# Local Near-Field Scattering Data Enables
Unique Reconstruction of Rough Electric Potentials

Manuel Cañizares  Manuel Cañizares
Basque Center for Applied mathematics, Bilbao, Spain<EMAIL_ADDRESS>
###### Abstract.

The focus of this paper is the study of the inverse point-source scattering
problem, specifically in relation to a certain class of electric potentials.
Our research provides a novel uniqueness result for the inverse problem with
local data, obtained from the near field pattern. Our work improves the work
of Caro and Garcia, who investigated both the direct problem and the inverse
problem with global near field data for critically singular and $\delta$-shell
potentials. The primary contribution of our research is the introduction of a
Runge approximation result for the near field data on the scattering problem
which, in combination with an interior regularity argument, enables us to
establish a uniqueness result for the inverse problem with local data.
Additionaly, we manage to consider a slightly wider class of potentials.

##  1\. Introduction

Along the paper, we will consider real potentials in dimension $d\geq 3$ that
can be written as

(1) $V=V^{0}+\gamma^{s}+\alpha\,\mathrm{d}\sigma,$

where $V^{0}\in L^{d/2}(\mathbb{R}^{d};\mathbb{R}),$ $\,\mathrm{d}\sigma$
denotes the surface measure of a compact hypersurface $\Gamma$ which is
locally described by the graph of Lipschitz functions, $\alpha\in
L^{\infty}(\Gamma;\mathbb{R})$ and $\gamma^{s}$ is of the form

$\gamma^{s}=\chi^{2}\,D^{s}g,$

for some $1/2<s<1$, $g\in L^{\infty}(\mathbb{R}^{d};\mathbb{R})$, and
$\chi\,\in\mathcal{C}^{\infty}(\mathbb{R}^{d};[0,1])$ is a cut-off function.
Here $D^{s}$ denotes the Riesz derivative, defined as
$\widehat{D^{s}f}(\xi)=|\xi|^{s}\widehat{f}(\xi)$. The supports of $V^{0}$ and
$\chi$ will be assumed to be compact. In essence, this means that the whole
potential $V$ is compactly supported. Note that the novel potentials that we
introduce in this work are just those of the class of $\gamma^{s}$, while
$V^{0}$ and $\alpha\,\mathrm{d}\sigma$ are those introduced by Caro and García
in [4], which they call critically singular and $\delta$-shell potentials,
respectively.
The direct problem consists in finding the wave scattered by the potential,
when an incident wave is emitted at fixed energy by a point source away from
its support. Mathematically, this translates to finding $u_{\\_}{sc}$ solving

(S)
$\begin{cases}\left(\Delta+\lambda-V\right)u_{\\_}{sc}(\centerdot\,,y)=Vu_{\\_}{in}(\centerdot\,,y)&\textrm{in
}\mathbb{R}^{d},\\\ u_{\\_}{sc}(\centerdot\,,y)\textrm{ satisfying
SRC},\end{cases}$

Here, SRC stands for Sommerfeld Radiation Condition. A function $u$ is said to
satisfy SRC if

(2) $\lim_{\\_}{|x|\to\infty}|x|^{\frac{d-1}{2}}\left(\frac{x}{|x|}\cdot\nabla
u(x)-i\lambda^{1/2}u(x)\right)=0$

uniformly in every direction, and $u_{\\_}{in}(x,y)=\Phi_{\\_}\lambda(x-y)$
denotes the incident wave emitted from the point
$y\in\mathbb{R}^{d}\setminus\mathrm{supp}\,V$, where $\Phi_{\\_}\lambda$
represents the fundamental solution for the Helmholtz equation with SRC, which
solves the distributional problem:

(F)
$\begin{cases}\left(\Delta+\lambda\right)\Phi_{\\_}\lambda=\delta_{\\_}0&\textrm{in
}\mathbb{R}^{d},\\\ \Phi_{\\_}\lambda\textrm{ satisfying SRC}.\end{cases}$

The SRC is classically introduced so that the solution to the Helmholtz
equation is unique and physically corresponds to a radiating wave [26].

In section 2, we will arrive to the following theorem:

######  Theorem 1.

Suppose $V$ is of the form (1). Then, there exists
$\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for every
$\lambda\geq\lambda_{\\_}0$, there is an unique solution
$u_{\\_}{sc}(\centerdot\,,y)\in X_{\\_}\lambda^{*}$ to the problem (S) for
every $y\in\mathbb{R}^{d}\setminus\mathrm{supp}\,V$.
Moreover, the mapping $Vu_{\\_}{in}(\centerdot\,,y)\mapsto
u_{\\_}{sc}(\centerdot\,,y)$ is bounded from $X_{\\_}\lambda$ to
$X_{\\_}\lambda^{*}$.

The spaces $X_{\\_}\lambda$ and $X_{\\_}\lambda^{*}$ are introduced in [4] and
their definition will be given at the beginning of section 2. The elements in
the space $X_{\\_}\lambda^{*}$ can be thought of as functions with one
derivative in $L^{2}$ that exhibit a certain degree of integrability in
frequences $|\xi|\sim\lambda^{1/2}$, and $X_{\\_}\lambda$ will be the pre-dual
of $X_{\\_}\lambda^{*}$. To prove the theorem, we will follow the argument in
[4] and make use of many of their results. In Caro and Garcia’s work, the
first step was to deal with the critically singular part of the potential by
obtaining an inverse for the operator $\Delta+\lambda-V^{0}$ via a Neumann-
series approach.

If we denote the inverse above by $(\Delta+\lambda+i0-V^{0})^{-1}$, it can be
applied to both sides of (S) to turn the problem into finding the inverse of
the operator

$I-(\Delta+\lambda+i0-V^{0})^{-1}\circ(\alpha\,\mathrm{d}\sigma+\gamma^{s}).$

The appropiate framework to study this operator is the Fredholm theory, due to
the fact that the operator defined as multiplication by
$(\alpha\,\mathrm{d}\sigma+\gamma^{s})$ is compact from $X_{\\_}\lambda^{*}$
to $X_{\\_}\lambda$. Fredholm alternative assures the existence of such an
inverse as long as the operator
$I-(\Delta+\lambda+i0-V^{0})^{-1}\circ(\alpha\,\mathrm{d}\sigma+\gamma^{s})$
is injective. Proving this injectivity will be equivalent to obtaining
uniqueness for the problem

(H) $\begin{cases}\left(\Delta+\lambda-V\right)u=0&\textrm{in
}\mathbb{R}^{d},\\\ u\textrm{ satisfying SRC}.\end{cases}$

The fundamental ingredient here is an unique continuation argument for the
operator $\Delta+\lambda-V$. In our case, it will be based on a Carleman
estimate proved by Caro and Rogers in [5] for a modified family of Bourgain-
type spaces. They were themselves based on those introduced by Haberman and
Tataru in [14] to study the Calderón problem. As we will note below, this
Carleman estimate will prove to be crucial for the solution of the inverse
problem.

Caro and Garcia applied this framework to invert the operator when when
$\alpha$ is in $L^{\infty}(\Gamma)$. We noticed that there was a possibility
to consider a wider range of potentials here. The idea is that
$\alpha\,\mathrm{d}\sigma$ acts as multiplication operator from
$H^{s}(\Omega)$ to $H^{-s}_{\\_}0(\Omega)$ with $s<1$, where $\Omega$ is any
domain containing the support of $V$. Then, the fact that $X_{\\_}\lambda^{*}$
is related to $H^{1}$, along with $\alpha\,\mathrm{d}\sigma$ being compactly
supported, and the fact $H^{1}(\Omega)$ is compactly embedded in
$H^{s}(\Omega)$, give the necessary ideas to prove compactness of the
multiplication operator. In principle, one could hope to consider $\alpha\in
L^{r}(\Gamma)$, with $r>d-1$, since $\alpha\,\mathrm{d}\sigma$ would still
behave as a compact multiplication operator form $H^{1}(\Omega)$ to
$H^{-1}_{\\_}0(\Omega)$. Nonetheless, the Carleman estimate doesn’t seem to
work in that case, and therefore we did not manage to relax this condition in
the solution of the inverse problem.

On the other hand, we attempted to consider distributions of the form
$D^{s}g$, with $s<1$ and $g\in L^{p}(\mathbb{R}^{d})$, hoping to find a class
of such distributions that generalized those of the form
$\alpha\,\mathrm{d}\sigma$. However, in principle, the set of indexes $(s,p)$
that made the Carleman estimate work for these distributions did not allow
them to see the hypersurface $\Gamma$. Therefore, we opted to consider
potentials that could be decomposed as a sum of those of both classes.
The arguments to find the scattering solution via Fredholm theory can be found
in section 2. Here, the structure is akin to that in [4], but some proofs have
to be redone in order for them to work for our wider class of potentials.
Afterwards, in sections 3 and 4, we will devote ourselves to proving the
following theorem concerning uniqueness with partial data for the inverse
problem.

######  Theorem 2.

Consider $d\geq 3$. Let $\Sigma_{\\_}1,\Sigma_{\\_}2$ be two relatively open
sets of dimension $d-1$, separated from $\mathrm{supp}\,V$, and that can be
expressed as the graph of $\mathcal{C}^{2}$ functions, and let $V_{\\_}1$ and
$V_{\\_}2$ be two potentials of the form described in (1). Let also
$u_{\\_}{sc,1}$ and $u_{\\_}{sc_{\\_}2}$ be the scattering solutions to the
problem (S) with respective potentials $V_{\\_}1$ and $V_{\\_}2$. Then, there
exists $\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for all
$\lambda\geq\lambda_{\\_}0$ except for at most a countable set, it holds that

${\left.\kern-1.2ptu_{\\_}{sc,1}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}={\left.\kern-1.2ptu_{\\_}{sc,2}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}\implies
V_{\\_}1=V_{\\_}2.$

From a physics perspective, we are stating that the identifiability of the
potential is possible by placing sources and detectors only in small pieces of
hypersurfaces away from the support of $V$. These sources will be emitting
monochromatic waves with a fixed energy $\lambda$. Note that $\Sigma_{\\_}1$
and $\Sigma_{\\_}2$ could well be the same set, or be completely separate and
non-intersecting.
We will prove this theorem via an orthogonality relation in the spirit of
Alessandrini’s identity for the Calderón problem [2] and the construction of
CGO solutions as in [28, 5, 16]. This orthogonality relation will be proven in
section 3. The main ingredient will be a Runge approximation result, indexed
as proposition 3.4, given in section 3.1. We will take a bounded open domain
$\Omega$, of class $\mathcal{C}^{2}$, whose boundary contains both
$\Sigma_{\\_}1$ and $\Sigma_{\\_}2$, and the Runge approximation will allow us
to approximate solutions in $\Omega$ by single layer potentials with densities
that are supported in any subset $\Sigma\subset\partial\Omega$:

$\mathcal{S}f(x)=\int_{\\_}{\partial\Omega}f(y)\,u_{\\_}{to}(x,y)\,\mathrm{d}y,$

where $f\in\mathcal{C}(\partial\Omega)$, $\mathrm{supp}\,f\subset\Sigma$. Here
the total wave is defined as
$u_{\\_}{to}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=u_{\\_}{in}+u_{\\_}{sc}$.
The result will give an approximation in the $L^{2}$ norm of a smaller open
domain $\Omega^{\prime}$ strictly contained in $\Omega$ and containing the
support of $V$. The proof of this lemma was inspired by that of Isakov for the
Calderón problem for $C^{2}$ conductivities [19] and that of Harrach, Pohjola
and Salo for the recovery of an $L^{\infty}$ scattering coefficient function
in the Helmholtz equation [15]. Nonetheless, the argument is slightly
different, since we consider approximating solutions of a different kind.
Also, there are some technicalities concerning the singularities of the
fundamental solutions, which have to be treated with a little bit of care.
This argument will allow us to extend an orthogonality relation of the type

(3) $\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=0,$

with $v_{\\_}1$,$v_{\\_}2$ being solutions that only see $\Sigma_{\\_}j$, to
an orthogonality relation of this type for all solutions to the equation.

However, the approximation in $L^{2}$ won’t be enough, due to the low
integrability of our potentials. We will in fact need to approximate our
solutions in an $H^{1}$ norm so that the solutions can be integrated against
the potential. Therefore, we will provide in lemma 3.5 with an interior
regularity result. In particular, for any $u$ solution of (S), we will obtain

$\lVert u\rVert_{\\_}{H^{1}(\Omega)}\lesssim\lVert
u\rVert_{\\_}{L^{2}(\Omega^{\prime})},$

as long as $\Omega$ is a domain strictly contained in $\Omega^{\prime}$ and
also containing the support of $V$. We will prove this inequality following an
argument by Chen that appears in [6], who proved $H^{2}$ regularity for second
order elliptic equations with a degree of regularity that in our case would
ask for $V\in L^{\infty}$. We adapt the argument to work for our kind of
potentials, which can be done thanks to the fact that
$\alpha\,\mathrm{d}\sigma+\gamma^{s}$ acts as a bilinear form over $H^{s}$,
which is an interpolation space between $L^{2}$ and $H^{1}$.
Then, the obtention of the final orthogonality relation will be given in
section 3.2.

######  Remark.

It might be interesting to note that, in our case, the only sets that are
given by the problem and that have a clear physical meaning are the supports
of the potentials and the measuring sets $\Sigma_{\\_}1$ and $\Sigma_{\\_}2$.
We will construct ad hoc the domains in which we obtain the orthogonality
relation (3). To make the arguments work, we need the frequency $\lambda$ not
to be a Neumann eigenvalue for these domains. Since the set of Neuman
eigenvalues in each domain is countable, we can already assert that the proofs
go through for all $\lambda\geq\lambda_{\\_}0$ except for at most a countable
set, as in the statement of Theorem 2. In [27], Stefanov used the monotonicity
of Dirichlet eigenvalues with respect to domain inclusion to prove that, for
any value of $\lambda$, it is possible to construct a domain such that
$\lambda$ is not a Dirichlet eigenvalue. However, this monotonicity doesn’t
seem to hold for Neumann eigenvalues for the Laplacian [11], and we have no
reason to think that it holds for $-\Delta+V$. Because of this, we have not
been able to prove the existence of an appropiate domain for every value of
$\lambda$. Nonetheless, given the amount of freedom that one has when choosing
the domain, it would be reasonable to expect that such a domain exists. The
analyis of this question is left for a future work.

Finally, to end the proof of Theorem 2, we will test the aforementioned
identity with a special type of solutions, the so-called complex geometric
optics (CGO) solutions. This is a classical method that goes back to Sylvester
and Uhlmann’s work [28] in the Calderón problem. In our case, these special
solutions are of the form

$v_{\\_}j(x)=e^{\zeta_{\\_}j\cdot x}(1+w_{\\_}j(x)),$

where $\zeta_{\\_}j\in\mathbb{C}^{d}$ are chosen such that
$\zeta_{\\_}j\cdot\zeta_{\\_}j=-\lambda$ and
$\zeta_{\\_}1+\zeta_{\\_}2=-i\kappa$ for an arbitray $\kappa\in\mathbb{R}^{d}$
-which is possible in dimension $d\geq 3$\- and the correction term $w_{\\_}j$
vanishes in a certain sense when $|\zeta_{\\_}j|$ grows. To prove the
existence of these solutions, we will follow the construction of both Caro and
García in [4] and Caro and Rogers in [5]. The key ingredient again will be the
aforementioned Carleman estimate. In particular, applying the operator
$(\Delta+\lambda-V)$ to $v_{\\_}j$ yields

$(\Delta+2\zeta_{\\_}j\cdot\nabla-V)w_{\\_}j=V.$

Therefore, to construct the CGO solutions it is enough to prove injectivity of
the adjoint operator $(\Delta-2\zeta_{\\_}j\cdot\nabla-V)$, which can be done
via a priori estimates. We will define a family of Bourgain spaces
$X_{\\_}\zeta^{s}$ via the norm

$\lVert
u\rVert_{\\_}{X_{\\_}\zeta^{s}}=\lVert(M|\Re(\zeta)|^{2}+M^{-1}|p_{\\_}\zeta|^{2})^{s/2}\widehat{u}\rVert_{\\_}{L^{2}}$

with $M>1$, where $\Re$ denotes the real part and

$p_{\\_}\zeta(\xi)=-|\xi|^{2}+2i\zeta\cdot\xi+\zeta\cdot\zeta.$

The index $s={1/2}$ will play an important role. Realising that the dual of
$X_{\\_}\zeta^{s}$ is $X_{\\_}{-\zeta}^{-s}$, if we want to prove the
existence of solutions in $X_{\\_}\zeta^{1/2}$ with potentials $V\in
X_{\\_}\zeta^{-{1/2}}$, we will need to prove the injectivity of the operator
from $X_{\\_}{-\zeta}^{-{1/2}}$ to $X_{\\_}{-\zeta}^{1/2}$. Therefore, the a
priori estimate that we will seek will be of the form

$\lVert
u\rVert_{\\_}{X_{\\_}{-\zeta}^{1/2}}\lesssim\lVert(\Delta-2\zeta\cdot\nabla-V)\,u\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}$

This will be done in section 4.

######  Acknowledgements.

The author would like to thank his supervisor Pedro Caro for his guidance and
invaluable ideas. He would also like to thank Antti Kykkänen and María Ángeles
García Ferrero for their comments. The author is funded by FPI fellowship
PRE2019-091776 and the project PID2021-122156NB-I00.

##  2\. Direct Scattering

In this section, we follow the strategy by Caro and García to solve the direct
problem for the point-source scattering. As the name of the problem suggests,
we consider an incident wave emitted by a point source at fixed energy
$\lambda>0$. This means that the spatial part of the incident wave emitted at
a point $y\in\mathbb{R}^{d}$ and measured at another point
$x\in\mathbb{R}^{d}$ will be given by
$u_{\\_}{in}(x,y)=\Phi_{\\_}\lambda(x-y)$, where $\Phi_{\\_}\lambda$ is the
radiating fundamental equation to the Helmholtz equation, i.e.
$\Phi_{\\_}\lambda$ is the solution to the problem

(4)
$\begin{cases}\left(\Delta+\lambda\right)\Phi_{\\_}\lambda=\delta_{\\_}0&\textrm{in
}\mathbb{R}^{d},\\\ \Phi_{\\_}\lambda\textrm{ satisfying SRC}.\end{cases}$

We denote by $\left(\Delta+\lambda+i0\right)^{-1}$ the solution operator for
the Helmholtz equation with SRC (2). Under this notation, we can write
$\Phi_{\\_}\lambda=\left(\Delta+\lambda+i0\right)^{-1}\delta_{\\_}0$, which
for instance can be expressed in terms of the Fourier symbol for the operator
$\Delta+\lambda$. If we denote the modulus of this symbol by
$m_{\\_}\lambda(\xi)=|\lambda-|\xi|^{2}|$, it can be checked that the
fundamental solution $\Phi_{\\_}\lambda$ is given, in the distributional
sense, as

(5)
$\braket{\Phi_{\\_}\lambda,f}=\frac{1}{(2\pi)^{d/2}}\left[\lim_{\\_}{\varepsilon\to
0}\int_{\\_}{m_{\\_}\lambda>\varepsilon}\frac{\widehat{f}(\xi)}{\lambda-|\xi|^{2}}\,\mathrm{d}\xi-i\frac{\pi}{2\lambda^{1/2}}\int_{\\_}{S_{\\_}\lambda}\widehat{f}(\xi)\,\mathrm{d}S_{\\_}\lambda(\xi)\right],$

for $f\in\mathscr{S}(\mathbb{R}^{d})$. Here,
$S_{\\_}\lambda=\\{\xi\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|\xi|=\lambda^{1/2}\\}$
is the critical hypersurface of the symbol $m_{\\_}\lambda$, and
$\,\mathrm{d}S_{\\_}\lambda$ denotes its volume form.

###  2.1. The free resolvent and Neumann series

As mentioned in the introduction, the scattering solution is constructed in
certain spaces $X_{\\_}\lambda^{*}$ that were considered by Caro and García in
[4]. In those spaces, refinements of estimates by Agmon and Hörmander [1], by
Kenig, Ruiz and Sogge [22] and by Ruiz and Vega [25] give us a good estimate
for the free resolvent $(\Delta+\lambda+i0)^{-1}$. We are going to recall the
definition of these spaces. For this, we need to construct a partition of
unity. Indeed, choose a function $\phi\in\mathscr{S}(\mathbb{R}^{d})$
supported in
$\\{\xi\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|\xi|\leq 2\\}$
such that $\phi(\xi)=1$ whenever $|\xi|\leq 1$ and define, for
$k\in\mathbb{Z}$,
$\psi_{\\_}k(\xi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\phi(2^{-k}\xi)-\phi(2^{-k+1}\xi)$.
Note that $\psi_{\\_}k$ is supported in
${\\{\xi\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,2^{k-1}\leq|\xi|\leq
2^{k+1}\\}}$ and ${\sum_{\\_}{k\in\mathbb{Z}}\psi_{\\_}k(\xi)=1}$ for $\xi\neq
0$. We define the Littlewood-Paley projectors as

(6)
$\displaystyle\begin{split}\widehat{P_{\\_}kf}(\xi)&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\psi_{\\_}k(\xi)\widehat{f}(\xi),\\\
\widehat{P_{\\_}{\leq
k}f}(\xi)&\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\sum_{\\_}{j\leq
k}\widehat{P_{\\_}jf}(\xi)=\phi(2^{-k}\xi)\widehat{f}(\xi).\end{split}$

Now, for $\lambda>0$, let $k_{\\_}\lambda\in\mathbb{Z}$ be such that
$2^{k_{\\_}\lambda-1}<\lambda^{1/2}\leq 2^{k_{\\_}\lambda}$. Then, the set of
indices in the projectors above that see the critical frequencies is

$I=\\{k_{\\_}\lambda-2,k_{\\_}\lambda-1,k_{\\_}\lambda,k_{\\_}\lambda+1\\}.$

For simplicity, we will call
$P_{\\_}{<I}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=P_{\\_}{\leq
k_{\\_}\lambda-3}$. Define now the space $B$ and its dual $B^{*}$, as in [1],
via the norms

$\lVert
f\rVert_{\\_}{B}=\sum_{\\_}{j\in\mathbb{N}_{\\_}0}\,\left(2^{j/2}\lVert
f\rVert_{\\_}{L^{2}(D_{\\_}j)}\right),\qquad\lVert
u\rVert_{\\_}{B^{*}}=\sup_{\\_}{j\in\mathbb{N}_{\\_}0}\,\left(2^{-j/2}\lVert
u\rVert_{\\_}{L^{2}(D_{\\_}j)}\right),$

with
$D_{\\_}j=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,2^{j-1}<|x|\leq
2^{j}\\}$ for $j\in\mathbb{N}$ and
$D_{\\_}0=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|x|\leq
1\\}.$ From now on, denote by $q_{\\_}d$ the end-point index for the Stein-
Tomas trace theorem, $2/q_{\\_}d=(d-1)/(d+1)$, by $p_{\\_}d$ the end-point
index for the $\dot{H}^{1}$-Hardy-Littlewood-Sobolev embedding theorem,
$1/p_{\\_}d=1/2-1/d$ for $d\geq 3$, and by $q^{\prime}_{\\_}d$ and
$p^{\prime}_{\\_}d$ their respective Hölder conjugates. The space
$X_{\\_}\lambda$ can be defined as the sum of two spaces, $Y_{\\_}\lambda$ and
$Z_{\\_}\lambda$, which are defined as elements in
$f\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ with norms

$\lVert
f\rVert^{2}_{\\_}{Y_{\\_}\lambda}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lVert
m_{\\_}\lambda^{-{1/2}}\widehat{P_{\\_}{<I}f}\rVert^{2}_{\\_}{L^{2}}+\sum_{\\_}{k\in
I}\lambda^{-{1/2}}\lVert
P_{\\_}kf\rVert^{2}_{\\_}B+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{-{1/2}}\widehat{P_{\\_}kf}\rVert^{2}_{\\_}{L^{2}},$

and

$\lVert
f\rVert^{2}_{\\_}{Z_{\\_}\lambda}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lVert
m_{\\_}\lambda^{-{1/2}}\widehat{P_{\\_}{<I}f}\rVert^{2}_{\\_}{L^{2}}+\sum_{\\_}{k\in
I}\lambda^{d(\frac{1}{q_{\\_}d^{\prime}}-\frac{1}{p_{\\_}d^{\prime}})}\lVert
P_{\\_}kf\rVert^{2}_{\\_}{L^{q_{\\_}d^{\prime}}}+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{-{1/2}}\widehat{P_{\\_}kf}\rVert^{2}_{\\_}{L^{2}}.$

The norm in $X_{\\_}\lambda$ will be the usual for the sum of normed spaces:

$\lVert f\rVert_{\\_}{X_{\\_}\lambda}=\inf_{\\_}{g+h=f}\\{\lVert
g\rVert_{\\_}{Y_{\\_}\lambda}+\lVert h\rVert_{\\_}{Z_{\\_}\lambda}\\}.$

The spaces above have respective dual spaces $Y_{\\_}\lambda^{*}$ and
$Z_{\\_}\lambda^{*}$ defined by the norms

$\lVert
u\rVert^{2}_{\\_}{Y^{*}_{\\_}\lambda}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}{<I}f}\rVert^{2}_{\\_}{L^{2}}+\sum_{\\_}{k\in
I}\lambda^{{1/2}}\lVert
P_{\\_}kf\rVert^{2}_{\\_}{B^{*}}+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}kf}\rVert^{2}_{\\_}{L^{2}},$

and

$\lVert
u\rVert^{2}_{\\_}{Z^{*}_{\\_}\lambda}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}{<I}f}\rVert^{2}_{\\_}{L^{2}}+\sum_{\\_}{k\in
I}\lambda^{d(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
P_{\\_}kf\rVert^{2}_{\\_}{L^{q_{\\_}d}}+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}kf}\rVert^{2}_{\\_}{L^{2}}.$

Then, the space $X^{*}_{\\_}\lambda$ can be defined as the dual space of
$X_{\\_}\lambda$, which will be isomorphic to the intersection of
$Y_{\\_}\lambda^{*}$ and $Z_{\\_}\lambda^{*}$, with norm

$\displaystyle\lVert u\rVert^{2}_{\\_}{X^{*}_{\\_}\lambda}=$
$\displaystyle\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}{<I}f}\rVert^{2}_{\\_}{L^{2}}+\sum_{\\_}{k\in
I}\left(\lambda^{{1/2}}\lVert
P_{\\_}kf\rVert^{2}_{\\_}{B^{*}}+\lambda^{d(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
P_{\\_}kf\rVert^{2}_{\\_}{L^{q_{\\_}d}}\right)+$
$\displaystyle\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{{1/2}}\widehat{P_{\\_}kf}\rVert^{2}_{\\_}{L^{2}}\sim\lVert
u\rVert^{2}_{\\_}{Y^{*}_{\\_}\lambda}+\lVert
u\rVert^{2}_{\\_}{Z^{*}_{\\_}\lambda}.$

It will be interesting to note that the Schwartz class is dense in all the
above spaces with respect to their corresponding norms [4]. We will now obtain
an estimate for the solution operator $X_{\\_}\lambda$ and
$X_{\\_}\lambda^{*}$. However, let us first prove the following lemma.

######  Lemma 2.1.

Let $1\leq p\leq\infty$ and $s\in\mathbb{R}$. Define $D^{s}$ as the Fourier
multiplier with symbol $|\xi|^{s}$. Then, it holds that

$\lVert D^{s}P_{\\_}kf\rVert_{\\_}{L^{p}}\sim 2^{ks}\lVert
P_{\\_}kf\rVert_{\\_}{L^{p}}.$

######  Proof:.

Fix $s\in\mathbb{R}$. We have, by definition,

$\widehat{D^{s}P_{\\_}kf}(\xi)=|\xi|^{s}\psi_{\\_}k(\xi)\widehat{f}(\xi)=2^{ks}\left(\frac{|\xi|}{2^{k}}\right)^{s}\psi\left(\frac{\xi}{2^{k}}\right)\widehat{f}(\xi).$

If we take $\chi$ to be a smooth cut-off function such that $0\leq\chi\leq 1$,
$\chi=0$ around the origin and $\chi=1$ on $\mathrm{supp}\,\psi$, then

$\widehat{D^{s}P_{\\_}kf}(\xi)=2^{ks}\left(\frac{|\xi|}{2^{k}}\right)^{s}\chi\left(\frac{\xi}{2^{k}}\right)\psi\left(\frac{\xi}{2^{k}}\right)\widehat{f}(\xi).$

Note that the function
$\rho(\xi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=|\xi|^{s}\chi(\xi)$
is compactly supported away from $0$ and therefore
$\rho\in\mathcal{C}^{\infty}_{\\_}c(\mathbb{R}^{d})$ for any value of $s$.
Now,

$\widehat{D^{s}P_{\\_}kf}(\xi)=2^{ks}\rho\left(\frac{\xi}{2^{k}}\right)\widehat{P_{\\_}kf}(\xi),$

and thus

$D^{s}P_{\\_}kf\sim 2^{ks}\left[(P_{\\_}kf)\ast
2^{kd}\check{\rho}(2^{k}\centerdot)\right].$

Therefore, by Young’s inequality,

$\lVert D^{s}P_{\\_}kf\rVert_{\\_}{L^{p}}\lesssim 2^{ks}\lVert
P_{\\_}kf\rVert_{\\_}{L^{p}}\lVert
2^{kd}\check{\rho}(2^{k}\centerdot)\rVert_{\\_}{L^{1}}=2^{ks}\lVert
P_{\\_}kf\rVert_{\\_}{L^{p}}\lVert\check{\rho}\rVert_{\\_}{L^{1}}\lesssim
2^{ks}\lVert P_{\\_}kf\rVert_{\\_}{L^{p}}.$

Now, to get the reverse inequality, observe that

$\widehat{P_{\\_}kf}(\xi)=|\xi|^{-s}|\xi|^{s}\widehat{P_{\\_}kf}(\xi)=\mathcal{F}[D^{-s}D^{s}P_{\\_}kf](\xi),$

where $\mathcal{F}$ denotes the Fourier transform. Therefore,

$\lVert P_{\\_}kf\rVert_{\\_}{L^{p}}\sim\lVert
D^{-s}D^{s}P_{\\_}kf\rVert_{\\_}{L^{p}}\lesssim 2^{-ks}\lVert
D^{s}P_{\\_}kf\rVert_{\\_}{L^{p}},$

and thus

$2^{ks}\lVert P_{\\_}kf\rVert_{\\_}{L^{p}}\lesssim\lVert
D^{s}P_{\\_}kf\rVert_{\\_}{L^{p}}.$

∎

We are ready now for the desired estimate.

######  Theorem 2.2.

There exists a constant $C>0$ depending only on $d$ such that

$\lVert(\Delta+\lambda+i0)^{-1}f\rVert_{\\_}{X^{*}_{\\_}\lambda}\leq C\lVert
f\rVert_{\\_}{X_{\\_}\lambda}$

for every $f\in\mathscr{S}(\mathbb{R}^{d})$.

######  Proof:.

In [4], Caro and García proved the inequalities

$\lVert(\Delta+\lambda+i0)^{-1}f\rVert_{\\_}{Y^{*}_{\\_}\lambda}\lesssim_{\\_}d\lVert
f\rVert_{\\_}{Y_{\\_}\lambda},$

and

$\lVert(\Delta+\lambda+i0)^{-1}f\rVert_{\\_}{Z^{*}_{\\_}\lambda}\lesssim_{\\_}d\lVert
f\rVert_{\\_}{Z_{\\_}\lambda}.$

Now, for the diagonal inequalities, note that
$P_{\\_}k(\Delta+\lambda+i0)^{-1}f=(\Delta+\lambda+i0)^{-1}P_{\\_}kf$ and
therefore it follows that, for any $k\notin I$,

$\braket{P_{\\_}k(\Delta+\lambda+i0)^{-1}f,\overline{g}}=\frac{1}{(2\pi)^{d/2}}\int_{\\_}{\mathbb{R}^{d}}\frac{\widehat{P_{\\_}kf}(\xi)\,\overline{\widehat{g}(\xi)}}{\lambda-|\xi|^{2}}\,\mathrm{d}\xi,$

since the frequencies of $P_{\\_}kf$ are separated from $S_{\\_}\lambda$. This
implies, by Plancherel’s identity, that

(7) $\lVert
m_{\\_}\lambda^{{1/2}}\mathcal{F}[P_{\\_}k(\Delta+\lambda+i0)^{-1}f]\rVert_{\\_}{L^{2}}=\lVert
m_{\\_}\lambda^{-{1/2}}\widehat{P_{\\_}kf}\rVert_{\\_}{L^{2}},$

On the other hand, Theorem 3.1 in [25] states that, for
$u\in\mathscr{S}(\mathbb{R}^{d})$ a solution of $(\Delta+\lambda)\,u=f,$

$\lVert
D^{{1/2}}u\rVert_{\\_}{B^{*}}\lesssim_{\\_}d\lambda^{\frac{d}{2}(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
f\rVert_{\\_}{L^{q_{\\_}d^{\prime}}},$

where $D^{{1/2}}$ is the Fourier multiplier with symbol $|\xi|^{{1/2}}$. By
duality, we have that

$\lVert
D^{{1/2}}u\rVert_{\\_}{L^{q_{\\_}d}}\lesssim_{\\_}d\lambda^{\frac{d}{2}(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
f\rVert_{\\_}{B}.$

Note again that, if $u$ solves $(\Delta+\lambda)\,u=f,$ then
$(\Delta+\lambda)P_{\\_}ku=P_{\\_}kf,$ for any $k\in\mathbb{Z}$. Thus, for
$k\in I$, we have that, since $2^{k}\sim\lambda^{1/2}$,

$\lambda^{1/4}\lVert P_{\\_}ku\rVert_{\\_}{L^{q_{\\_}d}}\sim_{\\_}d\lVert
D^{1/2}P_{\\_}ku\rVert_{\\_}{L^{q_{\\_}d}}\lesssim_{\\_}d\lambda^{\frac{d}{2}(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
f\rVert_{\\_}{B},$

where we have used lemma 2.1. This, along with (7) gives us the estimate

$\lVert(\Delta+\lambda+i0)^{-1}f\rVert_{\\_}{Z^{*}_{\\_}\lambda}\lesssim_{\\_}d\lVert
f\rVert_{\\_}{Y_{\\_}\lambda},$

and, by duality,

$\lVert(\Delta+\lambda+i0)^{-1}f\rVert_{\\_}{Y^{*}_{\\_}\lambda}\lesssim_{\\_}d\lVert
f\rVert_{\\_}{Z_{\\_}\lambda}.$

Applying the definition of the spaces $X_{\\_}\lambda$ and
$X_{\\_}\lambda^{*}$ gives us the desired result. ∎

Next, the compact support of the potential $V^{0}$ lets us split it into an
$L^{\infty}$ component and a $L^{p_{\\_}d}$ component whose norm can be as
small as needed, which allows for an estimate of the type

$\lVert V^{0}\rVert_{\\_}{\mathcal{L}(X^{*}_{\\_}\lambda,X_{\\_}\lambda)}\leq
C(\lambda^{1/4}+\lVert\mathbbm{1}_{\\_}FV^{0}\rVert_{\\_}{L^{d/2}}),$

where
$F=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|V^{0}(x)|>\lambda^{1/4}\\}$.
With this estimate, one can construct the solution operator
$(\Delta+\lambda+i0-V^{0})^{-1}$ via Neumann series and prove its boundedness
from $X_{\\_}\lambda$ to $X_{\\_}\lambda^{*}$. All this was done by Caro and
Garcia in [4], and can be summarized in the following proposition. Note that
their work was done on a ball, but their argument would be identical for any
Lipschitz domain. Therefore, for the rest of the section, we will consider a
bounded domain $\Omega$ such that $\mathrm{supp}\,V\subset\Omega$.

######  Proposition 2.3 ([4]).

Let $\Omega$ be a bounded Lipschitz domain. The operator defined by

$(\Delta+\lambda+i0-V^{0})^{-1}f=\sum_{\\_}{n\in\mathbb{N}}[((\Delta+\lambda+i0)^{-1}\circ
V^{0})]^{n-1}\,((\Delta+\lambda+i0)^{-1}\,f)$

for any $f\in X_{\\_}\lambda$ is bounded from $X_{\\_}\lambda$ to
$X_{\\_}\lambda^{*}$. Moreover, $u=(\Delta+\lambda+i0-V^{0})^{-1}f$ solves the
equation

$(\Delta+\lambda-V^{0})\,u=f\textnormal{ in }\mathbb{R}^{d}$

and, if $f$ is compactly supported in $\Omega$, then $u$ satisfies SRC (2).

###  2.2. The Fredholm alternative

Now we will construct the scattering solution $u_{\\_}{sc}(\centerdot\,,y)$ as
the solution of the equation

(8)
$[I-(\Delta+\lambda+i0-V^{0})^{-1}\circ(\gamma^{s}+\alpha\,\mathrm{d}\sigma)]\,u_{\\_}{sc}(\centerdot\,,y)=f(\centerdot\,,y)\textnormal{
in }\mathbb{R}^{d}$

with
$f(\centerdot\,,y)=(\Delta+\lambda+i0-V^{0})^{-1}\,(Vu_{\\_}{in}(\centerdot\,,y)).$
Note that applying the operator ${(\Delta+\lambda-V^{0})}$ to both sides of
(8) and making use of proposition 2.3, we can see that if
$u_{\\_}{sc}(\centerdot\,,y)$ solves (8), then it solves the equation

$(\Delta+\lambda-V)\,u_{\\_}{sc}(\centerdot\,,y)=Vu_{\\_}{in}(\centerdot\,,y)\textnormal{
in }\mathbb{R}^{d}.$

Moreover, since

$u_{\\_}{sc}(\centerdot\,,y)=(\Delta+\lambda+i0-V^{0})^{-1}[(\gamma^{s}+\alpha\,\mathrm{d}\sigma)\,u_{\\_}{sc}(\centerdot\,,y)+V\,u_{\\_}{in}(\centerdot\,,y)]$

and
$(\gamma^{s}+\alpha\,\mathrm{d}\sigma)\,u_{\\_}{sc}(\centerdot\,,y)+V\,u_{\\_}{in}(\centerdot\,,y)\in
X_{\\_}\lambda$ is supported in $\Omega$, by proposition 2.3, we can conclude
that $u_{\\_}{sc}(\centerdot\,,y)$ satisifes SRC (2).
Now, we will make use of the Fredholm theory to solve the equation (8). The
Fredholm alternative theorem states that if $T$ a compact operator on a Banach
space $\mathcal{B}$, then $(I-T)$ is invertible in $\mathcal{B}$ if and only
if $(I-T)$ is injective. We will justify the use of this technnique in the
following propositions by proving that, in essence, multiplication by
$V^{F}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\gamma^{s}+\alpha\,\mathrm{d}\sigma$
defines a compact operator from $X_{\\_}\lambda^{*}$ to $X_{\\_}\lambda$.
Start by proving the following:

######  Proposition 2.4.

For any bounded open domain $\Omega\subset\mathbb{R}^{d}$, the restriction map

$\displaystyle
r_{\\_}\Omega\,\mathrel{\mathop{\ordinarycolon}}\,X_{\\_}\lambda^{*}$
$\displaystyle\longrightarrow H^{1}(\Omega)$ $\displaystyle u$
$\displaystyle\longmapsto{\left.\kern-1.2ptu\right|_{\Omega}}$

is a bounded operator.

######  Proof:.

Let $u\in X_{\\_}\lambda^{*}$ and denote

$u_{\\_}I=\sum_{\\_}{k\in I}P_{\\_}k\,u,\quad u_{\\_}{\mathbb{Z}\setminus
I}=u-u_{\\_}I.$

Note that $r_{\\_}\Omega u=r_{\\_}\Omega u_{\\_}I+r_{\\_}\Omega
u_{\\_}{\mathbb{Z}\setminus I}$, and therefore

$\lVert r_{\\_}\Omega u\rVert_{\\_}{H^{1}(\Omega)}\leq\lVert r_{\\_}\Omega
u_{\\_}I\rVert_{\\_}{H^{1}(\Omega)}+\lVert r_{\\_}\Omega
u_{\\_}{\mathbb{Z}\setminus I}\rVert_{\\_}{H^{1}(\Omega)}.$

On the one hand, let $\alpha$ be a multiindex such that $|\alpha|\leq 1$.
Then, by triangle, Hölder’s and Bernstein’s inequalities,

$\displaystyle\lVert\partial^{\alpha}u_{\\_}I\rVert_{\\_}{L^{2}(\Omega)}$
$\displaystyle\leq\sum_{\\_}{k\in
I}\lVert\partial^{\alpha}P_{\\_}k\,u\rVert_{\\_}{L^{2}(\Omega)}\lesssim\sum_{\\_}{k\in
I}\lVert\partial^{\alpha}P_{\\_}k\,u\rVert_{\\_}{L^{q_{\\_}d}}\lesssim\sum_{\\_}{k\in
I}2^{|\alpha|k}\lVert P_{\\_}k\,u\rVert_{\\_}{L^{q_{\\_}d}}$
$\displaystyle\lesssim\lambda^{\frac{1}{2}-\frac{d}{2}(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\left(\sum_{\\_}{k\in
I}\lambda^{\frac{d}{2}(\frac{1}{q_{\\_}d}-\frac{1}{p_{\\_}d})}\lVert
P_{\\_}k\,u\rVert_{\\_}{L^{q_{\\_}d}}^{2}\right)^{{1/2}}\lesssim_{\\_}\lambda\lVert
u\rVert_{\\_}{X^{*}_{\\_}\lambda},$

while

$\lVert u_{\\_}I\rVert_{\\_}{L^{2}(\Omega)}\leq\sum_{\\_}{k\in I}\lVert
P_{\\_}ku\rVert_{\\_}{L^{2}(\Omega)}\leq\sum_{\\_}{k\in I}\lVert
P_{\\_}ku\rVert_{\\_}{L^{q_{\\_}d}}\lesssim_{\\_}\lambda\lVert
u\rVert_{\\_}{X^{*}_{\\_}\lambda},$

so that

$\lVert r_{\\_}\Omega
u_{\\_}I\rVert_{\\_}{H^{1}(\Omega)}\lesssim_{\\_}\lambda\lVert
u\rVert_{\\_}{X^{*}_{\\_}\lambda}.$

On the other hand, by Plancherel’s identity and the triangle inequality,

$\displaystyle\lVert u_{\\_}{\mathbb{Z}\setminus
I}\rVert_{\\_}{H^{1}(\mathbb{R}^{d})}^{2}\leq$
$\displaystyle\lVert(I-\Delta)^{{1/2}}\,u_{\\_}{\mathbb{Z}\setminus
I}\rVert_{\\_}{L^{2}}^{2}=\lVert(1+|\centerdot|^{2})^{{1/2}}\,\widehat{u_{\\_}{\mathbb{Z}\setminus
I}}\rVert_{\\_}{L^{2}}^{2}$
$\displaystyle\leq\lVert(1+|\centerdot|^{2})^{{1/2}}\,\widehat{P_{\\_}{<I}\,u}\rVert_{\\_}{L^{2}}^{2}+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert(1+|\centerdot|^{2})^{{1/2}}\,\widehat{P_{\\_}k\,u}\rVert_{\\_}{L^{2}}^{2}$
$\displaystyle\lesssim_{\\_}\lambda\lVert
m_{\\_}\lambda^{{1/2}}\,\widehat{P_{\\_}{<I}\,u}\rVert_{\\_}{L^{2}}^{2}+\sum_{\\_}{k>k_{\\_}\lambda+1}\lVert
m_{\\_}\lambda^{{1/2}}\,\widehat{P_{\\_}k\,u}\rVert_{\\_}{L^{2}}^{2}\leq\lVert
u\rVert_{\\_}{X_{\\_}\lambda^{*}}^{2},$

where we have used the fact that, since

$\mathrm{supp}\,(\widehat{P_{\\_}{<I}\,u})\subset\\{\xi\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|\xi|\leq\lambda\\}$

and, for $k>k_{\\_}\lambda+1$,

${\mathrm{supp}\,(\widehat{P_{\\_}{k}\,u})\subset\\{\xi\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,2^{k-1}\leq|\xi|\leq
2^{k+1}\\}},$

it follows that

$(1+|\xi|^{2})^{{1/2}}\widehat{P_{\\_}{<I}\,u}\sim
m_{\\_}\lambda^{{1/2}}(\xi)\widehat{P_{\\_}{<I}\,u}\quad\text{and}\quad(1+|\xi|^{2})^{{1/2}}\widehat{P_{\\_}k\,u}\sim
m_{\\_}\lambda^{{1/2}}(\xi)\widehat{P_{\\_}k\,u}.$

Clearly,

$\lVert r_{\\_}\Omega
u_{\\_}{I\setminus\mathbb{Z}}\rVert_{\\_}{H^{1}(\Omega)}\leq\lVert
u_{\\_}{I\setminus\mathbb{Z}}\rVert_{\\_}{H^{1}(\mathbb{R}^{d})},$

which proves the proposition. ∎

######  Corollary 2.5.

Every $u\in X_{\\_}\lambda^{*}$ belongs to
$H^{1}_{\\_}{\textnormal{loc}}(\mathbb{R}^{d})$.

Recall now that, if for any $s\geq 0$ we define $H^{s}(\Omega)$ as the space
of restrictions of functions in $H^{s}(\mathbb{R}^{d})$ to $\Omega$, we can
then define $H^{-s}_{\\_}0(\Omega)$ as the dual space of $H^{s}(\Omega)$.

######  Proposition 2.6.

For any bounded open domain $\Omega\subset\mathbb{R}^{d}$, the embedding

$H^{-1}_{\\_}0(\Omega)\lhook\joinrel\xrightarrow{\;i\;}X_{\\_}\lambda$

is continuous.

######  Proof:.

We prove it by duality, using Hahn-Banach’s Theorem. Indeed, let $\phi\in
C^{\infty}_{\\_}0(\Omega)$, then there exists $u\in X_{\\_}\lambda^{*}$ such
that [3]

$\lVert\phi\rVert_{\\_}{X_{\\_}\lambda}=\frac{\braket{u,\phi}}{\lVert
u\rVert_{\\_}{X_{\\_}\lambda^{*}}}\lesssim_{\\_}\lambda\frac{\braket{u,\phi}}{\lVert
r_{\\_}\Omega\,u\rVert_{\\_}{H^{1}(\Omega)}}=\frac{\braket{r_{\\_}\Omega\,u,\phi}}{\lVert
r_{\\_}\Omega\,u\rVert_{\\_}{H^{1}(\Omega)}}\leq\frac{\lVert
r_{\\_}\Omega\,u\rVert_{\\_}{H^{1}(\Omega)}\lVert\phi\rVert_{\\_}{H^{-1}_{\\_}0(\Omega)}}{\lVert
r_{\\_}\Omega\,u\rVert_{\\_}{H^{1}(\Omega)}}=\lVert\phi\rVert_{\\_}{H^{-1}_{\\_}0(\Omega)},$

where we have used proposition 2.4. Now, $C^{\infty}_{\\_}0(\Omega)$ is dense
in $H^{-1}_{\\_}0(\Omega)$ (proposition 2.9 in [20]) and $X_{\\_}\lambda$ is a
Banach space [4], so the proof follows by a standard density argument. ∎

######  Proposition 2.7.

Let $1/2<s<1$ and define
$V^{F}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\gamma^{s}+\alpha\,\mathrm{d}\sigma$.
There exists $C>0$ such that, for any $u,v\in\mathcal{S}(\mathbb{R}^{d})$, it
holds that

(9) $\lvert\braket{\gamma^{s}u,\,v}\rvert\lesssim\lVert
g\rVert_{\\_}{L^{\infty}}\left(\lVert u\rVert_{\\_}{H^{s}(\Omega)}\lVert
v\rVert_{\\_}{L^{2}(\Omega)}+\lVert u\rVert_{\\_}{L^{2}(\Omega)}\lVert
v\rVert_{\\_}{H^{s}(\Omega)}\right),$

where $\Omega$ is any open domain such that $\mathrm{supp}\,V\subset\Omega$.
In particular, $V^{F}$ acts as a bounded multiplication operator from
$H^{s}(\Omega)$ to $H^{-s}_{\\_}0(\Omega)$.

######  Proof:.

We will use the homogeneous fractional Leibniz rule [7, 23]. It is also known
as Kato-Ponce differentiation rule, since its inhomogeneous version was first
given by Kato and Ponce in [21]. Indeed, for
$u,v\in\mathcal{S}(\mathbb{R}^{d})$,
$1/r=1/p_{\\_}1+1/q_{\\_}1=1/p_{\\_}2+1/q_{\\_}2$ and $s<1$ one has

(10) $\lVert D^{s}(uv)\rVert_{\\_}{L^{r}}\lesssim\lVert
D^{s}u\rVert_{\\_}{L^{p_{\\_}1}}\lVert v\rVert_{\\_}{L^{q_{\\_}1}}+\lVert
u\rVert_{\\_}{L^{p_{\\_}2}}\lVert D^{s}v\rVert_{\\_}{L^{q_{\\_}2}}.$

In particular, taking $r=1$, $p_{\\_}1=q_{\\_}1=p_{\\_}2=q_{\\_}2=2$, one
obtains

$\displaystyle\lvert\braket{\gamma^{s}u,\,v}\rvert=$
$\displaystyle\lvert\braket{\chi^{2}D^{s}g,\,uv}\rvert=\lvert\braket{D^{s}g,\,\chi
u\chi v}\rvert=$ $\displaystyle\lvert\braket{g,\,D^{s}(\chi u\chi
v)}\rvert\leq\lVert g\rVert_{\\_}{L^{\infty}}\lVert D^{s}(\chi u\chi
v)\rVert_{\\_}{L^{1}}\lesssim$ $\displaystyle\lVert
g\rVert_{\\_}{L^{\infty}}\left(\lVert D^{s}(\chi
u)\rVert_{\\_}{L^{2}}\lVert\chi v\rVert_{\\_}{L^{2}}+\lVert\chi
u\rVert_{\\_}{L^{2}}\lVert D^{s}(\chi v)\rVert_{\\_}{L^{2}}\right),$

where we have used Hölder’s inequality and the aforementioned Kato-Ponce rule.
On the one hand, since $\chi$ is supported in $\Omega$,

$\lVert\chi u\rVert_{\\_}{L^{2}}\lesssim\lVert
u\rVert_{\\_}{L^{2}(\Omega)}\lesssim\lVert u\rVert_{\\_}{H^{s}(\Omega)}.$

On the other hand, let any $\tilde{u}\in\mathcal{S}(\mathbb{R}^{d})$ be such
that
${\left.\kern-1.2pt\tilde{u}\right|_{\Omega}}={\left.\kern-1.2ptu\right|_{\Omega}}$.
Using again the fractional Leibniz rule we obtain

$\displaystyle\lVert D^{s}(\chi\tilde{u})\rVert_{\\_}{L^{2}}\lesssim$
$\displaystyle\lVert
D^{s}\chi\rVert_{\\_}{L^{\infty}}\lVert\tilde{u}\rVert_{\\_}{L^{2}}+\lVert\chi\rVert_{\\_}{L^{\infty}}\lVert
D^{s}\tilde{u}\rVert_{\\_}{L^{2}}\lesssim\lVert\tilde{u}\rVert_{\\_}{H^{s}}.$

Now, since $\chi u=\chi\tilde{u}$, we can take infimum to obtain

(11) $\lVert D^{s}(\chi
u)\rVert_{\\_}{L^{2}}\lesssim\inf\\{\lVert\tilde{u}\rVert_{\\_}{H^{s}}\mathrel{\mathop{\ordinarycolon}}{\left.\kern-1.2pt\tilde{u}\right|_{\Omega}}={\left.\kern-1.2ptu\right|_{\Omega}}\\}=\lVert
u\rVert_{\\_}{H^{s}(\Omega)}.$

Therefore,

(12) $\lvert\braket{\gamma^{s}u,\,v}\rvert\lesssim\lVert
g\rVert_{\\_}{L^{\infty}}\left(\lVert u\rVert_{\\_}{H^{s}(\Omega)}\lVert
v\rVert_{\\_}{L^{2}(\Omega)}+\lVert u\rVert_{\\_}{L^{2}(\Omega)}\lVert
v\rVert_{\\_}{H^{s}(\Omega)}\right)$

Note that in particular we have that

$\lvert\braket{\gamma^{s}u,\,v}\rvert\lesssim\lVert
g\rVert_{\\_}{L^{\infty}}\lVert u\rVert_{\\_}{H^{s}(\Omega)}\lVert
v\rVert_{\\_}{H^{s}(\Omega)},$

while for the $\alpha\,\mathrm{d}\sigma$ term we have that

$\left|\braket{\alpha\,\mathrm{d}\sigma
u,\,v}\right|=\left|\int_{\\_}{\Gamma}uv\alpha\,\mathrm{d}\sigma\right|\leq\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\lVert
u\rVert_{\\_}{L^{2}(\Gamma)}\lVert v\rVert_{\\_}{L^{2}(\Gamma)}.$

Observe that, by the trace theorem for Sobolev spaces (see, for example, [30],
section 4.4.2), for any $1/2<s<1$ it holds that

$\lVert u\rVert_{\\_}{L^{2}(\Gamma)}\lesssim\lVert
u\rVert_{\\_}{H^{s-1/2}(\Gamma)}\lesssim\lVert u\rVert_{\\_}{H^{s}(\Omega)},$

so that

(13) $\lvert\braket{V^{F}u,\,v}\rvert\lesssim\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right),$

which ends the proof of the proposition. ∎

######  Proposition 2.8.

The operator

$(\Delta+\lambda+i0-V^{0})^{-1}\circ(\gamma^{s}+\alpha\,\mathrm{d}\sigma)$

is compact in $X^{*}_{\\_}\lambda$.

######  Proof:.

By proposition 2.3, $(\Delta+\lambda+i0-V^{0})^{-1}$ is bounded from
$X_{\\_}\lambda$ to $X_{\\_}\lambda^{*}$, so we only have to show that $V^{F}$
is compact form $X_{\\_}\lambda^{*}$ to $X_{\\_}\lambda$. All the necessary
ingredients have been laid in the previous propositions. Just recall that the
inclusion

$H^{1}(\Omega)\lhook\joinrel\xhookrightarrow{\;j\;}H^{s}(\Omega)$

is known to be compact and $H_{\\_}0^{-s}(\Omega)$ is continuously embedded in
$H_{\\_}0^{-1}(\Omega)$. We summarize the argument in the following diagram:

${{X_{\\_}\lambda^{*}}}$${{\xrightarrow{\;r_{\\_}{\Omega}\;}}}$${{H^{1}(\Omega)}}$${{\lhook\joinrel\xhookrightarrow{\;j\;}}}$${{H^{s}(\Omega)}}$${{\xrightarrow{\;V^{F}\;}}}$${{H_{\\_}0^{-s}(\Omega)}}$${{\xhookrightarrow{\;j^{*}\;}}}$${{H_{\\_}0^{-1}(\Omega)}}$${{\xhookrightarrow{\;i\;}}}$${{X_{\\_}\lambda}}$

∎

Now, to prove existence of the scattering solution, we are only missing the
injectivity of the operator

$(\Delta+\lambda+i0-V^{0})^{-1}\circ(\gamma^{s}+\alpha\,\mathrm{d}\sigma).$

With the reader’s consent, we will borrow a lemma that will be proved later in
section 4, namely lemma 2.2 concerning a Carleman estimate. This estimate
works on a family of Bourgain spaces defined by the following norm, for
$\zeta\in\mathbb{C}^{d}$ and $s\in\mathbb{R}$,

(14) $\lVert
u\rVert_{\\_}{X_{\\_}\zeta^{s}}=\lVert(M|\Re(\zeta)|^{2}+M^{-1}|p_{\\_}\zeta|^{2})^{s/2}\,\widehat{u}\rVert_{\\_}{L^{2}},$

with $M>1$, where $\Re$ denotes the real part and

$p_{\\_}\zeta(\xi)=-|\xi|^{2}+2i\zeta\cdot\xi+\zeta\cdot\zeta.$

lemmacarlemanen Let $R_{\\_}0>0$ such that $\mathrm{supp}\,V\subset
B_{\\_}{R_{\\_}0}=\\{x\in\mathbb{R}^{d}\mathrel{\mathop{\ordinarycolon}}|x|<R_{\\_}0\\}$.
Take $\varphi_{\\_}\zeta(x)=M\frac{(x\cdot\theta)^{2}}{2}+x\cdot\zeta,$ with
$\zeta=\tau\theta+i\mathcal{I}$, $\theta\in\mathbb{S}^{d-1}$ and
$\mathcal{I}\in\mathbb{R}^{d}$. There exists $C>0$ and
$\tau_{\\_}0=\tau_{\\_}0(R_{\\_}0,V,\lambda)$ such that

(15) $\lVert u\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}\leq CR_{\\_}0\lVert
e^{\varphi_{\\_}{\zeta}}(\Delta+\lambda-V)\,(e^{-\varphi_{\\_}{\zeta}}u)\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}$

for all for $u\in\mathscr{S}(\mathbb{R}^{d})$ with $\mathrm{supp}\,u\subset
B_{\\_}{R_{\\_}0}$ and $\tau>\tau_{\\_}0$. Take $R_{\\_}0$ such that
$\Omega\subset B_{\\_}{R_{\\_}0}$. Now, we can check that the spaces
$X_{\\_}\zeta^{1/2}$ and $H^{1}(\mathbb{R}^{d})$ are equal as sets, and that,
for every $u\in H^{1}(\mathbb{R}^{d})$ such that
$\mathrm{supp}\,u\subset\overline{\Omega}$, we have that
$e^{\varphi_{\\_}{\zeta}}(\Delta+\lambda-V)\,(e^{-\varphi_{\\_}{\zeta}}u)$ is
in $X_{\\_}{-\zeta}^{-{1/2}}$. Therefore, by density, (15) also holds for
every $u\in H^{1}(\mathbb{R}^{d})$ such that
$\mathrm{supp}\,u\subset\overline{\Omega}$.

######  Lemma 2.9.

Consider $d\geq 3$. If $u\in H^{1}_{\\_}{\text{loc}}(\mathbb{R}^{d})$ is a
solution of

$(\Delta+\lambda-V)\,u=0\text{ in }\mathbb{R}^{d}$

that satisfies the SRC (2), then $u$ has to be identically zero.

######  Proof:.

Let $R>0$ and call
$B=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|x|<R\\}$. On the
one hand, the restriction of u to $\mathbb{R}^{d}\setminus\mathrm{supp}\,V$
solves $(\Delta+\lambda)\,u=0$. By Theorem 11.1.1 in [17] this restriction is
smooth, and we have that

(16) $\int_{\\_}{\partial B}|\partial_{\\_}\nu
u-i\lambda^{1/2}u|^{2}\,\,\mathrm{d}S=\int_{\\_}{\partial B}|\partial_{\\_}\nu
u|^{2}+\lambda|u|^{2}+i\lambda^{1/2}(\partial_{\\_}\nu
u\,\overline{u}-\overline{\partial_{\\_}\nu u}\,u)\,\mathrm{d}S,$

where $\mathcal{I}$ denotes the imaginary part and
$\partial_{\\_}\nu=\nu\cdot\nabla$ the normal derivative with respect to the
vector $\nu=x/|x|$. Using Green’s identity in $B\setminus\overline{\Omega}$ we
obtain that

$\displaystyle\int_{\\_}{\partial B}\partial_{\\_}\nu
u\,\overline{u}-\overline{\partial_{\\_}\nu
u}\,u\,\mathrm{d}S=-\int_{\\_}{\partial\Omega}\partial_{\\_}\nu
u\,\overline{u}-\overline{\partial_{\\_}\nu u}\,u\,\mathrm{d}S$

Now, by the SRC (2), identity (16) yields

$\lim_{\\_}{R\to\infty}\int_{\\_}{\partial B}|\partial_{\\_}\nu
u|^{2}+\lambda|u|^{2}=i\lambda^{1/2}\int_{\\_}{\partial\Omega}\partial_{\\_}\nu
u\,\overline{u}-\overline{\partial_{\\_}\nu u}\,u\,\mathrm{d}S.$

Since the potential $V$ is real-valued, Green’s identity in $\Omega$ gives us
that

$i\int_{\\_}{\partial\Omega}\partial_{\\_}\nu
u\,\overline{u}-\overline{\partial_{\\_}\nu u}\,u\,\mathrm{d}S=0,$

which implies that

$\lim_{\\_}{R\to\infty}\int_{\\_}{\partial B}\lambda|u|^{2}=0,$

and, consequently, by Rellich’s lemma,
$\mathrm{supp}\,u\subset\overline{\Omega}$ and $u\in H^{1}(\mathbb{R}^{d})$.
Then, we can apply inequality (15) to $v=e^{\varphi_{\\_}\zeta}u$, which
belongs to $H^{1}(\mathbb{R}^{d})$ and is supported in $\overline{\Omega}$:

$\lVert e^{\varphi_{\\_}{\zeta}}u\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}\leq
CR_{\\_}0\lVert
e^{\varphi_{\\_}{\zeta}}(\Delta+\lambda-V)\,u\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}},$

where $R_{\\_}0$ is such that $\Omega\subset B_{\\_}{R_{\\_}0}$. Finally,
since $(\Delta+\lambda-V)\,u=0$, we can conclude that $u=0$. ∎

With this last lemma, the Fredholm theory argument is completed. Therefore, we
have effectively proved the following theorem:

######  Theorem 1.

Suppose $d\geq 3$ and $V$ is of the form (1). Then, there exists
$\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for every
$\lambda\geq\lambda_{\\_}0$, there is an unique solution
$u_{\\_}{sc}(\centerdot\,,y)\in X_{\\_}\lambda^{*}$ to the problem (S) for
every $y\in\mathbb{R}^{d}\setminus\mathrm{supp}\,V$.
Moreover, the mapping $Vu_{\\_}{in}(\centerdot\,,y)\mapsto
u_{\\_}{sc}(\centerdot\,,y)$ is bounded from $X_{\\_}\lambda$ to
$X_{\\_}\lambda^{*}$.

##  3\. Runge approximation and Alessandrini identity

In this section, we aim to prove the following orthogonality relation, which
is crucial to prove the inverse uniqueness with local data result:

######  Proposition 3.1.

Consider $d\geq 3$. Let $V_{\\_}1$ and $V_{\\_}2$ be two potentials of the
form (1), and let $\Sigma_{\\_}1,\Sigma_{\\_}2$ be two relatively open sets of
dimension $d-1$, separated from $\mathrm{supp}\,V$, and that can be expressed
as the graph of $\mathcal{C}^{2}$ functions. Choose a bounded open domain
$\Omega$ of class $\mathcal{C}^{2}$ such that
$\Sigma_{\\_}1,\Sigma_{\\_}2\subset\partial\Omega$,
$\mathrm{supp}\,V_{\\_}j\subset\Omega$, $j=1,2$
Then, there exists $\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for every
$\lambda\geq\lambda_{\\_}0$ except for at most a countable set, it holds that

${\left.\kern-1.2ptu_{\\_}{sc,1}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}={\left.\kern-1.2ptu_{\\_}{sc,2}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}\implies\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=0,$
$\text{for all }v_{\\_}1,v_{\\_}2\in H^{1}(\Omega)\textrm{ such that
}\left(\Delta+\lambda-V_{\\_}j\right)v_{\\_}j=0\textrm{ in }\Omega.$

We will divide the proof of this proposition in two parts. We will prove that
the orthogonality relation is fullfiled for solutions that can be represented
as single layer potentials with densities supported in $\Sigma_{\\_}1$ and
$\Sigma_{\\_}2$ respectively, to then extend it to every solution by
approximating them by these single layer potentials. We will start by
providing with a Runge approximation result, that allows us to do this
approximation in a $L^{2}$ norm. The ideas for the proof take inspiration from
[19] and [15]. However, due to the low-regularity of the potential $V$, we
need an approximation in a space of higher regularity. Lemma 3.5 below will
provide us with an estimate to make the approximation in a $H^{1}$ norm. In
this section, we will constantly refer to appendix A. There, we remember some
classical results regarding layer potentials and their use in solving the
Neumann problem for the Helmholtz equation, which is a key ingredient in our
proof or the Runge approximation.

###  3.1. Runge approximation

We will start the section by providing with two technical lemmas that make the
proof of the Runge approximation more readable. The notation
$\Omega^{\prime}\subset\subset\Omega$ will mean here and throughout the paper
that $\overline{\Omega}^{\prime}\subset\Omega$. Also, note that, given the
nature of $\gamma^{s}$, the integration has to be understood at times as a
duality pairing.

######  Lemma 3.2.

Let $V$ be as in (1). Let $\Omega^{\prime}\subset\subset\Omega$ be two open
bounded domains of class $\mathcal{C}^{2}$ such that
$\mathrm{supp}\,V\subset\Omega^{\prime}$, and let $v\in L^{2}(\Omega)$ be also
supported in $\Omega^{\prime}$. Suppose that $\varphi\in H^{1}(\Omega)$ is a
solution to

$\displaystyle\begin{cases}\;(\Delta+\lambda-V)\,\varphi=v&\text{in
}\Omega,\\\ \;\partial_{\\_}\nu\varphi=0&\text{on }\partial\Omega.\end{cases}$

Then, for every $y\in\partial\Omega$, it holds that

(17)
$\int_{\\_}{\partial\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)=-\int_{\\_}{\Omega^{\prime}}\left(\varphi(x)\,V(x)+v(x)\right)u_{\\_}{in}(x,y)\,\mathrm{d}x.$

######  Proof:.

This lemma boils down to a careful integration by parts, analysing the
singularity of $u_{\\_}{in}(x,y)$ when $x=y$. Let $B_{\\_}{y,\varepsilon}$ be
the ball of radius $\varepsilon>0$ centered in $y\in\partial\Omega$. In
appendix A we give the following expression:

(18) $\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)=G(y,x)|x-y|^{2-d},$

with $G$ a bounded function on $\partial\Omega\times\partial\Omega$. This
means that $|\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}|$ is a weakly singular
kernel of order $d-2$, which in turn ensures the absolute integrability of
$\varphi(\centerdot)\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(\centerdot\,,y)$ in
$\partial\Omega$ (see lemma A.2 in Appendix A). Therefore, by the Dominated
Convergence Theorem and Green’s formula, we have

$\displaystyle\int_{\\_}{\partial\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)=\lim_{\\_}{\varepsilon\to
0}$
$\displaystyle\int_{\\_}{\partial\Omega}(1-\mathbbm{1}_{\\_}{B_{\\_}{y,\varepsilon}}(x))\,\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)$
$\displaystyle=\lim_{\\_}{\varepsilon\to 0}$
$\displaystyle\left(\int_{\\_}{\partial\left(\Omega\setminus
B_{\\_}{y,\varepsilon}\right)}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)\right.$
$\displaystyle\left.-\int_{\\_}{(\partial
B_{\\_}{y,\varepsilon})\cap\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)\right),$

First, we will see that the second term is negligible. Indeed, we can use
again the limiting expression for $\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}$ in
(18), that is
$|\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)|\lesssim|x-y|^{2-d}$ when $x$
approaches $y$. Changing to polar coordinates centered in $y$ we obtain, for
$\varepsilon$ sufficiently small,

(19) $\displaystyle\begin{split}\left|\int_{\\_}{(\partial
B_{\\_}{y,\varepsilon})\cap\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)\right|\leq&\,\varepsilon\int_{\\_}{\mathbb{S}^{d-1}_{\\_}\cap}|\varphi(\varepsilon,\theta)|\,\mathrm{d}S(\theta)\\\
\lesssim&\,\varepsilon\sup_{\\_}{x\in\Omega\setminus\Omega^{\prime}}|\varphi(x)|\xrightarrow{\varepsilon\to
0}0,\end{split}$

where we have denoted by $\mathbb{S}^{d-1}_{\\_}\cap$ the relevant half sphere
in the change of variables, and we have used the fact that, since $\varphi$
solves $(\Delta+\lambda)\varphi=0$ in $\Omega\setminus\Omega^{\prime}$, its
restriction to that domain is smooth by Theorem 11.1.1 in [17], and therefore
bounded. Next, we can use integration by parts in $\Omega\setminus
B_{\\_}{y,\varepsilon}$ to obtain

$\displaystyle\int_{\\_}{\partial\left(\Omega\setminus
B_{\\_}{y,\varepsilon}\right)}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)=$
$\displaystyle\int_{\\_}{\Omega\setminus
B_{\\_}{y,\varepsilon}}\varphi(x)\left(\Delta_{x}+\lambda-V(x)\right)u_{\\_}{in}(x,y)\,\mathrm{d}x$
$\displaystyle-\int_{\\_}{\Omega\setminus
B_{\\_}{y,\varepsilon}}\left(\Delta+\lambda-V(x)\right)\varphi(x)\,u_{\\_}{in}(x,y)\,\mathrm{d}x$
$\displaystyle+\int_{\\_}{\partial\left(\Omega\setminus
B_{\\_}{y,\varepsilon}\right)}\partial_{\\_}{\nu}\varphi(x)\,u_{\\_}{in}(x,y)\,\mathrm{d}S(x).$

On the one hand, since $\partial_{\\_}\nu\varphi=0$ on $\partial\Omega$, we
have that

$\int_{\\_}{\partial\left(\Omega\setminus
B_{\\_}{y,\varepsilon}\right)}\partial_{\\_}{\nu}\varphi(x)\,u_{\\_}{in}(x,y)\,\mathrm{d}S(x)=\int_{\\_}{(\partial
B_{\\_}{y,\varepsilon})\cap\Omega}\partial_{\\_}{\nu}\varphi(x)\,u_{\\_}{in}(x,y)\,\mathrm{d}S(x).$

Now, observing that the limiting expression for $u_{\\_}{in}$ is

$u_{\\_}{in}(x,y)=F(y,x)|x-y|^{2-d},$

with $F$ a bounded function on $\partial\Omega\times\partial\Omega$ (again
refer to appendix A), we can do an identical argument as in (19) to obtain
that

$\left|\int_{\\_}{(\partial
B_{\\_}{y,\varepsilon})\cap\Omega}\partial_{\\_}{\nu}\varphi(x)\,u_{\\_}{in}(x,y)\,\mathrm{d}S(x).\right|\xrightarrow{\varepsilon\to
0}0.$

On the other hand, we have that
$\left(\Delta_{\\_}x+\lambda\right)u_{\\_}{in}(x,y)=0$ for $x\neq y$, while
${\left(\Delta+\lambda-V(x)\right)\varphi(x)=v(x)}$. Therefore, since both $V$
and $v$ are supported in $\Omega^{\prime}$,

$\displaystyle\int_{\\_}{\partial\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,\mathrm{d}S(x)=\lim_{\\_}{\varepsilon\to
0}\int_{\\_}{\Omega^{\prime}}\left(-\varphi(x)\,V(x)-v(x)\right)u_{\\_}{in}(x,y)\,\mathrm{d}x.$

However, the integral in the limit does not depend on $\varepsilon$ in any
way, and the lemma is proved. ∎

Remember now that total wave is $u_{to}=u_{\\_}{in}+u_{\\_}{sc}$, and consider
the single layer potential with density $f\in\mathcal{C}(\partial\Omega)$ as
defined by (see Appendix A for details)

(20)
$\left(\mathcal{S}f\right)(x)=\int_{\\_}{\partial\Omega}f(y)\,u_{to}(x,y)\,\,\mathrm{d}S(y),\quad
x\in\mathbb{R}^{d},$

and the normal derivative operator as

(21)
$\left(\mathcal{N}f\right)(x)=\int_{\\_}{\partial\Omega}f(y)\,\partial_{\\_}{\nu_{\\_}x}u_{to}(x,y)\,\mathrm{d}S(y),\quad
x\in\partial\Omega.$

######  Lemma 3.3.

Let $V$ be as in (1). Let $\Omega^{\prime}\subset\subset\Omega$ be two open
bounded domains of class $\mathcal{C}^{2}$ such that
$\mathrm{supp}\,V\subset\Omega^{\prime}$, and let $v\in L^{2}(\Omega)$ also
supported in $\Omega^{\prime}$. Suppose $\varphi\in H^{1}(\Omega)$ is a
solution to

$\displaystyle\begin{cases}\;(\Delta+\lambda-V)\,\varphi=v&\text{in
}\Omega,\\\ \;\partial_{\\_}\nu\varphi=0&\text{on }\partial\Omega.\end{cases}$

Then, it holds that

$\int_{\\_}{\partial\Omega}(\mathcal{N}f)(x)\,\varphi(x)\,\mathrm{d}S(x)=-\int_{\\_}{\Omega^{\prime}}(\mathcal{S}f)(x)\,v(x)\,\mathrm{d}x.$

######  Proof:.

First note that scattering part $u_{\\_}{sc}$ solves
$\left(\Delta+\lambda\right)u_{\\_}{sc}(\centerdot\,,y)=0$ in
$\mathbb{R}^{d}\setminus\mathrm{supp}\,V$ as long as
$y\in\mathbb{R}^{d}\setminus\mathrm{supp}\,V$, so it is smooth away from the
potential, by Theorem 11.1.1 in [17], and in particular it is smooth on
$\partial\Omega$. On the other hand, expression (18) means that
$|\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}|$ is a weakly singular kernel of order
$d-2$, which in turn ensures the absolute integrability of
$\varphi(\centerdot)f(\circ)\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(\centerdot\,,\circ)$,
and allows to use Fubini’s Theorem (see lemma A.2 in Appendix A). Therefore,

(22)
$\displaystyle\begin{split}\int_{\\_}{\partial\Omega}(\mathcal{N}f)(x)\,\varphi(x)\,\mathrm{d}S(x)&=\int_{\\_}{\partial\Omega}\varphi(x)\left(\int_{\\_}{\partial\Omega}f(y)\,\partial_{\\_}{\nu_{\\_}x}u_{to}(x,y)\,\mathrm{d}S(y)\right)\,\mathrm{d}S(x)\\\
&=\int_{\\_}{\partial\Omega}f(y)\left(\int_{\\_}{\partial\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{to}(x,y)\,\mathrm{d}S(x)\right)\,\mathrm{d}S(y).\end{split}$

Now, for any $y\in\partial\Omega$ we obtain, by integrating by parts in
$\Omega$ with respect to $x$,

(23)
$\displaystyle\begin{split}\int_{\\_}{\partial\Omega}\varphi(x)\partial_{\\_}{\nu_{\\_}x}u_{\\_}{sc}(x,y)\,\mathrm{d}S(x)=&\int_{\\_}{\Omega}\varphi(x)\left(\Delta_{x}+\lambda-V(x)\right)u_{\\_}{sc}(x,y)\,\mathrm{d}x\\\
&-\int_{\\_}{\Omega}(\Delta+\lambda-V)\,\varphi(x)\,u_{\\_}{sc}(x,y)\,\mathrm{d}x\\\
&+\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu}\varphi(x)\,u_{\\_}{sc}(x,y)\,\mathrm{d}S(x)\\\
=&\int_{\\_}{\Omega^{\prime}}\left(\varphi(x)\,V(x)\,u_{\\_}{in}(x,y)-v(x)\,u_{\\_}{sc}(x,y)\right)\,\mathrm{d}x.\end{split}$

In the last identity we have used that
$\left(\Delta_{x}+\lambda-V(x)\right)u_{\\_}{sc}(\centerdot\,,y)=Vu_{\\_}{in}(\centerdot\,,y)$
in $\mathbb{R}^{d}$, and also that $(\Delta+\lambda-V)\,\varphi=v$ in $\Omega$
and $\partial_{\\_}\nu\varphi=0$ on $\partial\Omega$, as well as the fact that
both $V$ and $v$ are supported in $\Omega^{\prime}$. Joining (23) above with
(17) in lemma 3.2 yields

$\int_{\\_}{\partial\Omega}\varphi(x)\,\partial_{\\_}{\nu_{\\_}x}u_{to}(x,y)\,\mathrm{d}S(x)=-\int_{\\_}{\Omega^{\prime}}v(x)\,u_{\\_}{to}(x,y)\,\mathrm{d}x.$

Applying this identity in (22) and using Fubini’s Theorem again yields

$\displaystyle\int_{\\_}{\partial\Omega}(\mathcal{N}f)(x)\,\varphi(x)\,\mathrm{d}S(x)=$
$\displaystyle-\int_{\\_}{\Omega^{\prime}}v(x)\left(\int_{\\_}{\partial\Omega}f(y)\,u_{\\_}{to}(x,y)\,\mathrm{d}S(y)\right)\,\mathrm{d}x$
$\displaystyle=$
$\displaystyle-\int_{\\_}{\Omega^{\prime}}v(x)\mathcal{(}Sf)(x)\,\mathrm{d}x.$

∎

Note that the single-layer potential $\mathcal{S}f$ as defined in (20) belongs
to $H^{1}(\Omega)$ by lemma A.3 in appendix A. Therefore, if we take two open
bounded domains $\Omega^{\prime}\subset\subset\Omega$ of class
$\mathcal{C}^{2}$, and a relatively open subset $\Sigma\subset\partial\Omega$,
we can define the following spaces:

$\displaystyle X=\left\\{u\in
H^{1}(\Omega)\,\mathrel{\mathop{\ordinarycolon}}\,\left(\Delta+\lambda-V\right)u=0\textrm{
in }\Omega\right\\}$ $\displaystyle X^{\Sigma}=\left\\{u=\mathcal{S}f\in
H^{1}(\Omega)\;\mathrel{\mathop{\ordinarycolon}}\;f\in\mathcal{C}(\partial\Omega),\;\mathrm{supp}\,f\subset\Sigma\right\\}$
$\displaystyle
X_{\\_}{\Omega^{\prime}}=\left\\{{\left.\kern-1.2ptu\right|_{\Omega^{\prime}}}\,\mathrel{\mathop{\ordinarycolon}}\,u\in
X\right\\}\subset H^{1}(\Omega^{\prime})$ $\displaystyle
X^{\Sigma}_{\\_}{\Omega^{\prime}}=\left\\{{\left.\kern-1.2ptu\right|_{\Omega^{\prime}}}\,\mathrel{\mathop{\ordinarycolon}}\,u\in
X^{\Sigma}\right\\}\subset H^{1}(\Omega^{\prime})$

######  Proposition 3.4 (Runge approximation).

Let $V$ a potential of the form (1). Let $\Sigma$ be a relatively open set of
dimension $d-1$, separated from $\mathrm{supp}\,V$, and that can be expressed
as the graph of $\mathcal{C}^{2}$ functions.
Choose a bounded open domain $\Omega$ of class $\mathcal{C}^{2}$ such that
$\Sigma\subset\partial\Omega$ and $\mathrm{supp}\,V\subset\Omega$, and let
$\Omega^{\prime}\subset\subset\Omega$ be a smooth domain such that
$\mathrm{supp}\,V\subset\Omega^{\prime}$, and such that
$\Omega\setminus\overline{\Omega^{\prime}}$ is connected.
Then, there exists $\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for every
$\lambda\geq\lambda_{\\_}0$ except for at most a countable set,
$X^{\Sigma}_{\\_}{\Omega^{\prime}}$ is dense in $X_{\\_}{\Omega^{\prime}}$
under the $L^{2}(\Omega^{\prime})$ norm, this is, any function in
$X_{\\_}{\Omega^{\prime}}$ can be approximated by functions in
$X^{\Sigma}_{\\_}{\Omega^{\prime}}$ in this norm.

######  Proof:.

We will show that
$X_{\\_}{\Omega^{\prime}}\subset{({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}}^{\perp}=\overline{X_{\\_}{\Omega^{\prime}}^{\Sigma}}$,
where the orthogonal complement and closure are taken with respect to the
$L^{2}(\Omega^{\prime})$ inner product. Indeed, let $v\in
L^{2}(\Omega^{\prime})$, and let $\varphi\in H^{1}(\Omega^{\prime})$ such that

(24)
$\displaystyle\begin{cases}\;(\Delta+\lambda-V)\,\varphi=E_{\\_}0v&\text{in
}\Omega,\\\ \;\partial_{\\_}\nu\varphi=0&\text{on }\partial\Omega,\end{cases}$

where $E_{\\_}0v$ denotes the extension by $0$ of $v$ to $\Omega$. This
$\varphi$ will be guaranteed to exist as long as $\lambda$ is not a Neumann
eigenvalue for the operator $-\Delta+V$ in $\Omega$. The set of these
eigenvalues is a countable set. For a proof of these claims, see appendix A.
Now, we have that

$\displaystyle v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\iff
0=\braket{r_{\\_}{\Omega^{\prime}}u,v}_{\\_}{\Omega^{\prime}}=\braket{u,E_{\\_}0v}_{\\_}{\Omega}=\braket{u,(\Delta+\lambda-V)\,\varphi}_{\\_}{\Omega},\quad\forall
u\in X^{\Sigma},$

Applying Green’s identity, and the fact that $(\Delta+\lambda-V)\,u=0$ in
$\Omega$ if $u\in X^{\Sigma}$,

$\displaystyle v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\iff 0=$
$\displaystyle\braket{u,(\Delta+\lambda-V)\,\varphi}_{\\_}{\Omega}$
$\displaystyle=$
$\displaystyle\braket{(\Delta+\lambda-V)\,u,\varphi}_{\\_}{\Omega}+\braket{u,\partial_{\\_}\nu\varphi}_{\\_}{\partial\Omega}-\braket{\partial_{\\_}\nu
u,\varphi}_{\\_}{\partial\Omega},\quad\forall u\in X^{\Sigma}.$
$\displaystyle\iff 0=$ $\displaystyle\braket{\partial_{\\_}\nu
u,\varphi}_{\\_}{\partial\Omega},\quad\forall u\in X^{\Sigma}.$

Now, since $u=\mathcal{S}f$, we have that (see lemma A.3), for
$x\in\partial\Omega$,

$\partial_{\\_}\nu u(x)=\frac{1}{2}f(x)+\left(\mathcal{N}f\right)(x),$

with the operator $\mathcal{N}$ as defined in (21). Therefore,

$v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\iff
0=\frac{1}{2}\braket{f,\varphi}_{\\_}{\partial\Omega}+\braket{\mathcal{N}f,\varphi}_{\\_}{\partial\Omega},\quad\forall
f\in\mathcal{C}(\partial\Omega)\,\text{ s.th.
}\mathrm{supp}\,f\subset\Sigma.\\\ $

Now, by lemma 3.3, we have that, for $f\in\mathcal{C}(\partial\Omega)$
supported on $\Sigma$,

$\braket{\mathcal{N}f,\varphi}_{\\_}{\partial\Omega}=-\braket{\mathcal{S}f,v}_{\\_}{\Omega^{\prime}}=0,$

since $v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}$ by assumption and
${\left.\kern-1.2pt\mathcal{S}f\right|_{\Omega^{\prime}}}\in
X_{\\_}{\Omega^{\prime}}^{\Sigma}$ by definition. This yields

$v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\iff
0=\braket{f,\varphi}_{\\_}{\partial\Omega},\quad\forall
f\in\mathcal{C}(\partial\Omega)\,\text{ s.th.
}\mathrm{supp}\,f\subset\Sigma.\\\ $

Consequently,

(25)
$v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\iff\varphi=0\quad\textrm{on
}\Sigma$

Then, $\varphi$ solves

$\begin{cases}\;(\Delta+\lambda)\,\varphi=0&\text{in
}\Omega\setminus\overline{\Omega^{\prime}},\\\
\;\partial_{\\_}\nu\varphi=\varphi=0&\text{on }\Sigma,\end{cases}$

which, by the Unique Continuation property for the operator $(\Delta-\lambda)$
[18], implies that $\varphi=0$ in $\Omega\setminus\overline{\Omega^{\prime}}.$
Also, since $\varphi$ solves $(\Delta+\lambda)\varphi=v$ in
$\Omega\setminus\mathrm{supp}\,V$, its restriction to this set belongs to
$H^{2}(\Omega\setminus\mathrm{supp}\,V)$ which means that
$\varphi=\partial_{\\_}\nu\varphi=0$ on
${\partial\Omega^{\prime}\subset\Omega\setminus\mathrm{supp}\,V}$.

Finally, let $w\in X_{\\_}{\Omega^{\prime}}$. We want to show that
$w\in{\left(({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}\right)}^{\perp}$.
Indeed, let ${v\in({X_{\\_}{\Omega^{\prime}}^{\Sigma}})^{\perp}}$ and
$\varphi$ as in (24). Then, using Green’s Identity again,

$\displaystyle\braket{w,v}_{\\_}{\Omega^{\prime}}=$
$\displaystyle\braket{w,(\Delta+\lambda-V)\,\varphi}_{\\_}{\Omega^{\prime}}$
$\displaystyle=$
$\displaystyle\braket{(\Delta+\lambda-V)\,w,\varphi}_{\\_}{\Omega^{\prime}}+\braket{w,\partial_{\\_}\nu\varphi}_{\\_}{\partial\Omega^{\prime}}-\braket{\partial_{\\_}\nu
w,\varphi}_{\\_}{\partial\Omega^{\prime}}$ $\displaystyle=$
$\displaystyle\braket{w,\partial_{\\_}\nu\varphi}_{\\_}{\partial\Omega^{\prime}}-\braket{\partial_{\\_}\nu
w,\varphi}_{\\_}{\partial\Omega^{\prime}}=0,$

where we have used that $(\Delta+\lambda-V)\,w=0$ in $\Omega^{\prime}$. This
proves that
$X_{\\_}{\Omega^{\prime}}\subset\overline{X_{\\_}{\Omega^{\prime}}^{\Sigma}}$,
which ends the proof. ∎

As we commented above, we need to make the approximation of proposition 3.4 in
a stronger norm than $L^{2}$. This will be clear later in the proof of
proposition 3.1, but the main reason is that the potential $V$ doesn’t act as
a bilinear operator over $L^{2}$, but over $H^{1}$. The next lemma is an
interior regularity result, also called Caccioppoli inequality in the
literature [12]. It will allow us to get $H^{1}$ convergence in a smaller set
than the one in which we have proven the $L^{2}$ approximation. Both the
statement and the proof of the lemma are inspired by [6]. However, due to the
nature of our potential $V$, we will obtain less regularity, and we will need
to use both Sobolev embeddings and interpolation of Sobolev spaces to complete
the proof.

######  Lemma 3.5 (Interior regularity).

Consider $d\geq 3$. Let V be a potential of the form (1). Let
$\Omega^{\prime}\subset\subset\Omega$ be two Lipschitz open domains such that
$\mathrm{supp}\,V\subset\Omega^{\prime}$. Then, if $u\in H^{1}(\Omega)$ solves

(26) $(\Delta+\lambda-V)\,u=0\quad\textrm{in }\Omega,$

then it holds that

$\lVert u\rVert_{\\_}{H^{1}(\Omega^{\prime})}\lesssim\lVert
u\rVert_{\\_}{L^{2}(\Omega)}.$

######  Proof:.

We will assume that $u$ is a real function. For a complex $u$, both its real
and imaginary parts are real functions that satisfy (26), since $V$ is real,
thus the result will follow.
The weak formulation of equation (26) is

(27) $\int_{\\_}{\Omega}\nabla
u\nabla\phi=\int_{\\_}{\Omega}(V-\lambda)\,u\,\varphi,\quad\forall\varphi\in
H_{\\_}0^{1}(\Omega).$

Consider now a real cut-off function $\eta$ between $\Omega^{\prime}$ and
$\partial\Omega$, namely,
$\eta\in\mathcal{C}^{\infty}_{\\_}0(\Omega;\mathbb{R})$ such that $\eta\equiv
1$ in $\Omega^{\prime}$ and $0\leq\eta\leq 1$ in
$\Omega\setminus\Omega^{\prime}$. Consider then the test function
$\varphi=\eta^{2}u\in H_{\\_}0^{1}(\Omega)$. With this test function, eq. (27)
becomes

$\int_{\\_}{\Omega}\left|\nabla u\right|^{2}\eta^{2}+\int_{\\_}{\Omega}\nabla
u\,2\eta\nabla\eta\,u=\int_{\\_}{\Omega}(V-\lambda)\,u^{2}\eta^{2}.$

Therefore, since $\mathrm{supp}\,V\subset\Omega^{\prime}$,

(28) $\int_{\\_}{\Omega}\left|\nabla
u\right|^{2}\eta^{2}\leq\left|\int_{\\_}{\Omega}\nabla
u\,2\eta\,u\nabla\eta\right|+{\left|\int_{\\_}{\Omega}\lambda\,u^{2}\eta^{2}\right|}+\left|\int_{\\_}{\Omega}V\,u^{2}\right|.$

Now, let’s analyze these three summands separately. For the second one we
have, since $0\leq\eta\leq 1$,

(29)
$\left|\int_{\\_}{\Omega}\lambda\,u^{2}\eta^{2}\right|\leq\lambda\int_{\\_}{\Omega}u^{2}=\lambda\lVert
u\rVert^{2}_{\\_}{L^{2}(\Omega)}.$

For the first one, set any $0<\varepsilon<1$. By Young’s inequality for
products, we have that $2ab\leq\varepsilon a^{2}+\varepsilon^{-1}b^{2}$ for
any two $a,b\geq 0$. This, along with Cauchy-Schwartz inequality gives

(30) $\displaystyle\begin{split}\left|\int_{\\_}{\Omega}\nabla
u\,2\eta\,u\nabla\eta\right|\leq&\,\varepsilon\int_{\\_}{\Omega}\left|\nabla
u\right|^{2}\eta^{2}+\varepsilon^{-1}\int_{\\_}{\Omega}\left|\nabla\eta\right|^{2}u^{2}\\\
\leq&\,\varepsilon\int_{\\_}{\Omega}\left|\nabla
u\right|^{2}\eta^{2}+\varepsilon^{-1}\lVert\nabla\eta\rVert^{2}_{\\_}{L^{\infty}(\Omega)}\lVert
u\rVert^{2}_{\\_}{L^{2}(\Omega)}.\end{split}$

Finally, in the last summand of (28) we will analyze $V^{0}$ and
$V^{F}=\alpha\,\mathrm{d}\sigma+\gamma^{s}$ separately:

$\left|\int_{\\_}{\Omega^{\prime}}V\,u^{2}\right|\leq\left|\int_{\\_}{\Omega^{\prime}}V^{0}u^{2}\right|+\left|\int_{\\_}{\Omega^{\prime}}V^{F}u^{2}\right|.$

To estimate the second term, use (13) in the proof of proposition 2.7. Indeed,
there exists $K>0$ such that

$\left|\int_{\\_}{\Omega^{\prime}}V^{F}u^{2}\right|\leq K\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right)\lVert
u\rVert_{\\_}{H^{s}(\Omega^{\prime})}^{2}.$

Now, note that in a domain with smooth boundary, the fractional order Sobolev
space $H^{s}(\Omega^{\prime})$ is equivalent to the interpolation space
between $L^{2}(\Omega^{\prime})$ and $H^{1}(\Omega^{\prime})$ with
interpolation index $s$ (see, for example, [29]), that is

$\lVert u\rVert^{2}_{\\_}{H^{s}(\Omega^{\prime})}\leq\lVert
u\rVert^{2(1-s)}_{\\_}{L^{2}(\Omega^{\prime})}\lVert
u\rVert^{2s}_{\\_}{H^{1}(\Omega^{\prime})}.$

Also, if we take $\delta>0$, using again Young’s inequality for products, we
have that
$ab\leq\delta^{-p}\frac{a^{p}}{p}+\delta^{p^{\prime}}\frac{b^{p^{\prime}}}{p^{\prime}}$,
for any $a,b\geq 0$ and $\frac{1}{p}+\frac{1}{p^{\prime}}=1$. Choosing
$p=\frac{1}{1-s}$, $p^{\prime}=\frac{1}{s}$ gives

$\lVert
u\rVert^{2}_{\\_}{H^{s}(\Omega^{\prime})}\leq(1-s)\delta^{-\frac{1}{1-s}}\lVert
u\rVert^{2}_{\\_}{L^{2}(\Omega^{\prime})}+s\delta^{1/s}\lVert
u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})}.$

Therefore, since $\lVert u\rVert_{\\_}{L^{2}(\Omega^{\prime})}\leq\lVert
u\rVert_{\\_}{L^{2}(\Omega)}$, we have that

(31) $\displaystyle\left|\int_{\\_}{\Omega^{\prime}}V^{F}u^{2}\right|\leq
K\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right)\left[(1-s)\delta^{-\frac{1}{1-s}}\lVert
u\rVert^{2}_{\\_}{L^{2}(\Omega)}+s\delta^{1/s}\lVert
u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})}\right].$

Now, to estimate the $V^{0}$ term, let $N>0$ and consider the set

$F=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|V^{0}(x)|>N\\},$

and define $E=\mathbb{R}^{d}\setminus F$. Then,
$V^{0}=\mathbbm{1}_{\\_}EV_{\\_}0+\mathbbm{1}_{\\_}FV_{\\_}0$, with
$\lVert\mathbbm{1}_{\\_}EV_{\\_}0\rVert_{\\_}{L^{\infty}}=N$. Then, using
Hölder inequality and the Sobolev embedding $H^{1}(\Omega^{\prime})\subset
L^{p_{\\_}d}(\Omega^{\prime})$,

(32)
$\displaystyle\begin{split}\left|\int_{\\_}{\Omega^{\prime}}V^{0}u^{2}\right|\leq&\left|\int_{\\_}{\Omega^{\prime}}V^{0}\mathbbm{1}_{\\_}Eu^{2}\right|+\left|\int_{\\_}{\Omega^{\prime}}V^{0}\mathbbm{1}_{\\_}Fu^{2}\right|\\\
\leq&N\lVert u\rVert^{2}_{\\_}{L^{2}(\Omega^{\prime})}+\lVert
V^{0}\mathbbm{1}_{\\_}F\rVert_{\\_}{L^{d/2}(\Omega^{\prime})}\lVert
u\rVert^{2}_{\\_}{L^{p_{\\_}d}(\Omega^{\prime})}\\\ \leq&N\lVert
u\rVert^{2}_{\\_}{L^{2}(\Omega^{\prime})}+C\lVert
V^{0}\mathbbm{1}_{\\_}F\rVert_{\\_}{L^{d/2}(\Omega^{\prime})}\lVert
u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})},\end{split}$

where $C$ is the Sobolev embedding constant. Putting (29), (30), (31) and (32)
in inequality (28) yields

$(1-\varepsilon)\int_{\\_}{\Omega}\left|\nabla u\right|^{2}\eta^{2}\leq
C_{\\_}1\lVert u\rVert^{2}_{\\_}{L^{2}(\Omega)}+C_{\\_}2\lVert
u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})},$

where

$\displaystyle C_{\\_}1$
$\displaystyle=\lambda+N+\varepsilon^{-1}\lVert\nabla\eta\rVert^{2}_{\\_}{L^{\infty}(\Omega)}+\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right)(1-s)\delta^{-\frac{1}{1-s}},$
$\displaystyle C_{\\_}2$ $\displaystyle=K\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right)s\delta^{1/s}+C\lVert
V^{0}\mathbbm{1}_{\\_}F\rVert_{\\_}{L^{d/2}(\Omega^{\prime})}.$

Therefore we have, again since $0\leq\eta\leq 1$,

$(1-\varepsilon)\lVert u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})}\leq
C_{\\_}1\lVert u\rVert^{2}_{\\_}{L^{2}(\Omega)}+C_{\\_}2\lVert
u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})},$

so that

$(1-\varepsilon-C_{\\_}2)\lVert u\rVert^{2}_{\\_}{H^{1}(\Omega^{\prime})}\leq
C_{\\_}1\lVert u\rVert^{2}_{\\_}{L^{2}(\Omega)}.$

Finally, since we can take $\varepsilon$ and $\delta$ as small as we want, and
we can take $N$ as big as needed to make $\lVert
V^{0}\mathbbm{1}_{\\_}F\rVert_{\\_}{L^{d/2}}$ sufficiently small so that

$C_{\\_}2+\varepsilon<1.$

Therefore, we can conclude that

$\lVert u\rVert_{\\_}{H^{1}(\Omega^{\prime})}\lesssim\lVert
u\rVert_{\\_}{L^{2}(\Omega)},$

and the lemma is proved. ∎

###  3.2. Proof of proposition 3.1

To prove proposition 3.1 we will further need the following lemmas, whose
proof can be again be found in [4]. It might be of interest to note that,
although we introduce a wider class of potentials, the proof is based on
working away from its support, and therefore it goes through identically.

######  Lemma 3.6 ([4]).

Let $\Omega$ be a bounded open domain of class $\mathcal{C}^{2}$. The
scattering solution of (26) satisfies the following reciprocity relation

$u_{\\_}{sc}(x,y)=u_{\\_}{sc}(y,x),\quad\forall
x,y\in\mathbb{R}^{d}\setminus\mathrm{supp}\,V.$

In particular, the single layer potential $\mathcal{S}$ is symmetric, that is,

$\int_{\\_}{\partial\Omega}\mathcal{S}f\,g\,\,\mathrm{d}S=\int_{\\_}{\partial\Omega}f\,\mathcal{S}g\,\,\mathrm{d}S,\quad\forall
f,g\in\mathcal{C}(\partial\Omega).$

The following lemma is also proved in the appendix, lemma A.3. From now on,
when necessary, we will denote by $u_{\\_}+$ the trace on $\partial\Omega$ of
${\left.\kern-1.2ptu\right|_{\mathbb{R}^{d}\setminus\overline{\Omega}}}$, and
by $u_{\\_}-$ the trace on $\partial\Omega$ of
${\left.\kern-1.2ptu\right|_{\Omega}}$. As well, we will denote by
$\partial_{\\_}\nu u_{\\_}+$ and $\partial_{\\_}\nu u_{\\_}-$ the normal
derivative of those, always with respect to the outward-pointing normal vector
of $\partial\Omega$ (as seen from inside $\Omega$).

######  Lemma 3.7 ([4]).

Consider $d\geq 3$, and let $\Omega$ be a bounded open domain of class
$\mathcal{C}^{2}$. Let $f\in\mathcal{C}(\partial\Omega)$. Then,
$u=\mathcal{S}f$ is the unique solution in
$H^{1}_{\\_}{\text{loc}}(\mathbb{R}^{d})$ to the problem

$\begin{cases}(\Delta+\lambda-V)\,u=0&\textrm{in
}\mathbb{R}^{d}\setminus\partial\Omega,\\\ \partial_{\\_}\nu
u_{\\_}--\partial_{\\_}\nu u_{\\_}+=f&\textrm{on }\partial\Omega,\\\ u\textrm{
satisfying SRC}.\end{cases}$

After these considerations, we will proceed by first proving that the
orthogonality relation is fullfiled by solutions represented as single layer
potentials with densities supported in $\Sigma_{\\_}1$ and $\Sigma_{\\_}2$.
The proof is similar to that of proposition 3.3 in [4].

######  Lemma 3.8.

Consider $d\geq 3$. Let $V_{\\_}1$ and $V_{\\_}2$ be two potentials of the
form (1), and let $\Sigma_{\\_}1,\Sigma_{\\_}2$ be two relatively open sets of
dimension $d-1$, separated from $\mathrm{supp}\,V$, and that can be expressed
as the graph of $\mathcal{C}^{2}$ functions. Choose a bounded open domain
$\Omega$ of class $\mathcal{C}^{2}$ such that
$\Sigma_{\\_}1,\Sigma_{\\_}2\subset\partial\Omega$ and
$\mathrm{supp}\,V_{\\_}j\subset\Omega$, $j=1,2$,.
Then, there exists $\lambda_{\\_}0=\lambda_{\\_}0(V,d)$ such that, for every
$\lambda\geq\lambda_{\\_}0$, it holds that

${\left.\kern-1.2ptu_{\\_}{sc,1}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}={\left.\kern-1.2ptu_{\\_}{sc,2}\right|_{\Sigma_{\\_}1\times\Sigma_{\\_}2}}\implies\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=0,$
$\text{for all }v_{\\_}1,v_{\\_}2\in H^{1}(\Omega)\textrm{ such that
}v_{\\_}j=\mathcal{S}_{\\_}jf_{\\_}j,\textrm{ for some
}f_{\\_}j\in\mathcal{C}(\partial\Omega)\textrm{ supported in }\Sigma_{\\_}j.$

######  Proof:.

Integrating in $\Omega$ and using Green’s identity gives

(33)
$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=\int_{\\_}{\partial\Omega}(v_{\\_}2\partial_{\\_}\nu{v_{\\_}1}_{\\_}--v_{\\_}1\partial_{\\_}\nu{v_{\\_}2}_{\\_}-)\,\,\mathrm{d}S,$

while doing so in $B\setminus\overline{\Omega}$, where
$B\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,\left|x\right|<R\\}$
gives

$\displaystyle 0=$
$\displaystyle\int_{\\_}{B\setminus\overline{\Omega}}(\Delta+\lambda)v_{\\_}1v_{\\_}2=$
$\displaystyle=$ $\displaystyle\int_{\\_}{\partial
B}(v_{\\_}2\partial_{\\_}\nu
v_{\\_}1-v_{\\_}1\partial_{\nu}v_{\\_}2)\,\,\mathrm{d}S-\int_{\\_}{\partial\Omega}(v_{\\_}2\partial_{\\_}\nu{v_{\\_}1}_{\\_}+-v_{\\_}1\partial_{\nu}{v_{\\_}2}_{\\_}+)\,\,\mathrm{d}S,$

since $v_{\\_}1$ and $v_{\\_}2$ are solutions to $(\Delta+\lambda)\,u=0$ in
$\mathbb{R}^{d}\setminus\overline{\Omega}$ by lemma 3.7. Here, making
$R\to\infty$ and applying SRC (2) yields

(34)
$\int_{\\_}{\partial\Omega}(v_{\\_}2\partial_{\\_}\nu{v_{\\_}1}_{\\_}+-v_{\\_}1\partial_{\nu}{v_{\\_}2}_{\\_}+)\,\,\mathrm{d}S=0.$

Then, inserting (34) in (33),

(35)
$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=\int_{\\_}{\partial\Omega}\left[v_{\\_}2\,(\partial_{\nu}{v_{\\_}1}_{\\_}--\partial_{\nu}{v_{\\_}1}_{\\_}+)-v_{\\_}1\,(\partial_{\nu}{v_{\\_}2}_{\\_}--\partial_{\\_}\nu{v_{\\_}2}_{\\_}+)\right]\,\,\mathrm{d}S.$

Now, since $v_{\\_}j=\mathcal{S}_{\\_}j\,f_{\\_}j$, applying lemma 3.7,

$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=\int_{\\_}{\partial\Omega}(\mathcal{S}_{\\_}2f_{\\_}2\,f_{\\_}1-\mathcal{S}_{\\_}1f_{\\_}1\,f_{\\_}2)\,\,\mathrm{d}S,$

and, by lemma 3.6,

$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=\int_{\\_}{\partial\Omega}\left[\mathcal{S}_{\\_}2-\mathcal{S}_{\\_}1\right]f_{\\_}1\,f_{\\_}2\,\,\mathrm{d}S.$

Finally, since $u_{\\_}{sc,1}=u_{\\_}{sc,2}$ in
$\Sigma_{\\_}1\times\Sigma_{\\_}2$ by assumption, the kernel of
$\mathcal{S}_{\\_}2-\mathcal{S}_{\\_}1$ is supported in
$(\partial\Omega\times\partial\Omega)\setminus(\Sigma_{\\_}1\times\Sigma_{\\_}2)$,
but $\mathrm{supp}\,(f_{\\_}1\otimes
f_{\\_}2)\subset\Sigma_{\\_}1\times\Sigma_{\\_}2$, so we conclude that

$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=0,$

and the lemma is proved. ∎

Now, we are set for the proof of the orthogonality relation. We restate the
proposition again: See 3.1

######  Proof:.

Now let $v_{\\_}1,v_{\\_}2\in H^{1}(\Omega)$ be such that to
$\left(\Delta+\lambda-V_{\\_}j\right)v_{\\_}j=0\textrm{ in }\Omega$. Let
$\Omega^{\prime\prime}$ and $\Omega^{\prime}$ be two open domains such that

$\mathrm{supp}\,V_{\\_}j\subset\Omega^{\prime\prime}\subset\subset\Omega^{\prime}\subset\subset\Omega,\quad
j=1,2,$

and let $(v_{\\_}j^{(m)})_{\\_}{m\in\mathbb{N}}\subset X^{\Sigma_{\\_}j}$,
$j=1,2$, be two sequences such that
$v_{\\_}j^{(m)}\xrightarrow{L^{2}(\Omega^{\prime})}v_{\\_}j$, which exist in
virtue of proposition 3.4, meaning that
$v_{\\_}j^{(m)}=\mathcal{S}_{\\_}jf_{\\_}j^{(m)}$ for some $f_{\\_}j^{(m)}$
supported in $\Sigma_{\\_}j$. Then,

(36)
$\displaystyle\begin{split}\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=&\braket{(V_{\\_}1-V_{\\_}2)(v_{\\_}1-v_{\\_}1^{(m)}),v_{\\_}{2}}+\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1^{(m)},v_{\\_}2-v_{\\_}2^{(m)}}\\\
&+\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1^{(m)},v_{\\_}2^{(m)}}.\end{split}$

By lemma 3.8, we have that

(37) $\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1^{(m)},v_{\\_}2^{(m)}}=0,\quad\forall
m\in\mathbb{N}.$

We will now show that the two first summands converge to zero as $m\to\infty$.
Note first that, for $\phi,\varphi\in H^{1}(\Omega^{\prime\prime})$, it holds
that

$\braket{V^{0}_{\\_}j\,\phi,\varphi}\leq\lVert
V^{0}_{\\_}j\rVert_{\\_}{L^{d/2}}\,\lVert\phi\rVert_{\\_}{L^{p_{\\_}d}(\Omega^{\prime\prime})}\,\lVert\varphi\rVert_{\\_}{L^{p_{\\_}d}(\Omega^{\prime\prime})}\lesssim\lVert\phi\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}\lVert\varphi\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})},$

and

$\braket{V^{F}\,\phi,\varphi}\lesssim\left(\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\right)\lVert\phi\rVert_{\\_}{H^{s}(\Omega^{\prime\prime})}\,\lVert\varphi\rVert_{\\_}{H^{s}(\Omega^{\prime\prime})}\lesssim\lVert\phi\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}\lVert\varphi\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}.$

We have used in the estimates above the Hardy-Littlewood-Sobolev inequality
and inequality (13) in proposition 2.7. Therefore, lemma 3.5 yields

(38)
$\braket{(V_{\\_}1-V_{\\_}2)(v_{\\_}1-v_{\\_}1^{(m)}),v_{\\_}{2}}\lesssim\lVert
v_{\\_}1-v_{\\_}1^{(m)}\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}\,\lVert
v_{\\_}2\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}\lesssim\lVert
v_{\\_}1-v_{\\_}1^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})}.$

In the second summand we proceed as in the first one. Note that, since
$v_{\\_}1^{(m)}$ are also solutions in $H^{1}(\Omega)$ of
$(\Delta+\lambda-V)\,u=0$, by lemma 3.5,

$\lVert v_{\\_}1^{(m)}\rVert_{\\_}{H^{1}(\Omega^{\prime\prime})}\leq\lVert
v_{\\_}1^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})},\quad\forall
m\in\mathbb{N},$

and, since $(v_{\\_}1^{(m)})_{\\_}{m\in\mathbb{N}}$ is a convergent sequence,
it must also be bounded. This is, $\exists C\geq 0$ such that

$\lVert v_{\\_}1^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})}\leq C,\quad\forall
m\in\mathbb{N}.$

Thus, we obtain

(39)
$\lvert\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1^{(m)},v_{\\_}2-v_{\\_}2^{(m)}}\rvert\lesssim\lVert
v_{\\_}2-v_{\\_}2^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})}.$

By putting (37), (38) and (39) in (36) we obtain

$\lvert\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}\rvert\lesssim\lVert
v_{\\_}1-v_{\\_}1^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})}+\lVert
v_{\\_}2-v_{\\_}2^{(m)}\rVert_{\\_}{L^{2}(\Omega^{\prime})}\xrightarrow{m\to\infty}0,$

which proves our statement. ∎

##  4\. CGO solutions and proof of theorem 2

Now that we have proved the Alessandrini-type identity in proposition 3.1, we
can test it with the CGO solutions. These will be solutions of the form

(40) $v_{\\_}j(x)=e^{\zeta_{\\_}j\cdot x}(1+w_{\\_}j(x)),$

for some $\zeta_{\\_}j\in\mathbb{C}^{d}$. We will follow once again the
arguments in [4] and [5] for this section. In particular, applying the
operator $(\Delta+\lambda-V)$ in (40) yields

$(\Delta+2\zeta_{\\_}j\cdot\nabla+\zeta_{\\_}j\cdot\zeta_{\\_}j+\lambda-V)w_{\\_}j=V-\lambda-\zeta_{\\_}j\cdot\zeta_{\\_}j.$

Now if $\zeta_{\\_}j$ are chosen such that
$\zeta_{\\_}j\cdot\zeta_{\\_}j=-\lambda$, we obtain

$(\Delta+2\zeta_{\\_}j\cdot\nabla-V)w_{\\_}j=V.$

Therefore, to prove the existence of these solutions, it is enough to prove
injectivity of the formal adjoint $(\Delta-2\zeta_{\\_}j\cdot\nabla-V)$. This
can be done via a priori estimates. This estimate will be proved in section
4.1 in the relevant spaces, based on a Carleman estimate for the laplacian by
Caro and Rogers [5], that can be perturbed to include the potentials $V$. The
inequalities will be analogous to those in [4] and [5] but, besides adding the
potential $\gamma^{s}$, we will follow a slightly different order in rotating
the inequalities and adding the potentials.
Later, in section 4.2 we will end the proof of Theorem 2. We will further
choose $\zeta_{\\_}j$ to fulfill $\zeta_{\\_}1+\zeta_{\\_}2=-i\kappa$ for an
arbitray $\kappa\in\mathbb{R}^{d}$ -which is possible in dimension $d\geq 3$-,
and the correction term $w_{\\_}j$ will vanish in a certain sense when
$|\zeta_{\\_}j|$ grows. Here the approach will be based in [4, 5, 13].

###  4.1. Existence of CGO solutions

As we mentioned in the introduction, we will prove the existence of CGO
solutions via a priori estimates. For $s\in\mathbb{R}$ and
$\zeta\in\mathbb{C}^{d}$ we define the inhomogeneous Bourgain space
$X_{\\_}\zeta^{s}$ as the space of distributions
$u\in\mathscr{S}^{\prime}(\mathbb{R}^{d})$ such that $\widehat{u}\in
L^{2}_{\\_}\text{loc}(\mathbb{R}^{d})$ and

$\lVert
u\rVert_{\\_}{X_{\\_}\zeta^{s}}=\lVert(M|\Re(\zeta)|^{2}+M^{-1}|p_{\\_}\zeta|^{2})^{s/2}\,\widehat{u}\rVert_{\\_}{L^{2}}<\infty,$

endowed with the norm $\lVert\centerdot\rVert_{\\_}{X_{\\_}\zeta^{s}}$, with
$M>1$, where $\Re$ denotes the real part and

$p_{\\_}\zeta(\xi)=-|\xi|^{2}+2i\zeta\cdot\xi+\zeta\cdot\zeta.$

These spaces were originally considered by Haberman and Tataru in [14], then
by Haberman in [13], by Caro and Rogers in [5], and by Caro and García in [4].
To prove the a priori estimate, we draw from Theorem 2 in [5]. From now on,
set $\zeta=\tau\theta+i\mathcal{I}$, with $\tau>0$,
$\theta\in\mathbb{S}^{d-1}$, $\mathcal{I}\in\mathbb{R}^{d}$, and
$\zeta\cdot\zeta\leq 0$, and define

$\varphi_{\\_}\zeta(x)=M\frac{(x\cdot\theta)^{2}}{2}+x\cdot\zeta.$

Then, Theorem 2 in [5] is roughly equivalent to the following:

######  Theorem 4.1 ([5], Theorem 2).

Take $R_{\\_}0>0$. There is an absolute constant $C>0$ such that, if
$M>CR_{\\_}0^{2}$ then,

$\lVert u\rVert_{\\_}{X_{\\_}{-\tau e_{\\_}n}^{1/2}}\leq CR_{\\_}0\lVert
e^{\varphi_{\\_}{\tau e_{\\_}n}}\Delta(e^{-\varphi_{\\_}{\tau
e_{\\_}n}}u)\rVert_{\\_}{X_{\\_}{-\tau e_{\\_}n}^{-{1/2}}}$

for $u\in\mathscr{S}(\mathbb{R}^{d})$ with $\mathrm{supp}\,u\subset
B_{\\_}{R_{\\_}0}=\\{x\in\mathbb{R}^{d}\mathrel{\mathop{\ordinarycolon}}|x|<R_{\\_}0\\}$
and $\tau>8MR_{\\_}0$.

We will transform the inequality to make it work for an arbitrary $\zeta$. If
for $\zeta=\tau\theta+i\mathcal{I}$ as above we take $Q\in SO(d)$ to be a
rotation such that $Qe_{\\_}n=\theta$, and denote by $Q^{T}$ its traspose, and
by $Q_{\\_}*$ its pullback, then it is easy to check that $\varphi_{\\_}{\tau
e_{\\_}n}=Q_{\\_}*\varphi_{\\_}{\tau\theta}$ and $p_{\\_}{-\tau
e_{\\_}n}=Q_{\\_}*p_{\\_}{-\tau\theta}$. Thus, if
$u\in\mathscr{S}(\mathbb{R}^{d})$ with $\mathrm{supp}\,u\subset
B_{\\_}{R_{\\_}0}$ and $\tau>8MR_{\\_}0$, we have

$\displaystyle\lVert u\rVert_{\\_}{X^{1/2}_{\\_}{-\tau\theta}}=$
$\displaystyle\lVert Q_{\\_}*u\rVert_{\\_}{X^{1/2}_{\\_}{-\tau e_{\\_}n}}\leq
CR_{\\_}0\lVert e^{\varphi_{\\_}{\tau e_{\\_}n}}\Delta(e^{-\varphi_{\\_}{\tau
e_{\\_}n}}Q_{\\_}*u)\rVert_{\\_}{X_{\\_}{-\tau e_{\\_}n}^{-{1/2}}}=$
$\displaystyle CR_{\\_}0\lVert Q_{\\_}*[e^{\varphi_{\\_}{\tau
e_{\\_}n}}\Delta(e^{-\varphi_{\\_}{\tau
e_{\\_}n}}u)]\rVert_{\\_}{X_{\\_}{-\tau e_{\\_}n}^{-{1/2}}}=CR_{\\_}0\lVert
e^{\varphi_{\\_}{\tau\theta}}\Delta(e^{-\varphi_{\\_}{\tau\theta}}u)\rVert_{\\_}{X_{\\_}{-\tau\theta}^{-{1/2}}},$

where we have used Theorem 4.1 and the fact that rotations commute both with
the laplacian and the Fourier transform. On the other hand, if
$\zeta=\tau\theta+i\mathcal{I}$ and we denote by
$\mathcal{N}_{\\_}\mathcal{I}$ the translation operator by
$\mathcal{I}\in\mathbb{R}^{d}$, it’s again easy to check that
$p_{\\_}{-\zeta}=\mathcal{N}_{\\_}\mathcal{I}p_{\\_}{-\tau\theta}$. Therefore,
keeping in mind that
$\mathcal{N}_{\\_}{-\mathcal{I}}\widehat{u}(\xi)=\widehat{e^{-i\mathcal{I}\cdot}u}(\xi)$,
we obtain

(41) $\displaystyle\begin{split}\lVert
u\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}&=\lVert
e^{-i\mathcal{I}\centerdot}u\rVert_{\\_}{X^{1/2}_{\\_}{-\tau\theta}}\leq
CR_{\\_}0\lVert
e^{\varphi_{\\_}{\tau\theta}}\Delta(e^{-\varphi_{\\_}{\tau\theta}}e^{-i\mathcal{I}\centerdot}u)\rVert_{\\_}{X_{\\_}{-\tau\theta}^{-{1/2}}}\\\
&=CR_{\\_}0\lVert
e^{-i\mathcal{I}\centerdot}e^{\varphi_{\\_}{\zeta}}\Delta(e^{-\varphi_{\\_}{\zeta}}u)\rVert_{\\_}{X_{\\_}{-\tau\theta}^{-{1/2}}}=CR_{\\_}0\lVert
e^{\varphi_{\\_}{\zeta}}\Delta(e^{-\varphi_{\\_}{\zeta}}u)\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}\end{split}$

Now we can perturb this inequality with the term $\lambda-V$ in the operator.
The following inequalities will be of interest during the whole argument. For
any $\zeta\in\mathbb{C}^{d}$ such that $\zeta\cdot\zeta\leq 0$ and $u$
compactly supported, we have the following

(42) $\displaystyle\lVert u\rVert_{\\_}{L^{2}}\leq
M^{-1/4}|\Re(\zeta)|^{-{1/2}}\lVert u\rVert_{\\_}{X^{1/2}_{\\_}\zeta},$ (43)
$\displaystyle\lVert u\rVert_{\\_}{L^{p_{\\_}d}}\lesssim M^{1/4}\lVert
u\rVert_{\\_}{X^{1/2}_{\\_}\zeta}.$

Inequality (43) is a direct consequence of Haberman’s embedding [13]

(44) $\lVert u\rVert_{\\_}{L^{p_{\\_}d}}\lesssim\lVert
u\rVert_{\\_}{\dot{X}^{1/2}_{\\_}\zeta},$

where the space $\dot{X}^{s}_{\\_}\zeta$ is defined by the norm $\lVert
u\rVert_{\\_}{\dot{X}_{\\_}\zeta^{s}}=\lVert|p_{\\_}\zeta|^{s}\,\widehat{u}\rVert_{\\_}{L^{2}}.$
We are going to quantify $(\lambda-V)\,u$ in $X^{-{1/2}}_{\\_}\zeta$ by
duality. Remember that in our case $|\Re(\zeta)|=\tau$. Start by estimating
$\lambda u$ with $\mathscr{S}(\mathbb{R}^{d})$. Let
$v\in\mathscr{S}(\mathbb{R}^{d})$, then by Cauchy-Schwartz inequality,

(45) $\lvert\braket{\lambda\,u,v}\rvert\leq\lambda\lVert
u\rVert_{\\_}{L^{2}}\lVert v\rVert_{\\_}{L^{2}}\leq\lambda
M^{-{1/2}}\tau^{-1}\lVert u\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}\lVert
v\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}$

Next, to estimate the term $V^{0}u$, we can split the potential as in the
proof of lemma 3.5. Indeed, consider the set
$F=\\{x\in\mathbb{R}^{d}\,\mathrel{\mathop{\ordinarycolon}}\,|V^{0}(x)|>N\\}$,
and define $E=\mathbb{R}^{d}\setminus F$. Then,
$V^{0}=\mathbbm{1}_{\\_}EV_{\\_}0+\mathbbm{1}_{\\_}FV_{\\_}0$, with
$\lVert\mathbbm{1}_{\\_}EV_{\\_}0\rVert_{\\_}{L^{\infty}}=N$, and therefore by
Cauchy-Schwartz and Hölder inequalities, as well as (42) and (43),

(46) $\displaystyle\begin{split}\lvert\braket{V^{0}\,u,v}\rvert&\leq N\lVert
u\rVert_{\\_}{L^{2}}\lVert
v\rVert_{\\_}{L^{2}}+\lVert\mathbbm{1}_{\\_}FV_{\\_}0\rVert_{\\_}{L^{d/2}}\lVert
u\rVert_{\\_}{L^{p_{\\_}d}}\lVert v\rVert_{\\_}{L^{p_{\\_}d}}.\\\
&\leq(NM^{-{1/2}}\tau^{-1}+M^{1/2}\lVert\mathbbm{1}_{\\_}FV_{\\_}0\rVert_{\\_}{L^{d/2}})\,\lVert
u\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}\lVert
v\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}.\end{split}$

For the term $\alpha\,\,\mathrm{d}\sigma$, we need to use the Besov space
version of Theorem 14.1.1 in [17], which gives us a trace boundedness $\lVert
u\rVert_{\\_}{L^{2}(\Gamma)}\leq\lVert
u\rVert_{\\_}{\dot{B}^{1/2}_{\\_}{2,1}}$. Remember that the Besov spaces
$\dot{B}_{\\_}{p,q}^{s}$ are given by the following norms, using Littlewood-
Paley projectors as defined in (6):

$\lVert
u\rVert_{\\_}{\dot{B}_{\\_}{p,q}^{s}}=\left(\sum_{\\_}{k\in\mathbb{Z}}2^{kqs}\lVert
P_{\\_}ku\rVert^{q}_{\\_}{L^{p}}\right)^{1/q}.$

Indeed, by Cauchy-Schwartz and Hölder inequalities,

$\displaystyle\begin{split}\lvert\braket{\alpha\,\mathrm{d}\sigma\,u,v}\rvert&\leq\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\lVert
u\rVert_{\\_}{L^{2}(\Gamma)}\lVert
v\rVert_{\\_}{L^{2}(\Gamma)}\leq\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\lVert
u\rVert_{\\_}{\dot{B}^{1/2}_{\\_}{2,1}}\lVert
v\rVert_{\\_}{\dot{B}^{1/2}_{\\_}{2,1}}\end{split}$

Now, estimate separately high and low frequencies. Let
$k_{\\_}\tau\in\mathbb{Z}$ be such that $2^{k_{\\_}\tau-1}<\tau\leq
2^{k_{\\_}\tau}$. Then, if $k>k_{\\_}\tau+1$, we have that
$2^{k/2}|\widehat{P_{\\_}ku(\xi)}|\sim
2^{-k/2}|p_{\\_}\zeta(\xi)|^{1/2}|\widehat{P_{\\_}ku(\xi)}|$, so that for the
high frequencies we have that, by Plancherel’s identity,

$\displaystyle\sum_{\\_}{k>k_{\\_}\tau+1}2^{k/2}\lVert
P_{\\_}ku\rVert_{\\_}{L^{2}}\sim\sum_{\\_}{k>k_{\\_}\tau+1}2^{-k/2}\lVert|p_{\\_}\zeta|^{1/2}\widehat{P_{\\_}ku}\rVert_{\\_}{L^{2}}\leq\tau^{-{1/2}}M^{1/4}\lVert
u\rVert_{\\_}{X_{\\_}\zeta^{1/2}},$

while for the low frequencies

$\sum_{\\_}{k\leq k_{\\_}\tau+1}2^{k/2}\lVert
P_{\\_}ku\rVert_{\\_}{L^{2}}\lesssim\tau^{1/2}\lVert u\rVert_{\\_}{L^{2}}\leq
M^{-1/4}\lVert u\rVert_{\\_}{X^{1/2}_{\\_}\zeta}.$

Therefore, combining the previous inequalities, we have that there exist a
constant $C^{\prime}>0$ such that

(47) $\lvert\braket{\alpha\,\mathrm{d}\sigma\,u,v}\rvert\leq
C^{\prime}\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}(M^{-{1/2}}+\tau^{-{1/2}}+\tau^{-1}M^{1/2})\lVert
u\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}\lVert
v\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}.$

Finally, for $\gamma^{s}$, use proposition 2.7 to obtain

$\displaystyle\lvert\braket{\gamma^{s}u,v}\rvert\leq C^{\prime\prime}\lVert
g\rVert_{\\_}{L^{\infty}}\left(\lVert D^{s}u\rVert_{\\_}{L^{2}}\lVert
v\rVert_{\\_}{L^{2}}+\lVert u\rVert_{\\_}{L^{2}}\lVert
D^{s}v\rVert_{\\_}{L^{2}}\right).$

Split again in high and low frequencies:

$\displaystyle\lVert D^{s}u\rVert_{\\_}{L^{2}}\leq$
$\displaystyle\int_{\\_}{\mathbb{R}^{d}}|\xi|^{2s}\,|\widehat{u}(\xi)|^{2}\,\mathrm{d}\xi=\int_{\\_}{|\xi|<\tau}|\xi|^{2s}\,|\widehat{u}(\xi)|^{2}\,\mathrm{d}\xi+\int_{\\_}{|\xi|\geq\tau}|\xi|^{2s}\,|\widehat{u}(\xi)|^{2}\,\mathrm{d}\xi.$

On the one hand, for the low frequencies, clearly

$\int_{\\_}{|\xi|<\tau}|\xi|^{2s}\,|\widehat{u}(\xi)|^{2}\,\mathrm{d}\xi\leq\,\tau^{2s}\,\lVert
u\rVert_{\\_}{L^{2}}^{2}\leq\,\tau^{2s-1}M^{-{1/2}}\,\lVert
u\rVert^{2}_{\\_}{X_{\\_}\zeta^{1/2}},$

while for the high frequencies, having in mind that we assumed $s<1$,

$\displaystyle\int_{\\_}{|\xi|\geq\tau}|\xi|^{2s}\,|\widehat{u}(\xi)|^{2}\,\mathrm{d}\xi\leq$
$\displaystyle\int_{\\_}{|\xi|\geq\tau}|\xi|^{2(s-1)}\,|p_{\\_}\zeta(\xi)|\,\widehat{u}(\xi)\,\mathrm{d}\xi\leq\tau^{2(s-1)}\,\lVert|p_{\\_}\zeta|^{1/2}\,\widehat{u}\rVert^{2}_{\\_}{L^{2}}$
$\displaystyle\ \leq$ $\displaystyle\,\tau^{2(s-1)}M^{1/2}\,\lVert
u\rVert^{2}_{\\_}{X_{\\_}\zeta^{1/2}}.$

Therefore, we have that

$\lVert
D^{s}u\rVert_{\\_}{L^{2}}\leq\left(\tau^{s-{1/2}}M^{-1/4}+\tau^{s-1}M^{1/4}\right)\lVert
u\rVert_{\\_}{X_{\\_}\zeta^{1/2}},$

which yields

(48) $\lvert\braket{\gamma^{s}u,v}\rvert\leq C^{\prime\prime}\lVert
g\rVert_{\\_}{L^{\infty}}\left(\tau^{s-1}M^{-{1/2}}+\tau^{s-3/2}\right)\lVert
u\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}\lVert
v\rVert_{\\_}{X^{{1/2}}_{\\_}\zeta}.$

Now, sum inequalities (45), (46), (47) and (48), and choose $M>CR_{\\_}0^{2}$
so that
$CR_{\\_}0C^{\prime}M^{-{1/2}}\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\leq
1/4$, then choose $N$ such that
$CR_{\\_}0M^{{1/2}}\lVert\mathbbm{1}_{\\_}FV^{0}\rVert_{\\_}{L^{d/2}}\leq
1/4,$ and finally choose $\tau>8MR_{\\_}0$ so that

$\displaystyle CR_{\\_}0\Bigl{[}\left(\lambda+N\right)M^{-{1/2}}\tau^{-1}$
$\displaystyle+C^{\prime}\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}\left(\tau^{-{1/2}}+\tau^{-1}M^{1/2}\right)$
$\displaystyle+C^{\prime\prime}\lVert
g\rVert_{\\_}{L^{\infty}}\left(\tau^{s-1}M^{-{1/2}}+\tau^{-3/2}\right)\Bigr{]}<1/4.$

With this previous discussion we can prove what is summarized in the following
lemma: Next, note that

$\Delta(e^{-\zeta\cdot x}u)=e^{-\zeta\cdot
u}(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta)\,u,$

and thus lemma 2.2 yields

$\lVert u\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}\leq CR_{\\_}0\lVert
e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta+\lambda-V)(e^{-M\frac{(\centerdot\cdot\theta)^{2}}{2}}u)\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}.$

Now we procceed to remove the remaining exponential factors. Take
$u=e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}v$ with
$v\in\mathscr{S}(\mathbb{R}^{d})$ supported in $B_{\\_}{R_{\\_}0}$, then

(49) $\lVert
e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}v\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}\leq
CR_{\\_}0\lVert
e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta+\lambda-V)v\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}.$

Additionaly, if we prove that

(50) $\lVert e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}\chi
w\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}}\lesssim\lVert
w\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}},$

it will follow by duality that

$\lVert
e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}v\rVert_{\\_}{X^{1/2}_{\\_}{-\zeta}}\lesssim\lVert(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta+\lambda-V)v\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}},$

while if

(51) $\lVert
e^{-M\frac{(\centerdot\cdot\theta)^{2}}{2}}w\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}}\lesssim\lVert
w\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}},$

then again by duality

$\lVert v\rVert_{\\_}{X_{\\_}{-\zeta}^{1/2}}\lesssim\lVert
e^{M\frac{(\centerdot\cdot\theta)^{2}}{2}}v\rVert_{\\_}{X_{\\_}{-\zeta}^{1/2}}.$

Above $\chi(x\cdot\theta)=\chi_{\\_}0(x\cdot\theta/R)$ with $\chi_{\\_}0$ a
cutoff $C_{\\_}0^{\infty}(\mathbb{R};[0,1])$ function such that
$\chi_{\\_}0(t)=1$ for $|t|\leq 2$ and $\chi_{\\_}0(t)=0$ for $|t|>4$. Putting
inequalities (49), (50) and (51) we will obtain

$\lVert v\rVert_{\\_}{X_{\\_}{-\zeta}^{1/2}}\leq
C^{\prime}R_{\\_}0\lVert(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta+\lambda-V)v\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}},$

with a new constant $C^{\prime}>0$. To indeed prove (50) and (51) we draw next
lemma from [5]:

######  Lemma 4.2 ([5], lemma 2.2).

Let $f\in\mathscr{S}(\mathbb{R})$ be a function of the $x_{\\_}n$ variable. If
$u\in\mathscr{S}(\mathbb{R}^{d})$ and $\tau>M>1,$ then

$\lVert fu\rVert_{\\_}{X^{1/2}_{\\_}{\tau e_{\\_}n}}\lesssim\lVert
p\widehat{f}\rVert_{\\_}{L^{1}(\mathbb{R})}\lVert
u\rVert_{\\_}{X^{1/2}_{\\_}{\tau e_{\\_}n}},$

where $p(\xi)=(M^{-1}|\xi|+1)^{2}$.

As before, we can transform the inequality to make it work for an arbitrary
$\zeta$ as we did above, and for a $f$ that depends on the variable
$x\cdot\theta$, which can be represented as $Q^{T}_{\\_}*f$, with
$\theta=Qe_{\\_}n$. On the one hand,

$\lVert Q^{T}_{\\_}*fu\rVert_{\\_}{X^{1/2}_{\\_}{\tau\theta}}=\lVert
fQ_{\\_}*u\rVert_{\\_}{X^{1/2}_{\\_}{\tau e_{\\_}n}}\lesssim\lVert
p\widehat{f}\rVert_{\\_}{L^{1}(\mathbb{R})}\lVert
Q_{\\_}*u\rVert_{\\_}{X^{1/2}_{\\_}{\tau e_{\\_}n}}=\lVert
p\widehat{f}\rVert_{\\_}{L^{1}(\mathbb{R})}\lVert
u\rVert_{\\_}{X^{1/2}_{\\_}{\tau\theta}},$

while if $\zeta=\tau\theta+i\mathcal{I}$, then

$\lVert fu\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}}=\lVert
e^{i\mathcal{I}\centerdot}fu\rVert_{\\_}{X^{1/2}_{\\_}{\tau\theta}}\lesssim\lVert
p\widehat{f}\rVert_{\\_}{L^{1}(\mathbb{R})}\lVert
e^{i\mathcal{I}\centerdot}u\rVert_{\\_}{X^{1/2}_{\\_}{\tau\theta}}=\lVert
p\widehat{f}\rVert_{\\_}{L^{1}(\mathbb{R})}\lVert
u\rVert_{\\_}{X^{1/2}_{\\_}{\zeta}}.$

We summarize all this discussion in the following lemma:

######  Lemma 4.3.

Take $R_{\\_}0>0$ such that $\mathrm{supp}\,V\subset B_{\\_}{R_{\\_}0}$. There
exists $C>0$ and $\tau_{\\_}0=\tau_{\\_}0(R_{\\_}0,V,\lambda)$ such that

$\lVert u\rVert_{\\_}{X_{\\_}{-\zeta}^{1/2}}\leq
CR_{\\_}0\lVert(\Delta-2\zeta\cdot\nabla+\zeta\cdot\zeta+\lambda-V)u\rVert_{\\_}{X_{\\_}{-\zeta}^{-{1/2}}}$

for all for $u\in\mathscr{S}(\mathbb{R}^{d})$ with $\mathrm{supp}\,u\subset
B_{\\_}{R_{\\_}0}$ and $\tau>\tau_{\\_}0$.

To prove the existence of CGO solutions we should introduce a couple of
natural spaces. Let $\Omega$ be a bounded open domain of class
$\mathcal{C}^{2}$ such that $\mathrm{supp}\,V_{\\_}j\subset\Omega$, $j=1,2$,
and, for $\zeta\in\mathbb{C}^{d}$ and $s>0$ define

$X_{\\_}\zeta^{s}(\Omega)=\\{{\left.\kern-1.2ptu\right|_{\Omega}}\,\mathrel{\mathop{\ordinarycolon}}\,u\in
X_{\\_}\zeta^{s}\\},$

endowed with the norm

$\lVert u\rVert_{\\_}{X_{\\_}\zeta^{s}(\Omega)}=\inf\\{\lVert
v\rVert_{\\_}{X_{\\_}\zeta^{s}}\,\mathrel{\mathop{\ordinarycolon}}\,{\left.\kern-1.2ptv\right|_{\Omega}}=u\\},$

and

$X_{\\_}{\zeta,c}^{s}(\Omega)=\\{u\in
X^{s}_{\\_}\zeta\,\mathrel{\mathop{\ordinarycolon}}\,\mathrm{supp}\,\,u\subset\overline{\Omega}\\},$

endowed as well with the norm
$\lVert\centerdot\rVert_{\\_}{X_{\\_}\zeta^{s}(\Omega)}$. The space
$X_{\\_}{-\zeta,c}^{-s}(\Omega)$ is defined as the dual space of
$X_{\\_}\zeta^{s}(\Omega)$, and it is easy to check that it can be identified
with the space of distributions in $X_{\\_}{-\zeta}^{-s}$ whose support lays
inside $\Omega$. In this context, the a priori estimate in lemma 4.3 works in
the space $X_{\\_}{-\zeta,c}^{-{1/2}}(\Omega)$, so that the solutions can be
constructed in $X_{\\_}\zeta^{1/2}(\Omega)$, as stated in the following
proposition.

######  Proposition 4.4.

Consider $d\geq 3$, let $\Omega$ be a bounded open domain of class
$\mathcal{C}^{2}$ amd let $R_{\\_}0>0$ such that
$\mathrm{supp}\,V\subset\Omega\subset B_{\\_}{R_{\\_}0}$. There exist a
constant $\tau_{\\_}0=\tau_{\\_}0(R_{\\_}0,V,\lambda)$ such that, for every
$\tau\geq\tau_{\\_}0$, and every
$\zeta=\Re(\zeta)+i\Im(\zeta)\in\mathbb{C}^{d}$ such that $|\Re(\zeta)|=\tau$,
$|\Im(\zeta)|=(\tau^{2}+\lambda)^{1/2}$ and $\Re(\zeta)\cdot\Im(\zeta)=0$,
there exist $w_{\\_}\zeta\in X_{\\_}\zeta^{1/2}(\Omega)$ so that
$v_{\\_}\zeta=e^{\zeta\cdot x}(1+w_{\\_}\zeta)$ is a solution to the equation
$(\Delta+\lambda-V)v_{\\_}\zeta=0$ in $\Omega$ and

(52) $\lVert
w_{\\_}\zeta\rVert_{\\_}{X_{\\_}\zeta^{1/2}(\Omega)}\lesssim\lVert
V\rVert_{\\_}{X_{\\_}\zeta^{-{1/2}}}.$

###  4.2. Proof of theorem 2

With proposition 4.4, we can construct the kind of special solutions that we
are looking for, with $\zeta_{\\_}j\cdot\zeta_{\\_}j=-\lambda$. Besides, we
need $\zeta_{\\_}j$ to satisfy that $\zeta_{\\_}1+\zeta_{\\_}2=-i\kappa$ for
an arbitrary given $\kappa\in\mathbb{R}^{d}$. We can explicitly construct
these $\zeta_{\\_}j$ by choosing $\eta,\theta\in\mathbb{S}^{d-1}$ such that
$\eta\cdot\theta=\eta\cdot\kappa=\theta\cdot\kappa=0$. Now, for
$\tau\geq\frac{|\kappa|^{2}}{4}-\lambda$ we can set

(53)
$\displaystyle\begin{split}\zeta_{\\_}1&=\tau\theta+i\left[-\frac{\kappa}{2}+(\tau^{2}+\lambda-\frac{|\kappa|^{2}}{4})^{1/2}\eta\right],\\\
\zeta_{\\_}2&=-\tau\theta+i\left[-\frac{\kappa}{2}-(\tau^{2}+\lambda-\frac{|\kappa|^{2}}{4})^{1/2}\eta\right],\end{split}$

which satisfy both $\zeta_{\\_}j\cdot\zeta_{\\_}j=-\lambda$ and
$\zeta_{\\_}1+\zeta_{\\_}2=-i\kappa$. Then, if we take
$\tau\geq\max\\{\tau_{\\_}0,(\frac{|\kappa|^{2}}{4}-\lambda)^{1/2}\\}$, these
$\zeta_{\\_}j$ satisfy the conditions of proposition 4.4. Let then $V_{\\_}1$
and $V_{\\_}2$ be two potentials of the form (1), and let $v_{\\_}j$ be the
CGO solutions corresponding to $\zeta_{\\_}j$ and $V_{\\_}j$, $j=1,2$. If we
consider any extension of $w_{\\_}j\in X_{\\_}{\zeta_{\\_}j}^{1/2}(\Omega)$ to
$X_{\\_}{\zeta_{\\_}j}^{1/2}$, this extension will be in
$H^{1}(\mathbb{R}^{d})$. Then, $w_{\\_}j$ belongs to $H^{1}(\Omega)$ and so
does $v_{\\_}j$. Therefore, we can apply proposition 3.1 and, plugging
$v_{\\_}1$ and $v_{\\_}2$ in the orthogonality relation

$\braket{(V_{\\_}1-V_{\\_}2)v_{\\_}1,v_{\\_}2}=0,$

we get

(54) $\displaystyle\begin{split}\braket{V_{\\_}1-V_{\\_}2,e^{-{i\kappa\cdot
x}}}=&-\braket{V_{\\_}1-V_{\\_}2,e^{-i\kappa\cdot
x}w_{\\_}1}-\braket{V_{\\_}1-V_{\\_}2,e^{-i\kappa\cdot x}w_{\\_}2}\\\
&-\braket{(V_{\\_}1-V_{\\_}2)w_{\\_}1,e^{-i\kappa\cdot
x}w_{\\_}2}.\end{split}$

We would like this terms to vanish. For the first two terms on the right hand
side we have that, by duality,

(55)
$\displaystyle\begin{split}\lvert\braket{V_{\\_}1-V_{\\_}2,e^{-i\kappa\cdot
x}w_{\\_}j}\rvert&\leq\lVert
V_{\\_}1-V_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j,c}^{-{1/2}}(\Omega)}\lVert
e^{-i\kappa\cdot
x}w_{\\_}j\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j}^{1/2}(\Omega)}\\\
&\lesssim(1+|\kappa|)\,\lVert
V_{\\_}1-V_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j}^{-{1/2}}}\lVert
V_{\\_}j\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j}^{-{1/2}}},\end{split}$

where in the last inequality we have used that
$\mathrm{supp}\,(V_{\\_}1-V_{\\_}2)\subset\Omega$, inequality (52) and the
following:

(56) $\lVert e^{-i\kappa\cdot
x}w_{\\_}j\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j}^{1/2}(\Omega)}\lesssim(1+|\kappa|)\,\lVert
w_{\\_}j\rVert_{\\_}{X_{\\_}{\zeta_{\\_}j}^{1/2}(\Omega)}.$

On the other hand, the third term can be bounded again by duality as

$\displaystyle\lvert\braket{(V_{\\_}1-V_{\\_}2)w_{\\_}1,e^{-i\kappa\cdot
x}w_{\\_}2}\rvert$
$\displaystyle\leq\lVert(V_{\\_}1-V_{\\_}2)w_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2,c}^{-{1/2}}(\Omega)}\lVert
e^{-i\kappa\cdot x}w_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{1/2}(\Omega)}$
$\displaystyle\lesssim(1+|\kappa|)\,\lVert(V_{\\_}1-V_{\\_}2)w_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{-{1/2}}}\lVert
V_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{-{1/2}}}$

where we have used again (52), (56) and the fact that
$\mathrm{supp}\,(V_{\\_}1-V_{\\_}2)w_{\\_}1\subset\Omega$. Now, to keep
estimating this term, we need boundedness of the operator multiplication by
$V_{\\_}1-V_{\\_}2$ from $X_{\\_}{\zeta_{\\_}1}^{1/2}(\Omega)$ to
$X_{\\_}{\zeta_{\\_}2}^{-{1/2}}$. In fact, let $V$ be a potential of the form
(1) and $w\in X_{\\_}{\zeta_{\\_}1}^{1/2}$. Proceeding just as in section 4.1,
we obtain that, for $\phi\in X_{\\_}{\zeta_{\\_}2}^{1/2}$,

$\lvert\braket{V\,w,\phi}\rvert\lesssim(\lVert
V^{0}\rVert_{\\_}{L^{d/2}}+\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)}+\lVert
g\rVert_{\\_}{L^{\infty}})\,\lVert
u\rVert_{\\_}{X_{\\_}{\zeta_{\\_}1}^{1/2}}\lVert\phi\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{1/2}},$

where the implicit constant depends only on $V$ and $d$, and $u\in
X^{1/2}_{\\_}\zeta$ is an arbitrary extension of $w$ to $\mathbb{R}^{d}$.
Taking the infimum over the norm of $u$ gives us the desired boundedness, i.e.

$\lVert(V_{\\_}1-V_{\\_}2)w_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{-{1/2}}}\lesssim\lVert
w_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}1}^{1/2}(\Omega)},$

to estimate the last summand in (54) as

(57)
$\displaystyle\begin{split}\lvert\braket{(V_{\\_}1-V_{\\_}2)w_{\\_}1,e^{-i\kappa\cdot
x}w_{\\_}2}\rvert&\lesssim(1+|\kappa|)\,\lVert
w_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}1}^{1/2}(\Omega)}\lVert
V_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{-{1/2}}}\\\
&\lesssim(1+|\kappa|)\,\lVert
V_{\\_}1\rVert_{\\_}{X_{\\_}{\zeta_{\\_}1}^{-{1/2}}}\lVert
V_{\\_}2\rVert_{\\_}{X_{\\_}{\zeta_{\\_}2}^{-{1/2}}},\end{split}$

where again we have used (52). Therefore, adding inequalities (55) and (57)
gives

(58) $\lvert\braket{V_{\\_}1-V_{\\_}2,e^{-{i\kappa\cdot
x}}}\rvert\lesssim(1+|\kappa|)\,\sum_{\\_}{j,k=1}^{2}\lVert
V_{\\_}j\rVert_{\\_}{X_{\\_}{\zeta_{\\_}k}^{-{1/2}}}\,\sum_{\\_}{l,m=1}^{2}\lVert
V_{\\_}l\rVert_{\\_}{X_{\\_}{\zeta_{\\_}m}^{-{1/2}}}$

We now want to show that the right-hand side of (58) goes to zero in some
sense as $\tau$ grows. This can be done in average, as showed by Haberman and
Tataru in [14]. We state it in the following lemma, almost identical to lemma
3.5 in [4], with the only addition of the new component of the potential
$\gamma^{s}$. We repeat the proof here just for the sake of completeness.

######  Lemma 4.5.

Let $V$ be a potential of the form (1) and fix $\nu\in\mathbb{S}^{d-1}$ and
$r>0$. For $\zeta\in\mathbb{C}^{d}$ of the form (53), denote
$\zeta=\zeta(\tau,T)$ with $\kappa=rT\nu$ for some $T\in SO(d)$. Then for
every $s\in(1/2,1)$ and $M>1$, it holds that

$\frac{1}{M}\int_{\\_}M^{2M}\int_{\\_}{SO(d)}\lVert
V\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau,T)}}^{2}\,\mathrm{d}\mu(T)\,\mathrm{d}\tau\lesssim
M^{-\omega}\lVert
V_{\\_}0\rVert^{2}_{\\_}{L^{d/2}}+M^{-2(1-s)}\left(\lVert\alpha\rVert^{2}_{\\_}{L^{\infty}(\Gamma)}+\lVert
g\rVert_{\\_}{L^{\infty}}^{2}\right),$

where

$\omega=\begin{cases}1/2&d=3\\\ 1/4&d\geq 4,\end{cases}$

the implicit constant depends on $V$ and $d$, and $\mu$ denotes the Haar
measure on $SO(d)$.

######  Proof:.

First, for the critically singular part, if $d\geq 4$,

$\lVert
V^{0}\rVert_{\\_}{X^{-{1/2}}_{\\_}{\zeta(\tau,T)}}\leq\tau^{-{1/2}}\lVert
V^{0}\rVert_{\\_}{L^{2}}\lesssim\tau^{-{1/2}}\lVert
V^{0}\rVert_{\\_}{L^{d/2}},$

since $V^{0}$ is compactly supported and $d/2\geq 2$ for $d\geq 4$. In the
case $d=3$, by the dual inequality to Haberman’s embedding (see corollaries
4.8 and 4.22 in [4]),

$\lVert
V^{0}\rVert_{\\_}{X^{-{1/2}}_{\\_}{\zeta(\tau,T)}}\leq\tau^{-d(1/p^{\prime}_{\\_}d-1/q^{\prime}_{\\_}d)}\lVert
V^{0}\rVert_{\\_}{L^{q^{\prime}_{\\_}d}}\lesssim\tau^{-1/4}\lVert
V^{0}\rVert_{\\_}{L^{d/2}},$

since $d/2\geq q^{\prime}_{\\_}d$ for $d=3$. For the remaining components, use
lemma 5.2 in [13], which states that for $f\in\dot{H}^{-1}$, it holds that

$\frac{1}{M}\int_{\\_}M^{2M}\int_{\\_}{SO(d)}\lVert
f\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau,T)}}^{2}\,\mathrm{d}\mu(T)\,\mathrm{d}\tau\lesssim
M^{-1}\lVert Lf\rVert^{2}_{\\_}{\dot{H}^{-{1/2}}}+\lVert
Hf\rVert^{2}_{\\_}{\dot{H}^{-1}},$

where $\widehat{Lf}=\mathbbm{1}_{\\_}{|\xi|<2M}\widehat{f}$ and $Hf=f-Lf$
stand the low and high frequencies of $f$, respectively. Now, for every
$1/2<s<1$, we obtain

$\frac{1}{M}\int_{\\_}M^{2M}\int_{\\_}{SO(d)}\lVert
f\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau,T)}}^{2}\,\mathrm{d}\mu(T)\,\mathrm{d}\tau\lesssim
M^{-2(1-s)}\lVert f\rVert^{2}_{\\_}{\dot{H}^{-s}}.$

On the one hand, since
$\mathrm{supp}\,(\alpha\,\mathrm{d}\sigma)\subset\Omega$,

$\lVert\alpha\,\mathrm{d}\sigma\rVert^{2}_{\\_}{\dot{H}^{-s}}\lesssim\lVert\alpha\,\mathrm{d}\sigma\rVert^{2}_{\\_}{H^{-s}}\lesssim\lVert\alpha\rVert_{\\_}{L^{2}(\Gamma)}\lesssim\lVert\alpha\rVert_{\\_}{L^{\infty}(\Gamma)},$

where we have used the dual inequality of the usual trace theorem for Sobolev
spaces, as well as the fact that $\Gamma$ has finite measure. On the other
hand, since $\gamma^{s}=\chi^{2}D^{s}\,g$ with
$\mathrm{supp}\,\chi\subset\Omega$, it follows that for
$u\in\mathcal{S}(\mathbb{R}^{d})$,

$\lvert\braket{\gamma^{s},u}\rvert\leq\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}\lVert
D^{s}(\chi^{2}u)\rVert_{\\_}{L^{2}}\lesssim\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})}\lVert u\rVert_{\\_}{\dot{H}^{s}},$

where the last inequality comes by proceededing as in (11). Therefore,

$\lVert\gamma^{s}\rVert_{\\_}{\dot{H}^{-s}}\lesssim\lVert
g\rVert_{\\_}{L^{\infty}(\mathbb{R}^{d})},$

which ends the proof of the lemma. ∎

To end the proof of Theorem 2, we use an argument that Haberman [13]
attributes to Nguyen and Spirn [24]. Indeed, if we fix $r>0$ and
$\nu\in\mathbb{S}^{d-1}$ and denote $\zeta=\zeta(\tau,T)$ as in lemma 4.5
above, we have that

$\lim_{\\_}{M\to\infty}\frac{1}{M}\int_{\\_}M^{2M}\int_{\\_}{SO(d)}\lVert
V\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau,T)}}^{2}\,\mathrm{d}\mu(T)\,\mathrm{d}\tau=0,$

and, for any $\varepsilon>0$, by simply restricting to
$B_{\\_}\varepsilon=\\{T\in SO(d)\mathrel{\mathop{\ordinarycolon}}\lVert
T-I\rVert\leq\varepsilon\\}$ with $I$ the identity map, we get that

$\lim_{\\_}{M\to\infty}\frac{1}{M\mu(B_{\\_}\varepsilon)}\int_{\\_}M^{2M}\int_{\\_}{B_{\\_}\epsilon}\lVert
V\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau,T)}}^{2}\,\mathrm{d}\mu(T)\,\mathrm{d}\tau=0.$

Now, take a sequence $M=M_{\\_}n$ such that $M_{\\_}n\to\infty$ as
$n\to\infty$. Since the quantity in the limit above is an average, we may
choose sequences $\tau_{\\_}{\varepsilon,n}$, $T_{\\_}{\varepsilon,n}$ and
$\delta_{\\_}{\varepsilon,n}>0$ such that

$\lVert
V\rVert_{\\_}{X^{-1/2}_{\\_}{\zeta(\tau_{\\_}{\varepsilon,n},T_{\\_}{\varepsilon,n})}}\leq\delta_{\\_}{\varepsilon,n},$

and such that $\delta_{\\_}{\varepsilon,n}\to 0$ as $n\to\infty$. Therefore,
going back to (58), we obtain

$|\widehat{V_{\\_}1-V_{\\_}2}(\kappa_{\\_}{\varepsilon,n})|=\lvert\braket{V_{\\_}1-V_{\\_}2,e^{-{i\kappa_{\\_}{\varepsilon,n}\cdot
x}}}\rvert\lesssim\delta_{\\_}{\varepsilon,n}^{2},$

where $\kappa_{\\_}{\varepsilon,n}=rT_{\\_}{\varepsilon,n}\nu$. Since
$B_{\\_}\varepsilon$ is compact, there exists a subsequence
$T_{\\_}{\varepsilon,n_{\\_}m}$ converging to some $T_{\\_}\varepsilon$. Thus,

$\lim_{\\_}{m\to\infty}|\widehat{V_{\\_}1-V_{\\_}2}(\kappa_{\\_}{\varepsilon,n_{\\_}m})|\lesssim\lim_{\\_}{m\to\infty}\delta_{\\_}{\varepsilon,n_{\\_}m}^{2}=0,$

and, since $V_{\\_}j$ are compactly supported, $\widehat{V_{\\_}1-V_{\\_}2}$
is continuous, which means that

$\widehat{V_{\\_}1-V_{\\_}2}(rT_{\\_}\varepsilon\nu)=0.$

Finally, since necessarily $T_{\\_}\varepsilon\to I$ as $\varepsilon\to 0$, we
conclude that

$\widehat{V_{\\_}1-V_{\\_}2}(r\nu)=0$

for any $r>0$ and $\nu\in\mathbb{S}^{d-1}$ and, by Fourier inversion we obtain
that $V_{\\_}1=V_{\\_}2$, which ends the proof of Theorem 2.

##  Appendix A Solution of the Neumann problem

Along this section, we aim to prove the following result, which is used in the
proof of the Runge approximation, proposition 3.4. Throughout this appendix,
consider a bounded open domain $\Omega$ of class $\mathcal{C}^{2}$ such that
$\mathrm{supp}\,V\subset\Omega$.

######  Theorem A.1.

Suppose $\lambda>0$ is not a Neumann eigenvalue for the operator $-\Delta+V$
in $\Omega$. Let $f\in L^{2}(\Omega)$ be such that
$\mathrm{supp}\,f\subset\Omega$. Then, there exist $u\in H^{1}(\Omega)$
solving the problem

(59) $\begin{cases}\left(\Delta+\lambda-V\right)u=f&\textrm{in }\Omega,\\\
\partial_{\\_}{\nu}u=0&\textrm{on }\partial\Omega.\end{cases}$

Remember that we say that $\lambda$ is a Neumann eigenvalue for $-\Delta+V$ in
$\Omega$ if there exists $\phi$ not identically zero solving the homogeneuos
Neumann problem.

(60) $\begin{cases}\left(\Delta+\lambda-V\right)\phi=0&\textrm{in }\Omega,\\\
\partial_{\\_}{\nu}\phi=0&\textrm{on }\partial\Omega.\end{cases}$

These eigenvalues in fact form a countable set. To see this, define the
unbounded operator $\left(T_{\\_}N,D(T_{\\_}N)\right)$ over $L^{2}(\Omega)$ as
$T_{\\_}Nu=\left(-\Delta+V\right)u$, with domain

$D(T_{\\_}N)=\\{u\in
L^{2}(\Omega)\,\mathrel{\mathop{\ordinarycolon}}\,\left(-\Delta+V\right)u\in
L^{2}(\Omega),\;\exists\,\partial_{\\_}\nu u\text{ on }\partial\Omega\text{
and }\partial_{\\_}\nu{\left.\kern-1.2ptu\right|_{\partial\Omega}}=0\\}.$

Observe now that a Neumann eigenvalue for $-\Delta+V$ on $\Omega$ will be an
eigenvalue for $\left(T_{\\_}N,D(T_{\\_}N)\right)$. The domain $D(T_{\\_}N)$
is a separable Hilbert space, since it is a subspace of $L^{2}(\Omega)$, and
$T_{\\_}N$ is symmetric over $D(T_{\\_}N)$, which ensures that the set of its
eigenvalues must be countable.
Indeed, suppose that there is an uncountable set of such eigenvalues. Let
$\lambda$ and $\mu$ be any two distinct eigenvalues, and $u$ and $v$ be
corresponding distinct eigenfunctions. Then,

$\lambda\braket{u,v}=\braket{T_{\\_}Nu,v}=\braket{u,T_{\\_}Nv}=\mu\braket{u,v},$

and thus $u\perp v$, which contradicts the fact that $D(T_{\\_}N)$ is
separable.
We will now prove Theorem A.1 using the method of layer potentials. Most of
the arguments that we will follow come from [9], [10] and [8]. The first step
in the argument is to realise that, to solve (59), we only need to be able to
solve the following problem, for $g\in\mathcal{C}^{\infty}(\partial\Omega)$:

(61) $\begin{cases}\left(\Delta+\lambda-V\right)v=0&\textrm{in }\Omega,\\\
\partial_{\\_}{\nu}v=g&\textrm{on }\partial\Omega.\end{cases}$

This can be done by taking a function $w$ that satisties the equation

$(\Delta+\lambda-V)\,w=f.$

This function can be constructed in $X_{\\_}\lambda^{*}$, for instance, by
using the techniques in section 2 after observing that, if $f\in
L^{2}(\Omega)$, then its extension by $0$ to $\mathbb{R}^{d}$ is in
$X_{\\_}\lambda$.
Also, since $\mathrm{supp}\,f\subset\Omega$, we have by Theorem 11.1.1 in [17]
that $w$ is smooth near $\partial\Omega$. Therefore, we can define its outward
normal derivative in $\partial\Omega$, which will belong to
$\mathcal{C}^{\infty}(\partial\Omega)$. Now, if we can solve problem (61), set
$g=\partial_{\\_}\nu w$, and denote by $v$ the solution to this problem. Then,
it is easy to check that $u={\left.\kern-1.2ptw\right|_{\Omega}}-v$ solves the
problem (59).
Note that ${\left.\kern-1.2ptw\right|_{\Omega}}\in H^{1}(\Omega)$ by
proposition 2.4, so we will only need to prove that $v$ belongs to
$H^{1}(\Omega)$ to conclude that $u$ belongs too.
In this case it will be useful to think of the fundamental solution for the
operator $\Delta+\lambda$ in $\mathbb{R}^{d}$, denoted by $\Phi_{\\_}\lambda$,
not in the distributional sense as in (5), but as a Hankel function. In fact,
$\Phi_{\\_}\lambda$ will take the form

(62)
$\Phi_{\\_}\lambda(x)=\frac{i}{4}\left(\frac{\lambda^{1/2}}{2\pi|x|}\right)^{d/2-1}H_{\\_}{d/2-1}^{(1)}\left(\lambda^{1/2}|x|\right),$

where $H_{\\_}\nu^{(1)}$ denotes the Hankel function of the first kind (or
Bessel function of the third kind). If we define
$u_{\\_}{in}(x,y)=\Phi_{\\_}\lambda(x-y)$ and recall the limiting properties
of the Hankel functions, it’s relatively easy to check that

(63) $\displaystyle\begin{split}u_{\\_}{in}(x,y)&=F(x,y)|x-y|^{2-d},\\\
\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)&=G(x,y)|x-y|^{2-d},\\\
\partial_{\\_}{\nu_{\\_}y}u_{\\_}{in}(x,y)&=\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(y,x)=G(y,x)|x-y|^{2-d},\end{split}$

with $F$ and $G$ being two bounded functions on
$\partial\Omega\times\partial\Omega$. Then, $u_{\\_}{in}$,
$\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}$ and
$\partial_{\\_}{\nu_{\\_}y}u_{\\_}{in}$ are, by definition, weakly singular
kernels of order $d-2$ on $\partial\Omega\times\partial\Omega$. The following
lemma is a combination of those in [10], chapter 3B, and is an important piece
to solve the problem (61).

######  Lemma A.2 ([10]).

If we denote by $T$ the integral operator over $\partial\Omega$ defined by a
weakly singular kernel $K$ of order $\alpha$ on
$\partial\Omega\times\partial\Omega$, with $0<\alpha<d-1$, as

$\left(Tf\right)(x)=\int_{\\_}{\partial\Omega}T(x,y)\,f(y)\,\mathrm{d}S(y),$

then the following statements hold:

  1. (1)

$T$ is compact on $L^{2}(\partial\Omega)$,

  2. (2)

$T$ transforms bounded functions into continuous functions, and

  3. (3)

if $f\in L^{2}(\partial\Omega)$ and $f+Tf\in\mathcal{C}(\partial\Omega)$, then
$f\in\mathcal{C}(\partial\Omega)$.

Take now $u_{\\_}{to}=u_{\\_}{in}+u_{\\_}{sc}$, where $u_{\\_}{sc}$ is the
scattering solution defined in (S) and constructed in section 2, and define
for $f$ continuous on $\partial\Omega$ and
$x\in\mathbb{R}^{d}\setminus\partial\Omega$ the single layer potential with
moment $f$ as

$\left(\mathcal{S}f\right)(x)=\int_{\\_}{\partial\Omega}u_{\\_}{to}(x,y)\,f(y)\,\mathrm{d}S(y),$

and the double layer potential as

$\left(\mathcal{D}f\right)(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}y}u_{\\_}{to}(x,y)\,f(y)\,\mathrm{d}S(y),.$

Define further the operator $\mathcal{N}$, which is the adjoint of
$\mathcal{D}$ over $\partial\Omega$, as

$\left(\mathcal{N}f\right)(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}x}u_{\\_}{to}(x,y)\,f(y)\,\mathrm{d}S(y),\quad
x\in\partial\Omega,$

that must be understood as an improper integral. Now, we have the following
lemmas, which are similar to classical results as in [9], [10] and [8] for the
Helmholtz and Laplace equations. We denote by $u_{\\_}+$ the trace on
$\partial\Omega$ of
${\left.\kern-1.2ptu\right|_{\mathbb{R}^{d}\setminus\overline{\Omega}}}$, and
by $u_{\\_}-$ the trace on $\partial\Omega$ of
${\left.\kern-1.2ptu\right|_{\Omega}}$. As well, we denote by
$\partial_{\\_}\nu u_{\\_}+$ and $\partial_{\\_}\nu u_{\\_}-$ the normal
derivative of those, always with respect to the outward-pointing normal vector
of $\partial\Omega$ (as seen from inside $\Omega$).

######  Lemma A.3.

Consider $d\geq 3$. Lef $f\in\mathcal{C}(\partial\Omega)$. Then, the single
layer potential $u=\mathcal{S}f$ is continuous throughout $\mathbb{R}^{d}$,
and we have the limiting values

(64) $\partial_{\\_}\nu
u_{\\_}{\pm}(x)=\left(\mathcal{N}f\right)(x)\mp\frac{1}{2}f(x),\quad
x\in\partial\Omega,$

where the integral exists as an improper integral. Consequently, we have the
jump relation $\partial_{\\_}\nu u_{\\_}--\partial_{\\_}\nu u_{\\_}+=f$ on
$\partial\Omega$. Furthermore, $u$ it is a solution in
$H^{1}_{\\_}\text{loc}(\mathbb{R}^{d})$ to $(\Delta+\lambda-V)\,u=0$ in
$\mathbb{R}^{d}\setminus\partial\Omega$ and fullfils SRC (2).

######  Proof:.

First, note that the single layer potential for the homogeneuos Helmholtz
equation

$v(x)=\int_{\\_}{\partial\Omega}u_{\\_}{in}(x,y)f(y)\,\mathrm{d}S(y)$

can be extended to the boundary, is a solution in
$H^{1}_{\\_}\text{loc}(\mathbb{R}^{d})$ to $\left(\Delta+\lambda\right)v=0$ in
$\mathbb{R}^{d}\setminus\partial\Omega$, fullfils SRC (2) and has boundary
values

$\partial_{\\_}\nu
v_{\\_}{\pm}(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}x}u_{\\_}{in}(x,y)\,f(y)\,\mathrm{d}S(y)\mp\frac{1}{2}f(x),\quad
x\in\partial\Omega,$

which is a classical result, see for example [9]. Now, define

$w(x)=\int_{\\_}{\partial\Omega}u_{\\_}{sc}(x,y)f(y)\,\mathrm{d}S(y).$

To see that $w$ is in $H^{1}_{\\_}\text{loc}(\mathbb{R}^{d})$, take
$K\in\mathbb{R}^{d}$ compact, and observe that, by proposition 2.4 and Theorem
1,

$\lVert w\rVert_{\\_}{H^{1}(K)}\lesssim\sup_{\\_}{y\in\partial\Omega}\lVert
u_{\\_}{sc}({\centerdot\,,y})\rVert_{\\_}{H^{1}(K)}\lesssim\sup_{\\_}{y\in\partial\Omega}\lVert
u_{\\_}{sc}({\centerdot\,,y})\rVert_{\\_}{X_{\\_}\lambda^{*}}\lesssim\sup_{\\_}{y\in\partial\Omega}\lVert
Vu_{\\_}{in}({\centerdot\,,y})\rVert_{\\_}{X_{\\_}\lambda}.$

If we take a smooth cut-off function $\eta$ such that $\eta\equiv 1$ in
$\mathrm{supp}\,V$ and $\eta\equiv 0$ in $\partial\Omega$, we have that
$V(x)u_{\\_}{in}(x,y)=V(x)\eta(x)u_{\\_}{in}(x,y)$ and

$\lVert
Vu_{\\_}{in}({\centerdot\,,y})\rVert_{\\_}{X_{\\_}\lambda}\lesssim\lVert\eta
u_{\\_}{in}(\centerdot\,,y)\rVert_{\\_}{X_{\\_}\lambda^{*}}\lesssim 1,$

where we have used that multiplication by $V$ is bounded from
$X_{\\_}\lambda^{*}$ to $X_{\\_}\lambda$, as showed in section 2, and that
$u_{\\_}{in}(\centerdot\,,y)$ is smooth away from $y$ by Theorem 11.1.1 in
[17], since it solves $(\Delta+\lambda)u_{\\_}{in}(\centerdot\,,y)=0$. This
proves that $w$ is in $H^{1}_{\\_}\text{loc}(\mathbb{R}^{d})$
Moreover, since $u_{\\_}{sc}$ solves the problem (S), it is easy to check that
$u=v+w$ solves $(\Delta+\lambda-V)\,u=0$ in
$\mathbb{R}^{d}\setminus\partial\Omega$ and fullfils the SRC (2). Also, since
for any $y\in\partial\Omega$, $u_{\\_}{sc}(\centerdot\,,y)$ solves
$\left(\Delta+\lambda\right)u_{\\_}{sc}(\centerdot\,y)=0$ in
$\mathbb{R}^{d}\setminus\mathrm{supp}\,V$, it is smooth in this set by Theorem
11.1.1 in [17], and in particular it is smooth near $\partial\Omega$.
Therefore, the limiting values of $w$ on the boundary are just

$\partial_{\\_}\nu
w_{\\_}{\pm}(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}x}u_{\\_}{sc}(x,y)\,f(y)\,\mathrm{d}S(y),\quad
x\in\partial\Omega,$

and therefore (64) is fullfiled. ∎

######  Lemma A.4.

Consider $d\geq 3$. Lef $f\in\mathcal{C}(\partial\Omega)$. Then, the double
layer potential $u=\mathcal{D}f$ can be extended continuosly to
$\partial\Omega$, and we have the limiting values

(65) $u_{\\_}{\pm}(x)=\left(\mathcal{D}f\right)(x)\pm\frac{1}{2}f(x),\quad
x\in\partial\Omega,$

where the integral exists as an improper integral. Consequently, we have the
jump relation $u_{\\_}+-u_{\\_}-=f$ on $\partial\Omega$. Furthermore, $u$ a
solution in $H^{1}_{\\_}{loc}(\mathbb{R}^{d})$ to $(\Delta+\lambda-V)\,u=0$ in
$\mathbb{R}^{d}\setminus\partial\Omega$, it fullfils SRC (2) and
$\partial_{\\_}\nu u_{\\_}--\partial_{\\_}\nu u_{\\_}+=0$ on $\partial\Omega$.

######  Proof:.

The proof goes exactly as the proof of lemma A.3 above, we just need to make a
comment on how to prove the last statement. Indeed, the fact that the double
layer potential for the homogeneous Helmholtz equation

$v(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}y}u_{\\_}{in}(x,y)f(y)\,\mathrm{d}S(y)$

fulfills that $\partial_{\\_}\nu v_{\\_}--\partial_{\\_}\nu v_{\\_}+=0$ on
$\partial\Omega$ is classical (see for example [9]). Meanwhile, the function

$w(x)=\int_{\\_}{\partial\Omega}\partial_{\\_}{\nu_{\\_}y}u_{\\_}{sc}(x,y)f(y)\,\mathrm{d}S(y)$

is smooth away from $\mathrm{supp}\,V$, since $u_{\\_}{sc}(\centerdot,y)$ is
smooth away from $\mathrm{supp}\,V$ as well. ∎

With lemma A.3, we can find an $H^{1}(\Omega)$ solution to the problem (61)
for $g\in L^{2}(\partial\Omega)$ if we can find $\varphi\in
L^{2}(\partial\Omega)$ solving the integral equation
$\mathcal{N}\varphi+\frac{1}{2}\varphi=g$. We will solve this equation via
Fredholm theory. For simplicity, define the operators

$\mathcal{K}=2\mathcal{D},\quad\mathcal{K}^{*}=2\mathcal{N},$

and the equation we are trying to solve can be written as
$\left(\mathcal{K}^{*}+I\right)\varphi=2g$. Note that $\mathcal{K}^{*}$ is the
adjoint operator of $\mathcal{K}$ and, by lemma A.2, both are compact
operators over $L^{2}(\partial\Omega)$. To prove the existence of a solution,
it is enough to prove that the operator $\mathcal{K}^{*}+I$ is surjective,
which by Fredholm alternative is equivalent to proving that $\mathcal{K}+I$ is
injective. Indeed, we have the following lemma:

######  Lemma A.5.

Suppose that $\lambda>0$ is not a Neumann eigenvalue for the operator
$-\Delta+V$ in $\Omega$. Then,

$\ker(\mathcal{K}+I)=\\{0\\}.$

######  Proof:.

Let $\psi\in\ker(\mathcal{K}+I)$, which, by lemma A.2, will be continuous on
$\partial\Omega$. Now define $v=\mathcal{D}\psi$. By lemma A.4, $v$ is a
solution to the problem

$\begin{cases}\left(\Delta+\lambda\right)v=0&\textrm{in
}\mathbb{R}^{d}\setminus\overline{\Omega},\\\ v_{\\_}+=0&\textrm{on
}\partial\Omega,\\\ v\textrm{ satisfying SRC}.\end{cases}$

Then, $v=0$ in $\mathbb{R}^{d}\setminus\Omega$, by uniqueness of the exterior
Dirichlet problem [8]. Therefore, $\partial_{\\_}\nu v_{\\_}+=0$ on
$\partial\Omega$ and, again by lemma A.4, $\partial_{\\_}\nu v_{\\_}-=0$. This
means that $v$ is as well a solution to the problem

$\begin{cases}\left(\Delta+\lambda-V\right)v=0&\textrm{in }\Omega,\\\
\partial_{\\_}\nu v_{\\_}-=0&\textrm{on }\partial\Omega,\\\ \end{cases}$

but, since $\lambda$ is not a Neumann eigenvalue for $-\Delta+V$ in $\Omega$,
we have that $v=0$ in $\Omega$, and in particular $v_{\\_}-=0$. Finally, by
lemma A.4, $\psi=u_{\\_}+-u_{\\_}-=0$, and the lemma is proved. ∎

With this last lemma, the proof of Theorem A.1 is concluded.

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