\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 160, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/160\hfil Solvability in the sense of sequences]
{Solvability in the sense of sequences to some non-Fredholm operators}

\author[V. Volpert, V. Vougalter \hfil EJDE-2013/160\hfilneg]
{Vitaly Volpert, Vitali Vougalter}  % in alphabetical order

\address{Vitaly Volpert \newline
Institute Camille Jordan, UMR 5208 CNRS,
University Lyon 1 \\
Villeurbanne, 69622, France.\newline
Department of Mathematics, Mechanics and Computer Science\\
Southern Federal University Rostov-on-Don, Russia}
\email{volpert@math.univ-lyon1.fr}

\address{Vitali Vougalter \newline
Department of Mathematics and Applied Mathematics,
University of Cape Town\\
Private Bag, Rondebosch 7701, South Africa}
\email{Vitali.Vougalter@uct.ac.za}

\thanks{Submitted March 8, 2013. Published July 12, 2013.}
\subjclass[2000]{35J10, 35P10, 47F05}
\keywords{Solvability conditions; non Fredholm operators; Sobolev spaces}

\begin{abstract}
 We study the solvability of certain linear nonhomogeneous
 elliptic problems and show that under reasonable technical conditions the
 convergence in $L^2(\mathbb{R}^d)$ of their right sides implies the
 existence and the convergence in $H^2(\mathbb{R}^d)$ of the solutions. The
 equations involve second order differential operators without Fredholm
 property and we use the methods of spectral and scattering theory for
 Schr\"odinger type operators analogously to our preceding work \cite{VV08}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{assumption}[theorem]{Assumption}
\allowdisplaybreaks

\section{Introduction}

Consider the equation
\begin{equation} \label{eq1}
 -\Delta u + V(x) u - a u=f,
\end{equation}
where $u \in E= H^2(\mathbb{R}^d)$ and  $f \in F=L^2(\mathbb{R}^d)$,
$d\in \mathbb{N}$, $a$ is a constant and
$V(x)$ is a function converging to $0$ at infinity. If $a \geq 0$,
then the essential spectrum of the operator $A : E \to F$
corresponding to the left-hand side of equation \eqref{eq1}
contains the origin. As a consequence, the operator does not
satisfy the Fredholm property. Its image is not closed, for $d>1$
the dimensions of its kernel and the codimension of its image are
not finite. In this work we will study some properties of such
operators.
 Let us note that elliptic problems involving non-Fredholm operators were
studied extensively in recent years (see  
\cite{VKMP02,VV08,VV09,VV10,VV11,VV12,VV13,VV14,B88}) along with their
potential applications to the theory of reaction-diffusion
equations (see  \cite{DMV05,DMV08}). In the particular
case where $a=0$ the operator $A$ satisfies the Fredholm property
in some properly chosen weighted spaces 
\cite{Amrouche1997,Amrouche2008,Bolley1993,Bolley2001,B88}.
 However, the case with $a \neq 0$ is essentially
different and the approach developed in these works cannot be
applied.

 One of the important questions about equations with non-Fredholm operators
 concerns their solvability. We will study it in the following
 setting. Let $f_n$ be a sequence of
 functions in the image of the operator $A$, such that
$f_n\to f$ in $L^2(\mathbb{R}^d)$ as $n\to \infty$. Denote
 by $u_n$  a sequence of functions from $H^2(\mathbb{R}^d)$ such that
$$
Au_n=f_n, \quad n\in \mathbb{N}.
$$
 Since the operator $A$ does not satisfy the Fredholm property, then
 the sequence $u_n$ may not be convergent.
 We will call a sequence $u_n$ such that $A u_n \to f$  a solution in
 the sense of sequences of equation $Au=f$ (see \cite{V2011}). If this
 sequence converges to a function $u_0$ in the norm of the space
 $E$, then $u_0$ is a solution of this equation. Solution in
 the sense of sequences is equivalent in this sense to the
 usual solution. However, in the case of non-Fredholm
 operators this convergence may not hold or it can occur in some
 weaker sense. In this case, solution in the sense of sequences may
 not imply the existence of the usual solution. In this work we
 will find sufficient conditions of equivalence of solutions in the
 sense of sequences and the usual solutions. In the other
 words, the conditions on sequences $f_n$ under which the
 corresponding sequences $u_n$ are strongly convergent.

In the first part of the article we consider the equation
\begin{equation} \label{eq2}
-\Delta u-au=f(x), \quad x\in \mathbb{R}^d, \; d\in \mathbb{N},
\end{equation}
where $a\geq 0$ is a constant and the right side is square integrable.
Note that for the operator $-\Delta-a$ on $L^2(\mathbb{R}^d)$ the
essential spectrum fills the semi-axis $[-a, \ \infty)$ such that its
inverse from $L^2(\mathbb{R}^d)$ to $H^2(\mathbb{R}^d)$ is not
bounded. Let us write down the corresponding sequence of equations with
$n\in \mathbb{N}$ as
\begin{equation} \label{eq3}
-\Delta u_n-au_n=f_n(x), \quad x\in \mathbb{R}^d, \; d\in \mathbb{N},
\end{equation}
with the right sides convergent to the right side of \eqref{eq2} in
$L^2(\mathbb{R}^d)$ as $n\to \infty$. The inner product of two functions
$$
(f(x),g(x))_{L^2(\mathbb{R}^d)}:=\int_{\mathbb{R}^d}f(x)\bar{g}(x)dx,
$$
with a slight abuse of notations when these functions are
not square integrable. Indeed, if $f(x)\in L^1(\mathbb{R}^d)$ and
$g(x)$ is bounded, then clearly the integral considered above makes sense,
like for instance in the case of functions involved in the orthogonality
conditions of Theorems \ref{thm1} and \ref{thm2} below. In the space of three dimensions for
some $A(x)=(A_{1}(x), A_{2}(x), A_{3}(x))$, the inner product
$(f(x),A(x))_{L^2(\mathbb{R}^3)}$ is the vector with the coordinates
$$
\int_{\mathbb{R}^3}f(x)\bar{A_{k}}(x)dx, \quad k=1,2,3.
$$
We start with formulating the proposition in one dimension.
We will consider the space $H^2(\mathbb{R}^d)$ with the norm
\begin{equation}\label{h2n} 
\|u\|_{H^2(\mathbb{R}^d)}^2:=\|u\|_{L^2(\mathbb{R}^d)}^2+ \|\Delta
u\|_{L^2(\mathbb{R}^d)}^2.
\end{equation}



\begin{theorem} \label{thm1} 
 Let $n\in \mathbb{N}$ and
$f_n(x)\in L^2(\mathbb{R})$, such that $f_n(x)\to f(x)$ in
$L^2(\mathbb{R})$ as $n\to \infty$.

(a) When $a>0$ let $xf_n(x)\in L^1(\mathbb{R})$, such that
$xf_n(x)\to xf(x)$  in $L^1(\mathbb{R})$ as $n\to \infty$ and the
orthogonality conditions
\begin{equation} \label{or1}
\Big(f_n(x),\frac{e^{\pm i\sqrt{a}x}}{\sqrt{2\pi}}\Big)_{L^2(\mathbb{R})}=0
\end{equation}
hold for all $n\in \mathbb{N}$. Then equations \eqref{eq2} and \eqref{eq3}
admit unique solutions
$u(x)\in H^2(\mathbb{R})$ and $u_n(x)\in H^2(\mathbb{R})$ respectively,
such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R})$ as $n\to \infty$.

(b) When $a=0$ let $x^2f_n(x)\in L^1(\mathbb{R})$, such that
$x^2f_n(x)\to x^2f(x)$  in $L^1(\mathbb{R})$ as $n\to \infty$ and the
orthogonality relations
\begin{equation} \label{or2}
(f_n(x),1)_{L^2(\mathbb{R})}=0, \quad
 (f_n(x),x)_{L^2(\mathbb{R})}=0
\end{equation}
hold for all $n\in \mathbb{N}$ . Then problems \eqref{eq2} and \eqref{eq3}
possess unique solutions
$u(x)\in H^2(\mathbb{R})$ and $u_n(x)\in H^2(\mathbb{R})$ respectively,
where $u_n(x)\to u(x)$ in $H^2(\mathbb{R})$ as $n\to \infty$.
\end{theorem}

Then we turn our attention to the issue in dimensions two and higher. The
sphere of radius $r>0$ in $\mathbb{R}^d$ centered at the origin will
be denoted by $S_{r}^d$, of radius $r=1$ as $S^d$ and its Lebesgue
measure by $|S^d|$. The notation $B^d$ will stand for the unit ball
in the space of $d$ dimensions with the center at the origin and $|B^d|$
for its Lebesgue measure.


\begin{theorem} \label{thm2} 
Let $d\geq 2$, $n\in \mathbb{N}$ and
$f_n(x)\in L^2(\mathbb{R}^d)$, such that $f_n(x)\to f(x)$ in
$L^2(\mathbb{R}^d)$ as $n\to \infty$.

(a) When $a>0$ let $|x|f_n(x)\in L^1(\mathbb{R}^d)$, such that
$|x|f_n(x)\to |x|f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$ and the
orthogonality conditions
\begin{equation}
\label{or3}
\Big(f_n(x),\frac{e^{ipx}}{(2\pi)^{d/2}}\Big)_{L^2(\mathbb{R}^d)}=0,
\quad p\in S_{\sqrt{a}}^d \text{ a.e., } d\geq 2
\end{equation}
hold for all $n\in \mathbb{N}$. Then equations \eqref{eq2} and \eqref{eq3}
admit unique solutions
$u(x)\in H^2(\mathbb{R}^d)$ and $u_n(x)\in H^2(\mathbb{R}^d)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^d)$ as
$n\to \infty$.

(b) When $a=0$ and $d=2$ let $|x|^2f_n(x)\in L^1(\mathbb{R}^2)$, such
that $|x|^2f_n(x)\to |x|^2f(x)$ in $L^1(\mathbb{R}^2)$ as $n\to \infty$
and the orthogonality relations
\begin{equation}\label{or4}
(f_n(x),1)_{L^2(\mathbb{R}^2)}=0, \quad
(f_n(x), x_{m})_{L^2(\mathbb{R}^2)}=0, \quad
m=1,2
\end{equation}
hold for all ${n\in \mathbb N}$. Then problems \eqref{eq2} and \eqref{eq3}
have unique solutions
$u(x)\in H^2(\mathbb{R}^2)$ and $u_n(x)\in H^2(\mathbb{R}^2)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^2)$ as
$n\to \infty$.

(c) When $a=0$ and $d=3,4$ let $|x|f_n(x)\in L^1(\mathbb{R}^d)$, such
that $|x|f_n(x)\to |x|f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$ and
the orthogonality condition
\begin{equation}
\label{or5}
(f_n(x),1)_{L^2(\mathbb{R}^d)}=0, \quad d=3,4
\end{equation}
holds  for all ${n\in \mathbb N}$. Then problems \eqref{eq2} and \eqref{eq3}
admit unique solutions
$u(x)\in H^2(\mathbb{R}^d)$ and $u_n(x)\in H^2(\mathbb{R}^d)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^d)$ as
$n\to \infty$.

(d) When $a=0$ and $d\geq 5$ let $f_n(x)\in L^1(\mathbb{R}^d)$, such
that $f_n(x)\to f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$. Then
equations \eqref{eq2} and \eqref{eq3} have unique solutions
$u(x)\in H^2(\mathbb{R}^d)$ and $u_n(x)\in H^2(\mathbb{R}^d)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^d)$ as
$n\to \infty$.
\end{theorem}

Note that when $a=0$ and the dimension of the problem is at least five,
orthogonality conditions in the Theorem above are not required
(see e.g. \cite[Lemmas 6 and 7]{VV14}).
We will be using the hat symbol to denote the standard Fourier transform
\begin{equation}
\label{ft}
\widehat{f}(p):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^d}f(x)
e^{-ipx}dx, \ p\in \mathbb{R}^d, \quad d\in \mathbb{N}.
\end{equation}
In the second part of the work we study the equation
\begin{equation}
\label{eq4}
-\Delta u+V(x)u-au=f(x), \quad x\in \mathbb{R}^3, \; a\geq 0,
\end{equation}
where the right side is square integrable. The correspondent sequence
of equations for $n\in \mathbb{N}$ will be
\begin{equation}
\label{eq5}
-\Delta u_n+V(x)u_n-au_n=f_n(x), \quad x\in \mathbb{R}^3, \; a\geq 0,
\end{equation}
where the right sides converge to the right side of \eqref{eq4} in
$L^2(\mathbb{R}^3)$ as $n\to \infty$. Let us make the following
technical assumptions on the scalar potential involved in equations above.
Note that the conditions on $V(x)$, which is shallow and short-range will
be analogous to those stated in \cite[Assumption 1.1]{VV08}
(see also  \cite{VV09,VV10}). The essential spectrum of such a
Schr\"odinger operator fills the nonnegative semi-axis (see e.g.  \cite{JMST}).



\begin{assumption} \label{assump3} \rm
 The potential function
$V(x):\mathbb{R}^3\to \mathbb{R}$ satisfies the estimate
$$
|V(x)|\leq
{\frac{C}{1+{|x|}^{3.5+\delta}}}
$$ 
for some $\delta>0$ and  $x=(x_{1},x_{2},x_{3})\in \mathbb{R}^3$ a.e. such that
$$
{4^{1/9}{\frac{9}{8}}(4\pi)^{-2/3}
\|V\|_{L^{\infty}(\mathbb{R}^3)}^{1/9}\|V\|_{L^{4/3}
(\mathbb{R}^3)}^{8/9}<1 \quad \text{and} \quad
\sqrt{c_{HLS}}\|V\|_{L^{3/2}(\mathbb{R}^3)}<4\pi}.
$$
\end{assumption}
Here and further down $C$ stands for a finite positive constant and
$c_{HLS}$  given on p.98 of \cite{LL97} is the constant in the
Hardy-Littlewood-Sobolev inequality
$$
\big|\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{f_{1}(x)f_{1}(y)}
{|x-y|^2}\,dx\,dy \big|\leq c_{HLS}\|f_{1}\|_{L^{3/2}(\mathbb{R}^3)}^2,
\quad f_{1}\in L^{3/2}(\mathbb{R}^3).
$$

According to \cite[Lemma 2.3]{VV08}, under Assumption \ref{assump3} above on
the potential function, the operator $-\Delta+V(x)-a$ on
$L^2(\mathbb{R}^3)$ is self-adjoint and unitarily equivalent to $-\Delta-a$
via the wave operators (see  \cite{K65},  \cite{RS04})
$$
\Omega^{\pm}:=\operatorname{s-lim}_{t\to \mp\infty}e^{it(-\Delta+V)}
e^{it \Delta},
$$
where the limit is understood in the strong $L^2$ sense (see e.g.
 \cite{RS79} p.34,   \cite{CFKS87} p.90). Hence
$-\Delta+V(x)-a$ on $L^2(\mathbb{R}^3)$ has only the
essential spectrum $\sigma_{ess}(-\Delta+V(x)-a)=[-a, \ \infty)$.
By means of the spectral theorem, its functions of the continuous spectrum
satisfying
\begin{equation}
\label{phik}
[-\Delta+V(x)]\varphi_{k}(x)=k^2\varphi_{k}(x), \quad k\in \mathbb{R}^3,
\end{equation}
  in the integral formulation the Lippmann-Schwinger equation
for the perturbed plane waves (see e.g.  \cite{RS79} p.98)
\begin{equation}
\label{LS} \varphi_{k}(x)=\frac{e^{ikx}}{(2\pi)^{3/2}}-\frac{1}
{4\pi} \int_{\mathbb{R}^3}\frac{e^{i|k||x-y|}}
{|x-y|}(V\varphi_{k})(y)dy
\end{equation}
and the orthogonality relations
\begin{equation}
\label{or}
(\varphi_{k}(x), \varphi_{q}(x))_{L^2(\mathbb{R}^3)}=\delta (k-q), \quad
k,q \in \mathbb{R}^3
\end{equation}
form the complete system in $L^2(\mathbb{R}^3)$. In particular, when
the vector $k=0$, we have $\varphi_{0}(x)$.
Let us denote the generalized Fourier transform with respect to these
functions using the tilde symbol as
$$
\tilde{f}(k):=(f(x),\varphi_{k}(x))_{L^2(\mathbb{R}^3)}, \quad k\in \mathbb{R}^3.
$$
The integral operator involved in \eqref{LS} is being denoted as
$$
(Q\varphi)(x):=-\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{e^{i|k||x-y|}}{|x-y|}
(V\varphi)(y)dy, \quad \varphi\in L^{\infty}(\mathbb{R}^3).
$$
Let us consider $Q: L^{\infty}(\mathbb{R}^3)\to L^{\infty}(\mathbb{R}^3)$.
Under Assumption \ref{assump3}, according to \cite[Lemma 2.1]{VV08} the operator norm
$\|Q\|_{\infty}<1$, in fact it is bounded above by a quantity independent of
$k$ which is expressed in terms of the appropriate $L^{p}(\mathbb{R}^3)$ norms
of the potential function $V(x)$. We have the following statement.

\begin{theorem} \label{thm4} 
 Let Assumption \ref{assump3} hold, ${n\in \mathbb N}$ and
$f_n(x)\in L^2(\mathbb{R}^3)$, such that $f_n(x)\to f(x)$ in
$L^2(\mathbb{R}^3)$ as $n\to \infty$. Assume also that
$|x|f_n(x)\in L^1(\mathbb{R}^3)$, such that
$|x|f_n(x)\to |x|f(x)$ in $L^1(\mathbb{R}^3)$ as $n\to \infty$.

(a) When $a>0$ let the orthogonality conditions
\begin{equation}
\label{or6}
(f_n(x), \varphi_{k}(x))_{L^2(\mathbb{R}^3)}=0, \ k\in S_{\sqrt{a}}^3 \ a.e.
\end{equation}
hold for all $n\in \mathbb{N}$. Then equations \eqref{eq4} and \eqref{eq5}
admit unique solutions
$u(x)\in H^2(\mathbb{R}^3)$ and $u_n(x)\in H^2(\mathbb{R}^3)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^3)$ as
$n\to \infty$.

(b) When $a=0$ let the orthogonality relation
\begin{equation}
\label{or7}
(f_n(x), \varphi_{0}(x))_{L^2(\mathbb{R}^3)}=0
\end{equation}
hold for all $n\in \mathbb{N}$. Then equations \eqref{eq4} and \eqref{eq5}
possess unique solutions $u(x)\in H^2(\mathbb{R}^3)$ and
$u_n(x)\in H^2(\mathbb{R}^3)$
respectively, such that $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^3)$ as
$n\to \infty$.
\end{theorem}

Note that \eqref{or6} and \eqref{or7} are the orthogonality conditions to
the functions of the continuous spectrum of our Schr\"odinger operator, as
distinct from the Limiting Absorption Principle in which one needs to
orthogonalize to the standard Fourier harmonics (see e.g.
\cite[Lemma 2.3 and Proposition 2.4]{GS04}).


\section{Proof of the generalization of the solvability in the sense
of sequences}

 Application of the standard Fourier transform \eqref{ft}
to both sides of equations \eqref{eq2} and \eqref{eq3} for $p\in
\mathbb{R}^d, \ d\in \mathbb{N}$ yields
$$
\widehat{u}(p)=\frac{\widehat{f}(p)}{p^2-a}, \quad
\widehat{u}_n(p)=\frac{\widehat{f}_n(p)}{p^2-a}, \quad
a\geq 0, \quad n\in \mathbb{N}.
$$
When $a=0$ we write their difference as
\begin{equation}
\label{d0}
\widehat{u}_n(p)-\widehat{u}(p)=\frac{\widehat{f}_n(p)-\widehat{f}(p)}
{p^2}\chi_{\{p\in \mathbb{R}^d: |p|\leq 1\}}+\frac{\widehat{f}_n(p)-\widehat{f}
(p)}{p^2}\chi_{\{p\in \mathbb{R}^d: |p|>1\}}.
\end{equation}
Here and further down $\chi_{A}$ will stand for the characteristic function
of a set $A\subseteq \mathbb{R}^d$. The complement of a set will be
designated as $A^{c}$. Denote the second term in the right
side of \eqref{d0} as $\xi_n^{d,  0}(p)$.

When $a>0$ and the dimension $d=1$ we introduce the following set as the union
of intervals on the real line
$$
I_{\delta}=I_{\delta}^{-}\cup I_{\delta}^{+}:=[-\sqrt{a}-\delta, -\sqrt{a}+\delta]\cup
[\sqrt{a}-\delta, \sqrt{a}+\delta], \quad 0<\delta<\sqrt{a},
$$
which enables us to express in this case
\begin{equation}
\label{1a}
\widehat{u}_n(p)-\widehat{u}(p)=\frac{\widehat{f}_n(p)-\widehat{f}(p)}
{p^2-a}\chi_{I_{\delta}^{-}}(p)+\frac{\widehat{f}_n(p)-\widehat{f}
(p)}{p^2-a}\chi_{I_{\delta}^{+}}(p)+\frac{\widehat{f}_n(p)
-\widehat{f}(p)}{p^2-a}
\chi_{I_{\delta}^{c}}(p).
\end{equation}
Denote the last term in the right side of \eqref{1a} as $\xi_n^{1, \ a}(p)$.

For $a>0$ and dimensions $d\geq 2$ we introduce the following set as the
layer in $\mathbb{R}^d$:
$$
A_{\sigma}:=\{p\in \mathbb{R}^d \ | \ \sqrt{a}-\sigma\leq |p|\leq \sqrt{a}+
\sigma \}, \quad 0<\sigma<\sqrt{a}
$$
and express
\begin{equation}
\label{da}
\widehat{u}_n(p)-\widehat{u}(p)=\frac{\widehat{f}_n(p)-\widehat{f}(p)}
{p^2-a}\chi_{A_{\sigma}}+\frac{\widehat{f}_n(p)-\widehat{f}(p)}{p^2-a}
\chi_{{A_{\sigma}}^{c}}.
\end{equation}
Denote the second term in the right side of \eqref{da} as $\xi_n^{d, a}(p)$.



\begin{proof}[Proof of Theorem \ref{thm1}]
(a) We express the first term in the right side of
\eqref{1a} as
\begin{equation}
\label{exp1}
\frac{\widehat{f}_n(-\sqrt{a})-\widehat{f}(-\sqrt{a})+\int_{-\sqrt{a}}^{p}
\frac{d}{dq}[\widehat{f}_n(q)-\widehat{f}(q)]dq }{p^2-a}\chi_{I_{\delta}^{-}}(p).
\end{equation}
Note that by means of orthogonality conditions \eqref{or1} and part (a) of
Lemma \ref{lem7} with
${w(x)=\frac{e^{\pm i\sqrt{a}x}}{\sqrt{2\pi}}}$, we have
\begin{equation}
\label{ocn1}
\Big(f_n(x),\frac{e^{\pm i\sqrt{a}x}}{\sqrt{2\pi}}\Big)_{L^2(\mathbb{R})}=0,
\quad n\in \mathbb{N}, \quad
\Big(f(x),\frac{e^{\pm i\sqrt{a}x}}{\sqrt{2\pi}}\Big)_{L^2(\mathbb{R})}=0,
\end{equation}
such that $\widehat{f}_n(\pm\sqrt{a})$ and $\widehat{f}(\pm\sqrt{a})$ vanish
and via \cite[Lemma 5]{VV14} equations \eqref{eq2} and \eqref{eq3}
considered in one dimension with $a>0$ admit unique solutions
$u(x)\in H^2(\mathbb{R})$ and $u_n(x)\in H^2(\mathbb{R})$ respectively.
By using the trivial estimate
\begin{equation}
\label{1qd}
\big|\frac{d}{dq}[\widehat{f}_n(q)-\widehat{f}(q)]\big|
\leq \frac{1} {\sqrt{2\pi}}\|x{f}_n-xf\|_{L^1(\mathbb{R})}, \ q\in \mathbb{R},
\end{equation}
we easily derive the upper bound on the absolute value of \eqref{exp1} as
$$
\frac{1}{\sqrt{2\pi}}\frac{\|x{f}_n-xf\|_{L^1(\mathbb{R})}}{2\sqrt{a}-\delta}
\chi_{I_{\delta}^{-}}(p).
$$
Therefore, the $L^2(\mathbb{R})$ norm of the first term in the right side
of \eqref{1a} can be estimated from above by
\begin{equation}
\label{ub1n0}
\sqrt{\frac{\delta}{\pi}}\frac{\|x{f}_n-xf\|_{L^1(\mathbb{R})}}
{2\sqrt{a}-\delta}\to 0, \ n\to \infty
\end{equation}
according to one of the assumptions of the theorem. Similarly to \eqref{exp1}
and using relations \eqref{ocn1}, we write the second term in the right side
of \eqref{1a} as
\begin{equation}
\label{exp2}
\frac{\int_{\sqrt{a}}^{p}\frac{d}{dq}[\widehat{f}_n(q)-\widehat{f}(q)]dq}
{p^2-a}\chi_{I_{\delta}^{+}}(p),
\end{equation}
which can be easily estimated from above in the absolute value by means of
\eqref{1qd} by
$$
\frac{1}{\sqrt{2\pi}}\frac{\|xf_n-xf\|_{L^1(\mathbb{R})}}{2\sqrt{a}-\delta}
\chi_{I_{\delta}^{+}}(p).
$$
Hence, the $L^2(\mathbb{R})$ norm of the second term in the right side
of \eqref{1a} admits the upper bound \eqref{ub1n0} as well.
Thus, via Lemma \ref{lem6}, which guarantees that
$$
\lim_{n\to \infty}\|\xi_n^{1, \ a}(p)\|_{L^2(\mathbb{R})}=0,
$$
we have $u_n(x)\to u(x)$ in $L^2(\mathbb{R})$ as
$n\to \infty$ and complete the proof of part a) of the theorem by means of
part (a) of Lemma \ref{lem5}.



(b) By means of orthogonality relations \eqref{or2} for all $n\in \mathbb{N}$
we have
\begin{equation}
\label{fn10}
\widehat{f}_n(0)=0, \quad {\frac{d\widehat{f}_n}{dp}(0)=0}.
\end{equation}
Then part (b) of Lemma \ref{lem7} yields
$$
(f(x),1)_{L^2(\mathbb{R})}=0, \quad (f(x),x)_{L^2(\mathbb{R})}=0,
$$
such that
\begin{equation}
\label{f10}
\widehat{f}(0)=0, \quad {\frac{d\widehat{f}}{dp}(0)=0}.
\end{equation}
Via part (b) of \cite[Lemma 5]{VV14} equations \eqref{eq2} and \eqref{eq3}
studied in one dimension with $a=0$ admit unique solutions
$u(x)\in H^2(\mathbb{R})$ and $u_n(x)\in H^2(\mathbb{R})$ respectively.
Identities \eqref{fn10} and \eqref{f10} yield the representation formula
$$
\widehat{f}_n(p)-\widehat{f}(p)=
\int_{0}^{p}\Big(\int_{0}^{s}\frac{d^2}{dq^2}[\widehat{f}_n(q)-
\widehat{f}(q)]dq \Big)ds, \quad p\in \mathbb{R},
$$
which we are going to use along with the inequality
$$
\big|\frac{d^2}{dq^2}[\widehat{f}_n(q)-\widehat{f}(q)]\big| \leq
\frac{1}{\sqrt{2\pi}}\|x^2f_n-x^2f\|_{L^1(\mathbb R)}, \ q\in \mathbb{R}.
$$
Thus, for the first term in the right side of \eqref{d0} in one dimension
we have the upper bound in the absolute value as
$$
\frac{1}{2\sqrt{2\pi}}\|x^2f_n-x^2f\|_{L^1(\mathbb R)}
\chi_{\{p\in \mathbb{R}: \ |p|\leq 1 \}}
$$
and in the $L^2(\mathbb R)$ norm as
$$
\frac{1}{2\sqrt{\pi}}\|x^2f_n-x^2f\|_{L^1(\mathbb R)}\to 0, \ n\to \infty
$$
according to one of the assumptions of the theorem. By means of Lemma \ref{lem6}
$$
\lim_{n\to \infty}\|\xi_n^{1, \ 0}(p)\|_{L^2(\mathbb R)}=0
$$
and we arrive at $u_n(x)\to u(x)$ in $L^2(\mathbb R)$ as $n\to \infty$.
We complete the proof of the theorem via part (a) of Lemma \ref{lem5}.
\end{proof}



\begin{proof}[Proof of Theorem \ref{thm2}]
(a) Orthogonality conditions \eqref{or3} along with
part (a) of Lemma \ref{lem7} with
$w(x)=\frac{e^{ipx}}{(2\pi)^{d/2}}$, $p\in S_{\sqrt{a}}^d$
a.e. imply
\begin{equation}
\label{oc3}
\Big(f(x), \frac{e^{ipx}}{(2\pi)^{d/2}}\Big)_{L^2(\mathbb{R}^d)}=0, \quad
p\in S_{\sqrt{a}}^d \ a.e.,
\end{equation}
such that by means of part a) of \cite[Lemma 6]{VV14} equations
\eqref{eq2} and  \eqref{eq3} with $a>0$ admit unique solutions
$u(x)\in H^2(\mathbb{R}^d)$ and $u_n(x)\in H^2(\mathbb{R}^d)$
respectively for $d\geq 2$.  Due to \eqref{or3} and \eqref{oc3}, we have
\begin{equation}
\label{ocnd}
\widehat{f}_n(\sqrt{a},\omega)=0, \quad \widehat{f}(\sqrt{a},\omega)=0 
\quad\text{a.e.}
\end{equation}
Here and below $\omega$ stands for the angle variables on the sphere centered
at the origin of a given radius. Via identities \eqref{ocnd} the first term in
the right side of \eqref{da} can be written as
\begin{equation}
\label{exp3}
\frac{\int_{\sqrt{a}}^{|p|}\frac{\partial}{\partial s}[\widehat{f}_n(s,
\omega)-\widehat{f}(s,\omega)]ds}{p^2-a}\chi_{A_{\sigma}}.
\end{equation}
Clearly, for $q\in \mathbb{R}^d$, $d\geq 2$ we have the inequality
\begin{equation}
\label{dqub}
\big|\frac{\partial}{\partial |q|}[\widehat{f}_n(|q|,\omega)-\widehat{f}(|q|,
\omega)]\big|\leq \frac{1}{(2\pi)^{d/2}}\||x|f_n-|x|f\|_
{L^1(\mathbb{R}^d)},
\end{equation}
such that expression \eqref{exp3} can be estimated from above in the absolute
value by
$$
\frac{1}{(2\pi)^{d/2}\sqrt{a}}\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}
\chi_{A_{\sigma}}
$$
and therefore in the $L^2(\mathbb{R}^d)$ norm by
$$
\frac{1}{(2\pi)^{d/2}\sqrt{a}}\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}
\sqrt{|B^d|[(\sqrt{a}+\sigma)^d-(\sqrt{a}-\sigma)^d]}\to 0, 
$$
as $n\to \infty$ 
due to one of the assumptions of the theorem. Hence, according to 
Lemma \ref{lem6}
$$
\lim_{n\to \infty}\|\xi_n^{d, a}(p)\|_{L^2(\mathbb{R}^d)}=0
$$
and we arrive at $u_n(x)\to u(x)$ in  $L^2(\mathbb{R}^d)$, $d\geq 2$
as $n\to \infty$. We complete the proof of part a) of the theorem via part 
(a) of Lemma \ref{lem5}.



(b) By means of orthogonality conditions \eqref{or4} along with part (b) of
Lemma \ref{lem7} we have
\begin{equation}
\label{oc4}
(f(x),1)_{L^2(\mathbb{R}^2)}=0, \quad (f(x),x_{m})_{L^2(\mathbb{R}^2)}=0, \quad
m=1,2.
\end{equation}
Thus via part b) of \cite[Lemma 6]{VV14} equations \eqref{eq2} and
\eqref{eq3} with $a=0$ considered in two dimensions admit unique solutions
$u(x)\in H^2(\mathbb{R}^2)$ and $u_n(x)\in H^2(\mathbb{R}^2)$
respectively. Identities \eqref{or4} and \eqref{oc4} imply
$\widehat{f}_n(0)=0, \ n\in \mathbb{N}$ and $\widehat{f}(0)=0$. Let
$\theta$ denote the angle between two vectors $p=(|p|,\theta_{p})$ and
$x=(|x|, \theta_{x})$ in
$\mathbb{R}^2$. Then
$$
\frac{\partial \widehat{f}_n}{\partial |p|}(0, \theta_{p})=
-\frac{i}{2\pi}\int_{\mathbb{R}^2}f_n(x)|x|cos\theta dx
$$
can be easily expressed as
$$
-\frac{i}{2\pi}\{\cos\theta_{p}\int_{\mathbb{R}^2}f_n(x)x_{1}dx+
\sin\theta_{p}\int_{\mathbb{R}^2}f_n(x)x_{2}dx\}=0
$$
due to orthogonality relations \eqref{or4}. Analogously, we can write
${\frac{\partial \widehat{f}}{\partial |p|}(0, \theta_{p})}$ as
$$
-\frac{i}{2\pi}\{\cos\theta_{p}\int_{\mathbb{R}^2}f(x)x_{1}dx+
\sin\theta_{p}\int_{\mathbb{R}^2}f(x)x_{2}dx\}=0
$$
via orthogonality conditions \eqref{oc4}. The argument above implies
$$
\widehat{f}_n(p)-\widehat{f}(p)=\int_{0}^{|p|}\Big(\int_{0}^{s}
\frac{\partial^2}{\partial \xi^2}[\widehat{f}_n(\xi,\theta_{p})-
\widehat{f}(\xi,\theta_{p})]d\xi\Big)ds.
$$
Clearly, for $p\in \mathbb{R}^2$ we have the inequality
$$
\big|\frac{\partial^2}{\partial |q|^2}[\widehat{f}_n(q)-\widehat{f}(q)]
\big|\leq \frac{1}{2\pi}\||x|^2f_n-|x|^2f\|_{L^1(\mathbb{R}^2)},
$$
which yields the upper bound
$$
|\widehat{f}_n(p)-\widehat{f}(p)|\leq \frac{1}{4\pi}
\||x|^2f_n-|x|^2f\|_{L^1(\mathbb{R}^2)}|p|^2, \quad p\in \mathbb{R}^2.
$$
Thus the first term in the right side of \eqref{d0} admits the estimate
from above in the absolute value as
$$
\frac{1}{4\pi}\||x|^2f_n-|x|^2f\|_{L^1(\mathbb{R}^2)}
\chi_{\{p\in \mathbb{R}^2: |p|\leq 1 \}}
$$
and in the $L^2(\mathbb{R}^2)$ norm as
$$
\frac{1}{2\sqrt{\pi}}\||x|^2f_n-|x|^2f\|_{L^1(\mathbb{R}^2)}\to 0, \quad n\to
\infty
$$
according to one of the assumptions of the theorem. By means of
 Lemma \ref{lem6} we have
$$
\lim_{n\to \infty}\|\xi_n^{2, \ 0}(p)\|_{L^2(\mathbb{R}^2)}=0
$$
and then via part a) of Lemma \ref{lem5} we obtain
$u_n(x)\to u(x)$ in $H^2(\mathbb{R}^2)$ as $n\to \infty$.



(c) Orthogonality condition \eqref{or5} and part a) of Lemma \ref{lem7} with
$w(x)=1, \ x\in \mathbb{R}^d$ yield
\begin{equation}
\label{oc5}
(f(x),1)_{L^2(\mathbb{R}^d)}=0, \quad d=3,4.
\end{equation}
Part (c) of \cite[Lemma 6]{VV14} implies that equations \eqref{eq2} and
\eqref{eq3} with $a=0$ in dimensions $d=3,4$ admit unique solutions
$u(x)\in H^2(\mathbb{R}^d)$ and $u_n(x)\in H^2(\mathbb{R}^d)$
respectively. Due to \eqref{or5} and \eqref{oc5}, we have
$\widehat{f}_n(0)=0$ and $\widehat{f}(0)=0$.
Hence we can write the first term in the right side of \eqref{d0} as
$$
\frac{\int_{0}^{|p|}\frac{\partial}{\partial |q|}[\widehat{f}_n(|q|,\omega)-
\widehat{f}(|q|,\omega)]d|q|}{p^2}\chi_{\{p\in \mathbb{R}^d: |p|\leq 1 \}}.
$$
By applying inequality \eqref{dqub} to the expression above
we easily obtain the upper bound for it in the absolute value as
$$
\frac{1}{(2\pi)^{d/2}}\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}
\frac{\chi_{\{p\in \mathbb{R}^d: |p|\leq 1 \}}}{|p|}
$$
and in the $L^2(\mathbb{R}^d)$ norm as
$$
\frac{1}{(2\pi)^{d/2}}\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}
\sqrt{\int_{0}^1|S^d||p|^{d-3}d|p|}\to 0, \ n\to \infty, \ d=3,4
$$
due to one of the assumptions of the theorem. By means of
Lemma \ref{lem6} we have
$$
\lim_{n\to \infty}\|\xi_n^{d, \ 0}(p)\|_{L^2(\mathbb{R}^d)}=0, \quad d=3,4.
$$
Part (a) of Lemma \ref{lem5} implies $u_n(x)\to u(x)$ in
$H^2(\mathbb{R}^d)$, $d=3,4$ as $n\to \infty$.



(d) In dimensions $d\geq 5$ equations \eqref{eq2} and \eqref{eq3} with
$a=0$ admit unique solutions $u(x)\in H^2(\mathbb{R}^d)$ and
$u_n(x)\in H^2(\mathbb{R}^d)$ respectively by means of 
\cite[Lemma 7]{VV14}. No orthogonality conditions are required in this case. We
have the following trivial inequality
$$
|\widehat{f}_n(p)-\widehat{f}(p)|\leq \frac{1}{(2\pi)^{d/2}}
\|f_n-f\|_{L^1(\mathbb{R}^d)}, \ p\in \mathbb{R}^d,
$$
which yields the upper bound in the absolute value on the first term
in the right side of \eqref{d0} as
$$
\frac{1}{(2\pi)^{d/2}}\|f_n-f\|_{L^1(\mathbb{R}^d)}
\frac{\chi_{\{p\in \mathbb{R}^d: |p|\leq 1 \}}}{p^2},
$$
such that we obtain the upper bound in the $L^2(\mathbb{R}^d)$
norm for it as
$$
\frac{1}{(2\pi)^{d/2}}\|f_n-f\|_{L^1(\mathbb{R}^d)}
\sqrt{\int_{0}^1|S^d||p|^{d-5}d|p|}\to 0, \ n\to \infty, \ d\geq 5
$$
due to one of the assumptions of the theorem. By means of Lemma \ref{lem6}
 we have
$$
\lim_{n\to \infty}\|\xi_n^{d, \ 0}(p)\|_{L^2(\mathbb{R}^d)}=0, \ d\geq 5.
$$
Part (a) of  Lemma \ref{lem5} yields $u_n(x)\to u(x)$ in
$H^2(\mathbb{R}^d)$, $d\geq 5$ as $n\to \infty$.
\end{proof}

Let us apply the generalized Fourier transform with respect to the functions
of the continuous spectrum of the Schr\"odinger operator to both sides of
equations \eqref{eq4} and \eqref{eq5}, which yields
$$
\tilde{u}(k)=\frac{\tilde{f}(k)}{k^2-a}, \quad
\tilde{u}_n(k)=\frac{\tilde{f}_n(k)}{k^2-a}, \quad k\in \mathbb{R}^3,
\; a\geq 0.
$$
For $a=0$ we express the difference of the transforms above as
\begin{equation}
\label{30}
\tilde{u}_n(k)-\tilde{u}(k)=\frac{\tilde{f}_n(k)-\tilde{f}(k)}{k^2}
\chi_{\{k\in \mathbb{R}^3: |k|\leq 1 \}}+
\frac{\tilde{f}_n(k)-\tilde{f}(k)}{k^2}\chi_{\{k\in \mathbb{R}^3: |k|>1 \}}.
\end{equation}
Let $\eta_n^{0}(k)$ stand for the second term in the right side of
\eqref{30}.

When $a>0$ we introduce the spherical layer in the space of three dimensions as
$$
B_{\sigma}:=\{k\in \mathbb{R}^3 : \sqrt{a}-\sigma\leq |k|\leq
\sqrt{a}+\sigma \}, \quad 0<\sigma<\sqrt{a},
$$
which enables us to write
\begin{equation}
\label{3a}
\tilde{u}_n(k)-\tilde{u}(k)=\frac{\tilde{f}_n(k)-\tilde{f}(k)}{k^2-a}
\chi_{B_{\sigma}}+\frac{\tilde{f}_n(k)-\tilde{f}(k)}{k^2-a}\chi_{B_{\sigma}^{c}}.
\end{equation}
The second term in the right side of \eqref{3a} is being designated as
$\eta_n^{a}(k)$.



\begin{proof}[Proof of Theorem \ref{thm4}] 
(a) Orthogonality conditions \eqref{or6} along with
\cite[Corollary 2.2]{VV08} and part (a) of Lemma \ref{lem7} with
$w(x)=\varphi_{k}(x)$, $k\in S_{\sqrt{a}}^3$  a.e. give us
\begin{equation}
\label{oc6}
(f(x),\varphi_{k}(x))_{L^2(\mathbb{R}^3)}=0, \quad k\in S_{\sqrt{a}}^3 
\text{ a.e.}
\end{equation}
Then by means of \cite[Theorem 1.2]{VV08} equations \eqref{eq4} and
\eqref{eq5} with a bounded potential function $V(x)$ and $a>0$ admit
unique solutions $u(x)\in H^2(\mathbb{R}^3)$ and
$u_n(x)\in H^2(\mathbb{R}^3)$ respectively.
Via orthogonality relations \eqref{or6} and \eqref{oc6} discussed above we
have on $S_{\sqrt{a}}^3$  a.e.
$$
\tilde{f}_n(\sqrt{a},\omega)=0, \quad \tilde{f}(\sqrt{a},\omega)=0,
$$
which enables us to express the first term in the right side of
\eqref{3a} as
$$
\frac{\int_{\sqrt{a}}^{|k|}\frac{\partial}{\partial|q|}[\tilde{f_n}(|q|, \omega)
-\tilde{f}(|q|, \omega)]d|q|}{k^2-a}\chi_{B_{\sigma}}.
$$
For the expression above we easily obtain the upper bound in the absolute
value as
$$
\|\nabla_{q}(\tilde{f_n}(q)-\tilde{f}(q))\|_{L^{\infty}(\mathbb{R}^3)}
\frac{\chi_{B_{\sigma}}}{\sqrt{a}}
$$
and in the $L^2(\mathbb{R}^3)$ norm as
$$
\frac{\|\nabla_{q}(\tilde{f_n}(q)-\tilde{f}(q))\|_{L^{\infty}(\mathbb{R}^3)}}
{\sqrt{a}}\sqrt{\frac{4\pi}{3}((\sqrt{a}+\sigma)^3-(\sqrt{a}-\sigma)^3)}\to
0, \quad n\to \infty
$$
via Lemma \ref{lem8}. By means of Lemma \ref{lem6} we have
$$
\lim_{n\to \infty}\|\eta_n^{a}(k)\|_{L^2(\mathbb{R}^3)}=0.
$$
Part (b) of Lemma \ref{lem5} yields
$u_n(x)\to u(x)$ in $H^2(\mathbb{R}^3)$ as $n\to \infty$.


(b) Orthogonality relations \eqref{or7}, Corollary 2.2 of  \cite{VV08} and
part (a) of Lemma \ref{lem7} with $w(x)=\varphi_{0}(x)$  imply
\begin{equation}
\label{oc7}
(f(x), \varphi_{0}(x))_{L^2(\mathbb{R}^3)}=0.
\end{equation}
We deduce from part (b) of \cite[Theorem 1.2]{VV08} that equations
\eqref{eq4} and \eqref{eq5} with $V(x)$ satisfying Assumption \ref{assump3} and $a=0$
possess unique solutions $u(x)\in H^2(\mathbb{R}^3)$ and
$u_n(x)\in H^2(\mathbb{R}^3)$ respectively.
Since orthogonality conditions \eqref{or7} and \eqref{oc7}  yield
$$
\tilde{f}_n(0)=0, \quad \tilde{f}(0)=0,
$$
we can express the first term in the right side of \eqref{30} as
$$
\frac{\int_{0}^{|k|}\frac{\partial}{\partial |q|}[\tilde{f}_n(|q|,\omega)-
\tilde{f}(|q|,\omega)]d|q|}{k^2}\chi_{\{k\in \mathbb{R}^3: |k|\leq 1 \}}.
$$
Obviously, for the quantity above there is an upper bound in the
absolute value as
$$
\|\nabla_{q}(\tilde{f}_n(q)-\tilde{f}(q))\|_{L^{\infty}(\mathbb{R}^3)}
\frac{\chi_{\{k\in \mathbb{R}^3: |k|\leq 1 \}}}{|k|}
$$
and therefore, in the $L^2(\mathbb{R}^3)$ norm simply as
$$
\sqrt{4\pi}\|\nabla_{q}(\tilde{f}_n(q)-\tilde{f}(q))\|_{L^{\infty}(\mathbb{R}^3)}
\to 0, \quad n\to \infty
$$
due to Lemma \ref{lem8}, Lemmas \ref{lem6} yields
$$
\lim_{n\to \infty}\|\eta_n^{0}(k)\|_{L^2(\mathbb{R}^3)}=0.
$$
Then by means of part (b) of Lemma \ref{lem5} we arrive at
$u_n(x)\to u(x)$ in $H^2(\mathbb{R}^3)$ as  $n\to \infty$.
\end{proof}


\subsection{Remarks}
 Denote by $F$ a space of functions which belong to 
$L^2(\mathbb{R}^d) \cap L^1(\mathbb{R}^d)$ and for which the norm
$$ 
\|f\|_F = \|f\|_{L^2(\mathbb{R}^d)} +
 \||x|f\|_{L^1(\mathbb{R}^d)}  
$$
is bounded.
 A sequence $f_n \in F$ such that $f_n \to f$ in the norm of the
 space satisfies conditions of Theorems \ref{thm1}, \ref{thm2}, \ref{thm4}. 
Hence if we  introduce a space $E$ in such a way that the operator $A$ acts
 from $E$ into $F$, then its image is closed. The functionals in
 solvability conditions are linear bounded functionals over $F$.

 The space $E$ can be defined as a closure of infinitely
 differentiable functions with compact supports in the norm
 $$ 
\|u\|_E = \|u\|_{H^2(\mathbb{R}^d)} + \|Au\|_F . 
$$
 The operator $A : E \to F$ is semi-Fredholm.

 Similar construction can be considered in the case where $|x|^2 f
 \in L^1(\mathbb{R}^d)$ (Theorems \ref{thm1}, b and \ref{thm2}, b).


\section{Auxiliary results}


The following elementary lemma shows that to conclude the proofs of 
Theorems \ref{thm1}, \ref{thm2} and \ref{thm4} it is sufficient to show the convergence in $L^2$ of the
solutions of the studied equations as $n\to \infty$.



\begin{lemma} \label{lem5}  
(a) Let the conditions of Theorem \ref{thm1} hold when $d=1$,
of Theorem \ref{thm2} when $d\geq 2$, such that $u(x),u_n(x)\in H^2(\mathbb{R}^d)$
are the unique solutions of equations \eqref{eq2} and \eqref{eq3}
respectively and $u_n(x)\to u(x)$ in $L^2(\mathbb{R}^d)$ as
$n\to \infty$. Then $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^d)$ as
$n\to \infty$.

(b) Let the conditions of Theorem \ref{thm4} hold, such that $u(x),u_n(x)\in
H^2(\mathbb{R}^3)$ are the unique solutions of equations \eqref{eq4} and
\eqref{eq5} respectively and $u_n(x)\to u(x)$ in $L^2(\mathbb{R}^3)$ as
$n\to \infty$. Then $u_n(x)\to u(x)$ in $H^2(\mathbb{R}^3)$ as
$n\to \infty$.
\end{lemma}


\begin{proof} (a) From equations \eqref{eq2} and \eqref{eq3} with $a\geq 0$ we
easily deduce
$$
\|\Delta u_n-\Delta u\|_{L^2(\mathbb{R}^d)}\leq a \|u_n-u\|_{L^2(\mathbb{R}^d)}
+\|f_n-f\|_{L^2(\mathbb{R}^d)}\to 0, \quad n\to \infty.
$$
By means of definition \eqref{h2n} we have $u_n(x)\to u(x)$ in
$H^2(\mathbb{R}^d)$ as $n\to \infty$.

(b) From equations \eqref{eq4} and \eqref{eq5}, for $a\geq 0$, we
easily obtain
$$
\|\Delta u_n-\Delta u\|_{L^2(\mathbb{R}^3)}\leq (a+\|V\|_{L^{\infty}(\mathbb{R}^3)})
\|u_n-u\|_{L^2(\mathbb{R}^3)}+\|f_n-f\|_{L^2(\mathbb{R}^3)}\to 0, \
n\to \infty.
$$
Therefore, definition \eqref{h2n} yields $u_n(x)\to u(x)$ in
$H^2(\mathbb{R}^3)$ as $n\to \infty$. 
\end{proof}


The auxiliary statement below will be helpful in establishing the convergence
in $L^2$ of the solutions of the equations discussed above as $n\to \infty$.

\begin{lemma} \label{lem6}  Let $n\in \mathbb{N}$ and
$f_n(x)\in L^2(\mathbb{R}^d)$, such that
$f_n(x)\to f(x)$ in $L^2(\mathbb{R}^d)$ as
$n\to \infty$. Then the expressions $\xi_n^{d,0}(p)$, $\xi_n^{1,a}(p)$, 
$\xi_n^{d, a}(p)$, $\eta_n^{0}(k)$, $\eta_n^{a}(k)$ defined in formulas
\eqref{d0}, \eqref{1a}, \eqref{da}, \eqref{30} and \eqref{3a} respectively
tend to zero in the corresponding $L^2(\mathbb{R}^d)$ norms as
$n\to \infty$.
\end{lemma}


\begin{proof} 
Clearly, $|\xi_n^{d, \ 0}(p)|\leq |\widehat{f}_n(p)-\widehat{f}(p)|$, 
$p\in \mathbb{R}^d$, such that
$$
\|\xi_n^{d, \ 0}(p)\|_{L^2(\mathbb{R}^d)}\leq \|f_n-f\|_{L^2(\mathbb{R}^d)}
\to 0, \quad  n\to \infty.
$$
The definition of this expression yields
$|\xi_n^{1, \ a}(p)|\leq \frac{|\widehat{f}_n
(p)-\widehat{f}(p)|}{{\delta}^2}$, $p\in  \mathbb{R}$. Hence
$$
\|\xi_n^{1, \ a}(p)\|_{L^2(\mathbb{R})}\leq
\frac{\|f_n-f\|_{L^2(\mathbb{R})}}{{\delta}^2}\to 0, \quad n\to \infty.
$$
Finally, in the no potential case
$|\xi_n^{d, a}(p)|\leq \frac{|\widehat{f}_n
(p)-\widehat{f}(p)|}{\sqrt{a}\sigma}$, $p\in  \mathbb{R}^d$, $d\geq 2$.
Thus
$$
\|\xi_n^{d, a}(p)\|_{L^2(\mathbb{R}^d)}\leq
\frac{\|f_n-f\|_{L^2(\mathbb{R}^d)}}{\sqrt{a}\sigma}\to 0, \quad n\to \infty.
$$
We easily estimate
$$
|\eta_n^{0}(k)|\leq |\tilde{f}_n(k)-\tilde{f}(k)|, \quad k\in \mathbb{R}^3,
$$
which implies
$$
\|\eta_n^{0}(k)\|_{L^2(\mathbb{R}^3)}\leq \|f_n-f\|_{L^2(\mathbb{R}^3)} \to 0,
\quad n\to \infty.
$$
The trivial inequality
$$
|\eta_n^{a}(k)|\leq \frac{|\tilde{f}_n(k)-\tilde{f}(k)|}{\sqrt{a}\sigma}, \
k\in \mathbb{R}^3
$$
yields
$$
\|\eta_n^{a}(k)\|_{L^2(\mathbb{R}^3)}\leq
\frac{\|f_n-f\|_{L^2(\mathbb{R}^3)}}{\sqrt{a}\sigma} \to 0, \ n\to \infty,
$$
which completes the proof of the lemma. 
\end{proof}


The following lemma provides better information on the convergence as
$n\to \infty$  of the right sides of the nonhomogeneous elliptic problems
studied in the article.

\begin{lemma} \label{lem7} 
 Let $n\in \mathbb{N}$ and
$f_n(x)\in L^2(\mathbb{R}^d), \ d\in \mathbb{N}$, such that
$f_n(x)\to f(x)$ in $L^2(\mathbb{R}^d)$ as $n\to \infty$.

(a) If $|x|f_n(x)\in L^1(\mathbb{R}^d)$, such that
$|x|f_n(x)\to |x|f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$ then
$f_n(x)\to f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$.
Moreover, if $(f_n(x),w(x))_{L^2(\mathbb{R}^d)}=0, \ n\in \mathbb{N}$, with
some $w(x)\in L^{\infty}(\mathbb{R}^d)$  then
$(f(x),w(x))_{L^2(\mathbb{R}^d)}=0$ as well.

(b) If $|x|^2f_n(x)\in L^1(\mathbb{R}^d)$, such that
$|x|^2f_n(x)\to |x|^2f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$ then
$|x|f_n(x)\to |x|f(x)$ in $L^1(\mathbb{R}^d)$ and
$f_n(x)\to f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$.
Moreover, if $(f_n(x),1)_{L^2(\mathbb{R}^d)}=0$ and
$(f_n(x),x_{k})_{L^2(\mathbb{R}^d)}=0$ for $n\in \mathbb{N}$ and  $k=1,...,d$
then $(f(x),1)_{L^2(\mathbb{R}^d)}=0$ and
$(f(x),x_{k})_{L^2(\mathbb{R}^d)}=0$ for $k=1,\dots,d$
as well.
\end{lemma}

\begin{proof} (a) 
Note that $f_n(x)\in L^1(\mathbb{R}^d)$, $n\in \mathbb{N}$ via the trivial 
argument analogous to the one of Fact 1 of  \cite{VV08}. We easily estimate 
the norm using the Schwarz inequality as
\begin{align*}
\|f_n-f\|_{L^1(\mathbb{R}^d)}
&\leq \sqrt{\int_{|x|\leq 1}|f_n-f|^2dx}
\sqrt{\int_{|x|\leq 1}dx}+\int_{|x|>1}|x||f_n-f|dx\\
&\leq \|f_n-f\|_{L^2(\mathbb{R}^d)}\sqrt{|B^d|}+
\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}\to 0, \quad n\to \infty.
\end{align*}
Then for $w(x)$, which is bounded by one of the assumptions of the lemma, we
obtain
\begin{align*}
|(f(x),w(x))_{L^2(\mathbb{R}^d)}|
&=|(f(x)-f_n(x),w(x))_{L^2(\mathbb{R}^d)}|\\
&\leq \|f_n-f\|_{L^1(\mathbb{R}^d)}\|w\|_{L^{\infty}(\mathbb{R}^d)}\to 0, \quad
n\to \infty,
\end{align*}
which completes the proof of part (a) of the lemma.

(b) By means of the argument, which relies on the Schwarz inequality and the
assumptions that $f_n(x)\in L^2(\mathbb{R}^d)$ and
$|x|^2f_n(x)\in L^1(\mathbb{R}^d)$, we easily obtain
$$
|x|f_n(x)\in L^1(\mathbb{R}^d), \quad n\in \mathbb{N}.
$$
Let us apply the Schwarz inequality again to arrive at the bound
\begin{align*}
\||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}
&\leq \int_{|x|\leq 1}|f_n-f|dx+ \int_{|x|>1}|x|^2|f_n-f|dx\\
&\leq \|f_n-f\|_{L^2(\mathbb{R}^d)}\sqrt{|B^d|}+
\||x|^2f_n-|x|^2f\|_{L^1(\mathbb{R}^d)}\to 0, 
\end{align*}
as $n\to \infty$.
Hence $f_n(x)\to f(x)$ in $L^1(\mathbb{R}^d)$ as $n\to \infty$ and
$(f(x),1)_{L^2(\mathbb{R}^d)}=0$ according to the argument above of part
(a) of the lemma. Here $w(x)=1$, $x\in \mathbb{R}^d$. Finally, for
$k=1,\dots,d$ we arrive at
$$
|(f(x),x_{k})_{L^2(\mathbb{R}^d)}|
=|(f(x)-f_n(x),x_{k})_{L^2(\mathbb{R}^d)}|
\leq \||x|f_n-|x|f\|_{L^1(\mathbb{R}^d)}\to 0
$$
as $n\to \infty$,  which completes the proof of part (b) of the lemma.
\end{proof}

The $L^{\infty}(\mathbb{R}^3)$ norm studied in the lemma below is finite
due to \cite[Lemma 2.4]{VV08}. We go further by proving that it tends
to zero.

\begin{lemma} \label{lem8} 
 Let the conditions of Theorem \ref{thm4} hold. Then we have
$$
\|\nabla_{k}(\tilde{f}_n(k)-\tilde{f}(k))\|_{L^{\infty}(\mathbb{R}^3)}\to 0,
 \quad n\to \infty.
$$
\end{lemma}

\begin{proof}
 Clearly, we need to estimate the quantity
\begin{equation} \label{l8}
\nabla_{k}(\tilde{f}_n(k)-\tilde{f}(k))=
(f_n(x)-f(x),\nabla_{k}\varphi_{k}(x))_{L^2(\mathbb{R}^3)}.
\end{equation}
It easily follows from the Lippmann-Schwinger equation \eqref{LS} that
$$
\nabla_{k}\varphi_{k}(x)=\frac{e^{ikx}}{(2\pi)^{3/2}}ix+
(I-Q)^{-1}Q\frac{e^{ikx}}{(2\pi)^{3/2}}ix+
(I-Q)^{-1}(\nabla_{k}Q)(I-Q)^{-1}\frac{e^{ikx}}{(2\pi)^{3/2}}.
$$
Here the operator
$\nabla_{k}Q: L^{\infty}(\mathbb{R}^3)\to L^{\infty}(\mathbb{R}^3;
{\mathbb C}^3)$ possesses the integral kernel
$$
\nabla_{k}Q(x,y,k)=-\frac{i}{4\pi}e^{i|k||x-y|}\frac{k}{|k|}V(y).
$$
Evidently, for the operator norm
\begin{equation}
\label{grad}
\|\nabla_{k}Q\|_{\infty}\leq \frac{1}{4\pi}\|V\|_{L^1(\mathbb{R}^3)}<\infty
\end{equation}
due to the rate of decay of the potential function $V(x)$ stated in
Assumption \ref{assump3}. Therefore, in order to prove the convergence to zero as
$n\to \infty$ of the $L^{\infty}(\mathbb{R}^3)$ norm of expression
\eqref{l8}, we need to estimate the three terms defined below. The first one
is given by
$$
R_{1}^{n}(k):=\Big(f_n(x)-f(x),\frac{e^{ikx}}{(2\pi)^{3/2}}ix\Big)_
{L^2(\mathbb{R}^3)}, \quad  k\in \mathbb{R}^3.
$$
We easily arrive at
$$
\|R_{1}^{n}(k)\|_{L^{\infty}(\mathbb{R}^3)}\leq \frac{1}{(2\pi)^{3/2}}
\||x|f_n-|x|f\|_{L^1(\mathbb{R}^3)}\to 0, \quad n\to \infty
$$
according to one of our assumptions. The second term which we need to estimate
is
$$
R_{2}^{n}(k):=\Big(f_n(x)-f(x),(I-Q)^{-1}Q\frac{e^{ikx}}{(2\pi)^{3/2}}ix
\Big)_{L^2(\mathbb{R}^3)}, \quad k\in \mathbb{R}^3.
$$
Let us use the upper bound
$$
\|R_{2}^{n}(k)\|_{L^{\infty}(\mathbb{R}^3)} \leq \frac{1}{(2\pi)^{3/2}}
\frac{1}{1-\|Q\|_{\infty}}\|Qe^{ikx}x\|_{L^{\infty}(\mathbb{R}^3)}\|f_n-f\|_
{L^1(\mathbb{R}^3)}.
$$
In the proof of \cite[Lemma 2.4]{VV08} it was established that
the norm $\|Qe^{ikx}x\|_{L^{\infty}(\mathbb{R}^3)}$ is bounded above by a
finite quantity independent of $k$. According to the part (a) of 
Lemma \ref{lem7} when
$n\to \infty$, $f_n\to f$ in $L^1(\mathbb{R}^3)$. Therefore,
$$
\|R_{2}^{n}(k)\|_{L^{\infty}(\mathbb{R}^3)}\to 0, \quad n\to \infty.
$$
Finally, it remains to estimate the expression
$$
R_{3}^{n}(k):=\Big(f_n(x)-f(x),(I-Q)^{-1}(\nabla_{k}Q)(I-Q)^{-1}
\frac{e^{ikx}}{(2\pi)^{3/2}}\Big)_{L^2(\mathbb{R}^3)}, \quad
 k\in \mathbb{R}^3.
$$
Using \eqref{grad}, we easily deduce the inequality
$$
\|R_{3}^{n}(k)\|_{L^{\infty}(\mathbb{R}^3)}\leq \frac{1}{4\pi (2\pi)^{3/2}}
\frac{\|V\|_{L^1(\mathbb{R}^3)}}{(1-\|Q\|_{\infty})^2}\|f_n-f\|
_{L^1(\mathbb{R}^3)}\to 0, \quad n\to \infty
$$
via the statement of the part (a) of Lemma \ref{lem7}. 
\end{proof}


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\end{document}