\mathrm{d}\xi$

with the test function $\phi\in\mathcal{D}$. Notice that
$v_{1}(x,y):=\mathcal{F}^{-1}(\chi
c(x,\cdot))(y)\in\mathcal{S}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ with
$supp\,v\subset D\times\mathbb{R}^{d}$ and the bound

(4.8)
$\displaystyle\|v_{1}(x,y)\|_{L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})}\lesssim
M\int_{supp\,\chi}|\chi|(1+|\xi|)^{-m}\,\mathrm{d}\xi\lesssim M.$

Denote $v_{2}(x,y)=\mathcal{F}^{-1}((1-\chi)a(x,\cdot))(y)$. Obviously $v_{2}$
is smooth and compactly supported with respect to $x$. Recalling $1-\chi$
vanishes when $|\xi|\leq 1$, then _assumption (A)_ implies

$|(1-\chi(\xi))a(x,\xi)|\lesssim M|\xi|^{-(m+1)}.$

Take $p,\,q$ such that $2\leq p\leq\infty$ and $\frac{1}{p}+\frac{1}{q}=1$.
Applying _assumption (A)_ and Hausdorff-Young inequality gives the inequality

$\displaystyle\|v_{2}(x,\cdot)\|_{W^{s,p}(\mathbb{R}^{d})}$
$\displaystyle=\|(I-\Delta)^{\frac{s}{2}}v_{2}(x,\cdot)\|_{L^{p}(\mathbb{R}^{d})}\lesssim\|(1+|\cdot|^{2})^{s/2}(1-\chi)a(x,\cdot)\|_{L^{q}(\mathbb{R}^{d})}$
$\displaystyle\lesssim M\left(\int_{|\xi|\geq
1}\frac{1}{|\xi|^{(m+1-s)q}}\,\mathrm{d}\xi\right)^{\frac{1}{q}}\lesssim M.$

The above inequality holds uniformly with respect to $x$ if and only if
$(m+1-s)q>n$ which can be satisfied by choosing $s>0$ and $s-{d}/{p}<m-(d-1)$.
Then using Sobolev embedding theorem gives $W^{s,p}(\mathbb{R}^{d})\subset
C^{k_{0},\alpha}(\mathbb{R}^{d})$ with $k_{0}\in\mathbb{N},\alpha\in(0,1]$
such that $k_{0}+\alpha=s-d/p$. When $a<m-(d-1)\leq a+1$ with
$a\in\mathbb{N}$, we can choose $p$ sufficiently large such that $s$ is close
to $m-(d-1)$ which implies $k_{0},\alpha$ can be chosen as $k_{0}=a$ and
$0<\alpha<m-(d-1)-a$. Furthermore, one has $v_{2}\in
C^{a,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ and

(4.9)
$\|v_{2}(x,y)\|_{L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})}\leq\|v_{2}(x,\cdot)\|_{C^{a,\alpha}(\mathbb{R}^{d})}\leq\|v_{2}(x,\cdot)\|_{W^{s,p}(\mathbb{R}^{d})}\lesssim
M.$

The third term in (4.7) can be rewritten as

$\displaystyle
h(x)\int_{\mathbb{R}^{d}}\mathcal{F}^{-1}((1-\chi)|\cdot|^{-m})(y)\phi(y)\,\mathrm{d}y.$

Now we claim when $m>d$,

(4.10) $\mathcal{F}^{-1}((1-\chi)|\cdot|^{-m})(y)=c_{1}|y|^{m-d}+g_{1}(y),$

where $c_{1}$ is a constant only dependent on $m,d$ and $g_{1}(y)\in
C^{\infty}(\mathbb{R}^{d})$. In fact, direct calculation gives

$\displaystyle\mathcal{F}^{-1}((1-\chi)|\cdot|^{-m})(y)$
$\displaystyle=|y|^{m-d}\mathcal{F}^{-1}((1-\chi)|\cdot|^{-m})(\hat{y})+(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{iy\cdot\xi}|\xi|^{-m}(\chi(|y|\xi)-\chi(\xi))\,\mathrm{d}\xi.$

Notice that $m>d$ and choose $\chi$ to be a radially symmetric function. We
have that

$c_{1}:=\mathcal{F}^{-1}((1-\chi)|\cdot|^{-m})(\hat{y})$

is a finite constant. As $\chi\in C_{0}^{\infty}$, we have that

$g_{1}(y):=(2\pi)^{-d}\int_{\mathbb{R}^{d}}e^{iy\cdot\xi}|\xi|^{-m}(\chi(|y|\xi)-\chi(\xi))\,\mathrm{d}\xi$

is smooth and bounded with respect to $y$. In conclusion, when $m>d$, by
combining (4.7)–(4.10) we have

$K(x,x-y)=c_{1}h(x)|y|^{m-d}+v_{1}(x,y)+v_{2}(x,y)+h(x)g_{1}(y),$

which gives

$K_{f}(x,y)=c_{m,d}h(x)|x-y|^{m-d}+F_{m}(x,y),\quad\|F_{m}\|_{L^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})}\lesssim
M$

with

$F_{m}(x,y):=v_{1}(x,x-y)+v_{2}(x,x-y)+h(x)g(x-y)\in
C^{a,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$

for $a<m-(d-1)\leq a+1$.

Then we consider the case $d-1<m<d$. Denote by $\Gamma$ the Gamma function
$\Gamma(\beta)=\int^{\infty}_{0}t^{\beta-1}e^{-t}\,\mathrm{d}t.$ One has the
identity

(4.11)
$\displaystyle|\xi|^{-m}=\frac{2^{-m/2}}{\Gamma({m}/{2})}\int_{0}^{\infty}t^{m/2-1}e^{{-t|\xi|^{2}}/{2}}\,\mathrm{d}t.$

One also has

(4.12)
$\mathcal{F}(e^{-t|\xi|^{2}/2})(y)=(2\pi)^{d/2}t^{-d/2}e^{-|y|^{2}/(2t)}.$

Applying (4.11)–(4.12) gives

$\displaystyle\int_{\mathbb{R}^{d}}(1-\chi(\xi))h(x)|\xi|^{-m}(\mathcal{F}^{-1}\phi)(\xi)\,\mathrm{d}\xi=c_{2}h(x)\int_{\mathbb{R}^{d}}|y|^{m-d}\mathcal{F}((1-\chi)\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y$
$\displaystyle=c_{2}h(x)\left(\int_{\mathbb{R}^{d}}|y|^{m-d}\phi(y)\,\mathrm{d}y+\int_{\mathbb{R}^{d}}|y|^{m-d}\mathcal{F}(\chi\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y\right)$
(4.13)
$\displaystyle=c_{2}h(x)\int_{\mathbb{R}^{d}}[|y|^{m-d}+(|\cdot|^{m-d}\ast\mathcal{F}^{-1}\chi)(y)]\phi(y)\,\mathrm{d}y,$

where $c_{2}=\frac{2^{-m+d/2}}{\Gamma(m/2)}\Gamma((d-m)/2)$. Moreover,
$g_{2}(y):=c_{2}[|\cdot|^{m-d}\ast\mathcal{F}^{-1}\chi)(y)]$ is smooth.
Combining (4.7)–(4.9) and (4.13) yields

$K_{f}(x,y)=c_{2}h(x)|x-y|^{m-d}+F(x,y),\quad\|F\|_{L^{\infty}(D\times
D)}\lesssim M$

with

$F(x,y):=v_{1}(x,x-y)+v_{2}(x,x-y)+h(x)g(x-y)\in
C^{0,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d}).$

for $d-1<m<d$.

At last, consider the case when $d=m$. Rewrite the right-hand side of (4.7) as

$\displaystyle\int_{\mathbb{R}^{d}}(1$
$\displaystyle-\chi(\xi))h(x)|\xi|^{-m}(\mathcal{F}^{-1}\phi)(\xi)\,\mathrm{d}\xi$
$\displaystyle=\frac{2^{-m+d/2}}{\Gamma(m/2)}\Gamma((d-m)/2+1)h(x)\int_{\mathbb{R}^{d}}\frac{|y|^{m-d}-1}{(d-m)/2}\mathcal{F}((1-\chi)\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y$
$\displaystyle+\frac{2^{-m+d/2}}{\Gamma(m/2)}\Gamma((d-m)/2)h(x)\int_{\mathbb{R}^{d}}\mathcal{F}((1-\chi)\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y.$

Letting $m\to d$ in, we obtain

$\displaystyle\int_{\mathbb{R}^{d}}(1-\chi(\xi))h(x)|\xi|^{-n}(\mathcal{F}^{-1}\phi)(\xi)\,\mathrm{d}\xi$
(4.14)
$\displaystyle=-c_{d}h(x)\int_{\mathbb{R}^{d}}\log{|y|}\mathcal{F}((1-\chi)\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y+\tilde{c}_{d}h(x)\int_{\mathbb{R}^{d}}\mathcal{F}((1-\chi)\mathcal{F}^{-1}\phi)(y)\,\mathrm{d}y$

with constants $c_{d},\tilde{c}_{d}>0$. Notice that

(4.15) $\displaystyle
g_{3}(y)=-c_{d}[\log{|\cdot|}\ast\mathcal{F}^{-1}\chi](y)+\tilde{c}_{d}\left\\{1-\int_{\mathbb{R}^{d}}\mathcal{F}^{-1}\chi(y)\,\mathrm{d}y\right\\}$

is smooth. Combining (4.7)–(4.9) and (4.14)–(4.15) yields

$K_{f}(x,y)=c_{d}h(x)\log|x-y|+F_{m}(x,y),\quad\|F\|_{L^{\infty}(D\times
D)}\lesssim M$

with

$F_{m}(x,y):=v_{1}(x,x-y)+v_{2}(x,x-y)+h(x)g_{3}(x-y)\in
C^{0,\alpha}(\mathbb{R}^{d}\times\mathbb{R}^{d})$

for $m=d$.

In what follows, we show that $I_{1}$ is bounded with respect to
$k\in\mathcal{R}$ and similar arguments apply to $I_{2},I_{3},I_{4}$. We
consider the following two situations.

_Case 1._ Let $m>d-1$ and $m\neq d$. We have from Lemma 4.1 that

$K_{f}(x,y)=ch(x)|x-y|^{m-d}+F_{m}(x,y),$

which yields that the kernel is continuous or weakly singular. Therefore,
(4.5) holds for $k\in\mathcal{R}$ and thus $I_{1}$ is analytic. Moreover, when
$k=k_{1}+ik_{2}\in\mathcal{R}$, we further obtain the following bound

$\displaystyle|I_{1}|$ $\displaystyle\lesssim e^{2k_{1}}\int_{\partial
B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}|\partial_{\nu(x)}\Phi(x,\tau)\partial_{\nu(y)}\Phi(y,t)|({h(\tau)}|\tau-t|^{m-d}+M)\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)$
(4.16) $\displaystyle\lesssim e^{2k_{1}}M\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}|\partial_{\nu(x)}\Phi(x,\tau)\partial_{\nu(y)}\Phi(y,t)|(|\tau-t|^{m-d}+1)\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y).$

To proceed, noting that the Green function $\Phi$ takes different forms for
$d=2$ and $d=3$, we discuss the following two cases.

If $d=3$, a direct calculation gives

(4.17)
$|\partial_{\nu(y)}\Phi(x,y)|\leq\frac{|k_{1}|e^{k_{1}|x-y|}}{4\pi|x-y|}+\frac{e^{k_{1}|x-y|}}{4\pi|x-y|^{2}}.$

We show that

(4.18) $\displaystyle|I_{1}|\lesssim e^{2(2R+1)|k_{1}|}MR^{4}(1+|k_{1}|)^{2}.$

In fact, we only need to estimate the term with highest singularity. To this
end, with the help of Lemma 3.1 we have the following estimate

$\displaystyle\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}\frac{1+|t-\tau|^{m-d}}{|x-\tau|^{2}|y-t|^{2}}\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)\lesssim\int_{\partial
B_{R}}\int_{\partial
B_{R}}\int_{D}\frac{1}{|x-\tau|^{2}|y-\tau|^{2}}\,\mathrm{d}\tau\mathrm{d}s(x)\mathrm{d}s(y)$
(4.19) $\displaystyle\lesssim\int_{\partial B_{R}}\int_{\partial
B_{R}}\frac{1}{|x-y|}\mathrm{d}s(x)\mathrm{d}s(y)\lesssim R^{4}.$

For other parts in right-hand side of (4.16), we have simialr estimates.
Combining (4.16) and (4.19) yields (4.18).

When $d=2$, by the following integral form of the Hankel function [13]

(4.20)
$H_{0}^{(1)}(z)=Ce^{iz}\int_{0}^{\infty}e^{-s}s^{-1/2}(s/2-iz)^{-1/2}\,\mathrm{d}s,$

we obtain

$\displaystyle|\partial_{\nu(y)}\Phi(x,y)|\lesssim\frac{|k_{1}|e^{k_{1}|x-y|}}{|k_{1}(x-y)|^{\frac{1}{2}}}+\frac{e^{k_{1}|x-y|}}{|k_{1}|^{\frac{1}{2}}|x-y|^{\frac{3}{2}}},\quad
k\in\mathcal{R},\,x\in D,\,y\in\partial B_{R}.$

Then in a similar way as the derivation of (4.18), we can obtain

(4.21)
$\displaystyle|I_{1}|\lesssim|k_{1}|^{-1}e^{2(2R+1)|k_{1}|}MR^{2}(1+|k_{1}|)^{2}.$

Combining (4.18) and (4.21) gives the estimate

(4.22) $|I_{1}|\lesssim
e^{2(2R+1)|k_{1}|}MR^{2d-2}|k_{1}|^{d-3}(1+|k_{1}|)^{2}.$

_Case 2._ When $m=d$ , Lemma 4.1 shows
$K_{f}(\tau,t)=ch(\tau)\log{|\tau-t|}+F_{m}(\tau,t)$ which is still weakly
singular. Therefore, we can apply the second inequality of Lemma 3.1 to verify
$I_{1}$ satisfies the inequalities (4.22).

Combining the above arguments, we arrive at the following lemma which provides
the analyticity and boundedness of the data with respect to $k\in\mathcal{R}$.

######  Lemma 4.2.

Suppose that the random function $f$ satisfies assumption (A). We have that
$I_{j}(k,\theta_{1},\theta_{2}),j=1,...,4,$ is analytic with respective to
$k\in\mathcal{R}$. Furthermore, the following estimates

$\displaystyle|I_{1}(k,\theta_{1},\theta_{2})|$ $\displaystyle\lesssim
e^{2(2R+1)|k_{1}|}MR^{2d-2}|k_{1}|^{d-3}(1+|k_{1}|)^{2},$
$\displaystyle|I_{2}(k,\theta_{1},\theta_{2})|$ $\displaystyle\lesssim
e^{2(2R+1)|k_{1}|}MR^{2d-2}|k_{1}|^{d-2}(1+|k_{1}|),$
$\displaystyle|I_{3}(k,\theta_{1},\theta_{2})|$ $\displaystyle\lesssim
e^{2(2R+1)|k_{1}|}MR^{2d-2}|k_{1}|^{d-2}(1+|k_{1}|),$
$\displaystyle|I_{4}(k,\theta_{1},\theta_{2})|$ $\displaystyle\lesssim
e^{2(2R+1)|k_{1}|}MR^{2d-2}|k_{1}|^{d-1},$

hold for
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}k=k_{1}+ik_{2}\in\mathcal{R}}$.

######  Remark 4.1.

(i) Conclusions like Lemma 4.2 also hold for
$I_{j}(-k,\theta_{1},\theta_{2})$, $j=1,2,3,4$ after analogous discussions.

(ii) We have to use Lemma 3.1 to derive inequality (4.19). Indeed, for
$x,y\in\partial B_{R}$ and $t,\tau\in D$, by a direction calculation one has

$\int_{\partial B_{R}}\int_{\partial
B_{R}}\int_{D}\int_{D}\frac{1+|t-\tau|^{m-d}}{|x-\tau|^{2}|y-t|^{2}}\,\mathrm{d}\tau\mathrm{d}t\mathrm{d}s(x)\mathrm{d}s(y)\lesssim
R^{4}/\text{dist}(\partial B_{R},D)^{4}.$

However, when $\text{dist}(\partial B_{R},D)\to 0$, the right-hand side of the
above inequality tends to infinity. Based on this reason, we use Lemma 3.1 to
estimate this integral.

The following unique continuation argument [11] is useful in the subsequent
analysis.

######  Lemma 4.3.

Let $U(z)$ be an analytic function in $\mathcal{R}$ and continuous in
$\bar{\mathcal{R}}$. If

$\displaystyle\left\\{\begin{array}[]{ccc}|U(z)|\leq\varepsilon,&z\in(0,L],\\\
|U(z)|\leq V,&z\in\mathcal{R},\\\ |U(z)|=0,&z=0,\end{array}\right.$

with constants $\varepsilon,L,V>0$, then there exists a function $\mu(z)$
satisfying

$\displaystyle\left\\{\begin{array}[]{cc}\mu(z)\geq 1/2,&z\in(L,2^{1/4}L),\\\
\mu(z)\geq\pi^{-1}((z/L)^{4}-1)^{-1/2},&z\in(2^{1/4}L,\infty)\end{array}\right.$

such that

$|U(z)|\leq V\varepsilon^{\mu(z)},\quad\forall z\in(L,\infty).$

Combining Lemma 4.2–4.3 yields the following conclusion.

######  Lemma 4.4.

Let $f$ satisfy assumption (A). Then we have the estimte

$|\epsilon^{2}(k,\theta_{1},\theta_{2})|\lesssim
K^{-1}M^{2}R^{4d-4}\epsilon^{2\mu(k)}e^{(8R+5)k},\quad k\in(K,\infty)$

with the function $\mu(k)$ satisfying

(4.25) $\displaystyle\left\\{\begin{array}[]{cc}\mu(k)\geq
1/2,&k\in(K,2^{1/4}K),\\\
\mu(k)\geq\pi^{-1}((k/K)^{4}-1)^{-1/2},&k\in(2^{1/4}K,\infty).\end{array}\right.$

.

###### Proof.

Denote $U(k)=k\epsilon^{2}(k,\theta_{1},\theta_{2})$. We have that $U(0)=0$
and $U(k)$ is analytic and continuous for $k\in\bar{\mathcal{R}}$. It follows
from Lemma 4.2 when $k\in\mathcal{R}$,

$\displaystyle|U(k)|\lesssim|k|\sum_{j=1}^{4}|I_{j}(k)|\sum_{l=1}^{4}|I_{l}(-k)|\lesssim
e^{8R+5|k_{1}|}M^{2}R^{4d-4},$

which yields

$\displaystyle|e^{-(8R+5)k}U(k)|\lesssim M^{2}R^{4d-4}.$

Obviously, for $k\in(0,K]$, one has $|e^{-(8R+5)k}U(k)|\leq\epsilon^{2}$.
Applying Lemma 4.4 to $e^{-(8R+5)k}U(k)$ gives

$|e^{-(8R+5)k}U(k)|\lesssim M^{2}R^{4d-4}\epsilon^{2\mu(k)},\quad
k\in(K,\infty)$

where $\mu(k)$ satisfies (4.25). Thus, we have

$|\epsilon^{2}(k,\theta_{1},\theta_{2})|\lesssim
K^{-1}M^{2}R^{4d-4}\epsilon^{2\mu(k)}e^{(8R+5)k},\quad k\in(K,\infty),$

which completes the proof.

Denote $\mathcal{C}_{Q}=\\{h\in
C_{0}^{\infty}(D):\|h\|_{H^{s}(\mathbb{R}^{d})}\leq Q\\}$ with $s>0$. We are
now in a position to state the increasing stability result of inverse random
source problem. The proof adopts the argument of analytic continuation
developed in [25].

######  Theorem 4.1.

Let the random function $f$ satisfy assumption (A) and assume
$h\in\mathcal{C}_{Q}$. We have the following stability estimate

(4.26) $\displaystyle\|h\|^{2}_{L^{2}(D)}\lesssim
K^{2m+\frac{2d}{d+2s}}\epsilon^{2}+(R+1)^{2d+\frac{2}{3}}\frac{Q^{2}+M^{2}}{K^{\frac{8s}{3(2s+d)}}E^{\frac{s}{2s+d}}},$

with $E=|\log{\epsilon}|$.

###### Proof.

Without losing generality, we assume $\epsilon<e^{-1}$. Then take

$\displaystyle
A=\left\\{\begin{array}[]{cc}\frac{1}{((8R+5)\pi)^{\frac{1}{3}}}K^{\frac{2}{3}}E^{\frac{1}{4}},&2^{\frac{1}{4}}((8R+5)\pi)^{\frac{1}{3}}K^{\frac{1}{3}}<E^{\frac{1}{4}},\\\
K,&E^{\frac{1}{4}}\leq
2^{\frac{1}{4}}((8R+5)\pi)^{\frac{1}{3}}K^{\frac{1}{3}}.\end{array}\right.$

If $2^{\frac{1}{4}}((8R+5)\pi)^{\frac{1}{3}}K^{\frac{1}{3}}<E^{\frac{1}{4}}$,
we have $A>2^{\frac{1}{4}}K>K$. Then for $k\in(K,A]$, applying Lemma 4.4
yields

$\displaystyle\epsilon^{2}(k,\theta_{1},\theta_{2})$ $\displaystyle\lesssim
K^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}M^{2}\epsilon^{2\mu(k)}e^{(8R+5)k}$
$\displaystyle\lesssim
K^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}M^{2}\exp\left\\{(8R+5)A-\frac{2E}{\pi}((k/K)^{4}-1)^{-\frac{1}{2}}\right\\}$
$\displaystyle\lesssim
K^{-1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}M^{2}\exp\left\\{(8R+5)\frac{K^{\frac{2}{3}}E^{\frac{1}{4}}}{((8R+5)\pi)^{\frac{1}{3}}}-\frac{2E}{\pi}(k/A)^{2}\right\\}$
$\displaystyle\lesssim
K^{-1}R^{4d-4}M^{2}\exp\left\\{-2\left(\frac{(8R+5)^{2}}{\pi}\right)^{\frac{1}{3}}K^{\frac{2}{3}}E^{\frac{1}{2}}\left(1-\frac{1}{2}E^{-\frac{1}{4}}\right)\right\\}.$

Noticing that $\epsilon<e^{-1}$ implies

$1-\frac{1}{2}E^{-\frac{1}{4}}>\frac{1}{2}.$

Then we get

(4.27) $\displaystyle\epsilon^{2}(k,\theta_{1},\theta_{2})$
$\displaystyle\lesssim
K^{-1}M^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}\exp\left\\{-(8R+5)^{\frac{2}{3}}K^{\frac{2}{3}}E^{\frac{1}{2}}\right\\}\lesssim
K^{-1}M^{2}R^{4d-4}\frac{1}{((8R+5)^{\frac{2}{3}}K^{\frac{2}{3}}E^{\frac{1}{2}})^{n}}.$

Here we have used $e^{-x}\leq n!/x^{n}$ where $n\in\mathbb{N}$. Combining
(4.4) and (4.27) gives the estimte

$\displaystyle|\hat{h}(\xi)|^{2}\lesssim
M^{2}\frac{R^{2d}}{A^{2}}+K^{-1}M^{2}A^{2m}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}\frac{1}{((8R+5)^{\frac{2}{3}}K^{\frac{2}{3}}E^{\frac{1}{2}})^{n}},\quad|\xi|\leq
2A,$

which yields

(4.28) $\int_{|\xi|\leq 2A^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi\lesssim
R^{2d}M^{2}A^{d\gamma-2}+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}A^{d\gamma+2m}K^{-1}M^{2}\frac{1}{((8R+5)^{\frac{2}{3}}K^{\frac{2}{3}}E^{\frac{1}{2}})^{n}}.$

By the condition $\|h\|_{H^{s}(\mathbb{R}^{d})}\leq Q$, we obtain

(4.29)
$\int_{|\xi|>2A^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi\lesssim\frac{Q^{2}}{A^{2\gamma
s}}.$

Take $n>m+1$ and $\gamma=\frac{2}{2s+d}$ such that
$\frac{n-m}{2}-\frac{1}{4}\gamma d>\frac{1}{4}(2-\gamma d)=\frac{1}{2}\gamma
s$. Therefore, together with (4.28)–(4.29) we have the inequality

$\displaystyle\|h\|^{2}_{L^{2}(D)}$ $\displaystyle=\int_{|\xi|\leq
2A^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi+\int_{|\xi|>2A^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi\lesssim
E^{-\frac{s}{2s+d}}\left(\frac{(Q^{2}+1)(R+1)^{2d+\frac{2}{3}}}{K^{\frac{8s}{3(2s+d)}}}+\frac{M^{2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}R^{4d-4}}}{K^{\beta}(1+R)^{\frac{2(m+n)}{3}}}\right),$

where $\beta=1+\frac{2}{3}n-\frac{2}{3}(2m+\frac{2d}{2s+d})$. Taking
$n>\max\\{3d-7-m,2m+\frac{1}{2}\\}$ and $\beta>\frac{8s}{3(2s+d)}$, we have

(4.30)
$\|h\|^{2}_{L^{2}(D)}\lesssim(R+1)^{2d+\frac{2}{3}}\frac{Q^{2}+M^{2}}{K^{\frac{8s}{3(2s+d)}}E^{\frac{s}{2s+d}}}.$

If $2^{\frac{1}{4}}((8R+5)\pi)^{\frac{1}{3}}K^{\frac{1}{3}}\geq
E^{\frac{1}{4}}$, it is straightforward to get

$\displaystyle\|h\|^{2}_{L^{2}(D)}$ $\displaystyle=\int_{|\xi|\leq
2K^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi+\int_{|\xi|>2K^{\gamma}}|\hat{h}(\xi)|^{2}\,\mathrm{d}\xi$
(4.31) $\displaystyle\lesssim
K^{2m+\frac{2d}{d+2s}}\epsilon^{2}+R^{2d}\frac{M^{2}+Q^{2}}{K^{\frac{4s}{2s+d}}}\lesssim
K^{2m+\frac{2d}{d+2s}}\epsilon^{2}+(R+1)^{2d-\frac{4s}{3(2s+d)}}\frac{M^{2}+Q^{2}}{K^{\frac{8s}{3(2s+d)}}E^{\frac{s}{2s+d}}}.$

Combining (4.30) and (4.31), we complete the proof.

The stability estimate (4.26) consists of the Lipschitz data discrepancy and a
logarithmic stability. The latter illustrates the ill-posedness of the inverse
source problem. Observe that as the upper bound $K$ of the frequency
increases, the logarithmic stability decreases which leads to the improvement
of the stability estimate. Clearly, the stability estimate (4.26) implies the
uniqueness of the inverse problem.

##  5 Inverse problem using far-field data data

In this section, we consider using the far-field data given by

(5.1)
$u^{\infty}(\hat{x},k)=-C_{d}k^{\frac{d-3}{2}}\int_{\mathbb{R}^{d}}e^{-{\rm
i}\hat{x}\cdot y}f(y){\rm d}y.$

Here $C_{d}$ is a constant dependent on the dimension $d$. According to [18],
we have for $\tau>0$ that

$\displaystyle\mathbb{E}[u^{\infty}(\hat{x},k+\tau)\overline{u^{\infty}(\hat{x},k)}]$
$\displaystyle=|C_{d}|^{2}(k+\tau)^{\frac{d-3}{2}}k^{\frac{d-3}{2}}\Big{[}\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-{\rm
i}(k+\tau)\hat{x}\cdot y}e^{{\rm i}k\hat{x}\cdot z}\mathbb{E}[f(y)f(z)]{\rm
d}y{\rm d}z$
$\displaystyle=|C_{d}|^{2}(k+\tau)^{\frac{d-3}{2}}k^{\frac{d-3}{2}}\int_{\mathbb{R}^{d}}\Big{[}\int_{\mathbb{R}^{d}}K_{f}(y,z)e^{-{\rm
i}k\hat{x}\cdot(y-z)}{\rm d}z\Big{]}e^{-{\rm i}\tau\hat{x}\cdot y}{\rm
d}y\Big{)}$
$\displaystyle=|C_{d}|^{2}(k+\tau)^{\frac{d-3}{2}}k^{\frac{d-3}{2}}\Big{[}\int_{\mathbb{R}^{d}}h(y)e^{-{\rm
i}\tau\hat{x}\cdot y}{\rm
d}y|k\hat{x}|^{-m}+\int_{\mathbb{R}^{2}}a(y,k\hat{x})e^{-{\rm
i}\tau\hat{x}\cdot y}{\rm d}y\Big{)}\Big{]}$
$\displaystyle=|C_{d}|^{2}\Big{(}\frac{k}{k+\tau}\Big{)}^{\frac{3-d}{2}}k^{d-3-m}\widehat{h}(\tau\hat{x})+M{O}(k^{d-4-m}).$

Hence we have

$\displaystyle|\widehat{h}(\tau\hat{x})|\lesssim\Big{|}k^{m+3-d}\mathbb{E}[u^{\infty}(\hat{x},k+\tau)\overline{u^{\infty}(\hat{x},k)}]\Big{|}+\frac{M}{k},$

which gives

$\displaystyle|\widehat{h}(\xi)|^{2}\lesssim\sup_{0<\eta<1,\hat{x}\in\mathbb{S}^{2}}\Big{|}k^{m+3-d}\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)\overline{u^{\infty}(\hat{x},k)}]\Big{|}^{2}+\frac{M^{2}}{k^{2}},\quad\text{for
all}~{}|\xi|\leq k.$

Based on the above estimate, we introduce the data discrepancy in a finite
interval $I=[0,K]$ with $0<K$ as follows

$\tilde{\epsilon}^{2}=\sup_{k\in
I,\eta\in(0,1),\hat{x}\in\mathbb{S}^{2}}\tilde{\epsilon}^{2}(k,\eta,\hat{x})$

where

$\tilde{\epsilon}^{2}(k,\eta,\hat{x}):=\Big{|}k^{m+3-d}\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)\overline{u^{\infty}(\hat{x},k)}]\Big{|}^{2}.$

As the source function is real-valued, we have $\overline{u(x,k)}=u(x,-k)$ and
then $\overline{u^{\infty}(x,k)}=u^{\infty}(x,-k)$. Hence, we can analytically
extend $\tilde{\epsilon}^{2}(\cdot,\eta,\hat{x})$ from $\mathbb{R}^{+}$ to
$\mathbb{C}$ as follows

$\tilde{\epsilon}^{2}(k,\eta,\hat{x})=k^{2(m+3-d)}\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)u^{\infty}(\hat{x},-k)]\mathbb{E}[u^{\infty}(\hat{x},-(1+\eta)k)u^{\infty}(\hat{x},k)],\quad
k\in\mathbb{C},$

by noticing that for $k>0$

$\tilde{\epsilon}^{2}(k,\eta,\hat{x})=\Big{|}k^{m+3-d}\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)\overline{u^{\infty}(\hat{x},k)}]\Big{|}^{2}.$

Recall $\mathcal{R}=\\{z\in\mathbb{C}:|\arg z|<\pi/4\\}$. In order to apply
Lemma 4.3 for $\tilde{\epsilon}^{2}(k,\eta,\hat{x})$ in $\mathcal{R}$, we
shall show $\tilde{\epsilon}^{2}(k,\eta,\hat{x})$ is analytic and bounded for
$k\in\mathcal{R}$. Recalling (5.1), for $k\in\mathcal{R}$, we have

$\displaystyle\Big{|}\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)u^{\infty}(\hat{x},-k)]\Big{|}\lesssim|k|^{d-3}\Big{|}\mathbb{E}\int_{D}e^{-ik(1+\eta)\hat{x}\cdot
y}f(y)\,\mathrm{d}y\int_{D}e^{ik\hat{x}\cdot z}f(z)\,\mathrm{d}z\Big{|}$
$\displaystyle\lesssim|k|^{d-3}\int_{D}\int_{D}|e^{-ik(1+\eta)\hat{x}}e^{ik\hat{x}\cdot
z}K(y,z)|\,\mathrm{d}y\,\mathrm{d}z$
$\displaystyle\lesssim|k|^{d-3}e^{3D_{0}|\Re k|}\|K(y,z)\|_{L^{1}(D\times D)}$

with $D_{0}=diam(D)$. Lemma 4.1 gives that the kernel is weakly singular and

$\|K(y,z)\|_{L^{1}(D\times D)}\lesssim M.$

Hence, we have the estimate

(5.2)
$|\mathbb{E}[u^{\infty}(\hat{x},(1+\eta)k)u^{\infty}(\hat{x},-k)]|\lesssim|k|^{d-3}e^{3D_{0}|\Re
k|}M.$

The above arguments guarantee the analyticity and boundedness of the data. In
summary, we arrive at the following lemma.

######  Lemma 5.1.

Suppose that the random field $f$ satisfies assumption (A). We have that
$\tilde{\epsilon}^{2}(k,\eta,\hat{x})$ is analytic with respect to
$k\in\mathcal{R}$ with the following upper bound

$\tilde{\epsilon}^{2}(k,\eta,\hat{x})\lesssim|k|^{2m}e^{6D_{0}|\Re k|}M^{2}.$

Using Lemma 4.3 and Lemma 5.1 we have the following lemma.

######  Lemma 5.2.

Let $f$ satisfy assumption (A). Then we have the estimte

$|\tilde{\epsilon}^{2}(k,\theta_{1},\theta_{2})|\lesssim
M^{2}\tilde{\epsilon}^{2\mu(k)}e^{(6D_{0}+1)k},\quad k\in(K,\infty)$

with the function $\mu(k)$ satisfying

$\displaystyle\left\\{\begin{array}[]{cc}\mu(k)\geq 1/2,&k\in(K,2^{1/4}K),\\\
\mu(k)\geq\pi^{-1}((k/K)^{4}-1)^{-1/2},&k\in(2^{1/4}K,\infty).\end{array}\right.$

.

Following the arguments in the proof of Theorem 4.1 in a straightforward way,
we have the following increasing stability estimate by far-field data. The
proof is omitted for brevity.

######  Theorem 5.1.

Let the random function $f$ satisfies assumption (A) and assume
$h\in\mathcal{C}_{Q}$. Then there holds the inequality

(5.3) $\displaystyle\|h\|^{2}_{L^{2}(D)}\lesssim
K^{\frac{2d}{d+2s}}\tilde{\epsilon}^{2}+\frac{Q^{2}+M^{2}}{K^{\frac{8s}{3(2s+d)}}E^{\frac{s}{2s+d}}},$

where $E=|\log{\tilde{\epsilon}}|$.

##  6 Conclusion

We establish an increasing stability of an inverse random source problem for
the Helmholtz equation. The analysis employs properties of the covariance
kernel and the explicit Green function. A possible extension of the current
work is to investigate the increasing stability of inverse random source
problem in an inhomogeneous media, where the explicit Green function is no
longer available. Another interesting topic is the stability for the nonlinear
inverse random potential problem. We hope to report the progress of these
problems in forthcoming works.

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#  Semi-infinite quantum wells in a position-dependent mass background

C. Quesne
Physique Nucléaire Théorique et Physique Mathématique, Université Libre de
Bruxelles,
Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
e-mail<EMAIL_ADDRESS>
###### Abstract

By using a point canonical transformation starting from the constant-mass
Schrödinger equation for the Morse potential, it is shown that a semi-infinite
quantum well model with a non-rectangular profile associated with a position-
dependent mass that becomes infinite for some negative value of the position,
while going to a constant for a large positive value of the latter, can be
easily derived. In addition, another type of semi-infinite quantum well
associated with the same position-dependent mass is constructed and solved by
starting from the Rosen-Morse II potential instead of the Morse one.

Keywords: Schrödinger equation; position-dependent mass; quantum well; point
canonical transformation

MSC Nos.: 81Q05; 81Q80

##  1 Introduction

Due to its many applications in several fields of physics, the Schrödinger
equation in a position-dependent mass (PDM) background has arisen much
interest over the years [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Finding exact
solutions of such a Schrödinger equation turns out to be very useful for
understanding some physical phenomena and testing some approximation methods.
Several techniques are available for generating PDM and potential pairs
leading to such exact solutions (see, e.g., [13, 14, 15] and references quoted
therein). Among them, one may quote the point canonical transformation (PCT)
applied to an exactly-solvable constant-mass Schrödinger equation [16], which
has proved very useful to achieve this goal (see, e.g., [17, 18, 19]).

In a series of papers, Jafarov, Nagiyev, and their collaborators studied
several variants of harmonic oscillator models in a PDM background [20, 21,
22, 23, 24, 25, 26, 27, 28]. They directly solved the corresponding
Schrödinger equations and proved that their solutions go over to the well-
known ones of the constant-mass harmonic oscillator in some appropriate
parameter limit. In a recent work [29], they considered a harmonic oscillator
potential well with a PDM that has the property of becoming infinite for some
negative value $-a$ of the position while going to a constant for a large
positive value of the latter. This resulted in a semi-infinite quantum well,
whose spectrum was determined by a direct approach.

The purpose of the present paper is twofold: first to demonstrate that the PCT
method provides a simple derivation of some of the results of Ref. [29], and
second to show that the same PDM may provide other exactly-solvable semi-
infinite quantum wells.

This paper is organized as follows. In Sect. 2, the model of Ref. [29] is
reviewed and shown to be derivable from the known results for the Morse
potential with a constant mass [30, 31]. In Sect. 3, another type of semi-
infinite quantum well is constructed and solved. Finally, Sect. 4 contains the
conclusion.

##  2 The Jafarov and Nagiyev semi-infinite quantum well and its derivation by
the PCT technique

In Ref. [29], Jafarov and Nagiyev considered the PDM Schrödinger equation

$\left(-\frac{d}{dx}\frac{1}{M(x)}\frac{d}{dx}+V_{\rm
eff}(x)\right)\psi(x)=E\psi(x),$ (1)

where $M(x)$ and $V_{\rm eff}(x)$ are defined by

$M(x)=\left(1+\frac{x}{a}\right)^{-2},\quad V_{\rm
eff}(x)=\frac{1}{4}M(x)\omega^{2}x^{2}=\frac{1}{4}\frac{a^{2}\omega^{2}x^{2}}{(x+a)^{2}},\quad-a<x<+\infty,$
(2)

with $a>0$.111Note that we have adopted here units wherein $\hbar=2m_{0}=1$ in
the original paper. Since the mass and the potential go to $+\infty$ for
$x\to-a$, the definition of the latter may be completed by setting $V_{\rm
eff}(x)=+\infty$ for $x\leq-a$. On the other hand, for $x\to+\infty$, it is
clear that $V_{\rm eff}(x)\to\frac{1}{4}\omega^{2}a^{2}$. The potential is
therefore characterized by an infinite wall on the left and a finite wall on
the right. Hence, it is a semi-infinite quantum well with a non-rectangular
profile.

At this stage, one may observe that the BenDaniel-Duke form [32] of the
kinetic energy operator adopted in (1) is only a special case of the von Roos
general two-parameter form [33]. Other orderings of the mass and the
differential operator, such as the Zhu-Kroemer [34] or the Mustafa-
Mazharimousavi [35, 36] ones, which pass the de Souza Dutra and Almeida test
[37], might have been chosen, but, as shown elsewhere [19], in general they do
not change the results much.

In Ref. [29], the bound-state solutions of eq. (1) were shown to correspond to
the energies

$E_{n}=\omega\left(n+\frac{1}{2}\right)-\frac{1}{a^{2}}n(n+1),\qquad
n=0,1,\ldots,N,\qquad N<\frac{1}{2}(\omega a^{2}-1).$ (3)

The corresponding wavefunctions were obtained in the form

$\psi_{n}(x)\propto(x+a)^{-\frac{1}{2}\omega a^{2}+n}\exp\left(-\frac{\omega
a^{3}}{2(x+a)}\right)L_{n}^{(\omega a^{2}-2n-1)}\left(\frac{\omega
a^{3}}{x+a}\right),$ (4)

expressed in terms of Laguerre polynomials $L_{n}^{(\alpha)}(z)$ and vanishing
at $x=-a$ and $x\to+\infty$. This form was then transformed into

$\psi_{n}(x)\propto(x+a)^{-\frac{1}{2}\omega a^{2}}\exp\left(-\frac{\omega
a^{3}}{2(x+a)}\right)y_{n}\left(2\frac{x+a}{\omega a^{3}};-\omega
a^{2}\right),$ (5)

where $y_{n}(z;\alpha)$ denotes a Bessel polynomial. It is this latter form
whose normalization coefficient was calculated and that was shown to reduce to
the well-known wavefunctions of the constant-mass harmonic oscillator in the
$a\to+\infty$ limit. For $E>\frac{1}{4}\omega^{2}a^{2}$, eq. (1) also has a
continuous spectrum, whose wavefunctions were briefly discussed in [29].

These results may be alternatively obtained by using the PCT technique applied
to a constant-mass Schrödinger equation [16]

$\left(-\frac{d^{2}}{du^{2}}+U(u)\right)\phi(u)=\epsilon\phi(u)$ (6)

for an appropriately chosen potential $U(u)$, defined on a finite or infinite
interval. Such a method uses a change of variable

$u(x)=\bar{a}v(x)+\bar{b},\qquad
v(x)=\int^{x}\sqrt{M(x^{\prime})}\,dx^{\prime},$ (7)

and a change of function

$\phi(u(x))\propto[M(x)]^{-1/4}\psi(x).$ (8)

Here, $\bar{a}$ and $\bar{b}$ are assumed to be two real parameters. The
potential $V_{\rm eff}(x)$, defined on a possibly different interval, and the
bound-state energies $E_{n}$ of the PDM Schrödinger equation are given in
terms of the potential $U(u)$ and the bound-state energies $\epsilon_{n}$ of
the constant-mass one by

$V_{\rm
eff}(x)=\bar{a}^{2}U(u(x))+\frac{M^{\prime\prime}}{4M^{2}}-\frac{7M^{\prime
2}}{16M^{3}}+\bar{c},$ (9)

and

$E_{n}=\bar{a}^{2}\epsilon_{n}+\bar{c},$ (10)

where a prime denotes derivation with respect to $x$ and $\bar{c}$ is some
additional real constant. From (8), the corresponding bound-state
wavefunctions are given by

$\psi_{n}(x)=\lambda[M(x)]^{1/4}\phi_{n}(u(x))$ (11)

provided they are normalizable on the definition interval of $x$. Here
$\lambda$ is some constant which may arise from the change of normalization
when going from $u$ to $x$.

For the mass chosen in (2), one finds that the mass-dependent term in (9) is
given by

$\frac{M^{\prime\prime}}{4M^{2}}-\frac{7M^{\prime
2}}{16M^{3}}=-\frac{1}{4a^{2}}.$ (12)

and that the change of variable (7) writes

$u(x)=\bar{a}a\log(x+a)+\bar{b}.$ (13)

On assuming

$\bar{a}=\frac{1}{a},\qquad\bar{b}=0,$ (14)

the latter reduces to

$u(x)=\log(x+a)\qquad\text{or}\qquad e^{u(x)}=x+a.$ (15)

From (15), it looks appropriate to assume for the potential $U(u)$ in the
starting constant-mass Schrödinger equation (6) some function of the real
exponential $e^{u}$. Let us consider the Morse potential [30], which may be
written as [31]

$U(u)=B^{2}e^{-2u}-B(2A+1)e^{-u},\qquad-\infty<u<+\infty,$ (16)

where $A$ and $B$ are two positive constants. Its bound-state eigenvalues are
given by

$\epsilon_{n}=-(A-n)^{2},\qquad n=0,1,\ldots,n_{\rm max},\qquad n_{\rm
max}<A,$ (17)

and the corresponding eigenfunctions are

$\phi_{n}(u)={\cal N}_{n}\exp[-(A-n)u-Be^{-u}]L_{n}^{(2A-2n)}(2Be^{-u}),$ (18)

with a normalization constant

${\cal N}_{n}=(2B)^{A-n}\left(\frac{n!(2A-2n)}{\Gamma(2A+1-n)}\right)^{1/2}.$
(19)

On applying eqs. (9), (12), (14), and (15), potential (16) gets transformed
into

$V_{\rm
eff}(x)=\frac{1}{a^{2}}\left(\frac{B^{2}}{(x+a)^{2}}-\frac{B(2A+1)}{x+a}\right)-\frac{1}{4a^{2}}+\bar{c},$
(20)

which reduces to the potential $V_{\rm eff}(x)$ given in (2), provided we
choose

$A=\frac{1}{2}(\omega a^{2}-1),\qquad B=\frac{1}{2}\omega
a^{3},\qquad\bar{c}=\frac{1}{4}\omega^{2}a^{2}+\frac{1}{4a^{2}}.$ (21)

From (10) and (17), its bound-state energies turn out to be given by (3) with
$N=n_{\rm max}$, as it should be. From (11), the corresponding wavefunctions
are obtained in the form

$\psi_{n}(x)=N_{n}(x+a)^{-\frac{1}{2}\omega a^{2}+n}\exp\left(-\frac{\omega
a^{3}}{2(x+a)}\right)L_{n}^{(\omega a^{2}-2n-1)}\left(\frac{\omega
a^{3}}{x+a}\right)$ (22)

with

$N_{n}=\lambda\sqrt{a}\,{\cal N}_{n}$ (23)

and

${\cal N}_{n}=(\omega a^{3})^{\frac{1}{2}(\omega
a^{2}-1)-n}\left(\frac{n!(\omega a^{2}-1-2n)}{\Gamma(\omega
a^{2}-n)}\right)^{1/2},$ (24)

coming from (19) and (21). The remaining undetermined constant $\lambda$ can
be easily found by making the change of variable $e^{u}=x+a$ in the
normalization integral for $\psi_{n}(x)$ and using the known normalization
coefficient ${\cal N}_{n}$ of $\psi_{n}(u(x))$. The result is
$\lambda=1/\sqrt{a}$, so that $N_{n}$ of eq. (22) simply reduces to ${\cal
N}_{n}$ given in (24).

We have therefore proved that the bound states of the semi-infinite quantum
well with a non-rectangular profile of [29] can be easily derived from those
of the constant-mass Morse potential. Its continuous spectrum eigenfunctions
might also be obtained from those of the latter.

##  3 Construction of another semi-infinite quantum well by the PCT technique

In the present section, we will assume the same mass $M(x)$ as in (2), which
implies that eqs. (12) and (13) remain valid. With the same choice as in (14),
we then get eq. (15).

Another potential for which the constant-mass Schrödinger equation is exactly
solvable and that can be written in terms of the exponential $e^{u}$ is the
Rosen-Morse II potential [38]

$U(u)=-A(A+1)\operatorname{sech}^{2}u+2B\tanh u,\qquad-\infty<u<+\infty,\qquad
B<A^{2},$ (25)

whose bound-state energies and wavefunctions are given by [31]

$\epsilon_{n}=-(A-n)^{2}-\frac{B^{2}}{(A-n)^{2}},\qquad n=0,1,\ldots,n_{\rm
max},\qquad n_{\rm max}<A-\sqrt{|B|},$ (26)

and

$\displaystyle\phi_{n}(x)$ $\displaystyle={\cal N}_{n}(1-\tanh
u)^{\left(A-n+\frac{B}{A-n}\right)/2}(1+\tanh
u)^{\left(A-n-\frac{B}{A-n}\right)/2}$ $\displaystyle\quad\times
P_{n}^{\left(A-n+\frac{B}{A-n},A-n-\frac{B}{A-n}\right)}(\tanh u),$ (27)

where $P_{n}^{(\alpha,\beta)}(z)$ denotes a Jacobi polynomial and ${\cal
N}_{n}$ is a normalization coefficient given by [39]

${\cal
N}_{n}=2^{n-A}\left(\frac{n!\Gamma(2A-n+1)[(A-n)^{2}-B^{2}/(A-n)^{2}]}{(A-n)\Gamma(A+1+B/(A-n))\Gamma(A+1-B/(A-n))}\right)^{1/2}.$
(28)

In the special case where $B=0$, eqs. (25)–(28) reduce to

$U(u)=-A(A+1)\operatorname{sech}^{2}u,\qquad-\infty<u<+\infty,$ (29)
$\epsilon_{n}=-(A-n)^{2},\qquad n=0,1,\ldots,n_{\rm max},\qquad n_{\rm
max}<A,$ (30) $\phi_{n}(u)=\bar{{\cal
N}}_{n}(\operatorname{sech}u)^{A-n}C_{n}^{\left(A-n+\frac{1}{2}\right)}(\tanh
u),$ (31) $\bar{{\cal
N}}_{n}=\frac{\Gamma(2A-2n+1)}{2^{A-n}\Gamma(A-n+1)}\left(\frac{(A-n)n!}{\Gamma(2A-n+1)}\right)^{1/2},$
(32)

where the Jacobi polynomials with equal parameters have been expressed in
terms of Gegenbauer ones using eq. (22.5.20) of Ref. [40] and the new
normalization factor $\bar{\cal N}_{n}$ combines the factor occurring in the
latter equation with the normalization factor (28).

On applying eqs. (9), (12), (14), and (15) to potential (29), we obtain

$V_{\rm
eff}(x)=V_{0}\left(\frac{1}{[(x+a)^{2}+1]^{2}}-\frac{1}{(x+a)^{2}+1}\right),\qquad-a<x<+\infty,$
(33)

where we have set

$\bar{c}=\frac{1}{4a^{2}},\qquad V_{0}=\frac{4A(A+1)}{a^{2}}>0.$ (34)

We note that

$\lim_{x\to-a}V_{\rm eff}(x)=\lim_{x\to+\infty}V_{\rm eff}(x)=0$ (35)

and we complete the definition of $V_{\rm eff}(x)$ by setting

$V_{\rm eff}(x)=+\infty\qquad\text{for}\qquad x<-a,$ (36)

which agrees with the behaviour of the mass $M(x)$. On the interval
$(-a,+\infty)$, the potential is negative and has a minimum given by

$V_{\rm eff}(x_{\rm min})=-\frac{V_{0}}{4}\qquad\text{for}\qquad x_{\rm
min}=-a+1.$ (37)

It is therefore another semi-infinite quantum well associated with the mass
defined in (2). An example of such a potential is displayed in Fig. 1.

Figure 1: Plot of the semi-infinite quantum well defined in (33) and (36) in
terms of $x$ for $a=1$ and $V_{0}=48$.

From eq. (34), it turns out that $A$ may be written as

$A=\nu-\frac{1}{2},\qquad\nu\equiv\frac{1}{2}\sqrt{1+a^{2}V_{0}},$ (38)

so that on using eqs. (10), (14), and (34), we obtain that the bound-state
spectrum of $V_{\rm eff}(x)$ is given by

$E_{n}=-\frac{1}{a^{2}}(\nu-n)(\nu-n-1),\qquad n=0,1,\ldots,n_{\rm max},$ (39)

while the corresponding wavefunctions can be expressed as

$\psi_{n}(x)=N_{n}\frac{(x+a)^{\nu-n-1}}{[(x+a)^{2}+1]^{\nu-n-\frac{1}{2}}}C_{n}^{(\nu-n)}\left(\frac{(x+a)^{2}-1}{(x+a)^{2}+1}\right),\qquad-a<x<+\infty.$
(40)

Provided $n_{\rm max}$ is restricted to values such that $n_{\rm max}<\nu-1$,
these wavefunctions vanish for $x\to-a$ and $x\to\infty$, as it should be.
Their normalization coefficient is given by

$N_{n}=\lambda\sqrt{a}2^{\nu-\frac{1}{2}-n}\bar{\cal N}_{n},$ (41)

where the remaining undetermined constant $\lambda$ can be obtained by making
the change of variable $e^{u}=x+a$ in the normalization integral of
$\psi_{n}(x)$ on $(-a,+\infty)$. The result reads $\lambda=1/\sqrt{a}$, so
that

$N_{n}=\frac{\Gamma(2\nu-2n)}{\Gamma(\nu-n-\frac{1}{2})}\left(\frac{n!}{(\nu-n-\frac{1}{2})\Gamma(2\nu-n)}\right)^{1/2}.$
(42)

As an example, we show in Fig. 2 the three bound-state wavefunctions of the
semi-infinite quantum well plotted in Fig. 1,

$\displaystyle\psi_{0}(x)$
$\displaystyle=2\sqrt{15}\frac{(x+1)^{5/2}}{[(x+1)^{2}+1]^{3}},$ (43)
$\displaystyle\psi_{1}(x)$
$\displaystyle=2\sqrt{15}\frac{(x+1)^{3/2}}{[(x+1)^{2}+1]^{3}}[(x+1)^{2}-1],$
(44) $\displaystyle\psi_{2}(x)$
$\displaystyle=2\sqrt{3}\frac{(x+1)^{1/2}}{[(x+1)^{2}+1]^{3}}[(x+1)^{4}-3(x+1)^{2}+1],$
(45)

corresponding to the energies $E_{0}=-\frac{35}{4}$, $E_{1}=-\frac{15}{4}$,
and $E_{2}=-\frac{3}{4}$, respectively.

Figure 2: Plot of $\psi_{n}(x)$, defined in (43), (44), and (45), in terms of
$x$ for $n=0$ (black line), $n=1$ (red line), and $n=2$ (green line).

##  4 Conclusion

In the present paper, we have first shown that the PCT method applied to the
constant-mass Schrödinger equation for the Morse potential allows us to easily
retrieve some of the results of Ref. [29]. In addition, we have proved that
normalized bound-state wavefunctions can be directly expressed in terms of
Laguerre polynomials, without having to transform them in terms of Bessel
polynomials.

In a second step, we have derived another example of semi-infinite quantum
well by starting from the Rosen-Morse II potential with $B=0$ and applying the
same technique.

Considering some other exactly solvable potentials with constant mass as
starting potential with the PDM defined in (2) may be an interesting topic for
future investigation.

## Acknowledgements

This work was supported by the Fonds de la Recherche Scientifique-FNRS under
Grant No. 4.45.10.08.

## Data availability statement

All data supporting the findings of this study are included in the article.

## Authors declarations

## Conflicts of interest

The author declares no conflict of interest.

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