\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 34, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE--2004/34\hfil An estimate for solutions]
{An estimate for solutions to the Schr\"{o}dinger equation} 

\author[Alexander Makin \& Bevan Thompson\hfil EJDE--2004/34\hfilneg]
{Alexander Makin \& Bevan Thompson}  % in alphabetical order


\address{Alexander Makin \hfill\break
Moscow State Academy of Instrument-Making and Informatics\\
Stromynka 20, 107846, Moscow, Russia}
\email{alexmakin@yandex.ru}

\address{Bevan Thompson \hfill\break
Department of Mathematics\\
The University of Queensland\\
Brisbane Qld 4072 Australia}
\email{hbt@maths.uq.edu.au}

\date{}
\thanks{Submitted May 29, 2003. Published March 10, 2004.}
\subjclass[2000]{35B45, 35J10}
\keywords{Schr\"{o}dinger operator, spectral parameter, eigenfunction}

\begin{abstract}
 In this note, we find a priori estimates in the $L_2$-norm for solutions
 to the Schr\"{o}dinger equation with a parameter. 
 It is shown that a constant occuring in the inequality does not depend 
 on the value of the parameter.
 In particular, the estimate is valid for eigenfunctions associated
 with the Schr\"{o}dinger operator with arbitrary boundary conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section] 

\section{Introduction}

Asymptotic properties of solutions to the Schr\"{o}dinger equation
of second order elliptic equation with a parameter have been
investigated in many papers;  see for instance \cite{i2,l3,m2,m3,t1}.
In particular, eigenfunctions and functions associated with the
Schr\"{o}dinger operator have been considered
with various boundary conditions.
The authors of these papers have studied the general elliptic operator
of second order
$$
\mathbb{L}u=\sum_{i,j=1}^n\frac{\partial}{\partial x_i}[a_{ij}(x)\frac{\partial u}{\partial x_j}]+\sum_{i=1}^nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u
$$
(or the Schr\"{o}dinger operator) in an arbitrary domain
$G$ of $\mathbb{R}^n$.

 Let $u^0(x)$ be a regular solution of the equation
$\mathbb{L}u^0+\lambda u^0=0$, and let $u^m(x)$ be a regular solution of the
equation $\mathbb{L}u^m+\lambda u^m=u^{m-1}$
(m=1,2,\dots). Under some smoothness conditions on the coefficients
and restrictions on the range of the spectral parameter $\lambda$,
the following estimated has been established: 
\begin{equation} \label{*}
\|u^m\|_{L_p(K)}\le C|\lambda|^{\delta(m,n,p,q)}\|u^m\|_{L_q(K')}
\end{equation}
where $1\le q\le p\le\infty$, the constant $C$ depends on the coefficients
of the operator and on compact sets $K$ and $K'$ 
(with $K,K'\subset G$). When $n>1$, $K$ must lie strictly inside $K'$. 
However, when $n=1$ this condition is  omitted \cite{l3,t1}; i.e. 
$K'$ can coincide with $K$ and  even $K'$ can lie inside $K$.
As done in \cite{i1}, estimates of the type \eqref{*} can be applied to
study the convergence of the spectral expansions corresponding to 
nonselfadjoint differential operators.

The main objective of the present paper is obtaining an estimate
of the type \eqref{*} in the multidimensional case when the
condition $K\subset K'$ is not satisfied. The methods used here
are different from those  used previously. To estimate a
solution of an elliptic equation, we investigate a corresponding
hyperbolic equation. This approach gives the possibility of using
energy estimates for solutions of hyperbolic equations, the
theorem on domain of dependence, and the generalized Kirchoff
formula for the solution of the Cauchy problem. 
As is well known,
the Kirchoff formula has a different form for $n$ odd and for $n$
even. When $n$ is odd, the domain of dependence is a sphere, and not
a ball. Since, we use this fact in our proof, the assumption that $n$ is odd
cannot be omitted. This leaves open the question of finding similar
estimates for  $n$ even.

\section{The main result}

We consider the Schr\"{o}dinger operator
$$
Lu= \Delta u- q(x)u
$$
defined on a bounded domain $\Omega\subset \mathbb R^n$ for odd $n>1$. Here
$q$ is a complex-valued function continuous on the closure of $\Omega$,
$\bar\Omega$.
Let $d$ be the diameter of $\Omega$, let $\lambda$ be a complex number, and let
$$
q_0=\max_{x\in\bar\Omega}|q(x)|.
$$
For $m=0,1,\dots$, let $u^m$ be twice continuously differentiable functions on
$\Omega$ satisfying
\begin{gather}
Lu^0+\lambda u^0=0, \label{e1}\\
Lu^m+\lambda u^m=u^{m-1}. \label{e2}
\end{gather}
Let $B(x,R)$ be the ball of radius $R$ and centered at $x$ in $\mathbb{R}^n$, and
let $\partial\Omega$ denote the boundary of $\Omega$.
For $x$ in $\Omega$ with $3R<\mathop{\rm dist}(x,\partial\Omega)$, we have the
following an a priori estimate.

\begin{theorem} \label{thm1}
For any complex number $\lambda$, and any positive real numbers 
$R,\varepsilon$  such that $3R+\varepsilon<\mathop{\rm dist}(x,\partial\Omega)$,
 we have
\begin{equation}
\|u^m\|_{L_2(B(x,R))}\le C\|u^m\|_{L_2(B(x,3R+\varepsilon)\setminus B(x,R))}
 \label{e3}
\end{equation}
for $m=0,1,\dots$ where $C=C(n, q_0, d, \varepsilon, m)$.
\end{theorem}

\begin{proof}
We proceed by induction. Consider $m=0$ and set $K=B(x,R)$,
$K_0=B(x, R+\varepsilon/8)$, $K_0'=B(x, R+3\varepsilon/8)$,
$K'=B(x, 3R+5\varepsilon/8)$,
$K''=B(x, 3R+7\varepsilon/8)$ and $\hat K=B(x, 3R+\varepsilon)$.
Let $\eta$ and $\eta_0$ be cut off functions satisfying
$$
\eta (x)=\begin{cases}
1, & \mbox{if }x\in K'\\
0, & \mbox{if }x\notin K'', \end{cases}\quad
\eta_0(x)=\begin{cases}
1, & \mbox{if }x\in K'\setminus K_0'\\
0, & \mbox{if }x\notin K''\mbox{ or }x\in K_0\,,\end{cases}
$$
$\eta(x)=\eta_0(x)$ if $x\in K''\setminus K'$,
$\eta, \eta_0\in C^{\infty}(\mathbb R^n)$, $0\le\eta, \eta_0\le1$.

Let $\mu=\sqrt\lambda$ where $-\pi/2<\arg\mu\le\pi/2$ and consider the
function
$$
\omega(t)=\begin{cases}
e^{-i\mu t}, & \mbox{if }\mathop{\rm Im}\mu\ge0\\
e^{i\mu t}, & \mbox{if }\mathop{\rm Im}\mu<0\,.\end{cases}
$$
 Clearly $\omega(0)=1$,
$|\omega'(0)|=|\mu|$ and $|\omega(t)|=e^{|Im\mu|t}$.
Define the operator $ \hat L$ by
\begin{equation}
\hat L\phi=\frac{\partial^2\phi}{\partial t^2}-\Delta\phi \label{e4}
\end{equation}
for $(x,t)\in \mathbb R^{n+1}$. From \eqref{e4} it follows that
for all $(x,t)\in\Omega\times\mathbb R$,
\begin{equation}
\hat L(\omega(t)u^0(x))=-q(x)u^0(x)\omega(t)\,. \label{e5}
\end{equation}
Consider the following three Cauchy problems:
\begin{equation} \label{e6}
\begin{gathered}
\hat L\phi=\hat L(\omega\eta u^0)\\
\phi(x,0)=\eta(x)u^0(x),\quad \phi_t(x,0)=\omega'(0)\eta(x)u^0(x)\,,
\end{gathered}
\end{equation}
\begin{equation} \label{e7}
\begin{gathered}
\hat L\phi=-q\eta u^0\omega\\
\phi(x,0)=\eta_0(x)u^0(x),\quad \phi_t(x,0)=\omega'(0)\eta_0(x)u^0(x)\,,
\end{gathered}
\end{equation}
\begin{equation} \label{e8}
\begin{gathered}
\hat L\phi=-q\eta u^0\omega\\
\phi(x,0)=\eta(x)u^0(x),\quad \phi_t(x,0)=\omega'(0)\eta(x)u^0(x)\,.
\end{gathered}
\end{equation}
Since $\hat L$ is the wave operator, from a results in \cite{l1},
solutions to problems \eqref{e6}, \eqref{e7}, and \eqref{e8} exist and are 
unique. Moreover the solution $\phi_2$
of problem \eqref{e7} satisfies the estimate
\begin{align*}
&\max_{0\le t\le T}(\|\phi_2(x,t)\|_{W_2^1(\mathbb R^n)}
+\|\frac{\partial}{\partial t} \phi_2(x,t)\|_{L_2(\mathbb R^n)})\\
&\le  e^{c_1T}(\|\phi_2(x,0)\|_{W_2^1(\mathbb R^n)}+
\|\frac{\partial}{\partial t}\phi_2(x,0)\|_{L_2(\mathbb R^n)}+
\int_0^T|\omega(t)|dt\|q\eta u^0\|_{L_2(\mathbb R^n)}).
\end{align*}
This inequality and the definition of the cut off functions imply that
\begin{equation} \label{e9}
\begin{aligned}
\max_{0\le t\le T}(\|\phi_2(x,t)\|_{W_2^1(\mathbb R^n)}&
+\|\frac{\partial}{\partial t}\phi_2(x,t)\|_{L_2(\mathbb R^n)})\\
&\le e^{c_1T}(\|\eta_0 u^0\|_{W_2^1(K''\setminus K_0)}+
|\mu\||u^0\|_{L_2(K''\setminus K_0)}\\&+Tq_0e^{|Im\mu|T}\|u^0\|_{L_2(K'')})
\end{aligned} 
\end{equation}
where $T>0$ is arbitrary.
Let $\phi_1$ and $\phi_3$ be the solutions of \eqref{e6} and \eqref{e8},
respectively. From \eqref{e5} and the theorem on domain of dependance
(see \cite{l1}) we have
\begin{equation}
\phi_1(x,t)=\phi_3(x,t) \label{e10}
\end{equation}
for all $(x,t)\in K_0\times[0, \mathop{\rm dist}(K_0, \partial K')]$. Since
$\omega(t)\eta(x)u^0(x)$ is the solution of problem \eqref{e6}
for $(x,t)\in\mathbb R^n\times[0,\infty)$, it follows that
\begin{equation}
\phi_3(x,t)=\omega(t)\eta(x)u^0(x) \label{e11}
\end{equation}
for all $(x,t)\in K_0\times[0, \mathop{\rm dist}(K_0,\partial K')]$.

 It is easy to see
that $\phi_3(x,t)-\phi_2(x,t)$ is a generalized solution of the Cauchy problem
\begin{equation} \label{e12}
\begin{gathered}
\hat L\phi=0\\
\phi(x,0)=(\eta(x)-\eta_0(x))u^0(x),\quad\phi_t(x,0)
=\omega'(0)(\eta(x)-\eta_0(x))u^0(x)
\end{gathered}
\end{equation}
in $\mathbb R^n\times[0, \infty)$.
Note that the function $\eta-\eta_0$ is non zero only on $K_0'$.
Let $u_h^0=h^{-n}\int_{\mathbb R^n}u^0(y)\gamma(\frac{x-y}{h})dy$ where
$\gamma\in C^\infty(\mathbb R^n)$, $0\le\gamma\le1$, $\gamma(y)=0$ for $|y|\ge1$,
$\int_{\mathbb R^n}\gamma(y)dy=1$, and we set $u^0(y)=0$ for all $y\not\in \hat K$.
Let $\tilde\phi_h$ be the solution of the Cauchy problem
\begin{equation}
\begin{gathered}
\hat L\phi=0\\
\phi(x,0)=(\eta(x)-\eta_0)u_h^0(x),\quad \phi_t(x,0)=\omega'(0)(\eta(x)-\eta_0(x))u_h^0(x)
\end{gathered}\label{e13}
\end{equation}
Since $(\eta-\eta_0)u_h^0\in C^\infty(\mathbb R^n)$ it follows (see \cite{m4}) that
$\tilde\phi_h$ is the classical solution of problem (13) and satisfies the
generalized Kirchoff formula (see \cite{m4})
$$
\tilde\phi_h(x,t)=\frac{\partial}{\partial t}Q_{\psi_0,h}(x,t)+Q_{\psi_1,h}(x,t)
$$
where
\begin{gather*}
\psi_{0,h}(x)=(\eta(x)-\eta_0(x))u_h^0(x),\quad
 \psi_{1,h}(x)=\omega'(0)(\eta(x)-\eta_0(x))u_h^0(x), \\
Q_\psi(x,t)=\frac{1}{2(2\pi)^{l+1}}\sum_{j=0}^l\frac{\partial^j}{\partial t^j}
\frac{P_l^{(l-j)}(1)}{t^{2l+1-j}}\int_{S_t}\psi(y)dS_y
\end{gather*}
where $S_t$ is the surface of the $n$--dimensional hypersphere of radius $t$
and centre $x$, $l=(n-3)/2$, and $P_l$ is the Legendre polynomial of degree $l$.
Hence $\tilde\phi_h(x,t)=0$ for all $(x,t)\in K_0\times[2R+\varepsilon/2, \infty)$.


It follows from \cite[Ch. 3, \S 18]{s1} that the solution $\phi_3-\phi_2$ of problem
\eqref{e12} is the
limit of $\tilde\phi_h$ as $h\to0$ and thus $\phi_3(x,t)=\phi_2(x,t)$ for all
$(x,t)\in K_0\times[2R+\varepsilon/2, \infty)$. From this, \eqref{e10},
and since $\mathop{\rm dist}(K_0, \partial K')=2R+\varepsilon/2$
it follows that $\phi_2(x,t_0)=\phi_1(x,t_0)$
for all $x\in K_0$ and $t_0=2R+\varepsilon/2$. From this, \eqref{e9} and
\eqref{e11} it follows that
\begin{equation} \label{e14}
\begin{aligned}
|\mu|e^{|Im\mu|t_0}\|u^0\|_{L_2(K_0)}\\
&\le e^{c_1t_0}(\|\eta_0 u^0\|_{W_2^1(K''\setminus K_0)}
+|\mu\||u^0\|_{L_2(K''\setminus K_0)}\\&+
q_0t_0e^{|Im\mu|t_0}\|u^0\|_{L_2(K'')}).
\end{aligned}
\end{equation}
Since the compact set $K''\setminus(K_0\setminus\partial K_0)$ lies strictly
inside the compact set $\hat K\setminus(K\setminus\partial K)$ it follows
from \cite[page 95]{m1} that
\begin{equation}
\|\eta_0 u^0\|_{W_2^1(K''\setminus K_0)}\le
\|\eta_0\|_{C^1(K''\setminus K_0)}\|u^0\|_{W_2^1(K''\setminus K_0)}\le
c_2|\mu\||u^0\|_{L_2(\hat K\setminus K)}.
\label{e15}
\end{equation}
Moreover, from \eqref{e14} and \eqref{e15}
 we obtain
\begin{align*}
\|u^0\|_{L_2(K)}
&\le e^{c_1d}(c_3\|u^0\|_{L_2(\hat K\setminus K)}
+\frac{dq_0}{|\mu|}\|u^0\|_{L_2(\hat K)})\\
&\le e^{c_1d}(c_3\|u^0\|_{L_2(\hat K\setminus K)}+\frac{dq_0}{|\mu|}\|u^0\|_{L_2(K)}
+\frac{dq_0}{|\mu|}\|u^0\|_{L_2(\hat K\setminus K)}).
\end{align*}
If $|\mu|>2dq_0e^{c_1d}$ then \eqref{e3} follows.

Now assume that the conclusion of Theorem \ref{thm1}
holds for $0\le k\le m-1$, $|\mu|> M$
where $M$ is a constant. We show that it holds if $k=m$, $|\mu|>\tilde M$
where $\tilde M$ is
sufficiently large. Consider the case $\mathop{\rm Im}\mu<0$.
We define the function
$$
\omega^m(x,t)=\sum_{j=0}^me^{i\mu t}P_j(t,\mu)u^{m-j}(x),
$$
where
$$
P_j(t,\mu)=\frac{1}{2^{j+1}j!\mu^j}\sum_{k=0}^{j-1}
\frac{i^{k-j}(j+k-1)!t^{j-k}}{2^kk!(j-k-1)!\mu^k},
$$
for $1\le j\le m$, and $P_0(t,\mu)=1/2$. It is easy to show that
$\omega^m(x,0)=u^m(x)/2$, $\omega_t^m(x,0)=\mu F(x)$, where
$$
F(x)=\sum_{j=0}^m\frac{(2j-2)!u^{m-j}(x)}{i4^jj!(j-1)!\mu^{2j}};
$$
here we formally set $(-1)!=-1$, and $(-2)!=1/2$. An easy calculation gives
$$
\hat L(\omega^m)=-q\omega^m,
$$
for all $x\in\Omega$ and $t>0$.

An argument similar to that used in the derivation of \eqref{e14} gives the
estimate
\begin{equation} \label{e16}
\begin{aligned}
\|\omega_t^m(x,t_0)\|_{L_2(K_0)}
&\le e^{c_1t_0}(\|\eta_0 u^m\|_{W_2^1(K''\setminus K_0)}
 +|\mu\||F\|_{L_2(K''\setminus K_0)}\\
&\quad +q_0\int_0^{t_0}\|\omega^m(x,t)\|_{L_2(K'')}dt)
\end{aligned}
\end{equation}
where $t_0=2R+\varepsilon/2$. It follows from \cite[Th. 1]{k1} that
\begin{equation}\label{e17}
\begin{aligned}
q_0\int_0^{t_0}\|\omega^m(x,t)\|_{L_2(K'')}dt
&\le q_0t_0e^{|Im\mu|t_0}\max_{0\le t\le t_0}\|\sum_{j=0}^m
P_j(t,\mu)u^{m-j}(x)\|_{L_2(K'')}\\
&\le c_2q_0t_0e^{|Im\mu|t_0}\|u^m\|_{L_2(\hat K)}.
\end{aligned}
\end{equation}
Moreover, as in \cite{k1}, we obtain
\begin{equation}
\|F\|_{L_2(K''\setminus K_0)}\le c_3\|u^m\|_{L_2(\hat K\setminus K)}.\label{e18}
\end{equation}
Since the compact set $K''\setminus(K_0\setminus\partial K_0)$ lies strictly
inside the compact set $\hat K\setminus(K\setminus\partial K)$ 
it follows as in \cite{m1} that
\begin{equation} \label{e19}
\|\eta_0 u^m\|_{W_2^1(K''\setminus K_0)}\le
\|\eta_0\|_{C^1(K''\setminus K_0)}\|u^m\|_{W_2^1(K''\setminus K_0)}\le
 c_4|\mu\||u^m\|_{L_2(\hat K\setminus K)}.
\end{equation}
Using \eqref{e16}--\eqref{e19}, we obtain
\begin{equation} \label{e20}
\|\omega_t^m(x,t_0)\|_{L_2(K_0)}\le
e^{c_1t_0}(c_5|\mu\||u^m\|_{L_2(\hat K\setminus K)}+c_6q_0t_0e^{|Im\mu|t_0}\|u^m\|_{L_2(\hat K)}).
\end{equation}
An easy calculation gives
\begin{equation}
\omega_t^m(x,t)=\sum_{j=0}^me^{i\mu t}R_j(t)u^{m-j}(x)\label{e21}
\end{equation}
where
$$
R_j(t)=\frac{1}{2^{j+1}j!\mu^j}\sum_{k=0}^j\frac{i^{k-j-1}(j+k-2)!(k+j-(j-k)^2)t^{t-k}}
{2^kk!(j-k)!\mu^{k-1}}
$$
for $1\le j\le m$, and $R_0(t)=i\mu/2$.
From \eqref{e20} and \eqref{e21} it follows that
\begin{equation} \label{e22}
\begin{aligned}
\|u^m\|_{L_2(K_0)}
&\le c_6\sum_{j=1}^m|\mu|^{-j}\|u^{m-j}\|_{L_2(K_0)}\\
&+e^{c_1t_0}(c_8\|u^m\|_{L_2(\hat K\setminus K)}
+c_8q_0t_0|\mu|^{-j}\|u^m\|_{L_2(\hat K)}).
\end{aligned}
\end{equation}
 From the induction hypothesis and the a posteriori estimates in \cite{k1} 
 we obtain
\begin{equation}
\sum_{j=1}^m|\mu|^{-j}\|u^{m-j}\|_{L_2(K_0)}
\le c_9\sum_{j=1}^m|\mu|^{-j}\|u^{m-j}\|_{L_2(K''\setminus K_0)}
\le c_{10}\|u^m\|_{L_2(\hat K\setminus K)}.
\label{e23}
\end{equation}
Clearly
\begin{equation}
\|u^m\|_{L_2(\hat K)}\le \|u^m\|_{L_2(K)}+\|u^m\|_{L_2(\hat K\setminus K)}.\label{e24}
\end{equation}
 From \eqref{e22}--\eqref{e24}, it follows that
$$
\|u^m\|_{L_2(K)}\le e^{c_{11}d}(\|u^m\|_{L_2(\hat K\setminus K)}
+|\mu|^{-1}\|u^m\|_{L_2(K)}+|\mu|^{-1}\|u^m\|_{L_2(\hat K\setminus K)}).
$$
If $|\mu|>\max(2e^{c_{11}d},M)$ then \eqref{e3} follows.

The case $\mathop{\rm Im}\mu\ge0$ follows by a similar argument, provided 
we replace the function $\omega^m(x,t)$ by 
$$
\psi^m(x,t)=\sum_{j=0}^me^{-i\mu t}Q_j(t,\mu)u^{m-j}(x),
$$
where
$$
Q_j(t,\mu)=\frac{1}{2^{j+1}j!\mu^j}\sum_{k=0}^{j-1}\frac{(-i)^{k-j}(j+k-1)!t^{j-k}}
{2^kk!(j-k-1)!\mu^k}
$$
for $1\le j\le m$; here $Q_0(t,\mu)=1/2$.

We now prove Theorem \ref{thm1} for $|\lambda|\le\lambda_0$ for arbitrary positive 
$\lambda_0$. Assume the conclusion of Theorem \ref{thm1} fails for some $m$. 
Then there exists $\varepsilon>0$ such that there exist sequences $C_k>0$, 
$x_k\in\Omega$ and $R_k>0$ with the following properties: 
$\lim_{k\to\infty}C_k=\infty$; $3R_k+\varepsilon<\mathop{\rm dist}(x_k,\partial\Omega)$;
for each $k$ there exist functions $u_k^l(x)$, $0\le l\le m$, satisfying
\eqref{e1} and \eqref{e2} and for which there exists $\lambda_k$ with 
$|\lambda_k|\le\lambda_0$ and
\begin{equation}
\|u_k^m\|_{L_2(B(x,R_k)}>C_k\|u_k^m\|_{L_2(B(x_k,3R_k+\varepsilon)
\setminus B(x_k,R_k))}.\label{e25}
\end{equation}
Since the domain $\Omega$ is bounded and $0<R_k\le d/3$, there exists a
subsequence $C_{k_p}$, $p=1,2,\cdots$, such that 
$\lim_{p\to\infty}x_{k_p}=\tilde x$
and $\lim_{p\to\infty}R_{k_p}=\tilde R$, where $0\le\tilde R\le d/3$ and
$3\tilde R+\varepsilon\le \mathop{\rm dist}(\tilde x, \partial \Omega)$.
 From this it follows
that $\mathop{\rm dist}(\tilde x,\partial\Omega)\ge\varepsilon$.

Consider the balls $B(\tilde x,\tilde R+\sigma)$ and 
$B(\tilde x,3(\tilde R+\sigma)+\varepsilon/2)$
where $\sigma=\varepsilon/100$. It is easy to see that for sufficiently large
$p_0$ and any $p\ge p_0$ $B(x_{k_p},R_{k_p})\subset B(\tilde x,\tilde R+\sigma)$
and $B(\tilde x,3(\tilde R+\sigma)+\varepsilon/2)\subset B(x_{k_p},3R_{k_p}+\varepsilon)$.
 From this it follows that
$$
B(\tilde x, 3(\tilde R+\sigma)+\varepsilon/2)\setminus B(\tilde x,\tilde R+\sigma)
\subset B(x_{k_p},3R_{k_p}+\varepsilon)\setminus B(x_{k_p},R_{k_p}).
$$
Thus we obtain
$$
\|u_{k_p}^m\|_{L_2(B(\tilde x, \tilde R+\sigma))}
\ge\|u_{k_p}^m\|_{L_2(B(x_{k_p},R_{k_p}))}
$$
and
$$
\|u_{k_p}^m\|_{L_2(B(\tilde x,3(\tilde R+\sigma)+\varepsilon/2)\setminus B(\tilde x,\tilde R+\sigma))}
\le\|u_{k_p}^m\|_{L_2(B(x_{k_p},3R_{k_p}+\varepsilon)\setminus B(x_{k_p},R_{k_p}))}.
$$
From the last two inequalities and from \eqref{e25}, it follows that
$$
\|u_{k_p}^m\|_{L_2(B(\tilde x,\tilde R+\sigma))}
>C_{k_p}\|u_{k_p}^m\|_{L_2(B(\tilde x,3(\tilde R+\sigma)
+\varepsilon/2)\setminus B(\tilde x,\tilde R+\sigma))}
$$
for $p\ge p_0$.

Set $\tilde K=B(\tilde x,\tilde R+\sigma)$ and 
$\hat K=B(\tilde x,3(\tilde R+\sigma)+\varepsilon/2)$.
Thus there is a sequence of functions $u_i^m$, $i=1,2,\dots$ such that
\begin{equation}
\|u_i^m\|_{L_2(\tilde K)}^2>\hat C_i^2\|u_i^m\|_{L_2(\hat K
\setminus \tilde K)}^2,\label{e27}
\end{equation}
where $\lim_{i\to\infty}\hat C_i=\infty$. Without loss of generality 
we may assume that
\begin{equation}
\|u_i^m\|_{L_2(\hat K)}=1\label{e28}
\end{equation}
for $i=1,2,\dots$. Set $K_j=B(\tilde x,\tilde R+\sigma+j\varepsilon/20)$ for
$j=1,\dots,5$. Since $|\lambda_i|\le\lambda_0$, it follows from 
\eqref{e28} and \cite[page 95]{m1}
that
$$
\|u_i^l\|_{W^1_2(K_5)}\le c_1
$$
for all $i$ and $0\le l\le m$. In view of this and \cite[Ch. 3, \S7]{l2} 
it follows that
\begin{equation}
\|u_i^l\|_{W^2_2(K_4)}\le c_2.\label{e29}
\end{equation}
It follows from this, \cite[Th. 2.5.1]{m4}, and Rellich's Lemma \cite{r1}, 
that there exists a 
subsequence of $u_i^0$, $u_{i_j}^0$, $j=1,2,\dots$, such that
 $\lim_{j\to\infty}\lambda_{i_j}=\tilde \lambda$
and
$$
\lim_{j\to\infty}\|u_{i_j}^0-\tilde u^0\|_{W^1_2(K_3)}=0
$$
for some $\tilde u^0\in W_2^1(K_3)$. It follows from this that $\tilde u^0$ is
a generalized solution of
$$
L\tilde u^0+\tilde\lambda\tilde u^0=0
$$
in $K_3\setminus\partial K_3$. From this and \cite[Ch. 3, \S10]{l2} it follows that 
$\tilde u^0\in W_2^2(K_2)$.
Similarly we find a further subsequence of $i_j$ denoted $p_j$, such that
$\lim_{j\to\infty}\lambda_{p_j}=\tilde \lambda$ and
$$
\lim_{j\to\infty}\|u_{p_j}^1-\tilde u^1\|_{W^1_2(K_3)}=0
$$
for some $\tilde u^1\in W_2^1(K_3)$. By a similar argument to that above it follows
that $\tilde u^1$ is a generalized solution of
$$
L\tilde u^1+\tilde\lambda\tilde u^1=\tilde u^0
$$
in $K_3\setminus\partial K_3$ and that $\tilde u^1\in W_2^2(K_2)$. Repeating
this argument we obtain $\tilde u^l\in W_2^2(K_2)$, $0\le l\le m$ which are
generalized solutions of
\begin{equation} \label{e30}
\begin{gathered}
L\tilde u^{m-l}+\tilde\lambda\tilde u^{m-l}=\tilde u^{m-l-1},\quad 0\le l\le m-1,\\
L\tilde u^0+\tilde\lambda\tilde u^0=0
\end{gathered}
\end{equation}
in $K_3\setminus\partial K_3$.
 From \eqref{e27} and \eqref{e28}, it follows that
$$
\lim_{i\to\infty}\|u_i^m\|_{L_2(\hat K\setminus\tilde K)}=0.
$$
It follows that
$$
\lim_{i\to\infty}\|u_i^m\|_{L_2(K_1\setminus\tilde K)}=0
$$
and therefore $\|\tilde u^m\|_{L_2(K_1\setminus\tilde K)}=0$. From this and
\eqref{e30} it follows that
\begin{equation}
\|\tilde u^{m-l}\|_{L_2(K_1\setminus\tilde K)}=0,\label{e31}
\end{equation}
for $1\le l\le m$. Setting $l=m$ in \eqref{e31} and using results from 
\cite[pages 235-236]{a1} we obtain
$\|\tilde u^0\|_{L_2(K_1)}=0$.
Setting $l=m-1$ in \eqref{e31}, from a results of \cite[pages 235-236]{a1}, 
it follows that $\|\tilde u^1\|_{L_2(K_1)}=0$.
Repeating this argument we obtain
\begin{equation}
\|\tilde u^m\|_{L_2(K_1)}=0.\label{e32}
\end{equation}
 From \eqref{e27} and \eqref{e28}, it follows that
$\lim_{i\to\infty}\|u_i^m\|_{L_2(\tilde K)}=1$
and thus
$$
\|\tilde u^m\|_{L_2(\tilde K)}=1.
$$
This contradicts \eqref{e32}. The proof is complete. 
\end{proof}.

\subsection*{Remark} From \eqref{e3} it follows that
\begin{equation}
\|u^m\|_{L_2(B(x,3R+\varepsilon))}
\le\tilde C\|u^m\|_{L_2(B(x,3R+\varepsilon)\setminus B(x,R))}\label{e33}
\end{equation}
where $\tilde C=\sqrt{C^2+1}$; i.e. the $L_2$-norm of any solution to the 
Schr\"{o}dinger equation with a parameter on a ball
can be estimated by the $L_2$-norm of the same solution on some
compact subset of the ball. Furthermore, the constant $\tilde C$ in 
\eqref{e33} does not depend on the value of the parameter $\lambda$.


\subsection*{Acknowledgements}
This work was carried out while the first author was visiting the University
of Queensland. The first author was supported by a Raybould Fellowship.

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\end{document}