\documentclass[twoside]{article}
\usepackage{amsmath,amsthm,amssymb,epsf} 
\pagestyle{myheadings}
\markboth{\hfil Nonclassical Sturm-Liouville problems \hfil EJDE--2000/71}
{EJDE--2000/71\hfil Robert Carlson  \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.~{\bf 2000}(2000), No.~71, pp.~1--24. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
 \vspace{\bigskipamount} \\
 %
 Nonclassical Sturm-Liouville problems and 
 Schr\"odinger operators on radial trees 
 %
\thanks{ {\em Mathematics Subject Classifications:} 34B10, 47E05.
\hfil\break\indent
{\em Key words:} Schr\"odinger operators on graphs, 
graph spectral theory, \hfil\break\indent
boundary-value problems, interior point conditions.
\hfil\break\indent
\copyright 2000 Southwest Texas State University. \hfil\break\indent
Submitted June 23, 2000. Published November 28, 2000.} }
\date{}
%
\author{ Robert Carlson }
\maketitle

\begin{abstract} 
Schr\"odinger operators on graphs with weighted edges may be defined using
possibly infinite systems of ordinary differential operators.  
This work mainly considers radial trees, whose branching and edge lengths 
depend only on the distance from the root vertex. 
The analysis of operators with radial coefficients on radial trees is reduced,
by a method analogous to separation of variables, to nonclassical 
boundary-value problems on the line with interior point conditions. 
This reduction is used to study self adjoint problems requiring boundary 
conditions `at infinity'.
\end{abstract}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}

\newcommand{\thmref}[1]{Theorem~\ref{#1}}
\newcommand{\lemref}[1]{Lemma~\ref{#1}}
\newcommand{\corref}[1]{Corollary~\ref{#1}}
\newcommand{\propref}[1]{Proposition~\ref{#1}}

\newcommand{\real}{{\mathbb{R}}}
\newcommand{\complex}{{\mathbb{C}}}
\newcommand{\Z}{{\mathbb{Z}}}


\section{Introduction}

The consideration of differential operators on graphs has old roots in physics 
and physical chemistry \cite{Exner,Montroll1,Montroll2,Pauling,Ruedenberg}.  
More recently there have been mathematical studies, 
some concerned with the interpretation of differential operators on   
graphs as limits of partial differential operators on thin domains
\cite{Evans, Friedlin, Kuchment, Rubinstein, Saito}, while others 
focus on novel problems of spectral or scattering theory 
\cite{Carlson96a,Carlson97,Carlson98a,Gerasimenko,Naimark}.  
Additional mathematical work includes
applications to nerve impluse transmission \cite{Nicaise},
and the study of evolution equations on networks \cite{vonBelow,Lumer}.
There is also a large literature where discrete problems in probability, combinatorics, and
group theory lead to difference operators on graphs \cite{Chung,Koranyi,Novikov,Romanov}.  

Given a formally self adjoint differential operator on a graph ${\cal G}$,
one of the first problems is to describe the domains for which the operator is
self adjoint on $L^2({\cal G})$.  As in the study of classical ordinary differential operators, 
the domain description will typically involve boundary conditions.
When the graph has a finite set of edges the problem of characterizing 
self adjoint domains for Schr\"odinger differential operators may be interpreted 
as a classical boundary-value problem.  The domain description is also fairly
straightforward if the graph has infinitely many edges, 
but the set of edge lengths has a positive lower bound \cite{Carlson98a}.

There is an added complication when more general infinite graphs are considered.  
The completion of the graph as a metric space may introduce additional points 
whose neighborhoods contain infinitely many vertices.  In some cases
boundary conditions at these new points are needed.

A simple example constructed using the interval $(0,1)$ will illustrate the problem.  
Suppose the vertices of the graph are $x_j = 1-2^{-j}$ for $j=0,1,2,\dots $, while  
the edges are $ (x_j,x_{j+1})$.
At $0$ impose the boundary condition $f(0) = 0$.  At other vertices impose the conditions 
\[ f(x_j^+) = f(x_j^-), \quad f'(x_j^+) = f'(x_j^-), \quad j=1,2,3,\dots .\]
By classical theory the operator $-D^2$ is symmetric, but not essentially self adjoint,
with the domain consisting of compactly supported smooth functions satisfying these 
boundary and interior point conditions.
The self adjoint extensions are determined by an additional boundary condition 
of the form $c_1f(1) + c_2f'(1) = 0$.

To shed light on the problem of domain description and other problems of operator
theory, this work provides a detailed analysis of 
Schr\"odinger operators $-D^2+q$ and the associated eigenvalue equation
\begin{equation} \label{1.a}
 -y'' + qy = \lambda y 
\end{equation}
for certain highly symmetric trees which we call radial trees (see Figure 1).  
A radial tree will be a tree whose vertex degrees 
and edge lengths are functions of the distance in the graph from the root vertex.
In addition the coefficient $q$ will be assumed to be radial.

\begin{figure}[ht]
\begin{center}
\epsfxsize=7cm
\epsffile{tree1.eps}
\end{center}
\caption{A radial tree}
\end{figure}

Since the radial trees are highly symmetric, one expects some
corresponding simplification in the description of the invariant subspaces of 
radial operators ${\cal L} = -D^2 + q$.
Roughly speaking, this simplification comes through a `separation of variables',
which provides an orthogonal sum decomposition of $L^2$ for the tree 
into invariant subspaces $U_{v,k}$ for the Schr\"odinger operator. 
For each subspace $U_{v,k}$ there is an interval ${\cal I}_n \subset \real $     
and an isometry taking $U_{v,k}$ onto a weighted Hilbert space $L^2({\cal I}_n,w_n)$ 
which carries the restriction of ${\cal L}$ to a self adjoint operator ${\cal L}_n$
which is given by $-D^2 + q$ on its domain.  Functions in the domain of ${\cal L}_n$ 
satisfy a sequence of jump conditions at interior points of the interval.

Having reduced the study of radial Schr\"odinger operators on the radial tree
to a sequence of nonclassical boundary-value problems on intervals, these interval 
problems are then analyzed.  This analysis is most detailed for trees of finite volume,  
where separated boundary conditions at the interval endpoints determine the domains 
of self adjoint operators, much as in the classical Sturm-Liouville problems.
These operators have discrete spectrum.  The eigenvalues may be identified with the roots of 
an entire function.  Growth estimates for the entire function provide information about the
distribution of eigenvalues.  

The next section of the paper provides an overview of differential operators on graphs.
Some results are established in a general setting for a graph ${\cal G}$ whose
metric space completion is compact, or has finite volume.
The third section uses subspaces of radial functions with support on subtrees of the
initial radial tree ${\cal T}$, together with discrete Fourier transforms, to carry out
the separation of variables reduction of the Schr\"odinger operators.

The fourth section uses product formulas to analyze solutions of \eqref{1.a} on intervals 
${\cal I}_n \subset \real $, subject to jump conditions coming from the graph vertices. 
When the radial tree ${\cal T}$ has finite volume the asymptotic behaviour of the solutions 
as $x$ increases has a fairly simple description.  In particular one finds generalized 
boundary values at the right endpoint of the interval.  Finally, in the fifth section the 
boundary behaviour of solutions to \eqref{1.a} is used to describe Sturm-Liouville type 
boundary-value problems giving rise to self adjoint operators.  

The author gratefully acknowledges some improvements to the paper which were made by
an anonymous referee.

\section{Differential operators on graphs}

In this work a graph ${\cal G}$ will have a countable vertex set 
and a countable edge set.  Unless otherwise stated, graphs
are assumed to be connected, and each vertex appears in only finitely many edges.  
Each edge has a positive weight (length) $l_j$.

A topological graph ${\cal G}$ may be constructed from this data \cite[p. 190]{Massey}.  
For each edge $e_j$ let $[a_j,b_j]$ be a real interval of length $l_j$.
Identify interval endpoints if the corresponding edge endpoints are the same vertex $v$.
The Euclidean length on the intervals may be extended to paths consisting of finitely
many nonoverlapping intervals by addition, and a metric $d(p_1,p_2)$ on ${\cal G}$
is defined as the infimum of the lengths of paths joining $p_1$ and $p_2$.

Several results from the theory of metric spaces will be used;
\cite[pp.139--170]{Royden} may be consulted for the proofs.
As a metric space ${\cal G}$ has a completion $\overline {\cal G}$. 
Recall that a metric space $X$ is totally bounded if for every
$\epsilon > 0$ there is a finite set $x_1,\dots ,x_n \in X$ such
that $\bigcup _k B(x_k,\epsilon )$ covers $X$.
A metric space is compact if and only if it is complete and
totally bounded.  This gives a picture of graphs with compact completion.

\begin{prop}
A graph ${\cal G}$ has compact completion $\overline{\cal G}$ 
if and only if for every $\epsilon > 0$ there
is a finite set of edges $e_k$, $k=1,\dots ,n$ such that for every $y \in {\cal G}$
there is a edge $e_k$ and a point $x_k \in e_k$ such that $d(x_k,y) < \epsilon $.
\end{prop}

The identification of edges $e_j$ with intervals facilitates the discussion
of function spaces and differential operators.  Let $L^2({\cal G})$ denote the 
Hilbert space $\oplus _j L^2(e_j)$ with the inner product
$$\langle f, g \rangle = \int_{\cal G} f\overline g
= \sum_j \int_{a_j}^{b_j} f_j(x)\overline{g_j(x)} \ dx , 
\quad f = (f_1, f_2, \dots ).$$
A formal differential operator $L = -D^2 + q $ acts componentwise on functions 
$f \in L^2({\cal G})$ in its domain.  
In our initial discussion the functions $q$ are assumed to be real
valued, measurable, and bounded.  The boundedness requirement will be relaxed 
when radial operators are discussed.  

In this paper the differential operators on ${\cal G}$ will have a common
dense domain ${\cal D}_0$.  To describe ${\cal D}_0$ we first distinguish interior
vertices, which have more than one incident edge, from boundary vertices
which have a single incident edge.  Edges $e_k$ incident on a vertex $v$ are
denoted $e_k \sim v$.  The functions $f \in {\cal D}_0$  
are $C^{\infty}$ on each (closed) edge, vanish except on finitely many edges,
vanish in a neighborhood of each boundary vertex, and satisfy the continuity and derivative conditions
\begin{equation} \label{bc}
f_j(v) = f_k(v), \quad  e_j,e_k \sim v, 
\end{equation}
\[ \sum_{e_k \sim v} f_k'(v) = 0.\] 
The derivatives here are computed in local coordinates where $v$ corresponds to the
left endpoint of each edge interval. 

The operator $L_0 = -D^2 + q $ with domain ${\cal D}_0$ is symmetric on $L^2({\cal G})$.
By essentially classical calculations \cite{Carlson98a} one may show that the adjoint operator
$L_0^*$ is $-D^2 + q$ acting on a domain ${\cal D}_1$.
The domain ${\cal D}_1$ consists of those functions $f\in L^2({\cal G})$ 
for which the components $f_n$ and $f_n^{(1)}$ are continuous, 
$f_n^{(1)}$ is absolutely continuous on $[a_n,b_n]$, $Lf \in L^2({\cal G})$,
and the vertex conditions \eqref{bc} are satisfied at interior vertices.

Since multiplication by $q$ is a bounded self adjoint operator on $L^2(G)$, 
the operator $-D^2+q$ will be self adjoint on a domain ${\cal D} \supset {\cal D}_0$ 
if and only if $-D^2$ is self adjoint on ${\cal D}$ \cite[p. 162]{RS2}.  
For now we restrict our attention to the operator $-D^2$.  
Integration by parts shows that $-D^2$ on the domain ${\cal D}_0$ 
has the associated positive quadratic form
\[ Q(f,g) = \int_{\cal G} f'\overline{g'} .\]    
Symmetric operators with positive forms always have self adjoint
extensions (the Friedrich's extension).  

It will be convenient to have criteria which
insure that various self adjoint extensions of $L_0$ have compact resolvent,
so that the spectrum will consist of a discrete set of eigenvalues of finite multiplicity.
Some results in this direction may be achieved by employing the form $Q$.
We start with a compactness result in the space $C(\overline{\cal G})$ of continuous
functions on the metric completion of ${\cal G}$ with the sup norm.

\begin{thm} \label{context}
Suppose that ${\cal G}$ is a connected graph which has a compact metric completion 
$\overline{\cal G}$.  
Let $B$ denote the set of continuous functions on ${\cal G}$ which are absolutely continuous
on each edge, and satisfy
\[ \int_{\cal G} |f|^2 + | f' |^2 \le 1 .\]
Then each function $f \in B$ has a unique continuous extension to $\overline{\cal G}$,
and the (extended) set $B$ has compact closure in $C(\overline{\cal G})$.
\end{thm} 

\begin{proof}
Since ${\cal G}$ has a compact metric completion, it has a finite diameter $L$.
There is a simple path $\gamma \subset {\cal G}$ of length at least $L/2$.  
For any function $f \in B$ the Cauchy-Schwarz inequality gives the integral bound 
\[ \int_{\gamma } |f| \le (\int_{\cal G} |f|^2 )^{1/2} (L/2)^{1/2} 
\le (L/2)^{1/2} .\] 
Thus there is a point $x_0 \in {\cal G}$ such that $|f(x_0)| \le (L/2)^{-1/2}$ .

Pick any other point $x \in {\cal G}$ and connect $x_0$ and $x$ by a simple path
of length at most $L$.
Integrate along the path (using the continuity of $f$ across the vertices) to get
\[ |f(x) - f(x_0)|^2 = |\int_{x_0}^x f'(t) \ dt |^2 \le 
d(x,x_0) \int_{x_0}^x |f'(t)|^2 \ dt .  \]
This gives a uniform bound for each $f \in B$.
Replacing $x_0$ above by another point $y \in {\cal G}$ 
shows that the functions in $B$ are uniformly equicontinuous.

By \cite[p. 149]{Royden} the functions in $B$ extend by continuity to a uniformly equicontinuous
family on the completion of ${\cal G}$.  The Arzela-Ascoli theorem \cite[p. 169]{Royden} 
then gives the result.
\end{proof}

If ${\cal G}$ has finite volume then a uniformly convergent sequence also converges
in $L^2$.  Moreover the compactness of the set $B$ will imply compactness of the resolvent
for self adjoint extensions of $L_0$ whose associated quadratic form is $Q$ \cite[p. 245]{RS4}.  

\begin{cor} \label{compactness}
If ${\cal G}$ has finite volume then $B$ has compact closure in $L^2$.
If $L$ is a self adjoint extension of $L_0$ whose associated quadratic form is
\[ \langle Lf,g \rangle = \int_{\cal G} f'\overline{g'} ,\]    
then $L$ has a compact resolvent.
\end{cor}

When ${\cal G}$ has finite volume, explicit lower bounds on (nonconstant) eigenvalues 
may be obtained via the next lemma.

\begin{prop} \label{loweval}
Suppose that $f$ is real valued,
\[\int_{\cal G} f^2 = 1 \]
and $f(x) = 0$ for some $x \in {\cal G}$. 
Then 
\[ \int_{\cal G} (f')^2 \ge vol({\cal G})^{-2} . \]
\end{prop}

\begin{proof}
There is some point $y \in {\cal G}$ such that $ f^2(y) \ge vol({\cal G}) ^{-1} $.
Connect $y$ to $x$ by a simple path $\gamma $.
By the Cauchy-Schwarz inequality 
\[ vol({\cal G}) ^{-1} \le f(y)^2 = [f(y) - f(x)]^2 = [\int_{\gamma } f'(t) \ dt ]^2 
\le vol({\cal G}) \int_{\cal G} |f'|^2 \ dV . \]
\end{proof}

One may consider whether the finite volume hypothesis in \corref{compactness}
may be relaxed to the assumption that the diameter of ${\cal G}$ is finite, or 
that the completion of ${\cal G}$ is compact.  We will sketch the construction of
a counterexample.  Start with the half open interval $(0,1]$, and 
place vertices at the points $1/n$, $n \ge 2$.  At each of these vertices
attach $K_n$ loops of length $r_n$, with $\lim_{n \to \infty} r_n = 0$.
The resulting graph has finite diameter and compact completion.
Next, construct smooth functions which have a constant value $c_n > 0$
on the loops at $1/n$, and which vanish at $x$ if the distance from $x$ to
the set of loops at $1/n$ exceeds $\sigma _n > 0$.  
By a suitable selection of the constants $K_n$, $r_n$, $c_n$ and $\sigma _n$, 
one finds symmetric operators $L_0$ which are bounded by any positive number $\epsilon $
on a subspace of infinite dimension.  In particular no self adjoint extension can have compact
resolvent.  

\section{Decomposing $L^2({\cal T})$ of a radial tree}

A graph is a tree if it is connected and simply connected.
A weighted tree is a radial tree if there is a vertex $R$, the root, such that 
the degree of vertices and the lengths of edges are functions of the distance 
from $R$.  A (formal) Schr\"odinger operator $-D^2 + q$ on a
radial tree ${\cal T}$ will be called formally radial if $q$ is a function of the distance from $R$.

Since the formally radial operator $L_0 = -D^2 + q$ with the domain ${\cal D}_0$ is symmetric 
and bounded below, it has self adjoint extensions.  
Such a self adjoint Schr\"odinger operator on a radial tree 
will be called radial if the domain is invariant under the automorphisms of the tree which fix $R$. 
One of the main goals of this work is to describe in detail some radial 
Schr\"odinger operators.  In pursuit of this goal, the symmetries of the tree 
will be used to decompose $L^2({\cal T})$.  Similar decompositions appear in
\cite{Naimark} and \cite{Romanov}. 

Radial trees are closely associated to a class of abelian groups.
Given a finite or infinite sequence of positive integers $\delta (0),\delta (1),\dots $,
let $\Z _{\delta (i)}$ denote the additive group of integers modulo $\delta (j)$.
The group ${\cal Z} = \oplus_i \Z _{\delta (i)}$ will be the complete direct sum of 
the groups $\Z _{\delta (i)}$, whose elements are sequences with $i$-th component from 
$\Z _{\delta (i)}$.  Addition is performed componentwise in $\Z _{\delta (i)}$ .

It will help to establish some notation for the tree (see Figure 2).  If $u,w \in {\cal T}$,
say that $w$ is below $u$ if the simple path from $w$ to $R$ contains $u$.
Points $w \in {\cal T}$ have a metric depth, which is the distance from the root.  The vertices $v$ have a combinatorial depth $j$, which is  
the number of edges separating $v$ from the root $R$.  Below each vertex with combinatorial 
depth $j$ will be $\delta (j)$ incident edges.  The classical degree of the root is thus
$\delta (0)$, while the degree of vertices with combinatorial depth $j > 0$ is $\delta (j) + 1$.   
The vertices at combinatorial depth $j > 0$ may be identified with the elements of the 
group ${\cal Z}_j = \oplus_{i=0}^{j-1} \Z _{\delta (i)}$.  Similarly, edges may be indexed
by their vertex of greatest depth.  The number of edges extending from depth $j-1$ to $j$
is $N_j = \prod_{i=0}^{j-1} \delta (i)$.  The full group ${\cal Z}$
may be identified with the set of all simple paths of maximal length starting at the root.

\begin{figure}[ht]
\begin{center}
\epsfxsize=7cm
\epsffile{tree2.eps}
\end{center}
\caption{A radial tree with $\delta (0)$ = 3, $\delta (1) = 2$}
\end{figure}

With this identification the group ${\cal Z}$ acts on the tree by permuting vertices and edges.
In particular the components $(0,\dots ,0,\Z _{\delta (i)},0,\dots )$ rotate subtrees.
Using this group action the space $L^2({\cal T})$ will be decomposed into a countable 
orthogonal sum of invariant subspaces for the radial Schr\"odinger operators, 
with certain symmetries \cite{Romanov}.  The reduced Schr\"odinger operators may then be interpreted 
as differential operators defined on intervals of $\real $.

Two types of subtrees will be associated with vertices $v$ having incident edges below them.
$T_v$ will denote the subtree rooted at $v$ and consisting of all vertices and edges below $v$.
For $l \in \Z _{\delta (j)}$, let $S_{v,l}$ denote 
the tree rooted at $v$, but containing only the one edge $(v,l)$ immediately below
$v$ and all vertices and edges below that edge.   

Next we introduce a collection of subspaces $U_{v,k}$ of $L^2({\cal T})$, defined
for $k=0,\dots ,\delta (0) - 1$ if $v = R$, and defined 
for $k=1,\dots ,\delta (j) - 1$ if $v$ has depth $j > 0$.
To construct these subspaces, begin with the functions $f$ which are radial on the tree
$S_{v,0}$, and which vanish on the complement of $T_v$.      
For $k=1,\dots ,\delta (j) - 1$ the subspace $U_{v,k}$ is the set of functions satisfying
\begin{equation} \label{2.a}
f(t_l) = e^{2\pi i kl/\delta (j)}f(t_0), \quad t_l \in S_{v,l},
\end{equation}
\[l = 0,\dots ,\delta (j)-1, \quad k = 1,\dots ,\delta (j) - 1, \]
where the points $t_l \in S_{v,l}$ have the same metric depth as $t_0$.
In case $v = R$, the subspace $U_{R,0}$ consists of all radial functions
on ${\cal T}$.  

\begin{thm}
The distinct subspaces $U_{v,k}$ are orthogonal, and their linear span is dense in $L^2({\cal T})$.
\end{thm}

\begin{proof}
For two distinct vertices $v,w$, the trees $T_v$ and $T_w$ are either disjoint, in which case
the subspaces $U_{v,k}$ and $U_{w,m}$ are obviously orthogonal, or  
after a possible relabeling, $v$ lies above $w$.
Notice that each element $f$ of $U_{v,k}$ is radial when restricted to $T_w$.  
If $w$ has combinatorial depth $j$ and $g \in U_{w,m}$ the calculation
\[ \int_{T_w} f\bar g 
= \sum_{l=0}^{\delta (j)-1} \int_{S_{w,l}} f(x_l)\bar g(x_l) \]
\[ = \sum_{l=0}^{\delta (j)-1} \int_{S_{w,0}} f(x_0)\bar g(x_0)e^{-2\pi i lm/\delta (j)} = 0, 
\quad m = 1,\dots ,\delta (j)-1, \]
shows that $U_{w,m}$ is orthogonal to functions $f$ which are radial on $T_w$.

If $k \not= m$ the orthogonality of functions $f \in U_{v,k}$ and $g \in U_{v,m}$ is 
established with a similar computation,
\[ \int_{T_v} f\bar g 
 = \sum_{l=0}^{\delta (j)-1} \int_{S_{v,0}} f(x_0)e^{2\pi i lk/\delta (j)} 
\bar g(x_0)e^{-2\pi i lm/\delta (j)} \]
\[ = \int_{S_{v,0}} f(x_0)\bar g(x_0) \sum_{l=0}^{\delta (j)-1} e^{2\pi i l(k-m)/\delta (j)} = 0, 
\quad k-m \not= 0 \mod \delta (j). \]

Turning to the denseness of the linear span of the spaces $U_{v,k}$, define
\[ V_j = \oplus_{v,k} U_{v,k} , \quad {\rm depth}(v) \le j. \]
The main idea is to show that for each nonnegative integer $j$ the
subspace $V_j$ includes all functions vanishing below the vertices with combinatorial 
depth $j+1$ and all functions which are radial on each $S_{v,l}$ if the combinatorial 
depth of $v$ is $j$.  The proof is by induction.

For $j = 0$ we want to show that any function $f$ supported on an edge $(R,m)$
incident on $R$ may be written as a linear combination of functions $f_k \in U_{R,k}$, 
$k= 0,\dots \delta (0)-1$.  For $t_0$ in edge $(R,0)$ and $t_m$ in edge $(R,m)$ at the same
depth, define $f_k$ by $f_k(t_0) = f(t_m)$.  Another discrete 
Fourier transform calculation gives
\begin{equation} \label{DFT}
\frac{1}{\delta (0)} \sum_{k=0}^{\delta (0)-1} e^{-2\pi ikm/\delta (0)} f_k(t_l) 
\end{equation}
\[ = \frac{1}{\delta (0)} \sum_{k=0}^{\delta (0)-1} e^{-2\pi ikm/\delta (0)} 
e^{2\pi ikl/\delta (0)} f_k(t_0) 
 = \Bigl \{ \begin{matrix} f(t_m) & l = m \cr 0 & l \not= m \end{matrix} \Bigr \} .\]

Similarly, the linear combinations of functions $f_k \in U_{R,k}$
includes all functions which are radial on each $S_{R,l}$.

To complete the argument suppose the induction hypothesis is true for $i < j$.
For a vertex $v$ with combinatorial depth $j$ the radial functions on $T_v$
are in $V_{j-1}$ by the induction hypothesis.  With the addition of the subspaces
$U_{v,k}$ for $k = 1,\dots ,\delta (j) - 1$ the argument used for the root $R$
may be adopted with trivial modifications to handle the general case. 

\end{proof}

Consider next how the subspaces $U_{v,k}$ may be used to reduce 
certain differential operators on the tree ${\cal T}$ to a sequence of
differential operators with interior point conditions on intervals ${\cal I}_n$.
Take a vertex $v$ with combinatorial depth $n$ and metric depth $x_n$.
Let $l_{j+1}$ be the length of the edges joining 
vertices at combinatorial depth $j$ to vertices at combinatorial depth $j+1$.  
For $j \ge n$ define a sequence of real numbers $x_j$ by $x_{j+1} = x_j + l_{j+1}$, and take
${\cal I}_n = \cup_j [x_j,x_{j+1}]$.

For $x \in {\cal I}_n$, the mapping which sends $f \in U_{v,k}$ to its value $f(t)$ at a point
$t \in S_{v,0}$ with metric depth $x$ in ${\cal T}$ is an isometric bijection from $U_{v,k}$ to 
a weighted space $L^2({\cal I}_n,w_n)$.  The weight function $w_n(x)$ is equal to 
$N_n^{-1}N_{j+1}$ on the interval $[x_j,x_{j+1})$ where as before $N_j = \prod_{k=0}^{j-1} \delta (k)$. 
The weighted inner product is   
\[ \langle f, g \rangle = \sum_{j \ge n} \int_{x_j}^{x_{j+1}} w_n(x) f(x) \bar g(x) \ dx .\]

Self adjoint operators ${\cal L} = -D^2 + q$ on $L^2({\cal T})$ may be constructed in the following manner.
For the given radial potential $q$, find self adjoint operators ${\cal L}_n = -D^2 + q$ 
on $L^2({\cal I}_n,w_n)$.  Use the identification of $L^2({\cal I}_n,w_n)$ with the spaces
$U_{v,k}$ to map $f$ in the domain of ${\cal L}_n$ into $L^2({\cal T})$, and similarly
identify ${\cal L}_nf$ with ${\cal L}f$.  To satisfy the vertex conditions \eqref{bc} 
the functions in the domain of ${\cal L}_n$ must satisfy the jump conditions
\[f(x_j^-) = f(x_j^+), \quad f'(x_j^-) = \delta (j)f'(x_j^+), \quad j > n. \] 

In addition there are vertex conditions at $v$ which must be satisfied.  For vertices $v$
other than the root, functions in $U_{v,k}$ vanish in the complement of $T_v$, so
we must have the boundary condition $f(x_n) = 0$.  The required vanishing of the sum
of the derivatives at $v$ is always satisfied in $U_{v,k}$ since 
\[ \sum_{l=0}^{\delta (n)-1} e^{2\pi ikl/\delta (n)} = 0, \quad k = 1,\dots ,\delta (n) - 1.\]

The same considerations apply at the root for the spaces $U_{R,k}$ if $k \not= 0$.
When $k=0$ there are two cases to consider.  If $\delta (0) = 1$ then
any of the classical Sturm-Liouville conditions $a_1f(x_0) + b_1f(x_0) = 0$
with $a_1,b_1 \in \real $ may be imposed.
If $\delta (0) > 1$ the interior vertex conditions \eqref{bc} must be satisfied at the root. 
This can be achieved for the subspace $U_{R,0}$ by imposing the 
condition $f'(x_0) = 0$.

\section{Solving $-y'' + qy = \lambda y $ with jump conditions}

The symmetries of a radial tree have provided a decomposition of $L^2({\cal T})$
into orthogonal subspaces $U_{v,k}$ which may be identified with a weighted Hilbert
space $L^2({\cal I}_n,w_n)$ on a real interval ${\cal I}_n$.  By means of this identification,
certain self adjoint operators ${\cal L}_n = -D^2 + q$ on $L^2({\cal I}_n,w_n)$ may be used to construct
self adjoint operators ${\cal L} = -D^2+q$ on $L^2({\cal T})$.  
Functions in the domain of ${\cal L}_n$ are required to satisfy the interior point jump conditions
\begin{equation} \label{jc}
f(x_j^-) = f(x_j^+), \quad f'(x_j^-) = \delta (j)f'(x_j^+), \quad j > n. 
\end{equation}
In addition, one of the left endpoint boundary conditions $a_1f(x_n) + b_1f(x_n) = 0$
is imposed.

This section will provide an analysis of solutions to \eqref{1.a} satisfying \eqref{jc}.  
It is convenient to define $x_{\infty} = \lim_j x_j$;
the value will be $+ \infty $ when $\sum_j l_j = \infty $.
The behaviour of solutions to \eqref{1.a} as $x \to x_\infty$ 
has implications for the explicit description of domains for the operators ${\cal L}_n$ 
in terms of generalized boundary conditions at $x_{\infty}$.
The growth of solutions as $|\lambda | \to \infty $ will be used to analyze
the distribution of eigenvalues. 
In the discussion of operators ${\cal L}_n = -D^2 + q$ on $L^2({\cal I}_n,w_n)$ 
the functions $q$ are still assumed to be real valued and measurable, 
but the previous boundedness requirement will be relaxed. 

For notational convenience the sequence $x_n,x_{n+1},\dots $ in ${\cal I}_n$
will be reindexed as $x_0,x_1,\dots $.  The same reindexing will apply to interval lengths
$l_j = x_j - x_{j-1}$, the branching numbers $\delta (j)$, and the edge counts $N_j$.
The weight $w_n$ will simply be denoted $w(x)$.

A piecewise linear rescaling of variables converts the operator
$-D^2 + q(x)$ subject to the jump conditions
\eqref{jc} into a more conventional form.  Define
\begin{equation} \label{defxi}
\xi = \frac{x}{N_{j+1}} + \sum_{k=1}^j \frac{x_k}{N_k}, \quad
x_j \le x < x_{j+1}, 
\quad N_j = \prod_{i=0}^{j-1} \delta (i).
\end{equation}
Let $\xi _j = \xi (x_j)$ and observe that 
$\xi _{j+1} - \xi _j = (x_{j+1}-x_j)/N_{j+1}$.
If $f$ satisfies the jump conditions \eqref{jc} then
$F(\xi ) = f(x)$ is continuous with a continuous derivative 
on $[\xi _0, \xi _{\infty})$.

Similarly, if $Y(\xi ) = y(x)$, $Q(\xi ) = q(x)$ and 
$$W(\xi ) = N_{j+1}, \quad \xi _j \le x < \xi _{j+1}, $$
then the equation
$$-y'' + q(x) y = \lambda y$$
becomes
\begin{equation} \label{xieqn}
-Y'' + W(\xi )^2Q(\xi ) Y = \lambda W(\xi )^2Y. 
\end{equation}
The usual reduction to an integral equation and the method of
successive approximations may be applied to the equation in this form.
If $z_1 = Y$, $z_2 = Y'$ then the integral equation is
$$\begin{pmatrix} z_1(\xi ) \cr z_2(\xi) \end{pmatrix} = 
\begin{pmatrix} z_1(\xi _0) \cr z_2(\xi _0) \end{pmatrix}  + \int_{\xi _0}^{\xi }
\begin{pmatrix} 0 & 1 \cr W(s )^2[Q(s ) - \lambda ] & 0 \end{pmatrix}
\begin{pmatrix}z_1(s) \cr z_2(s) \end{pmatrix} \ ds .$$

Notice in particular that 
$$\int_{\xi _0}^{\xi _j} W (s)^2 \ ds = \sum_{i=0}^{j-1} N_{i+1}^2(\xi _{i+1}-\xi _i)
= \sum_{i=0}^{j-1} N_{i+1} (x_{i+1}- x_i)
= \sum_{i=0}^{j-1} N_{i+1} l_{i+1},$$
so that the function $W^2$ will be integrable on $[\xi _0, \xi _{\infty })$
if the graph has finite volume.  
If in addition $W (\xi )^2Q(\xi )$ is integrable on $[\xi _0, \xi _{\infty })$ the usual 
Picard iteration method yields a sequence of successive approximations 
which converge uniformly to the desired solution
on $[\xi _0, \xi _{\infty })$. \cite[p. 97--98]{CL}  

\subsection{Basic description of solutions to \eqref{1.a}}

The jump conditions \eqref{jc} determine the initial data 
$y(x_j^+), y'(x_j^+)$ from the data $y(x_j^-), y'(x_j^-)$, so solutions
on one subinterval $[x_j,x_{j+1}]$ have a unique continuation to ${\cal I}_n$. 
In particular this shows that the space of solutions to \eqref{1.a}
on ${\cal I}_n$ satisfying \eqref{jc} has dimension $2$ as a complex vector space.

On each interval $[x_j,x_{j+1}]$ the space of solutions of \eqref{1.a} 
has a basis $c(x,x_j,\lambda )$, $s(x,x_j,\lambda )$ satisfying
\begin{equation} \label{initcond}
\begin{matrix} c(x_j,x_j,\lambda ) = 1, & s(x_j,x_j,\lambda ) = 0, \cr
c'(x_j,x_j,\lambda ) = 0, & s'(x_j,x_j,\lambda ) = 1.\end{matrix}
\end{equation}
In addition a basis $c(x,\lambda ),s(x,\lambda )$ may be obtained by continuation of 
the basis $c(x,x_0,\lambda ),s(x,x_0,\lambda )$ to the entire interval ${\cal I}_n$.

The continuation of solutions of \eqref{1.a} from $[x_0,x_1]$ to subsequent
intervals $[x_j,x_{j+1}]$ may be described using a sequence of transition matrices
\[ \tau _j = \begin{pmatrix} 1 & 0 \cr 0 & \delta ^{-1}(j)  \end{pmatrix}. \]
At $x_1^-$ the values and derivatives for the functions $c$ and $s$ are the columns of the 
$2\times 2$ matrix
\[\begin{pmatrix} c(x_1,x_0,\lambda ) & s(x_1,x_0,\lambda ) \cr 
c'(x_1,x_0,\lambda ) & s'(x_1,x_0,\lambda ) \end{pmatrix}.\]
The matrix $\tau _1$ takes the vector of initial data at $x_1^-$ to that at $x_1^+$
so that the jump conditions \eqref{jc} are satisfied:
\[ \begin{pmatrix} y(x_1^+) \cr y'(x_1^+) \end{pmatrix} 
= \tau _1\begin{pmatrix} y(x_1^-) \cr y'(x_1^-) \end{pmatrix}. \]

The solutions $c(x,\lambda )$ and $s(x,\lambda )$ on the interval $x_1 \le x < x_2$ are given by
$$(c(x,x_1,\lambda ), s(x,x_1,\lambda ))
\tau _1 \begin{pmatrix} c(x_1,x_0, \lambda ) & s(x_1,x_0,\lambda ) \cr 
c'(x_1,x_0, \lambda ) & s'(x_1,x_0,\lambda ) 
\end{pmatrix}.$$

By induction the next result is established.

\begin{lem} \label{4.1}
On the interval $x_j \le x < x_{j+1}$ the solution matrix $(c,s)$ for \eqref{1.a} 
has the form
\[ (c(x,\lambda ), s(x,\lambda ))\] 
\[ = (c(x,x_j,\lambda ), s(x,x_j,\lambda ))
\prod_{i=1}^{j} \tau _i\begin{pmatrix} 
c(x_i,x_{i-1}, \lambda ) & s(x_i,x_{i-1},\lambda ) \cr 
c'(x_i,x_{i-1}, \lambda ) & s'(x_ii,x_{i-1},\lambda ) 
\end{pmatrix}. \]
\end{lem}

Consistent with the usage in this lemma, matrix products are assumed to 
have factors whose indices decrease from left to right.

The solutions $c(x,x_j,\lambda ),s(x,x_j,\lambda )$ may be compared in a standard way
(\cite{Fulton}, \cite[p. 13]{Pos}) to the elementary functions 
$\cos(\omega [x-x_j]), \omega ^{-1} \sin(\omega [x-x_j])$, where $\omega =\sqrt{\lambda }$.    
Let $\Im (\omega ) $ denote the imaginary part of $\omega $.
Usually these estimates emphasize the $\lambda $ dependence, but we will also
need to make the $x$ dependence explicit.  For this reason a sketch of the proof is provided.

\begin{lem} \label{4.2}
Define 
\[ C_q(x) = \exp(\int_{x_j}^x |q(t)| \ dt) - 1, \quad  x_j \le x \le x_{j+1}.\]
If $|x-x_j| \le 1$ the solutions $c(x,x_j,\lambda ),s(x,x_j,\lambda )$ of \eqref{1.a} satisfy
\[ |c(x,x_j,\lambda ) - \cos(\omega [x-x_j]) | \le  
|\omega ^{-1}| e^{|\Im \omega |[x-x_j]} C_q(x), \]
\[ |c'(x,x_j,\lambda ) + \omega \sin(\omega [x-x_j])| \le  
 e^{|\Im \omega |[x-x_j]} C_q(x), \]
\[ |s(x,x_j,\lambda ) - \omega ^{-1}\sin(\omega [x-x_j]) | \le  
 |\omega ^{-2}| e^{|\Im \omega |[x-x_j]}C_q(x),\]
\[ |s'(x,x_j,\lambda ) - \cos(\omega [x-x_j])| \le 
 |\omega ^{-1}|e^{|\Im \omega |[x-x_j]} C_q(x).\]
\end{lem}

\begin{proof}
There is no loss of generality if we take $x_j = 0$.
By using the variation of parameters formula, a solution of \eqref{1.a} satisfying 
$y(0,\lambda ) = \alpha $, $y'(0,\lambda ) = \beta $,
with $\alpha ,\beta \in \complex $,
may be written as a solution of the integral equation
\begin{equation} \label{inteq1}
y(x,\lambda ) =  
\cos(\omega x)\alpha + \frac{\sin (\omega x)}{\omega }\beta + 
\int_0^x \frac{\sin(\omega [x-t])}{\omega } q(t)y(t,\lambda ) \ dt. 
\end{equation}
Differentiation with respect to $x$ gives
\begin{equation} \label{inteq2}
y'(x,\lambda ) =  
- \omega \sin(\omega x)\alpha + \cos (\omega x)\beta + 
\int_0^x \cos(\omega [x-t]) q(t)y(t,\lambda ) \ dt. 
\end{equation}

Start with the elementary estimates
\[|\sin(\omega x)|,|\cos(\omega x)| \le e^{|\Im \omega |x} , \quad 
|\omega ^{-1}\sin(\omega x)| = |\int_0^x \cos(\omega t) \ dt| \le x e^{|\Im \omega |x}.\]
For $c(x,0,\lambda )$ the integral equation \eqref{inteq1} and the assumption $|x| \le 1$ give 
\[ |e^{-|\Im \omega |x} c(x,0,\lambda )| 
\le 1 + \int_0^x |q(t)| e^{-|\Im \omega |t} |c(t,0,\lambda )| \ dt. \]
By Gronwall's inequality \cite[p. 24]{Har} 
\[ |e^{-|\Im \omega |x} c(x,0,\lambda )| \le \exp(\int_0^x |q(t)| \ dt) .\]
Thus \eqref{inteq1} implies that 
\[ |c(x,0,\lambda ) - \cos(\omega x) | \le 
 |\omega ^{-1}| e^{|\Im \omega |x} \int_0^x 
|q(t)| \exp(\int_0^t |q(s)| \ ds) \ dt \]
\[=  |\omega ^{-1}| e^{|\Im \omega |x}  
[ \exp(\int_0^x |q(t)| \ dt) - 1]. \]

There is a similar inequality for $s(x,0,\lambda )$ 
and \eqref{inteq2} leads to the inequalities for $|y'|$.  
\end{proof}

Elements of $\complex ^2 $ are given the Euclidean norm, and $2 \times 2$ matrices
$A$ will have the standard operator norm 
\[ \| A \| = \sup_{\| z \| \le 1 } \| A z \| , \quad z \in \complex ^2 .\]

Introduce the matrix
\[ \Omega = \begin{pmatrix} 1 & 0 \cr 0 & \omega \end{pmatrix} . \]
If $q$ is integrable and the interval ${\cal I}_n$ 
has finite length, then \lemref{4.2} implies 
\[ \| \Omega ^{-1} \begin{pmatrix} c(x,x_j, \lambda ) & s(x,x_j,\lambda ) \cr 
c'(x,x_j, \lambda ) & s'(x,x_j,\lambda ) \end{pmatrix} \Omega 
- \begin{pmatrix} \cos(\omega [x-x_j]) & \sin(\omega [x-x_j]) \cr 
-\sin (\omega [x-x_j]) & \cos(\omega [x-x_j]) \end{pmatrix} \| \]
\begin{equation} \label{standard estimates}
= O(\omega ^{-1} \int_{x_j}^{x_{j+1}} |q(x)| \ dx ), 
\end{equation}  
the estimates holding uniformly for $\lambda $ bounded.

\subsection{Asymptotics for $c(x,\lambda)$ and $s(x,\lambda )$}

The products arising in \lemref{4.1} may be simplified.  For brevity define 
\[ R_i(\omega ) = \Omega ^{-1} \begin{pmatrix} 
c(x_i,x_{i-1}, \lambda ) & s(x_i,x_{i-1},\lambda ) \cr 
c'(x_i,x_{i-1}, \lambda ) & s'(x_i,x_{i-1},\lambda ) 
\end{pmatrix} \Omega .\]
Since the matrices $\tau _j$ and $\Omega $ commute, we find that 
\[ \prod_{i=1}^j \tau _i\begin{pmatrix} 
c(x_i,x_{i-1}, \lambda ) & s(x_i,x_{i-1},\lambda ) \cr 
c'(x_i,x_{i-1}, \lambda ) & s'(x_i,x_{i-1},\lambda ) 
\end{pmatrix} 
 = \Omega \Bigl [ \prod_{i=1}^j \tau _i R_i(\omega ) \Bigr ] \Omega ^{-1} . \]

It will help to see that the matrix products have a limit as $j \to \infty$.  
The first result addresses the case when the tree ${\cal T}$ has finite metric depth.

\begin{lem} \label{4.3}
Suppose that $q$ is integrable and $\sum_j l_j < \infty $.
As $j \to \infty $ the product $ \prod_{i=1}^j \tau _i R_i(\omega ) $
converges uniformly on compact subsets of $ \complex ^* = \complex \setminus \{ 0 \} $ 
to a meromorphic matrix function $M_1(\omega )$.
If $\delta (j) >1$ for infinitely many $j$, then
for each $\omega $ we have $\det (M_1(\omega )) = 0$.
\end{lem}

\begin{proof} 

Let $K$ be a compact subset of $\complex ^*$, and
write $R_i(\omega )$ as a perturbation of the identity, $R_i(\omega ) = I + E_i(\omega )$.
Now consider a product
\[ \prod_{i=k}^{l} \tau _i R_i(\omega )  
= \prod _{i = k}^{l} [\tau _i + \tau _i E_i]. \]
Expand the product as a sum, with each summand the product of
$l-k+1$ matrices $\tau _i$ or $\tau _i E_ii$, and the first term being $ \prod_{i=k}^l \tau _i $. 

Using the fact that the matrix norm is subadditive and submultiplicative,  
the norm 
\[ \| \prod _{i = k}^{l} [\tau _i+ \tau _iE_i] - \prod_{i=k}^l \tau _i  \| \]
is bounded by the sum of the product of norms of the factors in 
the terms of the expanded sum.  Noting that $\| \tau _i\| \le 1$, 
these terms are individually no greater than the corresponding terms in 
the expansion of $\prod _{i=k}^l (1 + \| E_i \| ) - 1$. 
These observations lead to the estimate
\begin{equation} \label{prod1}
\| \prod _{i = k}^{l} [\tau _i + \tau _i E_i] - \prod_{i=k}^l \tau _i \| 
\le \prod _{i=k}^l (1 + \| E_i \| ) - 1, \quad k \le l 
\end{equation}

The estimate of \eqref{standard estimates} implies that 
\[  E_i  =  R_i - I  = O(l_i) + O(\int_{x_{j-1}}^{x_j} |q|) \]
for $\omega \in K $.
Since $\sum l_i < \infty $ and $q$ is integrable, the sum $\sum_i \| E_i(\omega ) \| $
converges uniformly for $\omega \in K$.
This implies \cite[p. 190]{Ahl} convergence of the products
\[ \lim_{l \to \infty } \prod _{i = k}^{l} [1  + \| E_i(\omega ) \| ] , \]
again uniformly for $\omega \in K$.

Based on these observations, \eqref{prod1} shows that the
products  $\prod _{i = k}^{l} [\tau _i + \tau _iE_i] $ are bounded
independent of $l \ge k$, and moreover the difference
\begin{equation} \label{3.a}
\prod _{i = k}^{l} [\tau _i + \tau _iE_i] - \prod_{i=k}^{l} \tau _i
\end{equation}
goes to $0$ as $k \to \infty $ independent of $l$ as long as $l \ge k$.
Notice that $\tau _i = I$ if $\delta (i)=1 $, while
\[ \lim_{j \to \infty} \prod_{i=1}^{j-1} \tau _i = 
\begin{pmatrix} 1 & 0 \cr 0 & 0 \end{pmatrix}, \quad  
{\rm if} \quad \delta (i) > 1 \quad {\rm infinitely \ often}. \]
The convergence argument is completed by considering
\[ \| \prod _{i = 1}^{l} [\tau _i + \tau _iE_i] - \prod _{i = 1}^{k} [\tau _i + \tau _iE_i] \| \]
\[\le 
\| \prod _{i = {\lceil k \rceil /2} +1}^{l} [\tau _i + \tau _iE_i] - 
\prod _{i = {\lceil k \rceil /2} +1}^{k} [\tau _i + \tau _iE_i] \| 
\ \| \prod _{i = 1}^{\lceil k \rceil /2} [\tau _i + \tau _iE_i] \|  .\]
The factor $\| \prod _{i = 1}^{\lceil k \rceil /2} [\tau _i + \tau _iE_i] \| $
is bounded independent of $k$, and the first factor on the right of the inequality 
goes to $0$ with $k$ by \eqref{3.a}.  The products thus form a Cauchy sequence of analytic 
functions uniformly for $\omega \in K$.

Since the determinant is continuous from $2 \times 2$ matrices to 
$\complex $, the limit matrix has determinant $0$ if $\delta (i) > 1$ infinitely often.  
\end{proof}

Notice that each of the matrix products 
$ \Omega \Bigl [ \prod_{i=1}^j \tau _i R_i(\omega ) \Bigr ] \Omega ^{-1} $
arising in \lemref{4.1} is an entire function of $\lambda $.  
It follows from \lemref{4.3} that these products converge uniformly 
on any circle of positive radius centered at $0$, so by the maximum principle
they converge uniformly on any compact set in $\complex $.
This establishes the next corollary.

\begin{lem} \label{4.4}
If $q$ is integrable and $\sum_j l_j < \infty $, the products
\[ \prod_{i=1}^j \tau _i\begin{pmatrix} 
c(x_i,x_{i-1}, \lambda ) & s(x_i,x_{i-1},\lambda ) \cr 
c'(x_i,x_{i-1}, \lambda ) & s'(x_i,x_{i-1},\lambda ) 
\end{pmatrix} 
 = \Omega \Bigl [ \prod_{i=1}^j \tau _i R_i(\omega ) \Bigr ] \Omega ^{-1} . \]
converge to an entire matrix function $M(\lambda )$ as $j \to \infty $. 
\end{lem}

\lemref{4.1} and \lemref{4.4} imply 
\begin{equation} \label{initdata}
\lim_{j \to \infty} \begin{pmatrix}
c(x_j^+,\lambda ) & s(x_j^+,\lambda ) \cr
c'(x_j^+,\lambda ) & s'(x_j^+,\lambda ) 
\end{pmatrix}
= M(\lambda ). 
\end{equation}
In case $ \sum_{i=1}^\infty l_i N_i < \infty $ and 
$W^2(\xi )Q(\xi )$ is integrable on $[\xi _0,\xi \infty )$, 
we may take advantage of the change of variables $x \to \xi $ discussed at the 
beginning of this section.  Let $C(\xi ,\lambda )$ and $S(\xi ,\lambda )$ 
be solutions of \eqref{xieqn} satisfying
\[ \begin{pmatrix} C(\xi _0,\lambda ) & S(\xi _0,\lambda ) \cr   
C'(\xi _0,\lambda ) & S'(\xi _0,\lambda ) \end{pmatrix} = I. \]   
Because \eqref{xieqn} is essentially regular on a finite interval, the matrix 
\[ \begin{pmatrix} C(\xi _{\infty},\lambda ) & S(\xi _{\infty},\lambda ) \cr   
C'(\xi _{\infty},\lambda ) & S'(\xi _{\infty},\lambda ) \end{pmatrix}  \]   
will be nonsingular.  Consequently, either $\lim_{j \to \infty} c(x_j, \lambda )$
or $\lim_{j \to \infty} s(x_j, \lambda )$ will be nonzero, and  
$M(\lambda )$ is not the zero function.
If $q = 0$ and $ \sum_{i=1}^\infty l_i < \infty $ the same conclusion may be established 
by direct computation of $M(0)$.    

\begin{thm} \label{4.5} 
Suppose that $\sum_j l_j < \infty $ and $q$ is integrable.
Then every solution of $-y'' + q(x)y = \lambda y $ on $[x_0,x_{\infty })$
satisfying the jump conditions \eqref{jc} is bounded.  
If in addition $M(\lambda )$ is not the zero function, then
except possibly for a discrete set of $\lambda \in \complex $
there are linearly independent solutions $y_1(x,\lambda ),y_2(x,\lambda )$ satisfying
\[ \lim_{x \to x_{\infty }} y_1(x,\lambda ) = \beta \not= 0, \quad 
\lim_{x \to x_{\infty }} y_2(x,\lambda ) = 0. \] 

If $\sum_j l_j < \infty $ and $\delta (j) > 1$ infinitely often, then every solution satisfies 
\[ \lim_{j \to \infty } y'(x_j^-,\lambda ) =  \lim_{j \to \infty } y'(x_j^+,\lambda ) = 0,\] 
and
\[ \lim_{x \to x_\infty } y'(x,\lambda ) = 0, \quad x \notin \{ x_j \} . \]

\end{thm}

\begin{proof}
By virtue of \eqref{initdata}, for every $\lambda \in \complex $ the functions 
$c(x,\lambda )$, $s(x,\lambda )$, $c'(x,\lambda )$, and $s'(x,\lambda )$ are bounded.
Since $\lim_{j \to \infty} l_j = 0$, we find that 
\[ \lim_{x \to {x_ \infty}}  (c(x,\lambda ), s(x,\lambda ) ) 
= \lim_{j \to \infty} (c(x_j^+,\lambda ), s(x_j^+,\lambda ) ) = (M_{11}(\lambda ),M_{12}(\lambda )).\]

Write
\begin{equation} \label{ftc}
c'(x_{j+1}^- ) - c'(x_j^+)
= \int_{x_j^+}^{x_{j+1}^-} c''(t) dt
=  \int_{x_j^+}^{x_{j+1}^-} [q(t)-\lambda ]c(t) dt. 
\end{equation}
Since $c(t,\lambda )$ is bounded and $q(t)$ is integrable, the condition
$\sum _j l_j < \infty $, implies that 
\[ \lim _{j \to \infty} |c'(x_{j+1}^- ) - c'(x_j^+)| = 0. \]
The jump condition gives $c'(x_{j}^+ ) = c'(x_j^-)/\delta (j)$, or
\[ \lim_{j \to \infty} c'(x_{j}^-)/\delta (j) = \lim_{j \to \infty} c'(x_j^+) 
= \lim_{j \to \infty} c'(x_{j+1}^-) = \lim_{j \to \infty} c'(x_{j}^-), \]
which forces $\lim _{j \to \infty} c'(x_j^-), = 0$ if $\delta (j) > 1$ infinitely often, 
and consequently  $\lim _{x \to x_{\infty}} c'(x) = 0$. 
The argument is the same for $s(x,\lambda )$, and so  \\
$ \lim _{x \to x_\infty } y'(x,\lambda ) = 0$ for any solution $y$. 

Thus $M_{21}(\lambda ) = 0 = M_{22}(\lambda )$.  By assumption $M(\lambda )$ is not
identically $0$.  If, for instance, $M_{11}(0) \not= 0$, the function $c(x,\lambda )$ satisfies
\[ \lim_{x \to x_{\infty }} c(x,\lambda ) = M_{11}(\lambda ) \not=0, \]
except possibly for a discrete set of $\lambda \in \complex $. 
Thus we may take $y_1(x,\lambda ) = c(x,\lambda )$, and $y_2(x,\lambda )$
may be selected from the null space of the functional $y(x_{\infty })$.

\end{proof}

An additional growth estimate will be useful when the distribution of eigenvalues is considered.

\begin{thm} \label{exptype}
Suppose that $\sum l_j < \infty $ and $q$ is integrable.  Then the matrix function
$M(\lambda )$ is entire of order $1/2$.
\end{thm}

\begin{proof}
It will suffice to establish the desired estimate for the function 
$M_1(\omega ) = \prod \tau _j R_j(\omega )$.
Let 
\[ F_j = \begin{pmatrix} \cos(\omega l_j) & \sin(\omega l_j) \cr  
-\sin(\omega l_j) & \cos(\omega l_j) \end{pmatrix} , \quad l_j = x_j - x_{j-1} ,\]
and define $G_j = R_j - F_j$.  Then we have
\[ \| M_1(\omega ) \| \le \prod \| \tau _j R_j(\omega ) \| 
\le \prod \| R_j(\omega ) \| \le \prod [ \| F_j \| + \| G_j \| ] . \] 

Notice that the matrix $F_j$ is normal, with orthonormal eigenvectors
\[ \begin{pmatrix} 1/\sqrt{2} \cr i/\sqrt{2} \end{pmatrix}, \quad 
\begin{pmatrix} 1/\sqrt{2} \cr -i/\sqrt{2} \end{pmatrix}, \]
and eigenvalues $\exp(\pm i \omega l_j)$.  Thus
\[ \| F_j \| = e^{|\Im \omega l_j|}, \]
while the estimates of \lemref{4.2} give 
\[ \| G_j \| \le |\omega |^{-1}[\exp(\int_{x_{j-1}}^{x_j} |q| ) - 1]
e^{|\Im \omega l_j|}. \]

It follows that 
\[ \prod [ \| F_j \| + \| G_j \| ] 
\le e^{|\Im \omega \sum_j l_j|}
\prod [ 1 + |\omega |^{-1}(\exp(\int_{x_{j-1}}^{x_j} |q| ) - 1) ] , \] 
and the product on the right is convergent uniformly for $|\omega | \ge 1$ since 
\[ |\exp(\int_{x_{j-1}}^{x_j} |q| ) - 1| = O(\int_{x_{j-1}}^{x_j} |q| ) , \] 
which is summable.  As desired, there is a constant $C_1$ such that 
\[ \| M_1(\omega ) \| \le C_1 e^{|\Im \omega \sum_j l_j|} . \]

\end{proof}

\section{Operator Theory}

\subsection{Deficiency Indices}

This section is concerned with the identification of self adjoint
boundary-value problems for $-D^2+q$ on the interval ${\cal I}_n$.  
By means of the separation of variables results this will also provide
self adjoint operators on the tree.  When the function $q$ is bounded   
the theory of deficiency indices \cite{Dunford} is helpful.
This approach is used first.  More singular cases are then treated
for trees with finite volume.

In case $q$ is bounded, consider the symmetric operator $S=-D^2+q$ whose domain consists of 
smooth functions on ${\cal I}_n$ satisfying the jump conditions \eqref{jc},
having support in a finite set of intervals, and vanishing, along with
their derivatives, at $x_0$ and $x_\infty $.  
Recall that the dimensions of the deficiency subspaces $N(S^* - \lambda I)$
are constant for $\lambda $ with positive, respectively negative imaginary part.

As one can see using the ideas in \cite{Carlson98a}, elements of the
deficiency subspaces must be classical solutions of the differential equation
$-y'' + qy = \lambda y$ on each subinterval $[x_k,x_{k+1}]$ satisfying the jump conditions
\eqref{jc}, hence the dimension of each deficiency subspace is no bigger than $2$.
Since $S$ is bounded below, the deficiency indices are the same.

The operator $S$ may be extended to a symmetric operator 
$S_1$ by replacing the requirement that functions and their derivatives vanish at $x_0$
with the classical boundary condition
\[ af(0) + bf'(0) = 0 , \quad a,b \in \real , \quad a^2 +  b^2 > 0. \]
Since $S_1$ is a proper symmetric extension of the closure of $S$,
the deficiency indices of $S_1$ must be either $(1,1)$ or $(0,0)$.
To determine whether the operator is essentially self adjoint, or requires an
additional boundary condition `at $\infty $', it is necessary to consider 
bounds on the solutions to \eqref{1.a}.

The fact that solutions of the equation $-y'' + qy = \lambda y$
have limits as $x \to x_\infty $ allows us to determine the deficiency indices of $S_1$.
First of all, in the finite volume case $\sum N_jl_j < \infty$,
all solutions of this equation are square integrable.  This means that the 
deficiency indices of $S$ are $(2,2)$, and those of $S_1$ are $(1,1)$.  Now consider the case 
$\sum l_j < \infty$, $\sum N_jl_j = \infty$.
If $q=0$ then $M(\lambda )$ is not the zero function, and \thmref{4.5} says that
for all but a discrete set of $\lambda \in \complex $
there is a solution of \eqref{1.a} which has a nonzero limit at $x_{\infty}$.
Such a solution cannot be in $L^2({\cal I}_n,w)$, 
so the deficiency indices of $S$ in this case must be either $(0,0)$ or $(1,1)$.  
Since $S_1$ is a proper symmetric extension of $S$, it must be
essentially self adjoint.  The addition of the bounded operator
multiplication by $q$ does not change the self adjointness.
We summarize with the next result.

\begin{thm}
Suppose that $q$ is bounded, and the edge lengths $l_j$ satisfy $\sum l_j < \infty$.
If $\sum N_jl_j < \infty$ the deficiency indices of $S$ are $(2,2)$.  
If $\sum N_jl_j = \infty$ the deficiency indices of $S$ are $(1,1)$.  
In case $\sum N_jl_j = \infty$ the operator ${\cal L}_n = -D^2 + q$ whose domain 
is the set of functions in the domain of $S^*$ satisfying the boundary conditions
\[ af(x_0) + bf'(x_0) = 0 , \quad a,b \in \real , \quad a^2 + b^2 > 0, \]
is a self adjoint operator on $L^2({\cal I}_n,w)$.
\end{thm}

\subsection{Trees with finite volume}

For trees with finite volume it is particularly convenient to use the change of 
variables $x \to \xi $ of \eqref{defxi}.  This change of variables 
provides a Hilbert space isometry from $L^2(I_n,w)$ onto 
$L^2([\xi _0,\xi _\infty ),W^2(\xi ))$ since 
$$\int_{I_n} f(x)\overline{g(x)} w(x) \ dx 
= \int_{\xi _0}^{\xi _\infty} F(\xi )\overline{G(\xi )} W^2(\xi ) \ d\xi .$$
The quadratic form for the operator $S$ becomes
$$\int_{I_n} (|f'|^2 + q(x)|f|^2) w(x) \ dx 
=  \int_{\xi _0}^{\xi _\infty} (W(\xi )^{-2}|F'(\xi )|^2 + Q(\xi )|F(\xi )|^2) W^2(\xi ) \ d\xi .$$
If $q(x)$ is merely integrable rather than being bounded, the description of operator domains
becomes more delicate.  The quadratic form approach for singular ordinary differential operators
may be found in \cite[p. 343]{Kato}.

For our purposes it will be convenient to directly construct the Green's function
for the boundary-value problem
\begin{equation} \label{bvp}
-Y'' + W(\xi )^2[Q(\xi )-\lambda ]Y = W(\xi )^2F(\xi ),
\end{equation}
$$a_1Y(\xi _0) + b_1Y'(\xi _0) = 0, \quad a_2Y(\xi _\infty ) + b_2Y'(\xi _\infty ) = 0.$$
where $a_i,b_i \in \real $ and $a_i^2 + b_i^2 > 0$.  

Let ${\cal D}$ denote the set of functions $G \in L^2([\xi _0,\xi _\infty ),W^2(\xi ))$
which are continuous, with absolutely continuous derivative, and such that
$[W(\xi )^{-2}D^2 + Q]G \in L^2([\xi _0,\xi _\infty ),W^2(\xi ))$.

\begin{thm} \label{regprob} 
Assume that $\sum N_jl_j < \infty$ and that $W^2(\xi )Q(\xi )$ 
is integrable on $[\xi _0,\xi _\infty )$.
The functions in ${\cal D}$ which also satisfy
a set of boundary conditions in \eqref{bvp} is a domain
on which the operator ${\cal L}_n = W(\xi )^{-2}D^2 + Q$ is self adjoint with compact resolvent
on $L^2([\xi _0,\xi _\infty ),W^2(\xi ))$. 
\end{thm}

\begin{proof}  Since the argument is straightforward the proof is merely outlined.
As noted earlier, \eqref{xieqn} may be treated as a regular problem on the finite
interval $[\xi _0,\xi _\infty )$.  If $U(\xi ,\lambda )$ and $V(\xi ,\lambda )$ 
are nontrivial solutions of \eqref{xieqn} satisfying the boundary conditions 
at $\xi _0$ and $\xi _\infty $ respectively, 
then the solution $Y(\xi ,\lambda )$ of \eqref{bvp} may be written as \cite[p.309]{BR} 
\begin{equation} \label{Green}
Y(\xi ,\lambda ) = 
\int_{\xi _0}^{\xi _\infty} G(\xi ,\eta ,\lambda )W^2(\eta )F(\eta ) \ d \eta ,
\end{equation}
with
$$G(\xi ,\eta ,\lambda ) = \Bigl \{ \begin{matrix}
U(\xi )V(\eta )/\sigma , & \xi _0 \le \xi \le \eta , \cr 
U(\eta )V(\xi )/\sigma , & \eta \le \xi \le \xi _\infty , 
\end{matrix} \Bigr \} \quad \sigma = VU' - UV'.$$

As in the classical case eigenvalues of \eqref{bvp} must be real, 
and are the roots of a nontrivial entire function.  Except at the eigenvalues
$\sigma \not= 0$, and the functions $U(\xi ,\lambda )$ and $V(\xi ,\lambda )$ 
are bounded on $[\xi _0,\xi _\infty )$.  The condition 
$\sum N_jl_j < \infty$ implies that bounded measurable functions are in
$L^2([\xi _0,\xi _\infty ),W^2(\xi ))$.
If $F \in L^2([\xi _0,\xi _\infty ),W^2(\xi ))$ then $Y(\xi )$ given by
\eqref{Green} is bounded by the Cauchy-Schwartz inequality.
That is, except at eigenvalues of \eqref{bvp} the integral operator ${\cal G}(\lambda )$ defined by
\eqref{Green} is bounded on $L^2([\xi _0,\xi _\infty ),W^2(\xi ))$, 
and is self adjoint for $\lambda \in \real $.  The range of ${\cal G}(\lambda )$
defines a domain on which $W(\xi )^{-2}D^2 + Q$ is self adjoint, and ${\cal G}(\lambda )$
is its resolvent, which is compact since the spectrum is discrete.
\end{proof}

Suppose the hypotheses of \thmref{regprob} hold.
The explicit formula shows that the resolvents for the boundary-value problems on
$[\xi _0, \xi _{\infty})$ are the strong limits of the resolvents \cite[pp. 284--290]{RS1} 
obtained by imposing the right endpoint conditions at $\xi _j$, and taking the limit as $j \to \infty $.  
This gives the sense in which radial Schr\"odinger operators on infinite trees are the limits
of finite tree operators.

To characterize the distribution of eigenvalues, let
$n(r)$ be the number of eigenvalues $\lambda _m$ with $|\lambda _m| \le r$.

\begin{thm}
The eigenvalues $\lambda _m$ of an operator ${\cal L}_n$ as described in
\thmref{regprob}, counted with multiplicity, satisfy
\[ n(r) \le O(r^{1/2 + \epsilon }) \]
for every $\epsilon > 0$.
\end{thm}

\begin{proof}
The function $a_2z(\xi _{\infty },\lambda ) + b_2 z'(\xi _{\infty},\lambda )$ 
whose roots are the eigenvalues of ${\cal L}$, is entire of order $1/2$ by \thmref{exptype}.
Since each eigenvalue has multiplicity at most $2$, the result follows from the analogous result for
the roots of an entire function of order $1/2$, \cite[p. 64]{Young}. 
\end{proof}

Eigenvalue distributions for a wide variety of weighted Laplacians on trees 
are studied in \cite{Naimark}.

Except for the implicit consequences of \propref{loweval}, we have not obtained any
description of the dependence of the eigenvalues of ${\cal L}_n$ on $n$. 
In some special cases there is a simple spectral mapping relating the eigenvalues of 
${\cal L}_n = -D^2$ and ${\cal L}_{n+k} = -D^2$.
Suppose that for some $k > 0$ and $0 < r < 1$ the branching indices and lengths satisfy 
\[ \delta (n+k) = \delta (n), \quad l_{n+k} = rl_n, \quad n = 1,2,\dots , \]
and that $\sum l_jN_j < \infty $.
Assume that functions $f$ in the domain of ${\cal L}_n$ 
satisfy the left endpoint boundary condition $f(x_n) = 0$. 
The conditions at $x_{\infty}$, chosen independent of $n$, are either $f(x_{\infty}) = 0$ or
$\lim_{j \to \infty} N_jf'(x_j) = 0$.  
The mapping 
\[ Y(x) = y(x_{n+k} + r(x - x_n)) \]
takes functions $y$ in the domain of ${\cal L}_{n+k}$ to $Y$
in the domain of ${\cal L}_n$.  If $y$ is an eigenfunction for ${\cal L}_n = -D^2$ with 
eigenvalue $\lambda $, then $Y$ is an eigenfunction for ${\cal L}_{n+k} = -D^2$ with   
eigenvalue $r^2\lambda $, and conversely.
For these cases we obtain the relation
\[{\rm spec}({\cal L}_n) = r^2{\rm spec}({\cal L}_{n+k}) .\] 

\subsection{Mixed boundary conditions on the tree}

In closing it is interesting to note that 
the techniques which have been developed here to reduce radial operators ${\cal L}$ on $L^2({\cal T})$ 
may also be employed to consider operators with mixed boundary conditions `at infinity'.
As before the tree ${\cal T}$ and the potential $q$ are radial.  Suppose that
$v(i)$, for $i = 1,\dots ,N_j,$ are the distinct vertices with combinatorial depth $j > 0$.
Assume that $\sum N_jl_j < \infty$, and that for each $i$ a boundary condition
\[ a_i F(\xi _{\infty} ) + b_i F'(\xi _{\infty}) = 0 , \quad a_i,b_i \in \real \]
is given.  

The determination of a domain for this operator begins with a change of the interior vertex
conditions at the vertices $v(i)$.  We impose the Dirichlet conditions
$f(v(i)) = 0$ rather than the conditions \eqref{bc}.  This new set of conditions decouples
the tree into a finite collection of $N_j + 1$ subtrees, each of which is a radial tree
with roots $v(i)$ or $R$.  The operator $-D^2 + q$ is now radial on each subtree,
so the previous analysis can be employed to identify self adjoint domains.

To recover the operator with mixed boundary conditions `at infinity',
first take the orthogonal direct sum of self adjoint operators
from the subtrees.  Now replace the Dirichlet boundary conditions 
at the vertices $v(i)$ with the original interior vertex conditions.
The two operators are finite symmetric extensions of a common operator,
and the techniques of \cite[p. 188]{Kato} show that they are both self adjoint.  

% \bibliography{refer}


\begin{thebibliography}{10}

\bibitem{Ahl}
L.~Ahlfors.
\newblock {\em Complex Analysis}.
\newblock McGraw-Hill, New York, 1966.

\bibitem{vonBelow}
J.~von~Below.
\newblock Classical solvability of linear parabolic equations on networks.
\newblock {\em Journal of Differential Equations}, 72:316--337, 1988.

\bibitem{BR}
G.~Birkhoff and G~Rota.
\newblock {\em Ordinary Differential Equations}.
\newblock Blaisdell, Waltham, 1969.

\bibitem{Carlson96a}
R.~Carlson.
\newblock Inverse eigenvalue problems on directed graphs.
\newblock {\em Transaction of the American Mathematical Society}, 351(10):4069--4088, 1999.

\bibitem{Carlson97}
R.~Carlson.
\newblock Hill's equation for a homogeneous tree.
\newblock {\em Electronic Journal of Differential Equations}, 1997(23):1--30, 1997.

\bibitem{Carlson98a}
R.~Carlson.
\newblock Adjoint and self adjoint differential operators on graphs.
\newblock {\em Electronic Journal of Differential Equations}, 1998(6):1--10, 1998.

\bibitem{Chung}
F.~Chung.
\newblock {\em Spectral Graph Theory}.
\newblock American Mathematical Society, Providence, 1997.

\bibitem{CL}
E.A.~Coddington and N.~Levinson.
\newblock {\em Theory of Ordinary Differential Equations}.
\newblock McGraw-Hill, New York, 1955.

\bibitem{Dunford}
N.~Dunford and J.~Schwartz.
\newblock {\em Linear Operators, Part II}.
\newblock Interscience, New York, 1964.

\bibitem{Evans}
W.D.~Evans and Y.~Saito.
\newblock Neumann Laplacians on domains and operators on associated trees.
\newblock {\em preprint}, 1998.

\bibitem{Exner}
P.~Exner and P.~Seba.
\newblock Electrons in semiconductor microstructures: a challenge to operator theorists.
\newblock {\em Proceedings of the Workshop on Schr\"odinger Operators, Standard and Nonstandard
(Dubna 1988)}.
\newblock World Scientific, Singapore 1989; pp. 79--100.

\bibitem{Friedlin}
M.~Friedlin and A.~Wentzell.
\newblock Diffusion processes on graphs and the averaging principle.
\newblock {\em Annals of Probability}, 21(4):2215--2245, 1993.

\bibitem{Fulton}
C.~Fulton and S.~Pruess.
\newblock Eigenvalue and eigenfunction asymptotics for regular
  {S}turm-{L}iouville problems.
\newblock {\em Journal of Mathematical Analysis and Applications},
  188(1):297--340, 1994.

\bibitem{Gerasimenko}
N.~Gerasimenko and B.~Pavlov.
\newblock Scattering problems on non-compact graphs.
\newblock {\em Theor. Math. Phys.}, 75:230--240, 1988.

\bibitem{Har}
P.~Hartman.
\newblock {\em Ordinary Differential Equations}.
\newblock John Wiley and Sons", 1973.

\bibitem{Kato}
T.~Kato.
\newblock {\em Perturbation Theory for Linear Operators}.
\newblock Springer-Verlag, New York, 1995.

\bibitem{Koranyi}
A.~Koranyi. (ed)
\newblock {\em Harmonic Functions on Trees and Buildings}.
\newblock American Mathematical Society, Providence, 1997.

\bibitem{Kuchment}
P.~Kuchment and H. Zeng.
\newblock Convergence of spectra of mesoscopic systems collapsing onto a graph.
\newblock {\em preprint}, 1999.

\bibitem{Lumer}
G.~Lumer.
\newblock Connecting of local operators and evolution equations on networks.
\newblock {\em in Potential Theory Copenhagen 1979}.
\newblock Lect. Notes Math 787, Springer, 1980; pp. 219--234.

\bibitem{Massey}
W.~Massey.
\newblock {\em Algebraic Topology: An Introduction}.
\newblock Harcourt, Brace and World, New York, 1967.

\bibitem{Montroll1}
E.~Montroll.
\newblock Quantum Theory on a Network I.
\newblock {\em J. Math. Phys.}, 11:635--648, 1970.

\bibitem{Montroll2}
R.~Mills and E.~Montroll.
\newblock Quantum Theory on a Network II.
\newblock {\em J. Math. Phys.}, 11:2525--2538, 1970.

\bibitem{Naimark}
K.~Naimark and M.~Solomyak.
\newblock Eigenvalue estimates for the weighted Laplacian on metric trees.
\newblock {\em Proc. London Math. Soc.}, 80(3):690--724, 2000.

\bibitem{Nicaise}
S.~Nicaise.
\newblock Some results on spectral theory over networks, applied to nerve impulse transmission.
\newblock {\em Polynomes orthogonaux et applications}.
\newblock Lect. Notes Math 1171, Springer, 1985; pp. 532--541.

\bibitem{Novikov}
S.~Novikov.
\newblock Schr\"odinger operators on graphs and symplectic geometry.
\newblock {\em The Arnoldfest: Proceedings of a Conference in Honor of V.I. Arnold
for his Sixtieth Birthday}.
\newblock AMS 1999.

\bibitem{Pauling}
L.~Pauling.
\newblock The diamagnetic anisotropy of aromatic molecules.
\newblock {\em J. Chem. Phys.}, 4:673--677, 1936.

\bibitem{Pos}
J.~P\"oschel and E.~Trubowitz.
\newblock {\em Inverse Spectral Theory}.
\newblock Academic Press, Orlando, 1987.

\bibitem{RS1}
M.~Reed and B.~Simon
\newblock {\em Methods of Modern Mathematical Physics, 1}.
\newblock Academic Press, New York, 1972.

\bibitem{RS2}
M.~Reed and B.~Simon
\newblock {\em Methods of Modern Mathematical Physics, 2}.
\newblock Academic Press, New York, 1975.

\bibitem{RS4}
M.~Reed and B.~Simon
\newblock {\em Methods of Modern Mathematical Physics, 4}.
\newblock Academic Press, New York, 1980.

\bibitem{Romanov}
R.~Romanov and G.~Rudin
\newblock Scattering on p-adic graphs.
\newblock {\em Computers Math. Applic.}, 34(5/6):587--597, 1997.

\bibitem{Royden}
H.~Royden.
\newblock {\em Real Analysis}.
\newblock Macmillan, New York, 1988.

\bibitem{Rubinstein}
J.~Rubinstein and M.~Schatzman.
\newblock Variational problems on multiply connected thin strips I: Basic estimates and 
convergence of the Laplacian spectrum.
\newblock {\em preprint}, 1999.

\bibitem{Ruedenberg}
K.~Ruedenberg and C.W.~Scherr.
\newblock Free-electron network model for conjugated systems I, Theory. 
\newblock {\em J. Chem. Physics}, 21(9):1565--1581, 1953.

\bibitem{Saito}
Y.~Saito.
\newblock The limiting equation of the Neumann Laplacians on shrinking domains.
\newblock {\em preprint}, 1999.

\bibitem{Young}
R.~Young.
\newblock {\em An Introduction to Nonharmonic Fourier Series}.
\newblock Macmillan, New York, 1988.

\end{thebibliography}

\noindent{\sc Robert Carlson }\\
University of Colorado at Colorado Springs  \\
Colorado Springs, CO 80933, USA\\
e-mail: carlson@math.uccs.edu  

\end{document}