\documentstyle[twoside]{article}
\pagestyle{myheadings}
\markboth{\hfil Adjoint and self-adjoint differential operators
\hfil EJDE--1998/06}{EJDE--1998/06\hfil Robert Carlson \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~06, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
Adjoint and self-adjoint differential operators on graphs
\thanks{ {\em 1991 Mathematics Subject Classifications:} 34B10, 47E05.
\hfil\break\indent
{\em Key words and phrases:} Graph, differential operator, adjoint,
self-adjoint extension.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted August 24, 1997. Published February 26, 1998.} }
\date{}
\author{Robert Carlson}
\maketitle
\begin{abstract}
A differential operator on a directed graph with weighted
edges is characterized as a system of ordinary differential
operators. A class of local operators is introduced to
clarify which operators should be considered as defined on the graph.
When the edge lengths have a positive lower bound,
all local self-adjoint extensions of the minimal symmetric
operator may be classified by boundary conditions at the vertices.
\end{abstract}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
Although there is a large body of literature on the spectral theory of linear
difference operators associated with a combinatorial graph \cite{Chung},
the study of differential operators on a topological graph has received
much less attention. This situation has begun to change, due in large
part to quantum-mechanical problems associated with advances in micro-electronic
fabrication \cite{Avron2,Exner1,Exner3,Ger1}. In developing physical models
one often needs to know when a differential operator is essentially self adjoint
on a given domain.
This paper provides a description of adjoints, and considers domains of essential
self adjointness for a class of differential operators on weighted directed graphs.
These differential operators ${\cal L}$ are actually a (possibly infinite) system of
ordinary differential operators on intervals whose lengths are given by the
edge weights of the graph ${\cal G}$. For regular ordinary differential
operators acting
on $L^2[a,b]$ there is a classical description of adjoints and self-adjoint
extensions in terms of boundary conditions \cite[pp. 284--297]{Cod}. This theory
has a close connection with the abstract treatment of self-adjoint extensions of symmetric
operators \cite[pp. 140--141]{RS2}. The general treatment is somewhat deficient
for differential operators on graphs, since the role of the vertices of the graph ${\cal G}$
is unclear. When there are infinitely many vertices the description of extensions
appears particularly awkward.
To remedy these problems, we will impose an additional restriction on the domain
of an operator ${\cal L}$. Let $\phi :{\cal G} \to {\cal C} $ denote a $C^\infty $
function which has compact support in ${\cal G}$ and is constant in an open neighborhood
of each vertex. We say that ${\cal L}$ is a local operator if for every $\phi $,
$\phi f$ is in the domain of ${\cal L}$ whenever $f$ is.
We will see that local operators have domains described
via boundary conditions which only compare boundary values at
endpoints which are identified with a single vertex of the
graph ${\cal G}$.
One result uses conditions at the vertices to characterize functions of
compact support in the domain of the adjoint of a local operator.
The main results assume that the edge lengths of ${\cal G}$ have a positive lower bound.
In this case there is a complete classification of local self-adjoint
operators ${\cal L}$ in terms of boundary conditions at the graph vertices
when the coefficients of the operator are bounded and satisfy some mild additional
regularity assumptions. A final application shows that Schr\"odinger operators
on a graph with $\delta -$ function interactions are essentially self adjoint
on a domain of functions of compact support.
\section{Local Differential Operators on Graphs}
In this work a graph ${\cal G}$ will have a countable vertex set ${\cal V}$
and a countable set of directed edges $e_n$.
Each edge has a positive weight (length) $w_n$.
Assume further that each vertex appears in at least one,
but only finitely many edges. The graph may have
loops and multiple edges with the same vertices.
A topological graph may be constructed using the graph data \cite[p.~190]{Massey}.
For each directed edge $e_n$ let $[a_n,b_n]$ be a real interval of length $w_n$,
and let $\alpha _m \in \{ a_n,b_n \} $.
Identify interval endpoints $\alpha _m$ if the corresponding edge endpoints are
the same vertex $v$, in which case we will write $\alpha _m \sim v $.
This topological graph, also denoted ${\cal G}$, is assumed to be connected.
The Euclidean metric on the intervals may be extended to a metric on ${\cal G}$
by taking the distance between two points to be the length of the shortest
(undirected) path joining them. Notice that every compact
set $K \subset {\cal G}$ is contained in a finite union of closed edges $e_n$,
since $K$ has a covering by open sets which hit only finitely many edges.
Let $L^2({\cal G})$ denote the Hilbert space $\oplus _n L^2(e_n)$
with the inner product
$$\langle f, g \rangle = \int_{\cal G} f\overline g
= \sum_n \int_{a_n}^{b_n} f_n(x)\overline{g_n(x)} \ dx ,
\quad f = (f_1, f_2, \dots ).$$
A differential operator ${\cal L}$ acts componentwise on functions $f \in L^2({\cal G})$
in its domain,
$${\cal L}f = \sum_{j=0}^M c_j(x)f^{(j)}(x).$$
The leading coefficient $c_M$ is nowhere $0$ and $c_j$ is a $j$ times continuously
differentiable complex valued function on each interval $[a_n,b_n]$.
The associated formal operator is
$$L = \sum_{j=0}^M c_j(x)D^j, \quad D = {d \over dx}.$$
The domain of ${\cal L}$, denoted ${\rm Dom}({\cal L})$, will always include
${\cal D}_{\rm min}$, the linear span of $C^\infty $ functions supported in the interior
of a single interval $(a_n,b_n)$.
The domain of ${\cal L}$ will be contained in ${\cal D}_{\rm max}$ (which depends on $L$),
the set of functions $f\in L^2({\cal G})$ with $f_n,\dots ,f_n^{(M-1)}$ continuous
and $f_n^{(M-1)}$ absolutely continuous on $[a_n,b_n]$, and $Lf \in L^2({\cal G})$.
A convenient reference for differential operators on $L^2[a,b]$
is \cite[pp.~1278--1310]{Dunford}.
The development there assumes that $c_j \in C^{\infty }$, but this distinction
is unimportant. In addition, these authors assume a somewhat larger minimal domain
for the operators. This is also inconsequential since ${\cal L}$ is
closable \cite[p. 168]{Kato}, and the closure of ${\cal L}$ will have a domain
\cite[pp. 169--171]{Kato} which includes the functions $f \in {\cal D}_{\rm max}$
which are supported on an interval $[a_n,b_n]$, and which satisfy
$$f_n^{(j)}(a_n) = 0 = f_n^{(j)}(b_n), \quad j=0,\dots ,M-1 .$$
If ${\cal L}_{\rm min}$ has the domain ${\cal D}_{\rm min}$, then the adjoint operator ${\cal L}_{\rm min}^*$
will again be a differential operator. By working on one interval $[a_n,b_n]$ at a time,
and using the classical theory \cite[p. 1294]{Dunford}, \cite[pp. 169--171]{Kato},
one may obtain the following result.
\begin{lemma} A function $f$ is in the domain of the adjoint operator ${\cal L}_{\rm min}^* $,
if and only if $f \in {\cal D}_{\rm max}$ for $L^+$, where
$$L^+ = \sum_{j=0}^M (-1)^jD^j\overline{c_j(x)}
= \sum_{j=0}^M (-1)^j \sum_{i=0}^j {j \choose i} \overline{c_j^{(j-i)}(x)}D^i .$$
If $f \in {\rm Dom}( {\cal L}_{\rm min}^*) $, then ${\cal L}_{\min}^*f = L^+f$.
\end{lemma}
If $\alpha _m \in \{ a_n,b_n \} $,
then the functionals $f^{(j)}(\alpha _m)$, for $j=0,\dots ,M-1$
are continuous \cite[pp. 1297--1301]{Dunford}
on ${\rm Dom}({\cal L})$ when the domain is given the norm
$\| f \| _{\cal L} = [\| f \| _2 + \| {\cal L}f \| _2 ] ^{1/2}$.
Say that $\beta _v$ is a vertex functional at $v$ if $\beta _v$ is a
linear combination of $f^{(j)}(\alpha _m)$ for $j=0,\dots ,M-1$, and $\alpha _m \sim v$.
A (homogeneous) vertex condition at $v$ is a equation of the form
$\beta _v(f) = 0$.
Whether or not ${\cal L}$ is local, there will always be a
(complex) vector space ${\cal B}_v$ of vertex functionals $\beta _v$ at $v$
such that every function $f$ in ${\rm Dom}({\cal L})$ satisfies $\beta _v(f) = 0$.
If ${\cal L}$ is local and closed, these vertex conditions will
give a local description of functions in ${\rm Dom}({\cal L})$.
Let ${\cal D}_{\rm com}$ be the set of functions of compact support
in ${\cal D}_{\rm max}$.
\begin{lemma} \it Suppose that ${\cal L}$ is local and closed. If
$f \in {\cal D}_{\rm com}$ and $\beta _v(f)= 0$ for all $\beta _v \in {\cal B}_v$
and all $v \in {\cal V}$, then $f$ is in the domain of ${\cal L}$.
\end{lemma}
\paragraph{Proof} Fix the vertex $v$, and let $\delta (v)$ be its degree.
Consider the range of the linear map from ${\rm Dom}({\cal L})$ to $C^{M\delta (v)}$,
which sends $g$ to boundary values
$$g^{(j)}(\alpha _m), \quad j = 0,\dots , M-1, \quad \alpha _m \sim v .$$
If this subspace did not include the vector of values $f^{(j)}(\alpha _m)$
there would be a vertex functional at $v$ which annihilated ${\rm Dom}({\cal L})$,
but not $f$. Since this contradicts the assumptions on $f$, there is
some $g_v \in {\rm Dom}({\cal L})$ satisfying
$$g_v^{(j)}(\alpha _m) = f^{(j)}(\alpha _m), \quad j=0,\dots ,M-1, \quad \alpha _m \sim v .$$
Since ${\cal L}$ is local, we may assume that $g_v$ has compact support and
vanishes in a neighborhood of every other vertex.
Since $f$ has compact support, there is a finite collection of vertices $v$
for which $f^{(j)}(\alpha _m) \not= 0$, for some $ 0 \le j < M$ , and $\alpha _m \sim v $.
Thus there is a function $g \in {\rm Dom}({\cal L})$ of compact support, such that
$f^{(j)}(\alpha _m)= g^{(j)}(\alpha _m)$ for $j=0,\dots ,M-1,$
at every endpoint $\alpha _m$. Since ${\cal L}$ is
closed and ${\cal D}_{\rm min} \subset {\rm Dom}({\cal L})$,
we find that $f - g$, and thus $f$, are in ${\rm Dom}({\cal L})$.
\hfill $\Box $\smallskip
Before turning to the description of the domain for the adjoint of a local
operator ${\cal L}$, some additional ideas are reviewed.
Suppose $f,g \in {\cal D}_{\rm max}$, with the support of $g$ in an open ball
containing at most one vertex $v$.
Then integration by parts \cite[p. 285]{Cod} leads to
$$\langle Lf,g \rangle - \langle f,L^+g \rangle = [f,g]_v$$
where $[f,g]_v$ is a nondegenerate form in the boundary values of $f$ and $g$ at the
$\alpha _m \sim v$.
Consider the second order case $Lf = f'' + c_1f' + c_0f$.
On $[a_n,b_n]$ we have, without restrictions on the support of $f$ and $g$,
\begin{eqnarray*}
\int_{a_n}^{b_n} \Bigl [ {\overline g}Lf - f\overline{L^+g} \Bigr ]
&=& f'(b_n){\overline g(b_n)} - f'(a_n){\overline g(a_n)}
+ f(a_n){\overline g'(a_n)} - f(b_n){\overline g'(b_n)} \\
&&+ f(b_n)c_1(b_n){\overline g(b_n)} - f(a_n)c_1(a_n){\overline g(a_n)}.
\end{eqnarray*}
If $g$ vanishes outside of a small neighborhood of $v$, and
$$\sigma _m = \left\{\begin{array}{ll}
0, & \alpha _m = b_m\,,\\
1, & \alpha _m = a_m\,, \end{array} \right. $$
then
$$[f,g]_v = \sum_m (-1)^{\sigma _m} \Bigl [
f'(\alpha _m){\overline g(\alpha _m)} - f(\alpha _m){\overline g'(\alpha _m)}
+ f(\alpha _m)c_1(\alpha _m){\overline g(\alpha _m)} \Bigr ]\,,$$
with $\alpha _m \sim v$.
At each $v$ pick an ordering $\alpha _1,\dots ,\alpha _{\delta (v)}$
of the $\alpha _m \sim v$, and for $f \in {\cal D}_{\rm max}$
let $\hat f \in C^{M\delta (v)}$ be the vector with components
$$\hat {f}_{j\delta (v) + k} = f^{(j)}(\alpha _k),
\quad j=0,\dots ,M-1, \quad k = 1,\dots , \delta (v).$$
With respect to this basis there is an invertible $M\delta (v) \times M\delta (v)$
matrix ${\cal S}_v$ such that
$$[f,g]_v = {\cal S}_v\hat f \bullet \hat g. \eqno (2.a) $$
where $\bullet $ denotes the usual dot product on $C^{M\delta (v)}$.
Single vertex conditions may now be written as
$$\sum b_{j,k}f^{(j)}(\alpha _k) = \sum b_{j,k}{\hat f}_{j\delta (v) + k} = 0,$$
and a maximal independent set of vertex conditions at $v$ may be written more
compactly as
$B_v\hat f = 0$, where $B_v$ is a $K(v) \times M\delta (v) $ matrix with
linearly independent rows.
Since the null space $N(B_v) \in {\cal C} ^{M\delta (v)}$ has dimension
$M\delta (v) - K(v)$, there is an $[M\delta (v) - K(v)] \times M\delta (v)$ matrix $B_v^+$,
such that
$$B_v^+X = 0 \quad {\rm if \ and \ only \ if} \quad {\cal S}_v^* X \in N(B_v)^{\perp }
, \quad X \in C^{M\delta (v)}. \eqno (2.b)$$
Call any such matrix $B_v^+$ a complementary matrix to $B_v$, and the vertex
conditions $B_v^+ \hat f = 0$ complementary boundary conditions.
\section{Domains of adjoint operators}
If ${\cal L}$ is local, functions in the domain of the adjoint operator ${\cal L}^*$
must also satisfy vertex conditions. The treatment of an operator defined
on a single interval may be found in \cite[pp.~284--297]{Cod}.
We have taken advantage of some refinements worked out in \cite{Cod1}.
Find a basis $z_1,\dots ,z_{M\delta - K(v)}$ for $N(B_v)$, and let
$Z_v$ be the $M\delta (v) \times [M\delta (v) - K(v)]$ matrix whose columns are $z_j$.
\begin{theorem} \it Suppose that ${\cal L}$ is local,
and that the vertex
conditions at $v$ annihilating the domain of ${\cal L}$ are written as
$$B_v \hat f = 0,$$
where $B_v$ is a $K(v) \times M\delta (v)$ matrix,
with linearly independent rows.
Then the adjoint ${\cal L}^*$ is local and closed.
A function $g \in {\cal D}_{\rm com}$ is in the domain of ${\cal L}^*$
if and only if $B_v^+\hat g = 0$ for a set of vertex conditions
complementary to the conditions $B_v \hat f=0$.
A matrix $B_v^+$ is complementary to $B_v$ if and only if
$B_v^+$ is $[M\delta (v) - K(v)] \times M\delta (v)$,
with linearly independent rows, and the equations
$$B_v^+ [{\cal S}_v^*]^{-1} (B_v^*) = 0 $$
are satisfied. One such matrix is $B_v^+ = ({\cal S}_vZ_v)^*$.
\end{theorem}
\paragraph{Proof} If $g \in {\rm Dom}({\cal L}^*)$ then $g \in {\rm Dom}({\cal L}_{\rm min}^*)$,
so by Lemma 2.1 ${\cal L}^*g = L^+g$, and
$$\langle Lf,g \rangle = \langle f,L^+g \rangle , \quad f \in {\rm Dom}({\cal L}).$$
Since ${\cal L}$ is local, any vertex values $\hat f$ at $v$
satisfying $B_v \hat f = 0$ are the vertex values of some $f \in {\rm Dom}({\cal L})$
which has compact support and $0$ is in an open neighborhood of every vertex except $v$.
For such $f$,
$$\langle Lf,g \rangle - \langle f, L^+g \rangle = 0 = [f,g]_v.$$
By (2.a) we have ${\cal S}_v^* \hat g \in N(B_v)^{\perp }$, and by
(2.b) the equations $B_v^+ \hat g = 0$ are satisfied for any matrix complementary
to $B_v$. Now if $\phi $ has compact support
and constant in neighborhood of each vertex, then $\phi g \in {\cal D}_{\rm com}$
with $B_v^+\hat \phi g = 0$. This implies that $\phi g \in {\rm Dom}({\cal L}^*)$ and
${\cal L}^*$ is local, and more generally
that $g \in {\cal D}_{\rm com}$ is in the domain of ${\cal L}^*$
if and only if $B^+\hat g = 0$.
In addition, adjoint operators are always closed.
What remains is to characterize the matrices $B_v^+$ complementary to $B_v$.
The vector $\hat g$ will satisfy the vertex conditions of a
function in ${\rm Dom}(L^*)$ if and only if ${\cal S}_v^*\hat g \in N(B_v)^\perp $.
Since
$${\rm Ran}(Z_v) = N(B_v), \quad N(B_v)^\perp = {\rm Ran}(Z_v)^\perp = N(Z_v^*),$$
the condition on $\hat g$ is equivalent to $Z_v^*{\cal S}_v^*\hat g = 0$.
Thus we may take $B_v^+ = ({\cal S}_vZ_v)^*$.
To recognize more generally when a matrix $B_v^+$ is complementary to $B$,
start with the fact that this is equivalent to requiring that
$\hat g \in N(B_v^+)$ if and only if ${\cal S}_v^*\hat g \in N(B_v)^\perp $,
or $\hat g \in [{\cal S}_v^*]^{-1} N(B_v)^\perp $.
Thus we want
$N(B_v^+) = [{\cal S}_v^*]^{-1} {\rm ran}(B_v^*) $,
or that $B_v^+$ is a $[M\delta (v) - K(v)] \times M\delta (v)$ matrix
with linearly independent rows such that the equation
$B_v^+ [{\cal S}_v^*]^{-1} (B_v^*) = 0 $ is satisfied.
\hfill$\Box$\smallskip
The following observation about self-adjoint operators is a corollary of the last result.
\begin{corollary} Suppose that ${\cal L}$ is self adjoint and local,
with vertex conditions $B_v \hat f_v = 0$ as in Theorem 3.1.
Then each $B_v$ is an $[M\delta (v)/2] \times M\delta (v)$ matrix, and
$$B_v [{\cal S}_v^*]^{-1} (B_v^*) = 0 \eqno (3.a).$$
Conversely, suppose that $L = L^+$, and that vertex conditions
$B_v \hat f_v = 0$ are given at each vertex so that (3.a) is satisfied.
If each $B_v$ is an $[M\delta (v)/2] \times M\delta (v)$ matrix
with linearly independent rows, then the operator
${\cal L}$ with
$${\rm Dom}({\cal L}) = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0, \quad v \in {\cal V} \} $$
is symmetric, and has no symmetric extensions whose domain is a subset of
${\cal D}_{\rm com}$. \end{corollary}
The next lemma will help identify formal operators $L = L^+$
and vertex conditions such that ${\cal L}$ will be essentially self adjoint if
${\rm Dom}({\cal L}) = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0 \} $.
We will need some hypotheses on the coefficients of $L$,
and will require that the lengths $w_n$ of the edges
have a positive lower bound.
\begin{lemma} Suppose that $w_n \ge C > 0 $ for all $n$,
and that vertex matrices $B_v$ with independent rows are given.
Assume that the leading coefficient $|c_M|$ of $L$ is bounded below by a positive
constant, and that all coefficients of $L^+$ are uniformly bounded on
${\cal G}$.
Let ${\rm Dom}({\cal L}) = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0, \quad v \in {\cal V} \} $,
and let ${\cal L}^+$ be the restriction of ${\cal L}^*$ to
${\rm Dom}({\cal L}^+) = \{ f \in {\cal D}_{\rm com} \ | \ B_v^+ \hat f = 0, \quad v \in {\cal V} \} $
for matrices $B_v^+$ complementary to $B_v$.
Assume that there is a positive constant $\epsilon $, and a complex number
$\lambda $ such that
$$\| ({\cal L}-\lambda )f \| \ge \epsilon \| f \| ,
\quad f \in {\rm Dom}({\cal L}), \eqno (3.b)$$
$$\| ({\cal L}^+- \overline{\lambda } ) \| \ge \epsilon \| f \| ,
\quad f \in {\rm Dom}({\cal L}^+). \eqno (3.c)$$
Then the closure of ${\cal L}-\lambda $ has a bounded inverse.
\end{lemma}
\paragraph{Proof} Part of the method of proof is adopted from \cite[p. 274]{Kato}.
The inequality (3.b) extends to the closure of ${\cal L}-\lambda $, which is therefore
injective and boundedly invertible on its range. If the range is not dense
there must be a nontrivial vector $\psi $ in $N({\cal L}^* - \overline {\lambda})$.
We will assume the existence of $\psi $, and obtain a contradiction.
Pick a $C^\infty $ function $\eta (x)$ on $(0,C)$ which is $1$ in a neighborhood of $0$
and vanishes identically for $x > C/4$.
Pick any edge $e_0$, and for $K = 1,2,3, \dots $ construct a $C^\infty $ cutoff function
$\phi _K$ on ${\cal G}$ as follows.
On the set $E_0$ of (closed) edges containing some point whose distance from a vertex of $e_0$
is less than or equal to $K$, let $\phi _K = 1$.
On edges $e = [a_n,b_n]$ not in $E_0$ which share a vertex $v\sim a_n$ (resp. $v\sim b_n$)
with an edge in $E_1$,
let $\phi _K = \eta (x - a_n)$ (resp. $\phi _K = \eta (b_n - x)$)
where $\eta $ is defined. Otherwise let $\phi _K = 0$.
Since ${\cal L}^*$ is local, $\phi _K \psi \in {\rm Dom}({\cal L}^+)$.
A computation gives
$$[{\cal L}^+ - \overline \lambda ]\phi _K \psi
= \phi _K [{\cal L}^+ - \overline \lambda ] \psi + R_K $$
where the first term on the right hand side is $0$.
The term $R_K$ is a sum, in which each summand has as a factor
$\phi _K^{(j)}$ for $j \ge 1$. Thus we may write
$$R_K = \sum_{j 0 $ for all $n$,
and that $L = L^+$.
Assume that $|c_M|$ is bounded below by a positive constant,
and that all coefficients of $L$ are uniformly bounded.
If $[M\delta (v)/2] \times M\delta (v)$ vertex matrices $B_v$ are given
with linearly independent rows, and satisfying (3.a), and if ${\cal L}$ has domain
$${\rm Dom}({\cal L}) = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0, \quad v \in {\cal V} \}, $$
then ${\cal L}$ is essentially self adjoint.
Conversely, every local self-adjoint operator ${\cal L}_1$ formally given by such an $L$
whose domain includes ${\cal D}_{\rm min}$ is the closure of one of the
operators ${\cal L}$.
\end{theorem}
\paragraph{Proof} Since the vertex matrices $B_v$ are self complementary,
${\rm Dom}({\cal L}) \subset {\rm Dom}({\cal L}^*)$ by Theorem 3.1.
Since $L = L^+$, ${\cal L}$ is symmetric.
It then follows \cite[p. 270]{Kato} that
$$\| ({\cal L} \pm i)f \| \ge \| f \| .$$
By Lemma 3.3 the closures of $({\cal L} \pm i)$
are boundedly invertible, so \cite[p. 256]{RS1}
${\cal L}$ is essentially self adjoint.
On the other hand, if ${\cal L}$ is local and self adjoint, with
${\cal D}_{\rm min} \subset {\rm Dom}({\cal L})$, then by Corollary 3.2
and the first part of this theorem
there are self complementary vertex matrices $B_v$, and a domain
$${\cal D}_1 = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0, \quad v \in {\cal V} \} $$
such that ${\cal D}_1 \subset {\rm Dom}({\cal L})$
and the restriction of ${\cal L}$ to ${\cal D}_1$ is essentially self adjoint.
\section{Schr\"odinger operators on graphs}
For many applications of physical interest, the functions in ${\rm Dom}({\cal L})$
will be continuous at the vertices. This condition can be express as a
set of $\delta (v) - 1$ independent conditions at each vertex,
$$f_{\alpha _m}(v) = f_{\alpha _{m+1}}(v), \quad m=1,\dots ,\delta (v) - 1 .$$
We turn to the example of Schr\"odinger operators
$L = D^2+p$ where one additional vertex
condition will be needed to define a self-adjoint operator.
An independent vertex condition may be written as
$$\sum_{n=1}^{\delta (v)} d_n f'(\alpha _m) = \rho (v) f(v), \eqno (3.d)$$
with not all coefficients equal to $0$, and
where $f(v)$ is the common value of the $f(\alpha _m)$.
The example considered after Lemma 2.2 shows that for $L=D^2+p$
$$[f,g]_v =
\sum_n (-1)^{\sigma _n} \Bigl [
f'(\alpha _m){\overline g(\alpha _m)} - f(\alpha _m){\overline g'(\alpha _m)} \Bigr ],
\quad \alpha _m \sim v. $$
Working directly with this form, it is a simple exercise to
characterize the additional vertex conditions
with the property that all functions satisfying the vertex conditions
are annihilated by the form. The following result is thus obtained.
\begin{corollary} \it Suppose that $w_n \ge C > 0 $ for all $n$, and that
$L = D^2$.
The operator ${\cal L}$ whose vertex conditions $B_v\hat f = 0$
include the continuity conditions $f(\alpha _m) - f(\alpha _{m+1}) = 0 $ for
$1 \le m \le \delta (v) -1$ at each vertex $v \in {\cal G}$,
and one additional boundary condition of the form
$$\gamma \sum_{n=1}^{\delta (v)} (-1)^{\sigma _n} f'(\alpha _m) - \rho f(v) = 0,
\quad \rho ,\gamma \in R, \quad \rho ^2 + \gamma ^2 \not= 0, $$
will be essentially self adjoint on
${\rm Dom}({\cal L}) = \{ f \in {\cal D}_{\rm com} \ | \ B_v \hat f = 0, \quad v \in {\cal V} \}$.
Conversely every local self-adjoint operator ${\cal L}_1 = D^2$
whose domain includes ${\cal D}_{\rm min}$ and satisfies the continuity conditions
at every vertex is the closure of one of the operators ${\cal L}$.
\end{corollary}
One may immediately extend this corollary to $L = D^2 + p$ for a real bounded
measurable function $p$ by a standard perturbation result \cite[p. 287]{Kato}.
For operators on the real axis, these vertex conditions are known as
$\delta $(function) interactions. See an extensive treatment of such operators
in \cite{Albeverio}.
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\smallskip
{\sc Robert Carlson}\\
University of Colorado at Colorado Springs
Colorado Springs, CO 80933 USA\\
Email address: carlson@castle.uccs.edu
\end{document}