 Research
 Open access
 Published:
Structures on selfadjoint vertex conditions of local SturmLiouville operators on graphs
Boundary Value Problems volumeÂ 2015, ArticleÂ number:Â 162 (2015)
Abstract
We study the vertex conditions of local SturmLiouville operators on metric graphs. Our aim is to give a new description of vertex conditions defining the selfadjoint SturmLiouville operators and to clarify the natural geometric structure on the space of complex vertex conditions. Based on this description, we give the selfadjointness results for local SturmLiouville operators on finite graphs and the PovznerWienholtztype selfadjointness results for local SturmLiouville operators on infinite graphs.
1 Introduction
A graph Î“ we consider in this paper is an ordered pair of disjoint sets \((V,E)\), where V is a countable vertex set and E is a countable edge set. Moreover, the graph is connected and each edge \(e_{n}\) has a positive length \(\vert e_{n}\vert \).
A SturmLiouville operator on the graph Î“ is actually a system of SturmLiouville operators on intervals complemented by appropriate matching conditions at inner vertices and some boundary conditions at the boundary vertices. The matching conditions and the boundary conditions are collectively called vertex conditions. The SturmLiouville operators on \(L_{w}^{2}(\Gamma,\mathbb{C} )\) are generated by the expression
where \(1/p,q,w\in L_{\mathrm{loc}}^{1}(\Gamma,\mathbb{R} )\) and \(w>0\) a.e. on Î“.
If V and E are finite sets, then the description of the selfadjoint vertex conditions can be treated as the description of the boundary conditions in selfadjoint multiinterval SturmLiouville problems. (The results about multiinterval SturmLiouville problems can be found in [1].) For example, in [2], Harmer described the selfadjoint boundary conditions for the SchrÃ¶dinger operators on the finite graphs in terms of a unitary matrix. When the graph has infinitely many vertices, the general treatments for SturmLiouville operators on intervals and SturmLiouville operators on finite graphs are deficient.
For regular SturmLiouville operators defined on \((a,b)\), in [3], Kong et al. clarified the natural geometric structure on the space of complex boundary conditions, which provides the basis for studying the dependence of SturmLiouville eigenvalues on boundary conditions. But for SturmLiouville operators on graphs, the geometric structure on the space of complex vertex conditions is not clear. Carlson provided a description of the domains of local essential selfadjoint differential operators on weighted directed graphs with the coefficients of the operators smooth enough [4]. And there are several other descriptions of the vertex conditions that one can add to the SchrÃ¶dinger expression in order to create a selfadjoint operator on graph Î“ (see, e.g., [5â€“7] for details). Based on the methods in [4] and [3], we give a new description of all vertex conditions defining the domains of local (essentially) selfadjoint SturmLiouville operators on weighted directed graphs. The new description allows us to clarify a natural geometric structure on the space of vertex conditions.
The paper is organized as follows. Some results of the differential operators on graphs are introduced in SectionÂ 2. In SectionÂ 3 we give some properties of selfadjoint vertex conditions for local SturmLiouville operators on the graph Î“. Based on these properties we get the necessary conditions for local SturmLiouville operators to be selfadjoint. Moreover, we prove that each selfadjoint complex vertex condition at vertex v has a normalized form and all the normalized forms are contained in a finite set with cardinal number \(2^{\delta(v)}\), where \(\delta(v)\) denotes the degree of the vertex v. In the fourth section we give the sufficient conditions for local SturmLiouville operators to be selfadjoint, which are PovznerWienholtztype selfadjointness results for SturmLiouville operators on graph. In the final section, we give an example to show how to obtain the proper selfadjoint restriction of a given SturmLiouville operator by using the results obtained in SectionÂ 3.
2 Notation and prerequisite results
Assume that the graph Î“ is connected and each vertex in Î“ appears in only finitely many edges. A metric graph Î“ may be constructed from the graph data as follows. For each directed edge \(e_{n}\), let \((a_{n},b_{n})\) be a real interval of length \(\vert e_{n}\vert \) with \(a_{n}< b_{n}\), then we could see \(e_{n}\) as \((a_{n},b_{n})\). Define the distance between two points in the graph as the length of the shortest path connecting them. For a function \(f\in L_{w}^{2}(\Gamma,\mathbb{C} )\), \(f_{n}\) denotes the restriction \(f\upharpoonright e_{n}\). Let \(L_{w}^{2}(\Gamma,\mathbb{C} )\) denote the Hilbert space \(\oplus_{n}L_{w_{n}}^{2}((a_{n},b_{n}),\mathbb{C} )\) with the inner product
In this paper, we restrict our considerations to local SturmLiouville operators, so that we could describe the adjoint and selfadjoint extensions of SturmLiouville operators on the graph Î“ in terms of appropriate conditions on each single vertex of the graph. The definition of local operator is given by â€‰Carlson in [4]. Let \(\phi:\Gamma \rightarrow\mathcal{C}\) denote a \(C^{\infty}\) function which has compact support in Î“ and is constant in an open neighborhood of each vertex. An operator \(\mathcal{L}\) is a local operator if for every Ï•, Ï•f is in the domain of \(\mathcal{L}\) whenever f is.
Fix the vertex v, and let \(\delta(v)\) be its degree. Identify interval endpoints as \(\alpha_{m}\) if the corresponding edge endpoints are the same vertex v, in which case we will write \(\alpha_{m}\sim v\), \(m=1,\ldots ,\delta(v)\). Since \(1/p,q,w\in L_{\mathrm{loc}}^{1}(\Gamma,\mathbb{R} )\), for any solution f of \(Lf=\lambda f\), f and \(pf^{\prime}\) are locally absolutely continuous in each edge \(e_{n}\) in Î“. Hence, for every vertex v, \(f(\alpha_{m})\) and \((pf^{\prime})(\alpha_{m})\), \(\alpha_{m}\sim v\), can be defined via appropriate limits. We define the maximal operator \(\mathcal{L}_{\max}\) as follows:
For \(f\in \operatorname{Dom}(\mathcal{L}_{\max})\), let \(\hat{f}_{v}\in \mathbb{C} ^{2\delta(v)}\) be the vector with the \((2k+j1)\)th component defined by
where \(f^{[0]}(x)=f(x)\) and \(f^{[1]}(x)=(pf^{\prime})(x)\). The vector \(\hat{f}_{v}\in \mathbb{C} ^{2\delta(v)}\) is called the boundary value of f at v, \(\alpha _{m}\sim v\).
It is easy to verify that the expression L is symmetric. Suppose \(f,g\in \operatorname{Dom}(\mathcal{L}_{\max})\), with the support of g in an open ball containing at most one vertex v. Then integration by parts leads to
where \([f,g]_{v}\) is a nondegenerate form in the boundary values of f and g at v, i.e., there is an invertible \(2\delta(v)\times 2\delta(v)\) matrix \(S_{v}\) such that
A maximal independent set of vertex conditions at v may be written as \(B_{v}\hat{f}_{v}=0\), where \(B_{v}\) is a \(K(v)\times2\delta(v)\) matrix with linearly independent rows.
Denote the local SturmLiouville operator \(\mathcal{L}\) as follows:
with domain \(\operatorname{Dom}(\mathcal{L})\subset \operatorname{Dom}(\mathcal{L}_{\max})\), and functions in \(\operatorname{Dom}(\mathcal{L})\) satisfy the vertex conditions \(B_{v}\hat {f}_{v}=0\) at each \(v\in V\). By working on one edge \(e_{i}\), using the classical theory in [8], pp.169171, and the results given in [4], we can get the following lemmas.
Lemma 1
Suppose that the operator \(\mathcal{L}\) defined by (2.1) is selfadjoint and local. The vertex conditions at v annihilating the domain of \(\mathcal{L}\) are written as
where \(B_{v}\) is a \(K(v)\times2\delta(v)\) matrix with linearly independent rows. Then each \(B_{v}\) is a \(\delta(v)\times2\delta(v)\) matrix, and
Lemma 2
Suppose that \(\inf_{e_{n}\in E}\vert e_{n}\vert >0\), \(w\equiv1\), \(p\in AC_{\mathrm{loc}}(\Gamma,\mathbb{R} )\), \(\vert p\vert \geqslant C>0\), \(p^{\prime}\) and q are uniformly bounded. The \(\delta(v)\times2\delta(v)\) matrix \(B_{v}\) is given with linearly independent rows and satisfies
at each \(v\in V\). If the operator \(\mathcal{L}\) induced by L has the domain
then \(\mathcal{L}\) is essentially selfadjoint. Conversely, every local selfadjoint operator \(\mathcal{L}_{1}\) formally given by L is the closure of one operator \(\mathcal{L}\).
3 Spaces of vertex conditions
A single vertex condition at v can be written as \(\sum b_{j,k}f^{[j]}(\alpha_{k})=0\). A maximal independent set of single vertex conditions at v can be written as \(B_{v}\hat{f}_{v}=0\), where \(B_{v}\) is a \(K(v)\times2\delta(v)\) matrix with linearly independent rows. In the following, we will call such an independent set of single vertex conditions at v a vertex condition at v. We introduce the notation
and write the matrix \(B_{v}\) into the block matrix \(( B_{1}\mid \cdots\mid B_{\delta(v)} ) \), in which \(B_{i}\) are \(K(v)\times2\) matrices, \(i=1,\ldots,\delta(v)\). Then the vertex conditions at v, \(B_{v}\hat {f}_{v}=0\) may be rewritten as
and will be denoted by \((B_{1}\mid\cdots\mid B_{\delta(v)})\). The systems
represent the same complex vertex conditions at vertex v if and only if there is a matrix \(T\in GL(K(v),\mathbb{C})\) such that
where \(GL(\delta (v),\mathbb{C})\) is the set of \(K(v)\times K(v)\) invertible matrices over \(\mathbb{C}\). Denote the space \(\mathfrak{B}_{v}^{\mathbb{C} }\) of complex vertex conditions at vertex v as the quotient space
where \(M_{\delta(v)\times2\delta(v)}(\mathbb{C} )\) stands for the set of \(\delta(v)\times2\delta(v)\) matrices over \(\mathbb{C} \), and \(M_{\delta(v)\times2\delta(v)}^{\ast}(\mathbb{C} )\) stands for the set of matrices in \(M_{\delta(v)\times2\delta(v)}(\mathbb{C} )\) with rank \(\delta(v)\). We give the space \(M_{\delta(v)\times 2\delta (v)}(\mathbb{C} )\) the usual topology on \(\mathbb{C} ^{\delta(v)\times2\delta(v)}\), then \(M_{\delta(v)\times2\delta (v)}^{\ast}(\mathbb{C} )\) is an open subset of \(M_{\delta(v)\times2\delta(v)}(\mathbb{C} )\). In this way, the quotient space \(\mathfrak{B}_{v}^{\mathbb{C} }\) inherits the quotient topology.
The complex vertex condition \(B_{v}\hat{f}_{v}=0\) in \(\mathfrak{B}_{v}^{ \mathbb{C} }\) could be represented by the \(\delta(v)\times2\delta(v)\) matrix \(B_{v}\) with rank \(\delta(v)\). Up to the elementary row transformations, the \(\delta(v)\)dimensional column vectors
are \(\delta(v)\) columns in \(B_{v}\). Then the space \(\mathfrak{B}_{v}^{\mathbb{C} }\) can be divided into \(\binom{2\delta(v)}{\delta(v)}\) canonical atlas of local coordinate systems on the Grassmann manifold of \(\delta^{2}(v)\)dimensional complex subspaces in \(\mathbb{C} ^{\delta^{2}(v)}\) through the origin [3], where \(\binom{2\delta (v)}{\delta(v)}\) is the binomial coefficient
Theorem 1
The space \(\mathfrak{B}_{v}^{\mathbb{C} }\) of complex vertex conditions at vertex v is a connected and compact complex manifold of complex dimension \(\delta^{2}(v)\). The space \(\mathfrak{B}_{v}^{\mathbb{C} }\) is a connected and real manifold of dimension \(2\delta^{2}(v)\) over the number field \(\mathbb{R} \).
Proof
The proof is similar to TheoremÂ 3.1 in [3].â€ƒâ–¡
For each edge \(e_{n}\), let \((a_{n},b_{n})\) be the corresponding real interval. Consider the operator \(\mathcal{L}_{\max}\) on \((a_{n},b_{n})\) with \(f,g\in \operatorname{Dom}(\mathcal{L}_{\max})\), we have
Then if \(f,g\in \operatorname{Dom}(\mathcal{L}_{\max})\) and g vanishes outside of a small neighborhood of v, we have
with
For \(f,g\in \operatorname{Dom}(\mathcal{L}_{\max})\), define \([f,g]_{v}\) as the righthand side of equality (3.2).
Assume that \(e_{k}=(a_{k},b_{k})\), \(k=1,\ldots,\delta(v)\), are the edges in which v is one endpoint, i.e., \(v=a_{k}\) or \(v=b_{k}\). For \(\alpha _{m}\sim v\), assume that \(\alpha_{1},\ldots,\alpha_{s}\) are \(b_{k}\) in the corresponding edges \(e_{k}=(a_{k},b_{k})\) respectively, \(k= 1,\ldots ,s \), and \(\alpha_{s+1},\ldots,\alpha_{\delta(v)}\) are \(a_{k}\) in the corresponding edges \(e_{k}=(a_{k},b_{k})\) respectively, \(k= s+1,\ldots ,\delta(v)\). Through a direct calculation we can obtain that
Based on LemmaÂ 1, we know that for the vertex condition \(( B_{1}\mid \cdots\mid B_{\delta(v)} ) \hat{f}_{v}=0\) of a selfadjoint operator \(\mathcal{L}\), the coefficient matrix is a \(\delta(v)\times2\delta(v)\) matrix and satisfies the condition \(B_{v}[S_{v}^{\ast }]^{1}B_{v}^{\ast }=0 \). Since \(S_{v}\) is the matrix given in (3.3), the condition \(B_{v}[S_{v}^{\ast}]^{1}B_{v}^{\ast}=0\) is equivalent to
If \(B_{v}= ( B_{1}\mid\cdots\mid B_{\delta(v)} ) \) is a \(\delta (v)\times2\delta(v)\) matrix with linearly independent rows and satisfies equality (3.4), the vertex condition \(B_{v}\hat{f}_{v}=0\) is called a selfadjoint vertex condition at v.
Obviously, the elementary row transformation on \(B_{v}\) does not change the vertex conditions at v, while the column transformations on \(B_{v}\) change the vertex conditions. But there is a class of column transformations that would not change the selfadjointness of the vertex conditions.
Lemma 3
For the local operator \(\mathcal{L}\) defined by (2.1) and
\(B_{v}\hat{f}_{v}=0\) is a selfadjoint vertex condition at v if and only if for any matrix A in the set
\(A\hat{f}_{v}=0\) is a selfadjoint vertex condition at v.
Proof
Since \(E_{2}^{\ast}=E_{2}\), \((E_{2})^{2}=I_{2}\), where \(I_{2}=\bigl( {\scriptsize\begin{matrix}{} 1 & 0 \cr 0 & 1 \end{matrix}} \bigr)\), then we have
Therefore, the matrix operationmultiplication by \(E_{2}\) does not change the rank of \(B_{i}\), and for A in the set (3.5), \(A\hat {f}_{v}=0\) is a selfadjoint vertex condition at v whenever \(B_{v}\hat{f}_{v}=0\) is a selfadjoint vertex condition.â€ƒâ–¡
Definition 1
The column transformations used in LemmaÂ 3 are called the selfadjoint column transformations.
Theorem 2
For the local operator \(\mathcal{L}\) defined by (2.1), up to the elementary row transformations and the selfadjoint column transformations, the coefficient matrices of selfadjoint vertex conditions at v are
where the complex matrix \((c_{ij})_{\delta(v)\times\delta(v)}\) consisting of the even numbered columns of matrix (3.6) has the following properties:

(1)
for \(i=j\), \(c_{ij}\in \mathbb{R} \),

(2)
for \(i< j\leqslant s\) or \(s< i< j\), \(c_{ij}=\overline{c_{ji}}\),

(3)
for \(i\leqslant s< j\), \((c_{ij})_{0\leqslant i\leqslant s,s< j\leqslant \delta(v)}=(c_{ji})_{0\leqslant i\leqslant s,s< j\leqslant\delta (v)}^{\ast}\).
Proof
For \((a_{ij})= ( A_{1}\mid\cdots\mid A_{s}\mid A_{s+1}\mid\cdots \mid A_{\delta(v)} ) \) is a coefficient matrix of selfadjoint vertex conditions at v, there must be one of the elements \(a_{i1}\) and \(a_{i2}\), \(i=1,\ldots,\delta(v)\), that is not zero. Through the elementary row transformations and the selfadjoint column transformations, the matrix \((a_{ij})\) becomes
For the convenience of writing, the elements changed are still written as \(a_{ij}\), \(1\leqslant i,j\leqslant\delta(v)\). Since \((a_{ij})\) is a matrix with rank \(\delta(v)\), up to the elementary row transformations, the \(\delta(v)1\) column vectors
are \((\delta(v)1)\) columns in (3.7). Then (3.7) could be written as
otherwise, matrix (3.7) is in the following two cases. (1) The elements \(a_{i3}\) and \(a_{i4}\) are all zero for all \(1\leqslant i\leqslant \delta(v)\). (2) One of the elements \(a_{13}\) and \(a_{14}\) is not zero, the rest of the elements in the third and fourth columns are zero. Then up to the elementary row transformations, the first two vectors in (3.8) appear in one block \(A_{i}\), \(i\geqslant3\), of (3.7). In these two cases, \((a_{ij})\) could not be the coefficient matrix of a selfadjoint vertex condition at v. Then \((a_{ij})\) could be changed into
The matrix \((a_{ij})\) is the coefficient matrix of a selfadjoint vertex condition at v, (3.9) is the coefficient matrix of the same selfadjoint vertex condition at v, then (3.9) must satisfy (3.4), i.e.,
After a direct calculation, we get that the matrix \((c_{ij})_{{\delta (v)\times\delta(v)}}\) has the following properties:

(1)
for \(i=j\), \(c_{ij}=\overline{c_{ji}}\), i.e., \(c_{ii}\in \mathbb{R} \);

(2)
for \(i< j\leqslant s\) or \(s< i< j\), \(c_{ij}=\overline{c_{ji}}\), i.e.,
$$ \operatorname {Re}c_{ij}=\operatorname {Re}c_{ji},\qquad \operatorname {Im}c_{ij}= \operatorname {Im}c_{ji}; $$ 
(3)
for \(i\leqslant s< j\), the block matrix \((c_{ij})_{0\leqslant i\leqslant s,s< j\leqslant\delta(v)}\) is equal to the \(s\times(\delta (v)s)\) matrix \((c_{ji})_{0\leqslant i\leqslant s,s< j\leqslant\delta (v)}^{\ast}\). In other words, the matrix \((c_{ij})_{{\delta(v)\times \delta(v)}}\) has the following form:
$$ \begin{pmatrix} \underset{s}{\underbrace{\textstyle\begin{array}{@{}cccc@{}} c_{11} & \overline{c_{21}} & \cdots& \overline{c_{s1}} \\ c_{21} & c_{22} & \cdots& \vdots\\ \vdots& \vdots& \ddots& \overline{c_{s2}} \\ c_{s1} & c_{s2} & \cdots& c_{ss}\end{array}\displaystyle }} & (c_{ji})^{\ast},i \leqslant s< j \\ (c_{ji}),i\leqslant s< j & \underset{\delta(v)s}{ \underbrace{\textstyle\begin{array}{@{}cccc@{}} c_{(s+1)(s+1)} & \overline{c_{(s+2)(s+1)}} & \cdots& \overline {c_{\delta (v)(s+1)}} \\ c_{(s+2)(s+1)} & c_{(s+2)(s+2)} & \cdots& \overline{c_{\delta (v)(s+2)}} \\ \vdots& \vdots& \ddots& \vdots\\ c_{\delta(v)(s+1)} & c_{\delta(v)(s+2)} & \cdots& c_{\delta (v)\delta(v)}\end{array}\displaystyle }} \end{pmatrix} . $$
â€ƒâ–¡
Write matrix (3.6) into the block matrix and denote it as \(( C_{1}\mid\cdots\mid C_{\delta(v)} ) \), then the set
is a set of matrices transformed from \(( C_{1}\mid\cdots\mid C_{\delta(v)} ) \) through the selfadjoint column transformations. For \(A\in S\), we call the columns which are even columns in \(( C_{1}\mid \cdots\mid C_{\delta(v)} ) \) the unnormalized columns in A. And we call the elements in the set (3.10) normalized forms. If \(( C_{1}\mid\cdots\mid C_{\delta(v)} ) \) is the coefficient matrix of a selfadjoint vertex condition at v, then the \(2^{\delta(v)}\) elements in the set S are also the coefficient matrices of selfadjoint vertex conditions.
Theorem 3
If we only allow the elementary row transformations on the matrix \(B_{v}= ( B_{1}\mid\cdots\mid B_{\delta(v)} ) \), where \(B_{v}\hat{f}_{v}=0\) is a selfadjoint vertex condition, then \(B_{v}\) can be normalized to one of the \(2^{\delta(v)}\) forms in (3.10), and the complex matrix consisting of the unnormalized columns has the properties (1)(3) in TheoremÂ 2.
Proof
From LemmaÂ 3 we can obtain that the vertex conditions in (3.10) satisfy (3.4) if and only if \(( C_{1}\mid\cdots\mid C_{\delta (v)} ) \) satisfies (3.4). According to the proof of TheoremÂ 2, we could reach the conclusion.â€ƒâ–¡
Definition 2
The \(2^{\delta(v)}\) elements in the set (3.10) with \(( C_{1}\mid \cdots\mid C_{\delta(v)} ) \) that have the properties (1)(3) in TheoremÂ 2 are called the normalized forms of the coefficient matrices of selfadjoint vertex conditions at v.
The elementary row transformations on the coefficient matrices of selfadjoint vertex conditions allow us to bring them into their corresponding normalized forms, and the selfadjoint column transformations on these matrices give us simple onetoone correspondences between the coefficient matrices in different normalized forms.
Theorem 4
For the local operator \(\mathcal{L}\) defined by (2.1), the space \(\mathfrak{S}_{v}^{\mathbb{C} }\) of selfadjoint complex vertex conditions at vertex v is a connected, closed and analytic real submanifold of \(\mathfrak{B}_{v}^{\mathbb{C} }\) and has dimension \(\delta^{2}(v)\) over the number field \(\mathbb{R} \). Therefore, \(\mathfrak{S}_{v}^{\mathbb{C} }\) is also compact.
Proof
For an arbitrary complex vertex condition \(( B_{1}\mid\cdots\mid B_{\delta(v)} ) \) at vertex v, it is in one equivalence class of quotient space \(M_{\delta(v)\times2\delta(v)}^{\ast}(\mathbb{C} )\diagup GL(\delta(v),\mathbb{C} )\). The set \(\mathfrak{S}_{v}^{\mathbb{C} }\) of selfadjoint complex vertex conditions at vertex v is a subset of \(\mathfrak{B}_{v}^{\mathbb{C} }\) which can be divided into \(2^{\delta(v)}\) canonical atlas of local coordinate systems. The proof of the analyticity and the connectivity of \(\mathfrak{S}_{v}^{\mathbb{C} }\) is similar to TheoremÂ 3.11 in [3]. Thus \(\mathfrak{S}_{v}^{\mathbb{C} }\) is an analytic real submanifold of \(\mathfrak{B}_{v}^{\mathbb{C} }\) and has dimension \(\delta^{2}(v)\) over the number field \(\mathbb{R} \). The canonical atlas of local coordinate systems of \(\mathfrak {S}_{v}^{\mathbb{C} }\) is internal connected and \(\mathfrak{S}_{v}^{\mathbb{C} }\) is connected.
Next we prove that the space \(\mathfrak{S}_{v}^{\mathbb{C} }\) is closed. Let \(\{ ( B_{1}^{(n)}\mid\cdots\mid B_{\delta (v)}^{(n)} ) \} _{n=1}^{+\infty}\) be a sequence in \(\mathfrak{S}_{v}^{\mathbb{C} }\) that converges to \(\tilde{B}= ( B_{1}^{\prime}\mid\cdots\mid B_{\delta(v)}^{\prime} ) \in\mathfrak{B}_{v}^{\mathbb{C} }\). Without loss of generality, we can assume that BÌƒ has the normalized form (3.9), i.e.,
Denote the matrix constructed by the even numbered columns of BÌƒ as D. Then we will show that D has the properties (1)(3) in TheoremÂ 2.
For sufficiently large n, \(B_{v}^{(n)}= ( B_{1}^{(n)}\mid\cdots \mid B_{\delta(v)}^{(n)} ) \in\mathcal{O}_{1}^{s}\), and hence
Denote the matrix constructed by the even numbered columns of \(B_{v}^{(n)}\) as \(D^{(n)}\). The matrix \(D^{(n)}\) has the properties (1)(3) in TheoremÂ 2, and
under the norm \(\Vert\cdot\Vert\) on \(M_{\delta(v)\times2\delta (v)}(\mathbb{C} )\). Therefore,
â€ƒâ–¡
Example 1
When \(\delta(v)=2\), \(s=1\), the canonical atlas of local coordinate systems of \(\mathfrak{B}_{v}^{\mathbb{C} }\) is as follows:
The canonical atlas of local coordinate systems of \(\mathfrak{S}_{v}^{\mathbb{C} }\) is as follows:
4 The selfadjoint SturmLiouville operators
Assume that the graph Î“ has a finite vertex set \(V= \{ v_{1},\ldots ,v_{n} \} \) and a finite edge set E. The vertex conditions at \(v_{i}\) are written as \(B_{v_{i}}\hat{f}_{v_{i}}=0\). Then, for the local operator \(\mathcal{L}\) defined by (2.1), the vertex conditions on Î“ are
Denote \(B_{\Gamma}\) as \(B_{\Gamma}= ( B_{v_{1}},\ldots ,B_{v_{n}} ) \), where \(B_{v_{i}}\) is a \(\delta(v_{i})\times 2\delta (v_{i})\) matrix with linearly independent rows. In the following, \(B_{\Gamma }\) represents (4.1), i.e., the vertex conditions on Î“. Define \(\mathfrak{B}_{\Gamma}^{\mathbb{C} }\) as
Then the complex selfadjoint vertex conditions space of the local operator \(\mathcal{L}\) on the graph Î“ is denoted as \(\mathfrak{S}_{\Gamma }^{\mathbb{C} }\),
Corollary 1
The space \(\mathfrak{B}_{\Gamma}^{\mathbb{C} }\) has dimension \(\sum_{i=1}^{n}\delta^{2}(v_{i})\) over the number field \(\mathbb{C} \), and has dimension \(\sum_{i=1}^{n}2\delta^{2}(v_{i})\) over the number field \(\mathbb{R} \). The space \(\mathfrak{S}_{\Gamma}^{\mathbb{C} }\) is a connected, closed and analytic real submanifold of \(\mathfrak{B} _{\Gamma}^{\mathbb{C} }\) with dimension \(\sum_{i=1}^{n}\delta^{2}(v_{i})\) over the number field \(\mathbb{R} \).
For \(x,y\in\Gamma\), denote the distance between x and y as \(d(x,y)\).
Lemma 4
Suppose that the graph Î“ has finitely many edges and
the operator \(\mathcal{L}\) defined by (2.1) with domain
is selfadjoint.
Proof
The proof is based on TheoremÂ 3.4 in [4].â€ƒâ–¡
Theorem 5
Suppose that the graph Î“ has infinitely many edges and vertices, \(\sup_{x,y\in\Gamma}d(x,y)=\infty\), and there is no finite accumulation point in V.
If the operator \(\mathcal{L}\) defined by (2.1) satisfies the following two conditions:

(1)
\(p>0\), \(p\in AC_{\mathrm{loc}}(\Gamma,\mathbb{R} )\),

(2)
\(\frac{1}{w},\frac{(p^{\prime})^{2}}{w}\in L_{\mathrm{loc}}(\Gamma,\mathbb{R} )\), \(\frac{p}{w}\) is essentially bounded on Î“,
and \(\mathcal{L}\) is lower bounded with domain
then \(\mathcal{L}\) is essentially selfadjoint. Conversely, every local selfadjoint operator \(\mathcal{L}_{1}\) formally given by L is the closure of one of the operators \(\mathcal{L}\).
Proof
Using the theory in [9] on one edge \(e_{i}\), we can get that the operator \(\mathcal{L}\) is symmetric and the domain of \(\mathcal {L}^{\ast}\) is contained in \(\operatorname{Dom}(\mathcal{L}_{\max})\). By TheoremÂ 3.1 and CorollaryÂ 3.2 in [4], we get that \(f\in \operatorname{Dom}(\mathcal{L}^{\ast})\) satisfies the same conditions \(B_{v}\hat{f}_{v}=0\) at the vertices. Then
Without loss of generality, we assume that \(\mathcal{L}\geqslant I\). Next we need to show that the equation
has only a trivial solution (derivative is understood in a distribution sense).
Fix a point \(o\in\Gamma\), we define a sequence of functions \(\{\chi _{n}\}\), \(\chi_{n}\in C_{\mathrm{comp}}^{\infty}(\Gamma)\) such that
and
where the function space \(C_{\mathrm{comp}}^{\infty}(\Gamma)\) contains the functions belonging to \(C^{\infty}(\Gamma)\) and having compact support on Î“. Assume that \(\tilde{f}\neq0\) is a solution of equation (4.2). Since fÌƒ satisfies the conditions
then for each v and \(g:=\tilde{f}\chi_{n}\),
Since \(p\in AC_{\mathrm{loc}}(\Gamma)\), one verifies that
Let \(\Gamma_{m}\) be a subtree of Î“ containing all \(x\in\Gamma \), \(x\leqslant m\).
and then the righthand side of equality (4.4) belongs to \(L_{w}^{2}(\Gamma,\mathbb{C} )\). Then we have \(\tilde{f}\chi_{n}\in \operatorname{Dom}(\mathcal{L})\) and
On the other hand,
Combining (4.5) with (4.6), we obtain
Since \(\tilde{f} \in L_{w}^{2}(\Gamma,\mathbb{C} )\), \(\tilde{f} =0\). This completes the proof.â€ƒâ–¡
Let \(\mathcal{L}\) be the operator defined by (2.1). Assume that the functions in \(\operatorname{Dom}(\mathcal{L})\) are continuous on Î“, i.e.,
at each vertex \(v\in V\). We get the following result.
Corollary 2
Suppose that the local operator \(\mathcal{L}\) defined by (2.1) is selfadjoint, with functions in \(\operatorname{Dom}(\mathcal{L})\) continuous on Î“. For a vertex v, let \(\alpha_{1},\ldots,\alpha_{s}\sim v\) be \(b_{k}\) in the corresponding edges \(e_{k}=(a_{k},b_{k})\), \(k= 1,\ldots,s\), and \(\alpha_{s+1},\ldots,\alpha_{\delta(v)}\) be \(a_{k}\) in the corresponding edges \(e_{k}=(a_{k},b_{k})\), \(k= s+1,\ldots,\delta(v)\). Then the functions f in \(\operatorname{Dom}(\mathcal{L})\) satisfy the condition
at each vertex v.
Proof
All the functions \(f\in \operatorname{Dom}(\mathcal{L})\) satisfy the continuity conditions, that means
at each vertex \(v\in\Gamma\). For a fix vertex v, \(B_{v}\) is a \(\delta (v)\times2\delta(v)\) matrix with linearly independent rows. Then
and \((c_{2\delta(v)1}^{\prime})^{2}+(c_{2\delta(v)}^{\prime })^{2}\neq0\). If \(c_{2\delta(v)}^{\prime}\neq0\), we can rewrite the vertex condition as \(A_{v}\hat{f}_{v}=0\), where
Therefore we can get that \(A_{v}\in\{ ( C_{1}\mid C_{2}\mid\cdots \mid C_{\delta(v)}E_{2} ) \}\), where the matrices \(( C_{1}\mid C_{2}\mid\cdots\mid C_{\delta(v)}E_{2} ) \) are introduced in (3.10). Since \(B_{v}\in\mathfrak{S}_{v}^{\mathbb{C} }\), the complex matrix
has the properties (1)(3) in TheoremÂ 2. Then we have
i.e.,
If \(c_{2\delta(v)}^{\prime}=0\) and \(c_{2\delta(v)1}^{\prime}\neq0\), through the elementary row transformations, the condition \(B_{v}\hat{f}_{v}=0 \) is equal to the condition \(A_{v}\hat{f}_{v}=0\), where
Then we have \(A_{v}\in\{ ( C_{1}\mid C_{2}\mid\cdots\mid C_{\delta (v)} ) \}\), the matrix \(( C_{1}\mid C_{2}\mid\cdots\mid C_{\delta(v)} ) \) is introduced in (3.6). Since \(B_{v}\in \mathfrak{S}_{v}^{\mathbb{C} }\), the complex matrix
has the properties (1)(3) in TheoremÂ 2. Then we have
Therefore the vertex conditions \(B_{v}\hat{f}_{v}=0\) are the conditions
That is, corresponding to the conditions
for \(r=\infty\).â€ƒâ–¡
Remark 1
For the complex selfadjoint conditions \(B_{v}\) including the continuity conditions, the space consisting of \(B_{v}\) is a real submanifold of \(\mathfrak{B}_{v}^{\mathbb{C} }\) with dimension 1.
References
Cao, X, Wang, Z, Wu, H: On the boundary conditions in selfadjoint multiinterval SturmLiouville problems. Linear Algebra Appl. 430(1112), 28772889 (2009)
Harmer, M: Hermitian symplectic geometry and extension theory. J. Phys. A 33(50), 91939203 (2000)
Kong, Q, Wu, H, Zettl, A: Geometric aspects of SturmLiouville problems. I. Structures on spaces of boundary conditions. Proc. R. Soc. Edinb., Sect. A 130(3), 561589 (2000)
Carlson, R: Adjoint and self adjoint differential operators on graphs. Electron. J. Differ. Equ. 1998, 6 (1998)
Kuchment, P: Quantum graphs. I. Some basic structures. Waves Random Media 14(1), S107S128 (2004)
Kostrykin, V, Schrader, R: Kirchhoffâ€™s rule for quantum wires. J. Phys. A 32(4), 595630 (1999)
Harmer, M: Hermitian symplectic geometry and extension theory. J. Phys. A 33(50), 91939203 (2000)
Kato, T: Perturbation Theory for Linear Operators. Springer, Berlin (1966)
Goriunov, A, Mikhailets, V: Regularization of singular SturmLiouville equation. Methods Funct. Anal. Topol. 16, 120130 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authorsâ€™ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhao, J., Shi, G. Structures on selfadjoint vertex conditions of local SturmLiouville operators on graphs. Bound Value Probl 2015, 162 (2015). https://doi.org/10.1186/s1366101504225
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366101504225