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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 126, pp. 1--15.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/126\hfil Spectral properties]
{Spectral properties of non-local uniformly-elliptic operators}

\author[F. A. Davidson, N. Dodds\hfil EJDE-2006/126\hfilneg]
{Fordyce A. Davidson, Niall Dodds}  % in alphabetical order

\address{Fordyce A. Davidson \newline
Division of Mathematics, University of Dundee, Dundee, DD1 4HN,
United Kingdom \newline
Tel. 01382 384692, fax 01382 385516}
\email{fdavidso@maths.dundee.ac.uk }

\address{Niall Dodds \newline
Division of Mathematics, University of Dundee, Dundee, DD1 4HN,
United Kingdom \newline
Tel. 01382 384471, fax 01382 385516}
\email{ndodds@maths.dundee.ac.uk}

\date{}
\thanks{Submitted July 6, 2006. Published October 11, 2006.}
\subjclass[2000]{34P05, 35P10, 47A75, 47G20}
\keywords{Non-local; uniformly elliptic; eigenvalues; multiplicities}

\begin{abstract}
 In this paper we consider the spectral properties of a class of
 non-local uniformly elliptic  operators, which arise from the study
 of  non-local uniformly elliptic partial differential equations.
 Such equations arise naturally in the study of
 a variety of physical and biological systems with examples
 ranging  from Ohmic heating to population dynamics.
 The operators studied here are  bounded perturbations of
 linear (local) differential operators, and the non-local perturbation
 is in the form of an integral term.
 We study the eigenvalues, the multiplicities of these eigenvalues,
 and the existence of corresponding positive eigenfunctions.
 It is shown here that the spectral properties of these non-local
 operators can differ considerably  from those of their local counterpart.
 However, we show that under suitable hypotheses, there still exists
 a principal eigenvalue of these operators.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{note}[theorem]{Note}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{exa}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

This paper studies the spectral properties of a class of linear
 integro-differential operators
\begin{equation}\label{Ldef}
[L_\epsilon u](x)=\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x))_{x_j}+b(x)u(x)+
\epsilon c(x)\int_U
 d(x)u(x)dx,\quad x\in U,
\end{equation}
where $U$ is a bounded connected subset of $\mathbb{R}^n$ with a suitably
smooth boundary, $\partial U$,
 and $-\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x))_{x_j}$ is uniformly elliptic
on $U$.

The operator
$L_\epsilon$ is defined on a domain that incorporates homogeneous
Dirichlet boundary conditions. By varying the real parameter
$\epsilon$,  the non-local
operator can be viewed as a continuous,
bounded perturbation of the  (local) differential
operator,
\begin{equation}\label{Adef}
[Au](x)=\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x))_{x_j}+b(x)u(x).
\end{equation}
In this paper this structure will be exploited to study the spectral
properties of  $L_\epsilon$.
Results will not be restricted to small $\epsilon$;
rather, $\epsilon$ should be viewed as a homotopy parameter from the
local operator $A$ to the general form $L_\epsilon$.

Operators of the type given in \eqref{Ldef} arise from the study of
non-local nonlinear parabolic problems of the form
\begin{equation}\label{nonlocalrd}
u_t(x,t)=\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x,t))_{x_j}+f(x,u(x,t),
\bar{u}(t)), \quad
\bar{u}(t)=\epsilon\int_U g(x,u(x,t))dx,
\end{equation}
for sufficiently differentiable  functions $f$ and $g$.
Here $\epsilon$ is a real parameter which can be viewed as a
measure of strength of the
non-local interactions. System
\eqref{nonlocalrd} requires to be
 augmented with appropriate boundary conditions,
for example homogeneous Dirichlet boundary conditions, as studied here, as
well as an initial condition.
Non-local boundary value problems of this type appear in
a wide variety of applications, including
Ohmic heating \cite{Freitas+G,Lacey}, the formation of shear
bands in materials
\cite{B+L+T}, heat transfer in thermistors \cite{F+F+H}, combustion
theory \cite{Pao}, the electric ballast resistor \cite{Chafee},
microwave heating of ceramic materials \cite{B+K2,Kriegsmann}, and
population dynamics \cite{F+G}.
An extensive survey of results, techniques and applications of non-local
reaction-diffusion equations of this form is given in  \cite{Freitas1}.

It is often desirable to identify steady states of
\eqref{nonlocalrd} and determine their stability, as stable steady states
represent possible asymptotic states of the system under consideration,
and are thus of most physical relevance. Let
us assume that a steady state, $u^*$, of \eqref{nonlocalrd}
exists. Then linearizing \eqref{nonlocalrd} around $u^*$, leads to
\begin{equation} \label{linear}
u_t(x,t)=\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x,t))_{x_j} +b(x)u(x,t)+
\epsilon c(x)\int_U d(x)u(x,t)dx,
\end{equation}
where  $b(x)=f_u(x,u^*(x),\bar{u^*}),\;c(x)=f_{\bar{u}}(x,u^*(x),\bar{u^*})$
and $d(x)=g_u(x, u^*(x))$. We will assume that $f$ and $g$, and consequently
$b$, $c$ and $d$ are real valued functions.
Formally at least, it is straightforward to see
that the values of $\lambda$ for which
\begin{equation}\label{evalue.eq}
\begin{gathered}
\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x))_{x_j}  +b(x)u(x)+
\epsilon c(x)\int_U d(x)u(x)dx=\lambda u(x),\quad x\in U;
\\
u(x)=0\quad \mbox{on } \partial U,
\end{gathered}
\end{equation}
has a solution, determine the  growth properties of the
solutions of \eqref{linear} and hence the (local asymptotic)
stability of the steady
state $u^*$ of \eqref{nonlocalrd}. This connection can be made rigorous
as detailed in \cite{Henry}. Indeed,
an operator equation $L_\epsilon u=\lambda u$ can be
defined on a suitable domain, which
is equivalent to \eqref{evalue.eq} and hence, the spectral properties
of $L_\epsilon$  determine the stability of
steady states of \eqref{nonlocalrd}.

The spectral properties of (\ref{Adef}) are well-known, and in
\cite{Freitas}, certain corresponding  properties   for the non-local
operator $L_\epsilon$ are derived using the  perturbation theory of linear
operators (see e.g. \cite{Kato}),
for the special case $n=1$.
Results in \cite{Freitas} deal with the structure of the set of eigenvalues
of $L_\epsilon$, $\sigma( L_\epsilon)$,
when considered as functions of the parameter $\epsilon$, and show that
the Fourier coefficients of the functions $c$ and $d$ in \eqref{Ldef}
with respect to the eigenfunctions of \eqref{Adef}, are fundamental
to determining the qualitative structure of $\sigma (L_\epsilon)$.
Further results concerning the spectrum of $L_\epsilon$, multiplicities
of the eigenvalues of $L_\epsilon$, and the nodal properties of the
associated eigenfunctions for the case $n=1$, are presented in \cite{D+D}.
In \cite{F+V}, some of the work done in \cite{Freitas} is extended
to $n\geq 1$ in the case where $A$ is of the form $Au=\Delta u+ a(x)u$,
where $\Delta \cdot$ denotes the Laplacian operator, as is standard.
Other papers that include related results on spectral properties of
non-local operators are
\cite{A+B, B+K, Bose, F+P, F+S, P+S}.

In this paper we extend results from
\cite{D+D} to the general $n\geq 1$
case and present
some further spectral results, which are new to all cases.
As is shown in the above references,
the presence of the non-local term in $L_\epsilon$
gives a  much wider variety of possible behaviour of the
spectrum, than that of the
corresponding local operator.

The following section
contains relevant definitions, along with some basic results.
In Section 3 we consider how the eigenvalues of $L_\epsilon$ change with $\epsilon$,
and results regarding multiplicities of eigenvalues of $L_\epsilon$ are
given in Section 4. The existence of a principal eigenvalue,
i.e. an eigenvalue to which there corresponds a positive eigenfunction, is
important to a number of results including those related to
determining stability and
the existence of positive solutions of
associated nonlinear problems (via bifurcation theory).
\cite[Theorem 5.5]{D+D},
\cite[Lemma 3.16]{Freitas} and  \cite[Proposition 6.1]{F+S} deal with
the existence of positive eigenfunctions of $L_\epsilon$.
We present further, more general results dealing with not only the existence,
but also the uniqueness of a principal eigenvalue of $L_\epsilon$ in Section 5.

\section{Preliminaries}\label{pre}

Let $U$ be an open
bounded connected subset of $\mathbb{R}^n$ where $n\geq 1$, and assume that
the boundary of $U$, $\partial U$ is
$C^{k+1}$ where $k:=\lfloor\frac{n+4}{2}\rfloor$.
Let $A,B,L_{\epsilon}: H^2(U)\cap H_0^1(U)\subset L^2(U)\to L^2(U)$
be defined by
\begin{equation}\label{Lep}
\begin{gathered}
\;[Au](x):=\sum_{i,j=1}^n (a_{ij}(x)u_{x_i}(x))_{x_j}+b(x)u(x),\;
\\
[Bu](x):=c(x)\int_Ud(x)u(x)dx,
\\
L_\epsilon :=A+\epsilon B,
\end{gathered}
\end{equation}
where $a_{ij},b\in C^k(\bar{U})$;
$c\in H^{k-1}(U)$; $d\in L^2(U)$; $c,d\not\equiv 0$;
and $\epsilon\in\mathbb{R}$. Note that following standard regularity
arguments, the condition on $c$ is necessary for the eigenfunctions
of $L_\epsilon$ to lie in $C^2(U)$. This in turn is necessary to ensure the
equivalence of the operator equation $L_\epsilon u =\lambda u$ and  \eqref{evalue.eq}.
 Also, assume that $-A$ is uniformly elliptic.

Then  $A$ is a densely defined, closed, self-adjoint operator with
compact resolvent. Its spectrum is real, bounded above and consists
entirely of isolated eigenvalues of finite multiplicity.
Denote these eigenvalues by
$\gamma_i$, $i=1,2,3,\dots $
Then it is well-known that
$\gamma_1>\gamma_2\geq\dots \geq\gamma_i\geq\gamma_{i+1}\geq\dots $ and $\gamma_i
\to -\infty$ as $i \to \infty$. The
largest eigenvalue of $A$, $\gamma_1$, is
simple and the
 eigenfunction $v_1$ corresponding to $\gamma_1$ can be chosen to be
strictly positive on $U$.
Furthermore this is the only eigenvalue of $A$ to which there corresponds a
positive eigenfunction,
and the set of eigenfunctions of $A$, $\{v_i\}_{i=1}^\infty$ can be
chosen to form
an orthonormal basis for $L^2(U)$.

Clearly $B$ is a bounded linear operator, and therefore it can be
shown that  for each fixed $\epsilon$,
$L_\epsilon$ is a densely defined, closed
operator with compact resolvent. Hence, for each fixed $\epsilon$, the spectrum,
$\sigma(L_\epsilon)$, consists entirely of isolated eigenvalues of
finite multiplicity. Denote these
eigenvalues by $\lambda_i(\epsilon)$
and for consistency let $\lambda_i(0)=\gamma_i$ for each $i\in\mathbb{N}$.
Denote the eigenfunctions of $L_\epsilon$ by $u_i(\epsilon)$,
where $u_i(\epsilon)$ corresponds to the eigenvalue $\lambda_i(\epsilon)$.
Then in this way, we generate a set
of functions, $\Sigma:=\{\lambda_i(\epsilon)\}_{i=1}^\infty$, which we shall also
refer to as eigenvalues of $L_\epsilon$. Similarly, the functions of $\epsilon$,
$u_i(\epsilon)$ will be referred to as eigenfunctions.
Then we may apply results contained in
\cite[Sections II-1, III-6.4, IV-3.5 and VII-1.3]{Kato} to our problem
to give:


\begin{lemma}\label{basic.lem}

\begin{itemize}

\item[(a)] For each $i$,  $\lambda_i(\epsilon)$
is a  continuous function of $\epsilon$, $\forall\epsilon\in\mathbb{R}$.

\item[(b)] Fix $j$. If  $\lambda_j(\epsilon) \neq \lambda_i(\epsilon)$ for
all $i \neq j$ and  $\forall\epsilon\in
(\epsilon_1, \epsilon_2)$, then $\lambda_j(\epsilon)$ is an analytic
function of $\epsilon$ $\forall\epsilon\in(\epsilon_1,\epsilon_2)$,
and the eigenprojection corresponding to $\lambda_j(\epsilon)$ is
an analytic function of $\epsilon$ $\forall\epsilon\in(\epsilon_1,
\epsilon_2)$.

\item[(c)] Let $S \subset \Sigma$  contain only a finite number of elements.
If
$\lambda_i(\epsilon) \neq \lambda_j(\epsilon)$ for any
$\lambda_i(\epsilon) \in S$ and $\lambda_j(\epsilon) \in
\Sigma\backslash S$,
  $\forall\epsilon\in
(\epsilon_1, \epsilon_2)$,
 then the sum of the eigenvalues in $S$ is an analytic
function of $\epsilon$ for all $\epsilon\in(\epsilon_1,\epsilon_2)$.
Furthermore, the total eigenprojection corresponding to
all the eigenvalues in $S$ is an
analytic function of $\epsilon$ $\forall\epsilon\in(\epsilon_1, \epsilon_2)$.
\end{itemize}
\end{lemma}

Also, by standard regularity theory, given the assumptions on
the coefficient functions $a_{ij}, b, c$, and the boundary $\partial U$,
it can be shown that any
eigenfunction of $L_\epsilon$ is actually in $C^2(\bar U)$, i.e. the spectral
properties of $L_\epsilon:H^2(U)\cap H_0^1(U)\subset L^2(U)\to L^2(U)$
are identical to the spectral properties of the corresponding
non-local differential equation.

It is useful to distinguish between those eigenvalues of $L_\epsilon$ which
change with $\epsilon$, and those which do not.

\begin{definition} \label{def2.2} \rm
We call $\lambda_i(\epsilon)$ a
\emph{ fixed eigenvalue } iff $\lambda_i(\epsilon) \equiv \gamma_i$.
If $\lambda_i(\epsilon)$ is not
fixed, then it is referred to as a \emph{ moving eigenvalue}.

Note that an eigenfunction $u_i(\epsilon)$ corresponding to a fixed
eigenvalue $\lambda_i(\epsilon)$, may or may not vary with $\epsilon$.
If the latter holds,
i.e. $u_i(\epsilon)\equiv v_i$, then
we refer to such an eigenfunction as being \emph{fixed}.

Let $\gamma_i$ be a fixed eigenvalue of $L_\epsilon$, and let
$X=\bigcap_{\epsilon\in\mathbb{R}} N(L_\epsilon-\gamma_i I)$. We call $X$
the \emph{fixed eigenspace} of $L_\epsilon$ corresponding to $\gamma_i$.
\end{definition}


Finally, the adjoint of $L_\epsilon$, denoted $L_\epsilon^*$ is
defined by
\[
L_\epsilon^*u =Au+\epsilon B^*u, \quad \epsilon \in \mathbb{R}, \quad u \in
H^2(U)\cap H_0^1(U)
\]
where
\[
[B^*u](x)=d(x)\int_Uc(x)u(x)dx.
\]


As already noted $A$ is self-adjoint, i.e.  $L_0$ is self-adjoint.
Moreover, $L_\epsilon$ is self-adjoint if and only if $c\equiv d$,
 and clearly if $L_{\epsilon^*}$ is self-adjoint for some $\epsilon^* \neq 0$,
then $L_\epsilon$ is self-adjoint for all $\epsilon \in \mathbb{R}$.

Whilst \cite{F+V} considered non-local perturbations of
$Au=\Delta u+a(x)u(x)$, it was
noted there that it is straightforward to extend all
of the spectral theory results in \cite{F+V} to a wider class of non-local
operators, including the operators given by \eqref{Lep}.
Hence, where we refer to relevant results from \cite{F+V}, we do
so in terms of
\eqref{Lep}.

Understanding the key differences in spectral structure of $L_\epsilon$
between the case $n=1$ studied in \cite{D+D, Freitas}
and the general case studied here is central to extending many
results from \cite{D+D} and \cite{Freitas}.
Hence we now proceed to highlight the important differences between the
structure of $\Sigma$ in these two cases.

One of the main differences between the case $n=1$ previously studied
in \cite{D+D, Freitas}, and the general
$n\geq 1$ case, is that for $n>1$, eigenvalues of $A$
can have arbitrarily large (but finite) multiplicities.
Therefore, although from Lemma \ref{basic.lem}
the eigenvalues and eigenprojections are
continuous,
it may be possible for an eigenvalue to ``split" at $\epsilon=0$ to form
multiple distinct eigenvalues
of $L_\epsilon$.
However in \cite{F+V},
it was shown that:

\begin{lemma}[\cite{F+V}]\label{same_moving.lem}
If an eigenvalue of $A$, $\gamma_i$, has geometric multiplicity
$m>1$, then $\gamma_i$ is an eigenvalue of $L_\epsilon$ $\forall
\epsilon\in\mathbb{R}$, and the fixed eigenspace, $X$, of $L_\epsilon$ corresponding
to $\gamma_i$ is
such that $\dim X\geq (m-1)$.
\end{lemma}

Note that by this result it follows that all moving eigenvalues
of $L_\epsilon$ are of geometric multiplicity 1 until they
``collide" with another eigenvalue
of $L_\epsilon$, (see Section 4 for a detailed discussion of multiplicities).
 The effect of $A$ having eigenvalues of higher multiplicity
is simply to add more (possibly high multiplicity)
fixed eigenvalues to $\Sigma$, or to increase the multiplicities of the fixed
eigenvalues already in $\Sigma$.
Therefore, for differential operators of the form given above,
the set of moving eigenvalues
has the same properties as the set of moving eigenvalues
of the operators studied in \cite{D+D, Freitas}, i.e. in the case $n=1$.
As shown in \cite{F+V},
these properties include that
$\lambda_j(\epsilon_1)=\lambda_j(\epsilon_2)$ for any $j \in \mathbb{N}$ and any
$\epsilon_1\neq \epsilon_2$ only if $\lambda_j(\epsilon_1)=\lambda_j(\epsilon_2)=\gamma_i$.
for some $i\in\mathbb{N}$.
Hence if a real moving eigenvalue starts moving in one direction
along the real line, then it can not turn back on itself as long as it remains
real.
Also,
$\lambda_j(\epsilon_1)=\lambda_k(\epsilon_2)$ for any $j\neq k \in \mathbb{N}$ and any
$\epsilon_1\neq \epsilon_2$ only if $\lambda_j(\epsilon_1)=\lambda_k(\epsilon_2)= \gamma_i$
for some $i\in\mathbb{N}$.
Hence if $\lambda_j$ and $\lambda_k$ are 2 real moving eigenvalues
which are real $\forall\epsilon\in\mathbb{R}$, and if $\gamma_j < \gamma_k$,
then $\lambda_j(\epsilon_1)\leq \lambda_k(\epsilon_2)$ for any $\epsilon_1, \epsilon_2\in\mathbb{R}$.


\section{The Spectrum of $L_\epsilon$}\label{pde_spectrum}

As noted above, in general the spectrum of $L_\epsilon$ will vary with
the parameter $\epsilon$. In this section we present results that deal
with precisely how the eigenvalues of $L_\epsilon$ change.

Ideally we would like to determine explicit formulae for
the functions $\{\lambda_i(\epsilon)\}_{i=1}^\infty$. We have been unable
to do this, but we have
obtained the following implicit (but nevertheless useful) expression.

\begin{lemma}\label{ep.lem}
Take any real number $\lambda \neq \gamma_i$ for any $i\in\mathbb{N}$.
Take an orthonormal basis of eigenfunctions of $A$, $\{v_i\}_{i=1}^\infty$,
and let
\[
c(x)=\sum_{i=1}^\infty c_i v_i(x),\quad d(x)=\sum_{i=1}^\infty d_i v_i(x),
\]
i.e. $\{c_i\}_{i=1}^\infty$ and
$\{d_i\}_{i=1}^\infty$ are the Fourier coefficients
of $c$ and $d$ respectively.
 Then the solution $\epsilon^*$ of the
equation $\lambda = \lambda_{k^*}(\epsilon^*)$ is unique if it exists,
and is given by
\begin{equation}\label{pde_ep_exp.eqn}
\epsilon^*=
\Big(\sum_{i=1}^\infty \frac{c_i d_i}{(\lambda-\gamma_i)}\Big)^{-1}.
\end{equation}
\end{lemma}


\begin{proof}
The uniqueness of the value $\epsilon^*$ follows from the arguments at the
end of the preceding section.
Let
\begin{equation}\label{beta.eqn}
u(x)=\sum_{i=1}^\infty \beta_i v_i(x).
\end{equation}
Then, substituting the above expressions for
$c,d$ and $u$ into the equation $L_\epsilon u=\lambda u$ and
comparing the coefficients of $v_i$ for each $i\in\mathbb{N}$  gives
\[
\beta_i (\gamma_i-\lambda)+\epsilon c_i \int_U d(x)u(x) dx=0.
\]
Hence, either $c_i = \beta_i=0$ or
\[
\epsilon=\frac{\beta_i(\lambda-\gamma_i)}{c_i \int_U d(x)u(x)dx}.
\]
But this will hold for all $i$ such that $c_i\neq 0$, and so in this case
\begin{equation}\label{repetitive.eqn}
\frac{\beta_i (\lambda-\gamma_i)}{c_i} = K,
\end{equation}
for some constant $K$ independent of $i$, and without loss of
generality we take $K = 1$. Hence,
\begin{equation}\label{u_exp.eq}
u(x)=\sum_{i=1}^\infty \frac{c_i}{(\lambda-\gamma_i)} v_i(x),
\end{equation}
and it follows directly from \eqref{repetitive.eqn} and \eqref{u_exp.eq} that
\[
\epsilon=\frac{1}{\int_Ud(x)u(x)dx}.
\]
By Parseval's formula
we can interchange the order of summation and integration
in $\int_U d(x)\sum_{i=1}^\infty \frac{c_i}{(\lambda-\gamma_i)}v_i(x) dx$,
as $\{v_i\}_{i=1}^\infty$ is an orthonormal basis
for $L^2(U)$, to give
\[
\epsilon=
\Big(\sum_{i=1}^\infty \frac{c_i d_i}{(\lambda-\gamma_i)}\Big)^{-1}.
\]
\end{proof}

Note that as observed in \cite{F+V},
\begin{equation}\label{fixed_results.eqn}
\begin{aligned}
\lambda_i(\epsilon)\equiv \gamma_i
&\; \Rightarrow\; \lambda_i'(0)=0 \\
&\;\Leftrightarrow \; \int_U c(x) v_i(x) dx \int_U d(x) v_i(x) dx =0 \\
&\;\Leftrightarrow \; \mbox{either } Bv_i\equiv 0\mbox{ or }B^*v_i\equiv 0,
\end{aligned}
\end{equation}
also
\[
B [u_i(\epsilon^*)]\equiv 0\;\mbox{for some}\;\epsilon^*\in\mathbb{R} \;\Rightarrow\;
\lambda_i(\epsilon)\equiv \gamma_i\;\mbox{and}\; u_i(\epsilon)\equiv v_i=u_i(\epsilon^*).
\]

Furthermore, as a result of Lemma \ref{ep.lem}

\begin{lemma}\label{derivative_fixed.lem}
No moving eigenvalue of $L_\epsilon$ emanates from $\gamma_j$ if and only if
\begin{equation}\label{intcd.eqn}
\int_U c(x)v(x)dx \int_U d(x)v(x)dx=0
\end{equation}
for all $v\in N(A-\gamma_j I)$.
\end{lemma}

\begin{proof}
If no moving eigenvalue of $L_{\epsilon}$ emanates from $\gamma_j$, then
\eqref{intcd.eqn} follows by the results listed immediately above.
Now, suppose that \eqref{intcd.eqn} holds
for all $v\in N(A-\gamma_j I)$, and assume that there exists a 
moving eigenvalue $\lambda_j(\epsilon)$ emanating from $\gamma_j$ at $\epsilon=0$,
with corresponding eigenfunction $u_j(\epsilon)$. Then by 
Lemma \ref{same_moving.lem}, there exists  an orthogonal basis
 $\{v_1,...,v_n\}$ of
$N(A-\gamma_j I)$, such that
\begin{equation}\label{null_gamma.eqn}
\mbox{Span}\{v_1,...,v_{n-1}\}= 
N(L_{\epsilon}-\gamma_j I),\quad\forall\epsilon\neq 0.
\end{equation}
Moreover, by Lemma \ref{basic.lem}(c), 
\[
\lim_{\epsilon\rightarrow 0} u_j(\epsilon)
=v_n.
\]
 Therefore putting $u=u_j(\epsilon)$ in \eqref{beta.eqn} gives $\beta_n\neq 0$ for
$\epsilon$ sufficiently small.
Hence, from \eqref{u_exp.eq}, $c_n\neq 0$, and noting that
\eqref{intcd.eqn} is equivalent to $c_id_i=0$ for $i=1,...,n$,
we have $d_n=0$. However, again from above 
$d_n=0\Rightarrow v_n\in N(L_{\epsilon}-\gamma_j I)\;\forall\epsilon\in\mathbb{R}$,
but this contradicts \eqref{null_gamma.eqn}.
\end{proof}

In \cite{Freitas} an expression for $\lambda_j'(0)$ was obtained
for the case $n=1$. In \cite{D+D} an expression was obtained for
$\lambda_j'(\epsilon)$, $\forall\epsilon\in\mathbb{R}$, for the case $n=1$
when $L_\epsilon$ is self-adjoint. The following theorem extends
these two results in the case $n=1$, and furthermore holds for general $n$.


\begin{theorem}\label{deriv.thm}
Let $\lambda_j(\epsilon)$ be a moving eigenvalue of $L_\epsilon$
of algebraic multiplicity 1, and let
$u_j(\epsilon)$ and $u_j^*(\epsilon)$ be eigenfunctions of $L_\epsilon$ and $L_\epsilon^*$
respectively, corresponding to $\lambda_j(\epsilon)$.
Then
\begin{equation}\label{deriv_exp.eqn}
\lambda_j'(\epsilon)=\frac{\int_U d(x)[u_j(\epsilon)](x)dx \int_U c(x)[u^*_j(\epsilon)](x)dx}
{\int_U [u_j(\epsilon)](x) [u^*_j(\epsilon)](x) dx}.
\end{equation}
\end{theorem}

\begin{proof}
First suppose that $\lambda_j(\epsilon)\neq\gamma_i\;\forall \: i\in \mathbb{N}$.
Then by Lemma \ref{ep.lem}
$[u_j(\epsilon)](x):=\sum_{i=1}^\infty \frac{c_i}{\lambda_j(\epsilon)-\gamma_i} v_i(x)$
 and
$[u^*_j(\epsilon)](x):=\sum_{i=1}^\infty \frac{d_i}{\lambda_j(\epsilon)-\gamma_i} v_i(x)$
are eigenfunctions of $L_\epsilon$ and
$L_\epsilon^*$ respectively  corresponding to the eigenvalue
 $\lambda_j(\epsilon)$.
Also by Lemma \ref{ep.lem},
\begin{equation}\label{epsilon.eqn}
\epsilon(\lambda_j)=\Big(\sum_{i=1}^\infty \frac{c_i d_i }{\lambda_j(\epsilon)-\gamma_i}
\Big)^{-1}.
\end{equation}
Now as $\lambda_j(\epsilon)\neq\gamma_i$ for any $i\in\mathbb{N}$, it is straightforward
to show that
$\sum_{i=1}^\infty \frac{c_id_i}{\lambda_j(\epsilon)-\gamma_i}$ is
uniformly convergent in a suitable neighbourhood of $\epsilon$.
Hence, we can differentiate the series in \eqref{epsilon.eqn}
term by term and use the chain rule
to obtain
\[
\frac{d\epsilon}{d\lambda}=\Big(\sum_{i=1}^\infty
\frac{c_i d_i }{\lambda_j (\epsilon)-\gamma_i}
\Big)^{-2}\Big(\sum_{i=1}^\infty \frac{c_i d_i }
{(\lambda_j(\epsilon)-\gamma_i)^2}\Big),
\]
and so
\begin{equation}\label{deriv_exp2.eqn}
\frac{d\lambda}{d\epsilon}=\Big(\sum_{i=1}^\infty \frac{c_i d_i }
{\lambda_j(\epsilon)-\gamma_i}
\Big)^{2}\Big(\sum_{i=1}^\infty \frac{c_i d_i }
{(\lambda_j(\epsilon)-\gamma_i)^2}\Big)^{-1}.
\end{equation}
If we substitute
the series expansions for
$c$, $d$, $u(\epsilon)$ and $u^*(\epsilon)$ into \eqref{deriv_exp.eqn},
and
interchange the order of summation and integration,
then the equivalence of
 \eqref{deriv_exp.eqn} and \eqref{deriv_exp2.eqn} is proven.

For $\lambda_j(\epsilon)=\gamma_j$ (i.e. when $\epsilon=0$), the result
follows by the analyticity of the eigenvalues and eigenfunctions, as
discussed in Lemma
\ref{basic.lem}.
\end{proof}

Hence, as in the case $n=1$, we have

\begin{corollary}\label{sa.cor}
If $L_\epsilon$ is self-adjoint, and if $\lambda_j(\epsilon)$ is an
eigenvalue of $L_\epsilon$ of geometric multiplicity 1, then
\[
\lambda_j'(\epsilon)=\frac{\Big(\int_U c(x) [u_j(\epsilon)](x) dx\Big)^2}
{\int_U [u_j(\epsilon)](x)^2 dx}.
\]
\end{corollary}

When deriving an expression for $\lambda_j'(0)$, we must deal with
the case where an eigenvalue of $A$ ``splits" to form a moving (simple)
eigenvalue and a fixed eigenvalue of $L_\epsilon$.

\begin{theorem}\label{pde_derivative_zero.thm}
Suppose that there exists an eigenvalue of $L_\epsilon$,
$\lambda_j(\epsilon)$ for which $\lambda_j(0)=\gamma_j$, but
$\lambda_j(\epsilon) \neq \gamma_j$ for $\epsilon\neq 0$.
If $\tilde{v}:=\lim_{\epsilon\to 0} u_j(\epsilon)$,
then
\[
\lambda_j'(0)=\frac{\int_U c(x)\tilde v (x) dx\int_U d(x)\tilde v (x) dx}
{\int_U \tilde v(x)^2 dx}.
\]
If in addition, $L_\epsilon$ is self-adjoint, then
\[
\lambda_j'(0)=\max_{u\in N(A-\gamma_j I)} \frac{\big(\int_Uc(x)u(x)dx\big)
^2}{\int_U u(x)^2 dx}.
\]
\end{theorem}

\begin{proof}
Even in the case where the moving eigenvalue $\lambda_j(\epsilon)$ intersects
a fixed eigenvalue at $\gamma_j$, by Lemma \ref{basic.lem}
$\lambda_j(\epsilon)$ is analytic at $\epsilon=0$, and hence as a consequence of
Theorem \ref{deriv.thm} and the analyticity of the eigenfunctions,
we have
\[
\lambda_j'(0)=\frac{\int_U c(x)\tilde{v}(x) dx\int_U d(x)\tilde{v}(x) dx}
{\int_U\tilde{v}(x)^2dx}.
\]
If $L_\epsilon$ is self-adjoint, $\tilde{v}$ will be
perpendicular to the fixed eigenspace $X$, corresponding
to $\gamma_j$,
whilst for any $v\in X$,
$\int_U c(x) v(x)dx=0$. Therefore it
follows that
\[
\max_{u\in N(A-\gamma_j I)} \frac{\big(\int_U c(x)u(x) dx\big)^2}{\int_U u(x)^2 dx}=
\frac{\big(\int_U c(x)\tilde{v}(x) dx\big)^2}
{\int_U\tilde{v}(x)^2dx}=\lambda_j'(0).
\]
\end{proof}
Note that the above result is consistent with Corollary 
\ref{derivative_fixed.lem} proved earlier.

Whilst
it is possible in general for all of the eigenvalues of $L_\epsilon$
to be fixed,
such behaviour is not possible if $L_\epsilon$ is self-adjoint
as illustrated by the following result.

\begin{theorem}\label{movingev.thm}
If $L_\epsilon$ is self-adjoint,
 then at least one of the eigenvalues of $L_\epsilon$
is not fixed.
\end{theorem}

\begin{proof}
The set of eigenfunctions of $A$,
$\{v_i\}_{i=1}^\infty$ forms a
basis for $L^2(U)$, and hence
$\exists\, v_j\in \{v_i\}_{i=1}^\infty$ such that
$\int_U c(x)v_j(x)dx=\int_U d(x)v_j(x)dx \neq 0$.
Then, by Corollary \ref{derivative_fixed.lem}, there exists a moving
eigenvalue emanating from $\gamma_j$.
\end{proof}

In the case where $\sigma(L_\epsilon)=\sigma(A)$ for all $\epsilon\in\mathbb{R}$, varying $\epsilon$
affects  the corresponding eigenfunctions as is now shown.

\begin{theorem}\label{movingefn.thm}
If $\sigma(L_\epsilon)=\sigma(A)$ for all $\epsilon\in\mathbb{R}$, then at least one of the
eigenfunctions of $L_\epsilon$ is not fixed.
\end{theorem}

\begin{proof}
Suppose that $\lambda_i(\epsilon) \equiv \gamma_i$, $\forall i\in\mathbb{N}$.
Then, similar to the proof of Theorem~\ref{movingev.thm}, there
is at least one eigenfunction of $A$, $v_j$ say, such that $Bv_j\not\equiv 0$.
But by  assumption $\lambda_j(\epsilon) \equiv \gamma_j$ and so it follows from
the equation
\[
Au_j(\epsilon)+\epsilon Bu_j(\epsilon)=\lambda_j(\epsilon) u_j(\epsilon),
\]
that $u_j(\epsilon)\not\equiv v_j$.
\end{proof}

As noted in Lemma \ref{basic.lem},
the eigenvalues of $L_\epsilon$ are continuous functions
of $\epsilon$. We shall prove that the eigenvalues
of $L_\epsilon$ are also \emph{equicontinuous} functions of $\epsilon$.
Equicontinuity is defined as follows.

\begin{definition}\label{uects.dfn} \rm
Let $X$ be a normed vector space, and let $(a,b)$ be a (possibly unbounded)
interval of the real line.
A sequence of functions, $f_n: (a,b)\to X$ $n=1,2,\dots $ is
\emph{uniformly equicontinuous} if for each $\epsilon >0$, there
exists $\delta >0$, such that
\[
|x-y|<\delta \;\Rightarrow\;
\|f_n(x)-f_n(y)\|<\epsilon,\;\forall x,y \in (a, b),\;\forall n\in\mathbb{N}.
\]
\end{definition}

We shall prove the equicontinuity of the eigenfunctions
with the aid of the following two lemmas.

\begin{lemma}\label{basis.thm}
If $L_\epsilon$ is self-adjoint, then for each $\epsilon\in\mathbb{R}$,
the eigenfunctions of $L_\epsilon$ can
be chosen to form an orthonormal basis for $L^2(U)$.
\end{lemma}

\begin{proof}
Fix $\epsilon$ and assume without loss of generality
 that $0$ is not an eigenvalue
of $L_\epsilon$. (If $0$ is an eigenvalue then simply consider the
operator $L_\epsilon + KI$ for some constant $K$ suitably chosen.) The only spectral values of $L_\epsilon$ are
eigenvalues, therefore $L_\epsilon^{-1}:L^2(U)\to L^2(U)$ is bounded.
Furthermore,
$L_\epsilon = L_\epsilon^*$ implies
$L_\epsilon^{-1}= (L_\epsilon^{-1})^*$, and as noted above, $L_\epsilon^{-1}$
is compact.
Also, $0$ is not an eigenvalue of $L_\epsilon^{-1}$. Hence
\[
\ker L_\epsilon^{-1}=\{0\}.
\]
Applying Corollary 6.35  in \cite{R+Y} and using the equivalence of
the eigenfunctions of $L_\epsilon$ and $L_\epsilon^{-1}$ concludes the proof.
\end{proof}

Now, define $\rho (L_\epsilon)$ to be the resolvant set of $L_\epsilon$,
i.e. $\rho (L_\epsilon):=\mathbb{C} \backslash \sigma (L_\epsilon)$. Then we have:

\begin{lemma}\label{rhombus2.lem}
Fix $\epsilon^*\in\mathbb{R}$. If $L_\epsilon$ is self-adjoint, and if $(\lambda, \epsilon)\in\mathbb{R}^2$
satisfies
\begin{equation}\label{rhombus2.eqn}
|\epsilon-\epsilon^*|\|B\|<\min_{i\in\mathbb{N}}|\lambda_i(\epsilon^*)-\lambda|,
\end{equation}
then $\lambda\in \rho(L_{\epsilon})$.
\end{lemma}

\begin{proof}
As noted in Section II-5.1 of \cite{Kato}, for any fixed $\epsilon^*$,
if $\lambda\in\rho(L_{\epsilon^*})$,
and if
\[
|\epsilon-\epsilon^*|\|B\|<\|(L_{\epsilon^*}-\lambda I)^{-1}\|^{-1},
\]
then $\lambda\in \rho(L_{\epsilon})$. As $L_\epsilon$ is self-adjoint, by
Lemma \ref{basis.thm}
the eigenfunctions of $L_\epsilon$ form an orthonormal basis for
$L^2(U)$. Hence, it is straightforward to show that
\[
\|(L_{\epsilon^*}-\lambda I)^{-1}\|=
\frac{1}{\min_{i\in\mathbb{N}}|\lambda_i(\epsilon^*)-\lambda |}.
\]
Therefore if
\[
|\epsilon-\epsilon^*|\|B\|<\min_{i\in\mathbb{N}}|\lambda_i(\epsilon^*)-\lambda |,
\]
then $\lambda\in \rho(L_{\epsilon})$.
\end{proof}


\begin{theorem}\label{uects_evs.thm}
If $L_\epsilon$ is self-adjoint, then the set of eigenvalues,
$\Sigma=\{\lambda_i(\epsilon)\}_{i=1}^\infty$, is uniformly
equicontinuous for $\epsilon\in\mathbb{R}$.
\end{theorem}

\begin{proof}
We first make some observations about the eigenvalues of $L_\epsilon$
in the self-adjoint case.
As a consequence of \eqref{deriv_exp2.eqn}, the moving eigenvalues of
$L_\epsilon$ all increase with respect to $\epsilon$. Then as noted in the final
sentence of Section \ref{pre}, as the eigenvalues of a
self-adjoint operator are real,
2 moving eigenvalues of $L_\epsilon$ never intersect.
Fixed eigenvalues are clearly
analytic $\forall\epsilon\in\mathbb{R}$.
In the case where a moving eigenvalue, $\lambda_i(\epsilon)$
intersects a fixed eigenvalue, $\lambda_j(\epsilon)\equiv\gamma_j$, of
$L_\epsilon$, $\lambda_i(\epsilon)+\gamma_j$ is analytic at the point of
intersection by Lemma \ref{basic.lem}(c), and hence by
Lemma \ref{basic.lem}(b), a moving eigenvalue $\lambda_i(\epsilon)$
is also analytic $\forall\epsilon\in\mathbb{R}$.

Now, consider an eigenvalue $\lambda_j(\epsilon)$, and take $\epsilon_1$, $\epsilon_2\in\mathbb{R}$.
As $\lambda_j(\epsilon_2)\notin \rho(L_{\epsilon_2})$, it follows from
Lemma \ref{rhombus2.lem} that
\begin{equation}\label{ects1.eqn}
|\epsilon_1-\epsilon_2|\|B\|\geq \min_{i\in\mathbb{N}}|\lambda_i(\epsilon_1)-\lambda_j(\epsilon_2)|.
\end{equation}
First suppose that $\lambda_j(\epsilon_2)$
is not a point of intersection of a moving eigenvalue and a fixed
eigenvalue of $L_\epsilon$. Then for
$|\epsilon_1-\epsilon_2|$ sufficiently small,
\[
\min_{i\in\mathbb{N}}|\lambda_i(\epsilon_1)-\lambda_j(\epsilon_2)|=
|\lambda_j(\epsilon_1)-\lambda_j(\epsilon_2)|.
\]
Hence from \eqref{ects1.eqn},
\[
\frac{|\lambda_j(\epsilon_1)-\lambda_j(\epsilon_2)|}{|\epsilon_1-\epsilon_2|}\leq\|B\|.
\]
This holds for any $j\in\mathbb{N},\;\forall \epsilon_1,\epsilon_2\in\mathbb{R}$,
such that $|\epsilon_1-\epsilon_2|$ is sufficiently small,
whenever $\lambda_j(\epsilon_2)\neq\gamma_i\;\forall i\in\mathbb{N}$.
Therefore
\[
|\lambda_j'(\epsilon)|\leq \|B\|\quad\forall j\in\mathbb{N},\;\forall\epsilon\in\mathbb{R},
\]
whenever $\lambda_j(\epsilon)\neq\gamma_i\;\forall i\in\mathbb{N}$.
But then since $\lambda_j(\epsilon)$ is analytic for all $\epsilon\in\mathbb{R}$,
it is certainly continuously differentiable, and therefore it follows that
the previous inequality also holds for $\lambda_j(\epsilon)=\gamma_i$ for some
$i\in\mathbb{N}$.
i.e.
\[
|\lambda_j'(\epsilon)|\leq \|B\|\quad\forall j\in\mathbb{N},\;\forall\epsilon\in\mathbb{R},
\]
and the result follows.
\end{proof}

\section{Algebraic and Geometric Multiplicity}

The geometric and algebraic multiplicities of the eigenvalues are of
importance in establishing conditions for results on nodal properties
of eigenfunctions, and for bifurcation in associated non-linear
problems, respectively.
Hence, we now consider whether the multiplicities of the
eigenvalues $\lambda_i(\epsilon)$ change as the parameter $\epsilon$ is varied.
Geometric multiplicity of an eigenvalue $\lambda$ of $L_\epsilon$,
can be defined in the usual way, i.e. $\dim (N(L_\epsilon-\lambda I))$.
Since $L_\epsilon$ is a closed
linear operator with compact resolvent, the algebraic
multiplicity of an eigenvalue, $\lambda$ of $L_\epsilon$ can be
defined to be
the algebraic multiplicity of the eigenvalue, $1/\lambda$ of
$L_\epsilon^{-1}$.
(Here we are assuming
without loss of generality
 that $L_\epsilon^{-1}$  does exist; if $L_\epsilon$ is not invertible,
then consider $L_\epsilon+K I$ for an appropriate constant, $K$.)
A \textit{simple eigenvalue} is defined to be an
eigenvalue of algebraic multiplicity 1, (see e.g. \cite{A+P}).
Note that since
$A$ is self-adjoint its eigenvalues  will
have equal algebraic and geometric multiplicities.


\subsection{Algebraic Multiplicity} \label{algmult}

As was noted earlier,
the algebraic multiplicity
of an eigenvalue of $A$, although finite,
can be arbitrarily large.
However,
the following theorem can be deduced from Section IV-3.5 of \cite{Kato}.

\begin{theorem}\label{algmult.thm}
Let $S \subset \Sigma$ contain only a finite number of elements. If
$\lambda_i(\epsilon) \neq \lambda_j(\epsilon)$ for any
$\lambda_i(\epsilon) \in S$ and $\lambda_j(\epsilon) \in
\Sigma\backslash S$
  $\forall\epsilon\in
(\epsilon_1, \epsilon_2)$,
then the sum of the algebraic
multiplicities of the eigenvalues in $S$ is constant with respect to $\epsilon$,
$\forall\epsilon\in(\epsilon_1,\epsilon_2)$.
\end{theorem}

\begin{remark} \label{rmk4.2} \rm
As a consequence of the above theorem and Lemma \ref{same_moving.lem},
a moving eigenvalue $\lambda_i(\epsilon)$ of $L_\epsilon$ is simple
for $0<|\epsilon|<\hat\epsilon$, where $\hat\epsilon=\min_{\epsilon\neq 0}\{\epsilon |
\lambda_i(\epsilon)=\lambda_j(\epsilon)\;\mbox{for}\;i\neq j\}$.
\end{remark}

\subsection{Geometric Multiplicity}

In the self-adjoint case, since it is well-known
that eigenvalues of self-adjoint operators
 have equal geometric and algebraic multiplicities, a direct consequence
of Theorem
\ref{algmult.thm} is

\begin{theorem} \label{thm4.3}
Suppose that $L_\epsilon$ is self-adjoint. If $\lambda_i(\epsilon)$ is a moving
eigenvalue of $L_\epsilon$, $\gamma_j$ is a fixed eigenvalue of $L_\epsilon$, and if
$\lambda_i(\epsilon^*)=\gamma_j$
then $\dim(N(L_{\epsilon^*}-\gamma_j I))\geq 2$.
\end{theorem}

However it is possible for geometric multiplicity to not be
preserved in the general case, even where no eigenvalues of $L_\epsilon$
intersect,
as shown by the following simple example.

\begin{exa}\label{1.exa}
Let $A$ and $\gamma$ be such that $\gamma$ is an eigenvalue of $A$
of multiplicity
2, and let $v_1$ and $v_2$ be orthogonal eigenfunctions corresponding
to $\gamma$. Let $c=v_1$ and $d=v_2$. Then, using Corollary
\emph{\ref{derivative_fixed.lem}}, it can be shown
that all of the eigenvalues of $L_\epsilon$ are fixed and
$N(L_0-\gamma I)=N(A-\gamma I)=\mbox{Span}\{v_1, v_2\}$,
whilst for any $\epsilon\neq 0$,
$N(L_\epsilon -\gamma I)=\mbox{Span}\{v_1\}$.
\end{exa}

In the remainder of this section we present results that restrict
the behaviour of the geometric multiplicities in the general
case.

\begin{theorem} \label{thm4.5}
An eigenvalue $\lambda_j(\epsilon)$ has geometric multiplicity 1 provided
$\lambda_j(\epsilon)\neq \gamma_i$  for any $i\in\mathbb{N} $.
\end{theorem}

\begin{proof}
Suppose that for some $\epsilon^*$ and
some $j$,
$N(L_{\epsilon^*}-\lambda_j(\epsilon^*))= \mbox{span}\{u,\,v\}$ with $u$ and
$v$ linearly independent where $\lambda_j(\epsilon^*)\neq \gamma_i$  for any
$i\in\mathbb{N}$.
Then, there
exist constants $a$ and $b$ with $|a|+|b| \neq 0$ such that
$B(au+bv)\equiv 0$.
Hence, $au+bv = v_i$  and  $\lambda_j(\epsilon^*) = \gamma_i$ for some $i\in\mathbb{N}$,
which is a contradiction and so the result is proved.
\end{proof}

Then since $\sigma (A)\subset \mathbb{R}$, we have

\begin{corollary} \label{coro4.6}
Any complex eigenvalue of $L_\epsilon$ has geometric multiplicity 1.
\end{corollary}


The following theorem is an extension of \cite[Theorem 4.8]{D+D}.
The proof of  \cite[Theorem 4.8]{D+D} uses results concerning a corresponding
initial value problem.
Such results are not available
here, so we require an alternative
method, which is similar to that used in the proof of a different result
(Lemma 3.9) in
\cite{Freitas}.

\begin{theorem}\label{geomult3.thm}
Suppose that  $\gamma_j$ is an eigenvalue of $A$ of geometric
multiplicity $m$. Then,
there exists at most one value, $\epsilon_j$, such that
$\mathrm{dim}(N(L_{\epsilon_j}-\gamma_j I)) =m+1$.
\end{theorem}

\begin{note} \rm
Let $S: \hat X\to X$ be a closed operator, where
$\hat X$ is a dense subset of a Banach space, $X$, and consider
$T: \hat X\to X$.
Then $T$ is said to be \emph{relatively degenerate with respect to $S$}
if and only if (i) $\exists\; k_1, k_2 \geq 0$, such that
\[
\|Tu\|\leq k_1 \|u\|+k_2 \|Su\|\quad \forall u\in \hat X,
\]
and (ii) $R(T)$ is finite dimensional.
\end{note}


\begin{proof}[Proof of Theorem \ref{geomult3.thm}]
This proof will use the Weinstein-Aronszajn (W-A) determinant
and the W-A formula for relatively
 degenerate perturbations, details of which can be
found in \cite[Section IV-6.1]{Kato}.

Note that $\epsilon B$ is a relatively degenerate perturbation
with respect to $(A-\lambda I)$, since it is bounded
and its range is finite dimensional.
In the same way as
shown in \cite{Freitas}, the W-A determinant, $\omega$, associated
with $\epsilon B$ and $(A-\lambda I)$ has the form
\begin{equation}\label{omega.eqn}
\omega(\lambda,\epsilon)=1+\epsilon G(\lambda),
\end{equation}
for some function $G$, not dependent upon $\epsilon$.
The W-A formula for degenerate perturbations gives
\[
M(\lambda, L_\epsilon)=M(\lambda, A)+N(\lambda, \omega),
\]
where for $T\in\{A,L_\epsilon\}$
\[
M(\lambda, T)=\begin{cases}
0 & \mbox{if }\lambda\in\rho(T) \\
\mbox{the algebraic multiplicity of } \lambda & \mbox{if }\lambda\in
\sigma(T),
\end{cases}
\]
and
\[
N(\lambda, \omega)=\begin{cases}
k & \mbox{if $\lambda$ is a zero of $\omega$ of order $k$}\\
-k & \mbox{if $\lambda$ is a pole of $\omega$ of order $k$}\\
0 & \mbox{otherwise}.
\end{cases}
\]
Hence, it follows that for
an eigenvalue $\lambda$ of $A$,
of multiplicity $m$, to be an eigenvalue of multiplicity $m+1$ of
$L_\epsilon$ requires  $N(\lambda,\omega)>0$ and this is true only
when $\omega(\lambda,\epsilon)$
has a zero. However, from the form of
\eqref{omega.eqn}, it can be seen that for any fixed $\lambda$,
$\omega(\lambda,\epsilon)$ equals $0$ for at most
one value of $\epsilon$.
\end{proof}

The case where a moving eigenvalue and a fixed eigenvalue emanate from the
same point $\gamma_i$, can be thought of as a moving eigenvalue
intersecting $\gamma_i$ at $\epsilon=0$. 
%Note that 
In this case,  by Lemma  \ref{same_moving.lem}
there exists an orthonormal basis, $\{w_1,\dots ,w_m\}$, for 
$N(A-\gamma_i I)$
 such that Span$\{w_1,\dots ,w_{m-1}\}\subset N(L_\epsilon-\gamma_i I)\;
\forall\epsilon\in\mathbb{R}$, and $w_m=\lim_{\epsilon\to 0} u_m(\epsilon)$, where
$u_m(\epsilon)$ is the eigenfunction corresponding to the moving eigenvalue.
Hence $\dim N(A-\gamma_i I)=1+\dim N(L_\epsilon-\gamma_i I)$, for any $\epsilon\neq 0$.

We now consider when a moving eigenvalue, $\lambda_j(\epsilon)$,
intersects a fixed eigenvalue, $\lambda_i(\epsilon)\equiv \gamma_i$,
of $L_\epsilon$ at $\epsilon\neq 0$.

\begin{theorem} \label{thm4.9}
Let $\lambda_i(\epsilon)\equiv\gamma_i$ be a fixed eigenvalue of $L_\epsilon$, such that no moving
eigenvalue of $L_\epsilon$ emanates from $\gamma_i$ at $\epsilon=0$.

\begin{itemize}

\item[(a)]
If either $Bv\not\equiv 0$ or $B^* v\not\equiv 0,$
for any $v\in N(A-\gamma_i I)$,
 then $\dim (N(L_\epsilon-\gamma_i I)) \leq
\dim (N(A-\gamma_i I))$ $\forall\epsilon\in\mathbb{R}$.
\item[(b)]
If $Bv= B^* v\equiv 0$, $\forall v\in N(A-\gamma_i I)$,
 and if a moving eigenvalue $\lambda_j(\epsilon_i) = \gamma_i$ for some
$\epsilon_i\in\mathbb{R}$, $\epsilon_i\neq 0$, then
$\dim (N(L_{\epsilon_i}-\gamma_i I)) =\dim (N(A-\gamma_i I))+1$.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk4.10} \rm
Whilst part (a) of the above result gives that
 $\dim (N(L_\epsilon-\gamma_i I)) \not >
\dim (N(A-\gamma_i I))$ $\forall\epsilon\in\mathbb{R}$,
we may have $\dim (N(L_\epsilon-\gamma_i I)) <
\dim (N(A-\gamma_i I))$ for some $\epsilon\neq 0$,
as in Example \ref{1.exa}.
\end{remark}

\begin{proof}
\textbf{(a)} If $\exists\; v\in N(A-\gamma_i I)$ such that $Bv\not\equiv
0$, then by Lemma \ref{same_moving.lem},
there exists an   orthonormal basis, $\{v_1, v_2,\dots  v_m\}$ for
$N(A-\gamma_i I)$, such that $B(v_1)=B(v_2)=\dots =B(v_{m-1})\equiv 0$,
whilst $B(v_m)\not\equiv 0$. Now suppose that
$\dim (N(L_{\epsilon_i}-\gamma_i I))=m+1$ for some $\epsilon_i\neq 0$,
i.e. suppose that there exists an orthonormal
basis of $N(L_{\epsilon_i}-\gamma_i I)$ of the form $\{v_1, v_2,\dots  v_{m-1}
,z_1, z_2\}$. Define the functions
\[
w_n:=\Big(\int_U d(x)v_m(x)dx\Big)
\; z_n+\Big(\int_U d(x)z_n(x)dx\Big)\;v_m,\quad  n=1,2.
\]
Then $\{v_1, v_2,\dots  v_{m-1}, w_1, w_2\}$ are linearly independent
and it can be shown that $w_1,\, w_2 \in N(L_{\epsilon_i/2}-\gamma_i I)$, i.e.
$\dim(N(L_{\epsilon_i/2}-\gamma_i I))=m+1$. But this
contradicts Theorem~\ref{geomult3.thm}.

If $B^* v\not\equiv 0$, for some $v\in N(A-\gamma_i I)$,
then by considering  the adjoint problem,
$(L_{\epsilon}^* -\lambda I)u=0$, and
using  the equivalence of the spectrums and multiplicities
 of $L_\epsilon$ and $L_\epsilon^*$, a similar
contradiction can be obtained and result (a) is proven.

\noindent
\textbf{(b)}
Since by assumption $Bv\equiv 0$, $N(A-\gamma_i I)\subseteq
N(L_\epsilon-\gamma_i I)$ $\forall\epsilon\in\mathbb{R}$.
As in the proof of Theorem 4.10 in \cite{D+D} it can be shown that
\[
u(x):=\sum_{j\neq i}\frac{c_i}{(\gamma_i-\gamma_j)}v_j(x)
\]
is in $N(L_{\epsilon_i}-\gamma_i I)$, where in the above summation,
the eigenvalues are repeated according to their geometric multiplicity.
The function $u$ is orthogonal to
$N(A-\gamma_i I)$, and hence the result is proven.
\end{proof}

\section{Existence and Uniqueness of Principal Eigenvalues}

As mentioned earlier, for differential operators of the type given in
 \eqref{Lep},
it is known that the eigenfunction corresponding to the first
eigenvalue of $A$ is unique and is of one sign on $U$, but in
general less is known
about the nodal properties of the eigenfunctions corresponding to the other
eigenvalues. By the continuity of the (1-dimensional) eigenprojection
corresponding to $\lambda_1(\epsilon)$, it follows that for $\epsilon$
sufficiently small, the eigenfunction $u_1(\epsilon)$, corresponding
to $\lambda_1(\epsilon)$
is of one sign on $U$, i.e.
$L_\epsilon$ has a principal eigenvalue for sufficiently small $\epsilon$.
In this section we are concerned with proving the
existence and in some cases the uniqueness of a principal eigenvalue
of $L_\epsilon$, without restriction on $|\epsilon|$.

\begin{theorem}
Suppose that either $c\geq 0$ on $U$ or $c\leq 0$ on $U$.
Then,
$L_\epsilon$ has a principal eigenvalue
$\lambda_p(\epsilon)\geq\gamma_1$ (i.e. an eigenvalue to which there
corresponds
a positive eigenfunction) either $\forall \epsilon \geq 0$ or $\forall \epsilon
\leq 0$.
\end{theorem}

\begin{proof}
Following similar methods to those used in the proof of
Theorem 5.5 in \cite{D+D}, it can be shown that
$L_\epsilon$ has an eigenvalue greater than or
equal to $\gamma_1$, either $\forall \epsilon \geq 0$ or $\forall \epsilon\leq 0$.

We now show that any eigenvalue, $\lambda$ of $L_\epsilon$, satisfying $\lambda\geq\gamma_1$ has
a corresponding eigenfunction which is positive on $U$.
If $\lambda > \gamma_1$, then
$(A-\lambda I)$ satisfies the strong maximum principle (see for
example Theorem 2.4 in \cite{A+LG}). Let $u(x)$ be
an eigenfunction corresponding to $\lambda > \gamma_1$. Then
we note that
$\int_U d(x)u(x) dx\neq 0$, as there exist no fixed eigenvalues
greater than $\gamma_1$. Hence, supposing without loss of generality
that $-\epsilon c(x) \int_U d(x) u(x) dx$ is a non-negative function which is
not identical to zero, applying the strong maximum principle yields
\[
u(x)=(A-\lambda I)^{-1}\Big[-\epsilon c(x) \int_U d(x) u(x) dx\Big] >0\quad 
\forall x\in U.
\]

We are left to consider
the case $\lambda_i(\epsilon)=\gamma_1$ for some $i\in\mathbb{N}$. If $\epsilon=0$ is the only
solution to $\lambda_i(\epsilon)=\gamma_1$, then as the corresponding
eigenfunction, $v_1$ has no interior zeros, and the result follows directly.
If $\lambda_i(\epsilon)=\gamma_1$ for some $i\in\mathbb{N}$ and some $\epsilon\neq 0$,
then by the arguments at the end of Section 2, $\lambda_1(\epsilon)\equiv\gamma_1$.
A corresponding eigenfunction does not change sign
as the following argument shows.
The function
 $c$ is either non-negative or non-positive, therefore $B^*(v_1)\not\equiv 0$,
and hence as $\lambda_1(\epsilon)\equiv\gamma_1$, $B(v_1)\equiv 0$.
Therefore $v_1$ is an eigenfunction corresponding to $\gamma_1$, $\forall
\epsilon\in\mathbb{R}$.
and the result is proved.
\end{proof}


Now, having established the existence of a principal eigenvalue of $L_\epsilon$
under certain hypotheses,
we finish this section by considering the uniqueness  of the principal eigenvalue.


\begin{theorem}\label{unique_pev_pde.thm}
If $L_\epsilon$ is self-adjoint, if $c$ is strictly of one sign on $U$ and if
$\epsilon >0$, then the eigenfunction corresponding
to $\lambda_1(\epsilon)$ is the only eigenfunction of $L_\epsilon$
with no interior zeros.
\end{theorem}


\begin{proof}
The case $\epsilon=0$ corresponds to the well studied local problem.
Now consider $\epsilon > 0$ and suppose that either $u_1(\epsilon)$
has an interior zero, or $u_k(\epsilon)$ has no interior zeros for $k\neq 1$.
As a consequence of Lemma \ref{basic.lem}, the eigenfunctions of $L_\epsilon$ can be
chosen to be continuous functions of $\epsilon$.
Hence $\exists \hat\epsilon > 0$, and $u_k\in C^2(\bar U)$ such that
$[u_k(\hat\epsilon)] (x)\geq 0\;\forall x\in \bar U$, whilst $\exists \hat x\in
\bar U $
such that $[u_k(\hat\epsilon)](\hat x)= [u_k(\hat\epsilon)]_{x_1}(\hat x)=
[u_k(\hat\epsilon)]_{x_2}(\hat x)=\dots = [u_k(\hat\epsilon)]_{x_n}(\hat x)=0$.
Then,
 it follows from standard properties of uniformly elliptic operators that
\[
\sum_{i,j=1}^n [a_{ij}(\hat x) u_{k\;x_i}(\hat x)]_{x_j} \geq 0,
\]
and also $(b(\hat x)-\lambda)u_k(\hat x)=0$, for any $\lambda\in \mathbb{R}$.
Note that by an appropriate Sobolev embedding, $c\in C(\bar U)$,
and then, $a_{ij}, u_k\in C^2(\bar U)$ and $b,c\in C(\bar U)$ implies that
the differential equation
\[
\sum_{i,j=1}^n [a_{ij}(\hat x) u_{k\;x_i}(\hat x)]_{x_j}
+(b(x)-\lambda)u_k(x) +\epsilon c(x)\int_U c(x)u_k(x) dx=0,
\]
holds not only for $x\in U$, but also for $x\in \bar U$.
Hence,
\[
\epsilon c(\hat x) \int_U c(x) u_k(x) dx \leq 0,
\]
which contradicts our assumptions on the
sign of $\epsilon$.
Hence the result is proven.
\end{proof}


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\end{document}