\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. 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