On the Decomposition of the Laplacian on Metric Graphs

TL;DR

This paper investigates how the Laplacian operator on symmetric metric graphs can be decomposed into simpler components, revealing structural insights and extending methods from metric trees to more general graphs.

Contribution

It introduces a decomposition technique for the Laplacian on family preserving metric graphs, generalizing existing methods from metric trees to broader graph structures.

Findings

Decomposition into one-dimensional operators is possible for family preserving metric graphs.

The structure of the graph directly influences the properties of the decomposed operators.

A correspondence between discrete and continuum decompositions is established.

Abstract

We study the Laplacian on family preserving metric graphs. These are graphs that have a certain symmetry that, as we show, allows for a decomposition into a direct sum of one-dimensional operators whose properties are explicitly related to the structure of the graph. Such decompositions have been extremely useful in the study of Schr\"odinger operators on metric trees. We show that the tree structure is not essential, and moreover, obtain a direct and simple correspondence between such decompositions in the discrete and continuum case.

Equations

$$h(\Gamma )=\sum _{k=0}^{\infty }{l}_k$$

$$Q=\left\{f\in L^2(\Gamma )\text{ s.t. f is continuous, f∣e∈H1(e) for every e∈E(Γ) f(o)=0, and ∫Γ∣f′(x)∣2dμ(x)<∞}\right\}$$

$$E_n:{\ell }^2(S_n)\to {\ell }^2(S_{n+1}),E_n(\phi )(v)=\sum _{x\in S_n\cap G_v}\phi (x)=\sum _{x\in S_n,x\sim v}\phi (x)$$

$${\Lambda }_{n,+j}:{\ell }^2(S_n)\to {\ell }^2(S_n),{\Lambda }_{n,+j}=E_n^T\cdots E_{n+j}^T\cdot E_{n+j}\cdots E_n$$

$${\Lambda }_{n,-j}:{\ell }^2(S_n)\to {\ell }^2(S_n),{\Lambda }_{n,-j}=E_{n-1}\cdots E_{n-j}\cdot E_{n-j}^T\cdots E_{n-1}^T$$

$$h^{\varphi }(x)=\frac{1}{\sqrt{g_n(|x|)\cdot |G_x\cap S_n|}}\sum _{v\in G_x\cap S_n}\varphi (v)$$

$$P_{\varphi }(f)(x)=⟨f_{|x|},h_{|x|}^{\varphi }⟩h^{\varphi }(x)$$

$$\begin{array}[]{c}\#\left\{e\in E\,:\,gen(e)=n\right\}=\#\left\{v\in V\,:\,gen(v)=n\right\}\\ \text{or}\\ \#\left\{e\in E\,:\,gen(e)=n\right\}=\#\left\{v\in V\,:\,gen(v)=n+1\right\}\end{array}$$

$$(i)(G_v\cap S_n)\bigcap (G_u\cap S_n)=\varnothing$$

$$(ii)G_v\cap S_n=G_u\cap S_n.$$

$$(E_{n+j}\cdots E_n)(\varphi )(u)=\sum _{x\in S_n\cap G_u}{k}_u(x)\varphi (x)$$

$$\begin{array}{rl}(E_{n+k+1}\cdots E_n)(\varphi )(u)=E_{n+k+1}(E_{n+k}\cdots E_n)(\varphi )(u)& =\sum _{y\in S_{n+k}\cap G_u}\sum _{x\in S_n\cap G_y}{k}_y(x)\varphi (x)\\ & =\sum _{x\in S_n\cap G_u}{k}_u(x)\varphi (x).\end{array}$$

$$\left(E_n^T\cdots E_{n+j}^T\right)(\varphi )(v)=\sum _{x\in S_{n+j}\cap G_v}{k}^v(x)\varphi (x)$$

$$G_v\cap S_{n+k}=G_u\cap S_{n+k}:=A_{v,u}.$$

$$G_{w_1}\cap S_n=G_{w_2}\cap S_n:=B_{v,u}.$$

$$\tilde{\varphi }(w)=\sum _{x\in B_{v,u}}{k}_w(x)\varphi (x).$$

$$\tilde{\varphi }(w)=c_1\sum _{x\in B_{v,u}}\varphi (x)$$

$${\Lambda }_{n,+k}(\varphi )(v)=E_n^T\cdots E_{n+k}^T\left(\tilde{\varphi }\right)(v)=\sum _{x\in A_{v,u}}{k}^v(x)\tilde{\varphi }(x).$$

$$\sum _{x\in A_{v,u}}{k}^v(x)\tilde{\varphi }(x)=\sum _{x\in A_{v,u}}{c}_2\tilde{\varphi }(x)=\sum _{x\in A_{v,u}}{k}^u(x)\tilde{\varphi }(x)=E_n^T\cdots E_{n+k}^T\left(\tilde{\varphi }\right)(u)={\Lambda }_{n,+k}\left(\varphi \right)(u)$$

$${\lambda }_{n,+k}{\varphi }_0^r(x)={\Lambda }_{n,+k}({\varphi }_0^r)(x)=c\sum _{y\in G_v\cap S_{n(r)}}{\varphi }_0^r(y)$$

$$⟨h_t^{r,k},h_t^{r,k}⟩$$

$$=\sum _{|x|=t}\frac{1}{g_{n(r)+k}(t)\cdot |G_x\cap S_{n(r)+k}|}\left(\sum _{v\in G_x\cap S_{n(r)+k}}{\varphi }_k^r(v)\right)\overline{\left(\sum _{u\in G_x\cap S_{n(r)+k}}{\varphi }_k^r(u)\right)}=$$

$$=\frac{1}{g_{n(r)+k}(t)}\sum _{|x|=t}|G_x\cap S_{n(r)+k}|{\varphi }_k^r(v_x)\overline{{\varphi }_k^r(v_x)}$$

$$1=⟨{\varphi }_k^r,{\varphi }_k^r{⟩}_{{\ell }^2}^2=\sum _{v\in S_{n(r)+k}}{\varphi }_k^r(v)\overline{{\varphi }_k^r(v)}$$

$$\sum _{v\in S_{n(r)+k}}{\varphi }_k^r(v)\overline{{\varphi }_k^r(v)}=\underset{i=1}{\sum ^k}|G_{x_i}\cap S_{n(r)+k}|{\varphi }_k^r(v_{x_i})\overline{{\varphi }_k^r(v_{x_i})}$$

$${⟨h_t^{r,k},h_t^{r,k}⟩}_{{\mathbb{C}}^{g_{\Gamma }(t)}}={⟨{\varphi }_k^r,{\varphi }_k^r⟩}_{{\ell }^2}=1$$

$$P_{r,k}(P_{r,k}(f))(x)=⟨⟨f_{|x|},h_{|x|}^{r,k}⟩h_{|x|}^{r,k},h_{|x|}^{r,k}⟩h^{r,k}(x)=⟨f_{|x|},h_{|x|}^{r,k}⟩\underset{=1}{\underbrace{⟨h_{|x|}^{r,k},h_{|x|}^{r,k}⟩}}{h}^{r,k}(x)=P_{r,k}(f)(x)$$

$$⟨P_{r,k}(f),f-P_{r,k}(f)⟩$$

$$={\int }_0^{h(\Gamma )}\sum _{|x|=t}{P}_{r,k}(f)(x)\overline{\left(f-P_{r,k}(f)(x)\right)}dt$$

$$={\int }_0^{h(\Gamma )}\sum _{|x|=t}⟨f_t,h_t^{r,k}⟩h^{r,k}(x)\overline{\left(f(x)-⟨f_t,h_t^{r,k}⟩h^{r,k}(x)\right)}dt$$

$$={\int }_0^{h(\Gamma )}⟨⟨f_t,h_t^{r,k}⟩h_t^{r,k},f_t-⟨f_t,h_t^{r,k}⟩h_t^{r,k}⟩dt$$

$$={\int }_0^{h(\Gamma )}0dt=0$$

$$\#\{e\in E:gen(e)=n\}=\#\{v\in V:gen(v)=n\}$$

Full text

1 Introduction

The study of Schrödinger operators on graphs has drawn a considerable amount of attention in the past few decades. So much so, that any attempt at a short comprehensive review is doomed to fail. We can only refer the reader to a few representative surveys and collections [6, 7, 18, 27]. Besides arising naturally in many physical contexts, the setting of graphs offers a wide array of examples where a variety of mathematical phenomena related to the effects of geometry on spectral properties may be studied. Of particular recent interest is the setting of continuum (aka ‘metric’) graphs. In this setting, a graph is seen as a one dimensional simplicial complex, where the line segments have lengths, and are glued to each other at the relevant vertices. Functions are defined on the line segments, and the operator studied is the one dimensional Laplacian on the line segments, with some prescribed boundary conditions at the vertices. This is the model that will be at the focus of our attention here. We shall describe it formally in the next section.

While originating in chemistry and physics in the study of molecules and mesoscopic systems (such as waveguides, see e.g. [7] and references therein), such continuum models (also known as quantum graphs) have drawn the attention of the spectral theory community and have served as a platform for the study of various topics. These include trace formulas in quantum chaos [19], isospectrality and its association with geometry [21], Anderson localization and extended states [1, 2, 11, 22, 24, 33], Hardy inequalities [16, 28], eigenvalue estimates [5, 8, 17] and others [14, 15, 26].

A useful method in the context of infinite metric trees (i.e. connected graphs with no cycles), that has been applied in several of the works mentioned above, was introduced by Naimark and Solomyak in [28]. This method requires the tree to be spherically symmetric around a particular vertex (the root) and involves a decomposition of the operator into a direct sum of one-dimensional operators whose structure is directly related to the structure of the tree and to the boundary conditions at the vertices. A similar method exists in the discrete case, where the graph is considered as a combinatorial object and the operator studied is the discrete Laplacian or the adjacency matrix (see, e.g., [3, 9, 10, 12]). While the similarity between the decomposition in the continuous and discrete case is clear and lies in the exploitation of the symmetry properties of the graph, it is important to note there are essential technical differences. Whereas the discrete case involves studying cyclic subspaces generated by specially chosen functions, the continuum case (as presented in [28, 32]) involves defining the relevant invariant subspaces directly and relies heavily on the structure of the tree.

It has recently been realized in [13] that in the discrete case, the tree structure is not essential for this decomposition. It is in fact possible to carry out this procedure for a more general class of graphs that we call ‘family preserving’ and whose definition we give below (see Definition 2.2)222[13] actually study a slightly more general class of graphs that they call ‘path commuting’, but as the definition is considerably more involved and less intuitive and since all relevant examples are family preserving we have decided to prefer here simplicity to generality and restrict our attention to family preserving graphs.. This class contains radially symmetric trees, antitrees (see Section 5 for the definition), and various other graphs. We refer the reader to [13] for examples and some graphics.

The objective of this work is to extend this decomposition for family preserving graphs to the continuum case of metric graphs as well. Since, as remarked above, the standard decomposition technique for metric trees relies heavily on the tree structure, one needs to take a different approach. A natural approach to this task, and the one that we shall take, is to try and obtain a direct translation of the discrete decomposition to the decomposition in the metric case. Such an approach has also the added bonus of making explicit the connection between the combinatorial and continuum decompositions, and as we shall show, can recover the original Naimark-Solomyak procedure from the procedure in the discrete case.

While employing the discrete decomposition scheme is indeed natural for getting a decomposition in the metric case, the results in this paper should by no means be viewed as a direct extension of those in [7]. First, as the functions in the metric case live on edges, whereas those in the discrete case live on vertices, it is a non-trivial task to adapt one procedure to the other. Second, as is evident in Section 4 (where we describe the decomposition algorithm) the discrete decomposition is only one of several steps towards obtaining the one-dimensional constituent operators in the direct sum, and a rather straightforward one at that. Finally, as the class of family preserving graphs is considerably larger than that of spherically symmetric rooted trees, we believe the results here open the door to constructing and studying a wide array of metric graphs with interesting and rich spectral properties.

Remark 1.1*.*

In the context of decomposing an operator using symmetries, we refer to the preprint [4], which discusses quotients of finite-dimensional operators with respect to a representation of some symmetry group. Specifically, [4, Section 6] discusses the case of compact quantum graphs, and the quotient operators are taken with respect to a representation of the underlying graph’s symmetry group. It is reasonable to expect that the methods there extend to the infinite dimensional case studied here. A natural question then is whether one can use them in this case to reproduce our results. We note that it seems that our approach is somewhat more direct as it does not involve going through representations of the relevant symmetry group, (which is infinite here). On the other hand, the role of the symmetry is made more explicit in the approach of [4]. It would be interesting to study the relation between these two approaches further.

The rest of this paper is structured as follows. After presenting some basic definitions, we describe our main abstract result (Theorem 2.13) in the next section. As the proof of Theorem 2.13 relies on a connection between the discrete and continuous case, we devote the first two subsections of Section 3 to some preliminary results on this connection, presenting the proof in Section 3.3. Although Theorem 2.13 is the main abstract result, the raison d’être of this paper lies in Section 4, where we describe the structure of the components in the decomposition and their relation to the structure of the graph. Section 5 presents a demonstration that our approach reduces to the Naimark-Solomyak approach when $\Gamma$ is a tree and an application to the spectral analysis of spherically homogeneous (see Definition 2.4) metric antitrees333The preprint [25] presents an extensive spectral analysis of spherically homogeneous metric antitrees, also using a decomposition inspired by [13]. Since spherically homogeneous metric antitrees are family preserving, our general setting includes such graphs as particular cases (although we restrict attention to the self-adjoint case). Our emphasis here, however, is on the decomposition method itself and not on the spectral analysis..

Acknowledgments We thank Rami Band, Gregory Berkolaiko, Aleksey Kostenko and the anonymous referees for useful comments.

2 Definitions and Statement of the Main Result

2.1 Some Definitions

We begin with some definitions pertaining to the combinatorial structure of a graph. We shall add the continuous structure later. A rooted graph is a graph $G=\left\langle V(G),E(G)\right\rangle$, with a special vertex $o\in V(G)$, which is called the root. We assume that $V(G),E(G)$ (the vertex and edge set, respectively) are infinite, and that for every $v\in V(G)$, $deg(v)<\infty$, where $\deg(v)$ is the degree of $v$(=the number of edges incident to $v$). We also assume throughout that our graphs are simple (i.e. there are no multiple edges or loops) and connected.

For every $v\in V(G)$, we denote the set of edges incident to $v$ by $E_{v}$. A path of length $k$ between two vertices, $v,w\in V(G)$ is a $k$-tuple of vertices $\left(v=v_{1},v_{2},v_{3}\ldots v_{k}=w\right)$ such that for each $j=1,\ldots,(k-1)$, $(v_{j},v_{j+1})\in E(G)$. The set of all shortest paths between two vertices $u,v\in V(G)$ will be denoted as $\langle u,v\rangle$. For $u,v,w\in V(G)$, we write $u\in\langle v,w\rangle$ if $u$ lies on some shortest path between $v$ and $w$. Given a graph, the relation $u\preceq v\iff u\in\langle o,v\rangle$ induces a partial order on the vertices, with $o$ being the minimal element. For $v\in V(G)$, we define $gen(v)$ to be the length (i.e. the number of steps) of some shortest path between $v$ and $o$ and $G_{v}$ to be the graph induced by the set of all vertices comparable with $v$ (with respect to $\preceq$). Namely, $G_{v}$ is the graph induced by the set of vertices, $w$, such that either $w\preceq v$ or $v\preceq w$. An illustration of some of these definitions is given in Figure 2.1.

Define $S_{n}=\left\{v\in V(G)\,:\,gen(v)=n\right\}$ (namely, $S_{n}$ is the set of vertices with distance $n$ from the root). We assume throughout that for any $n$, there are no edges between vertices in $S_{n}$. With this assumption and the partial order structure induced on a graph, an edge $(u,v)=e\in E(G)$ has an initial vertex, namely the vertex closer to $o$, which is denoted by $i(e)$, and a terminal vertex, denoted by $t(e)$. We also define $gen(e)=gen(i(e))$.

The discrete Laplacian on a graph $G$ is the densely defined operator $\Delta_{d}:\,D(\Delta_{d})\rightarrow\ell^{2}(G)$, defined by $\Delta_{d}\phi(x)=\underset{x\sim y}{\sum}\left(\phi(x)-\phi(y)\right)$, where $x\sim y$ means that $x$ and $y$ are neighbors, and $D(\Delta_{d})=\left\{\phi\in\ell^{2}(G)\,:\,\Delta\phi\in\ell^{2}(G)\right\}$. $D(\Delta_{d})$ is dense because it contains all of the functions with compact (finite) support.

In order to discuss some symmetry properties of a graph, one would want to be able to say when any two vertices of the same sphere (i.e. two vertices in $S_{n}$) “look alike”. The definition of a rooted graph automorphism serves that purpose.

Definition 2.1**.**

A rooted graph automorphism is a bijection $\tau:\,V(G)\rightarrow V(G)$ such that $\tau(o)=o$, and $\left(\tau(v),\tau(u)\right)\in E\iff\left(v,u\right)\in E$.

A rooted graph $G=\left\langle V(G),E(G)\right\rangle$ will be called spherically symmetric if for every $n\in\mathbb{N}$ and for every $v,u\in S_{n}$, there exists a rooted graph automorphism $\tau$, such that $\tau(v)=u$. In other words, the group of all rooted graph automorphisms on $G$ acts transitively on $S_{n}$.

Two vertices $u,v\in S_{n}$ will be called forward neighbors if there exists some $w\in S_{n+1}$ such that $(v,w),(u,w)\in E$. Similarly, $u,v\in S_{n}$ will be called backward neighbors if there exists some $w\in S_{n-1}$ such that $\left(v,w\right),\left(u,w\right)\in E$. We can now define the type of symmetry we need.

Definition 2.2**.**

A rooted graph $G$ will be called family preserving if the following conditions hold:

$(i)$ If $u,v\in S_{n}$ are backward neighbors, then there exists a rooted graph automorphism $\tau$, such that $\tau(u)=v$, and $\tau|_{S_{n-j}}=Id$ for all $n\geq j\geq 1$.

$(ii)$ if $u,v\in S_{n}$ are forward neighbors, then there exists a rooted graph automorphism $\tau$ such that $\tau(u)=v$, and $\tau|_{S_{n+j}}=Id$ for all $j\geq 1$.

Remark 2.3*.*

Family preserving graphs were introduced in [13] in the discrete context as graphs on which it is possible to obtain a constructive decomposition of the combinatorial Laplacian into operators over cyclic subspaces. Examples are given in Figure 2.1 above and in Section 4 below, where we describe the decomposition procedure. We shall also present more examples in Section 5 below. We refer the reader to [13] for more illustrations and further discussion. We remark here only that in the case that $G$ is a tree then Definition 2.2 is equivalent to spherical symmetry, whereas for general graphs, this is a strictly stronger property. In Figure 2.2 we give an example of a graph which is spherically symmetric but not family preserving.

Having presented the basic definitions we need for the underlying combinatorial structure, we can now add a continuous structure on the edges. Consider a rooted graph $G$ and identify each edge, $e\in E(G)$, with a non degenerate line segment$\subseteq\mathbb{R}$. That edge is now a metric space (with the usual metric on $\mathbb{R}$), and in particular its length, denoted $l(e)$, is that of the associated line segment. We assume444This assumption is made in order to avoid technical issues regarding the definition of the domain of the operator and we make it here for simplicity. The central ideas in our approach do not rely on this assumption. throughout that $\underset{e\in{E(G)}}{\inf}l(e)>0$. This metric structure on the edges together with the gluing at the vertices defines a length for any path on the resulting topological space. This allows one to define a metric $d$ on the graph, under which the distance between two points is the length of a minimal path between them. To avoid confusion with the underlying combinatorial structure, we denote this metric space $\Gamma$ and refer to such a structure as a metric graph. For $x\in\Gamma$ we denote $|x|=d(x,o)$ (note that $x$ here is not necessarily a vertex of $G$, the underlying combinatorial graph). In the sequel, when we want to refer to the combinatorial structure of a metric graph, $\Gamma$, we shall use the notation $G_{\Gamma}$ (though we may omit the subscript when there is no risk of confusion). For simplicity, we let $V(\Gamma)=V\left(G_{\Gamma}\right)$ and $E(\Gamma)=E\left(G_{\Gamma}\right)$.

Definition 2.4**.**

A rooted metric graph $\Gamma$ will be called spherically homogeneous if for every $e_{1},e_{2}\in E(\Gamma)$, $gen(e_{1})=gen(e_{2})\Rightarrow l(e_{1})=l(e_{2})$.

In this work, unless stated otherwise, a metric graph $\Gamma$ will always be spherically homogeneous. For any such graph, the length of an edge and the distance of a vertex from the root depend solely on their generation. Thus, we denote by $t_{k}$ the distance of a vertex in $S_{k}$ from the root. Also, letting $l_{k}=l(e)$ for some $e$ with $gen(e)=k$ and following [28, 32], we may define the height of $\Gamma$ by

[TABLE]

which may be finite or infinite.

Definition 2.5**.**

A rooted metric graph $\Gamma$ will be called family preserving if it is spherically homogeneous and if $G_{\Gamma}$ is family preserving.

The underlying metric structure induces a measurable structure on $\Gamma$. We denote the relevant $\sigma$-algebra by $\mathcal{B}$. The measure of a set $E\in\mathcal{B}$ is $\underset{e\in E(\Gamma)}{\sum}\lambda(e\cap E)$, where $\lambda$ is the Lebesgue measure on $e$. We will denote that measure by $\mu$. $L^{2}(\Gamma)$ is the space of all measurable functions (with respect to $\mathcal{B}$) $f:\,\Gamma\rightarrow\mathbb{C}$ such that $\int_{\Gamma}|f|^{2}d\mu<\infty$.

Given a metric graph $\Gamma$, we denote by $\mathcal{H}$ the space $L^{2}(\Gamma)$. Recall that given a domain $\Omega\subseteq\mathbb{R}^{d}$ and $k\in\mathbb{N}$, $H^{k}(\Omega)$ is the space of all functions $f\in L^{2}(\Omega)$ for which all of the weak derivatives up to order $k$ are in $L^{2}(\Omega)$.

We define a quadratic form $q:\,Q\times Q\rightarrow\mathbb{C}$ by

[TABLE]

and for $f,g\in Q$, $q(f,g)=\intop_{\Gamma}f^{\prime}\overline{g^{\prime}}d\mu$. The (Dirichlet-Kirchhoff) Laplacian, $\Delta_{c}^{\Gamma}$, on $\Gamma$ is the self adjoint operator associated with this quadratic form. By [7, Theorem 1.4.19], and under the assumption $\underset{e\in{E(\Gamma)}}{\inf}l(e)>0$, the domain of $\Delta_{c}^{\Gamma}$ is the space of all functions $f\in L^{2}(\Gamma)$ such that $f(o)=0$, $f$ is continuous, $f|_{e}\in H^{2}(e)$ for every $e\in E(\Gamma)$, $\int_{\Gamma}|f^{\prime\prime}(x)|^{2}dx<\infty$, and for every $v\in V(\Gamma)$, $\underset{u\sim v}{\sum}\left(f|_{[u,v]}\right)^{\prime}(v)=0$, where $[u,v]$ is the line segment connecting $u$ and $v$. These conditions are called the Kirchhoff boundary conditions [28]. When it is clear from the context, we will omit the subscript and superscript and just write $\Delta$.

Remark 2.6*.*

Although in this work we only consider the Kirchhoff boundary conditions on the vertices, there are other possible vertex conditions which can be defined in order to make the Laplacian on $\Gamma$ self adjoint (see e.g. [7, Chapter 1]). As long as these vertex conditions are chosen so as to preserve the underlying symmetry of the operator (i.e., in a spherically symmetric fashion), it is clear that our results, properly modified, extend to that case as well.

2.2 The Main Abstract Result

Our main result is a decomposition of the Laplacian into invariant subspaces described by the structure of the graph. These spaces are generated by functions arising in the decomposition of the corresponding discrete structure. The discrete decomposition provides the skeleton for the decomposition in the metric case. Thus, before describing our main result, we need to recall the result in the discrete case.

Let $G$ be a family preserving graph. Recall that $G_{v}$ set of all vertices comparable with $v$ with respect to the order relation $\preceq$ introduced in the previous subsection. Given $n,j\in\mathbb{N}$, introduce the following operators:

[TABLE]

[TABLE]

and if $j\leq{n}$, then we may also define

[TABLE]

A decomposition for the Laplacian on family preserving graphs was presented in [13].

Theorem 2.7**.**

([13, Theorem 2.6])* Let $G$ be a family preserving graph. Then $\ell^{2}(G)=\oplus_{r=0}^{\infty}H_{r}$ where $H_{r}$ is invariant under $\Delta_{d}$ and such that:*

$(i)$* For every $r$ there exists $n(r)$ and a vector $\phi_{0}^{r}$ such that $supp(\phi_{0}^{r})\subseteq S_{n(r)}$, and $H_{r}=\overline{span\left\{\phi_{0}^{r},\Delta_{d}\phi_{0}^{r},\Delta_{d}^{2}\phi_{0}^{r},\ldots\right\}}$.*

$(ii)$* The set $\{\phi_{0}^{r},\phi_{1}^{r},\ldots\}$ obtained from $\{\phi_{0}^{r},\Delta_{d}\phi_{0}^{r},\Delta_{d}^{2}\phi_{0}^{r},\ldots\}$ by applying the Gram-Schmidt process has the property that $supp(\phi_{k}^{r})\subseteq S_{n(r)+k}$ for every $k\geq 0$.*

Furthermore, for every $r,j\in\mathbb{N}$, $\phi_{0}^{r}$ is an eigenfunction of $\Lambda_{n(r),\pm{j}}$.

Remark 2.8*.*

Throughout this work, the notation $\phi_{n}^{r}$ will refer to the functions presented in Theorem 2.7, $(ii)$.

Remark 2.9*.*

The last statement, about $\phi_{0}^{r}$ being an eigenfunction of $\Lambda_{n(r),\pm{j}}$, is not part of the statement of Theorem 2.6 in [13]. However, it is shown in its proof. We include it here in the statement for simplicity of reference.

Remark 2.10*.*

It also follows from the proof of [13, Theorem 2.6] that for every $k\in\mathbb{N}$, there exists $c\in\mathbb{C}$ such that $\left(\Delta_{d}^{k}(\phi_{0}^{r})\right)|_{S_{n+k}}=c\phi_{k}^{r}$.

Now let $\Gamma$ be a family preserving metric graph and $G=G_{\Gamma}$ the associated discrete graph. For $x\in\Gamma$ define $G_{x}$ to be $G_{i(e)}\cap{G_{t(e)}}$ where $x$ lies on the edge $e$ (recall that $i(e)$ is the initial vertex of $e$, and $t(e)$ is the terminal vertex of $e$). Note that for any $v\in V(G)$ and $0<t<h(\Gamma)$ the set $G_{v}\cap\left\{x\in\Gamma\,:\,|x|=t\right\}$ is finite. Moreover, if $|u|=|v|$ then, by symmetry, the size of this set is equal to the size of $G_{u}\cap\left\{x\in\Gamma\,:\,|x|=t\right\}$. For $|v|=n$ we denote this number by $g_{n}(t)$. Finally, we let $g_{\Gamma}(t)=g_{0}(t)=\#\left\{x\in\Gamma\mid|x|=t\right\}$.

The first step in translating the discrete decomposition into a continuous one is defining a procedure to obtain a function on the metric graph from one defined on the discrete graph. We do this by ‘spreading’ the values of the function over the graph, taking into account the symmetry. To be precise, given $n\in\mathbb{N}$ and $\phi\in\ell^{2}(S_{n})$, let $h^{\phi}:\,\Gamma\rightarrow\mathbb{C}$ be defined by

[TABLE]

Next we want to use this procedure to define projection operators based on the functions $\phi_{n}^{r}$ above. For any $f:\,\Gamma\rightarrow\mathbb{C}$, let $f_{t}\in\mathbb{C}^{g_{\Gamma}(t)}$ be the vector whose entries are the values $f$ takes on points with distance $t$ from the root (with respect to some fixed ordering of these points). For $\phi\in\ell^{2}(S_{n})$, let $P_{\phi}:\,L^{2}(\Gamma)\rightarrow L^{2}(\Gamma)$ be defined by

[TABLE]

where $\langle\cdot,\cdot\rangle$ is the standard inner product in $\mathbb{C}^{g_{\Gamma}(|x|)}$. We shall show in the next section that for every $r,n\in\mathbb{N}$, $P_{\phi_{n}^{r}}$ is in fact an orthogonal projection (with image in $L^{2}(\Gamma)$). We define the space $F_{r,n}$ to be the image of the orthogonal projection $P_{r,n}\coloneqq P_{\phi_{n}^{r}}$. The decomposition presented in this work will include the image of $P_{r,0}$ for every $r\in\mathbb{N}$. To abbreviate, we shall denote $h^{\phi_{n}^{r}}=h^{r,n}$, $h^{r,0}=h^{r}$, $P_{r,0}=P_{r}$, and $F_{r,0}=F_{r}$.

We want to use the spaces $F_{r}$ to decompose $\Delta$ on $\Gamma$. There is, however, a ‘local dimension counting’ issue: while functions on $\Gamma$ are determined by the values they take on edges, functions on $G_{\Gamma}$ are determined by their values on vertices. If $G$ is a tree, this is not an issue since the number of edges of a particular generation is always equal to the number of vertices of the next generation. Generally, however, the local number of vertices and edges does not need to be the same (think of antitrees described in Section 5.2). In order to deal with this problem, in addition to the assumption that there are no edges between edges of the same generation, we need the following

Definition 2.11**.**

A metric graph $\Gamma$ will be called locally balanced if for every $n\in\mathbb{N}$,

[TABLE]

It is intuitively clear that any graph can be made into a locally balanced graph by adding vertices in the middle of ‘bad’ edges. Since such vertices come with Kirchhoff boundary conditions, this makes no difference as far as $\Delta$ is concerned. Thus

Proposition 2.12**.**

Let $\Gamma$ be a metric family preserving graph, then there exists another family preserving graph $\overline{\Gamma}$ such that the pair $\left\langle\Gamma,\Delta_{\Gamma}\right\rangle,\,\left\langle\overline{\Gamma},\Delta_{\overline{\Gamma}}\right\rangle$ are unitarily equivalent, and $\overline{\Gamma}$ is locally balanced.

A proof is given in the appendix. Now, let $\Gamma$ be a metric family preserving graph and let $\overline{\Gamma}$ be a corresponding locally balanced family preserving graph. Note that although $\Delta_{\Gamma}$ is unitarily equivalent to $\Delta_{\overline{\Gamma}}$, the spaces $F_{r}$ for each one of these graphs may be different.

We are finally ready to state our main result.

Theorem 2.13**.**

Let $\Gamma$ be a locally balanced family preserving metric graph satisfying $\underset{e\in{E(\Gamma)}}{\inf}l(e)>0$. Assume further that there are no edges between vertices of the same generation in $\Gamma$. Then the subspaces $(F_{r})_{r\in\mathbb{N}}$ form a decomposition of $L^{2}(\Gamma)$, which reduces the Laplacian. Meaning:

$(i)$* $L^{2}(\Gamma)=\underset{r}{\oplus}F_{r}$.*

$(ii)$* For every $r$, the projection $P_{r}$ onto $F_{r}$ commutes with the Laplacian, i.e. $P_{r}(D(\Delta))\subseteq D(\Delta)$, and $P_{r}\Delta=\Delta P_{r}$ on $D(\Delta)$.*

Furthermore, for every $r\in{\mathbb{N}}$, there exist $a_{r}\in{\left(0,\infty\right)}$ and $b_{r}\in{\left(0,\infty\right]}$, which depend on $\phi_{0}^{r}$, such that functions in $F_{r}$ are supported on $\left\{x\in{\Gamma}:a_{r}<|x|<b_{r}\right\}$ and $\Delta|_{F_{r}}$ is unitarily equivalent to a differential operator on $L^{2}\left(a_{r},b_{r}\right)$.

Remark 2.14*.*

The actual strength of Theorem 2.13 is the simple description of the spaces $\left(F_{r}\right)_{r\in{\mathbb{N}}}$. By this we mean that the resulting operator on $L^{2}\left(a_{r},b_{r}\right)$ is the Laplacian with boundary conditions along some set of points. The boundary onditions and the set on which they are defined can be relatively easily described via structural properties of the underlying graph. This, along with the proof of the second part of Theorem 2.13, is shown in Section 4.

3 Proof of Theorem 2.13

Before proving Theorem 2.13, we need some results regarding the discrete structure of a family preserving graph, and some properties of the functions presented in Theorem 2.7.

3.1 The Discrete Structure of a Family Preserving Graph

In order to express the type of symmetry family preserving graphs have, we will first need some definitions.

Definition 3.1**.**

Let $n,k\in\mathbb{N}$ and let $v,u\in S_{n}$. A $k$-forward path from $v$ to $u$ is a path $(x_{0},\ldots,x_{k},\ldots,x_{2k})$ of length $2k+1$ such that $x_{0}=v$, $x_{2k}=u$, and $gen(x_{j})=n+j$ for all $j\leq k$.

Similarly, a $k$-backward path from $v$ to $u$ is a path $(x_{0},\ldots,x_{k},\ldots,x_{2k})$ of length $2k+1$ such that $x_{0}=v$, $x_{2k}=u$, and $gen(x_{j})=n-j$ for all $j\leq k$.

A $k$-forward path between $x,y\in S_{n}$ is a path that starts at $x$, takes $k$ steps forward (i.e. away from the root), and then takes $k$ steps backwards and reaches $y$. A $k$-backward path starts at $x$, takes $k$ steps towards the root and then takes $k$ steps back to $y$.

Definition 3.2**.**

Let $n,k\in\mathbb{N}$ and let $u\in S_{n},\,v\in S_{n+k}$. A descending path between $u$ and $v$ is a path $(x_{0}=u,\ldots,x_{k}=v)$, where $gen(x_{i})=gen(x_{i-1})+1$ for every $1\leq i\leq k$. An ascending path is a path $(y_{0}=v,\ldots,y_{k}=u)$ where $gen(y_{i})=gen(y_{i-1})-1$ for every $1\leq i\leq k$.

We will use the following lemma, proved in [13].

Lemma 3.3**.**

([13, Lemma 3.4]) Let $G$ be a family preserving graph. Let $x,y\in S_{n}$, and assume there is a $k$-forward path between $x$ and $y$. Then there is a rooted graph automorphism $\tau$, such that $\tau(x)=y$, and $\tau|_{S_{n+k}}=Id$.

Similarly, assume there is a $k$-backward path between $x$ and $y$. Then there is a rooted graph automorphism $\tau$, such that $\tau(x)=y$, and $\tau|_{S_{n-k}}=Id$

Corollary 3.4**.**

Let $G$ be a family preserving graph. Let $x,y\in S_{n}$, and assume that there exists a $k$-forward path between $x$ and $y$. Assume also that $x_{k}=v\in S_{n+k}$. Then the number of descending paths between $x$ and $v$ is equal to the number of descending paths between y and v.

Similarly, assume there exists a $k$-backward path between $x$ and $y$ and assume that $x_{k}=v\in S_{n-k}$. Then the number of ascending paths between $x$ and $v$ is equal to the number of ascending paths between $y$ and $v$.

Proof.

We will prove the first part. The second one is analogous. Let $\tau$ be a rooted graph automorphism such that $\tau(x)=y$, and $\tau|_{S_{n+k}}=Id$. Then $\tau$ takes every descending path between $x$ and $v$ to a descending path between $y$ and $v$. The fact that $\tau$ is bijective implies that this correspondence is also bijective. ∎

Lemma 3.5**.**

Let $\Gamma$ be a family preserving graph, and let $n\in\mathbb{N}$. For every $k\in\{-n,-n+1,\ldots,0,1,2,\ldots\}$ and $v,u\in S_{n+k}$, either

[TABLE]

or

[TABLE]

Proof.

Let $v,u\in S_{n+k}$. Suppose $(G_{v}\cap S_{n})\bigcap(G_{u}\cap S_{n})\neq\emptyset$, and let $w\in(G_{v}\cap S_{n})\bigcap(G_{u}\cap S_{n})$. Let $x\in G_{v}\cap S_{n}$. If $k>0$ then, by the assumption, there is a $k$-forward path between $x$ and $w$. Thus by Lemma 3.3, there is a rooted graph automorphism $\tau$ such that $\tau(w)=x$, and $\tau|_{S_{n+k}}=Id$. There is a descending path $(w=x_{0},x_{1},\ldots,x_{k}=u)$, which translates under $\tau$ to a descending path $(x=\tau(x_{0}),\tau(x_{1}),\ldots,\tau(x_{k})=u)$, which implies that $x\in G_{u}\cap S_{n}$. Since $x$ was chosen arbitrarily from $G_{v}\cap S_{n}$, we have that $G_{v}\cap S_{n}\subseteq G_{u}\cap S_{n}$. It can be similarly proven that $G_{u}\cap S_{n}\subseteq G_{v}\cap S_{n}$, and we have $G_{v}\cap S_{n}=G_{u}\cap S_{n}$, as required.

If $k\leq 0$ the proof is similar. ∎

Lemma 3.5 is the first step in obtaining the orthogonality of the spaces $(F_{r})_{r\in\mathbb{N}}$ as a consequence of the orthogonality of the functions in the set $\{\phi_{n}^{r}\,:\,r,n\in\mathbb{N}\}$.

Lemma 3.6**.**

Let $G$ be a family preserving graph, $j,n\in\mathbb{N}$, and $\phi\in\ell^{2}(S_{n})$. Then $\forall u\in S_{n+j+1}$,

[TABLE]

where $k_{u}(x)$ is the number of different descending paths from $x$ to $u$.

Proof.

By induction on $j$. For $j=1$, the claim is trivial. Assume the claim is true for $j=k$. Then for $j=k+1$,

[TABLE]

∎

Lemma 3.7**.**

Let $G$ be a family preserving graph, $j,n\in\mathbb{N}$, and $\phi\in\ell^{2}(S_{n+j})$. Then $\forall v\in S_{n}$,

[TABLE]

where $k^{v}(x)$ is the number of different ascending paths from $x$ to $v$.

The proof is symmetric to the proof of Lemma 3.6.

Remark 3.8*.*

It is easy to see that by the definition of $\Delta_{d}$, given $\phi\in\ell^{2}(S_{n})$, $k\in\mathbb{N}$, for $v\in S_{n+k}$ we have $\Delta_{d}^{k}(\phi)(v)=(E_{n+k-1}\cdots E_{n})(\phi)(v)$.

The fact that for every $r,k\in\mathbb{N}$, $\phi_{0}^{r}$ is an eigenvector of $\Lambda_{n(r),+k}$ means that $F_{r}=F_{r,n}$ for every $n\in\mathbb{N}$. This is proved in the following lemmas, and is necessary in order to prove that for every family preserving metric graph $\Gamma$, the spaces $(F_{r})_{r\in\mathbb{N}}$ span $L^{2}(\Gamma)$. The main idea is that on family preserving graphs, for every $\phi\in\ell^{2}(S_{n})$, $\Lambda_{n,\pm{k}}(\phi)$ takes the same values on vertices that share a $k$-forward/backward path. This is shown in the following lemma.

Lemma 3.9**.**

Let $G$ be a family preserving graph and let $v,u\in S_{n}$ such that there exists a $k$-forward path between $v$ and $u$. Then for every $\phi\in\ell^{2}(S_{n})$, $\Lambda_{n,+k}(\phi)(v)=\Lambda_{n,+k}(\phi)(u)$. Furthermore, denote by $K$ the set of vertices in $S_{n}$ that share a $k$-forward path with $v$ (or u). Then $\Lambda_{n,+k}(\phi)|_{K}\equiv c\underset{x\in K}{\sum}\phi(x)$, where $c$ is some natural number greater than 1, which depends only on $n,k$. The analogous statement for $\Lambda_{n,-k}$ and for $v,u\in{S_{n}}$ that have a $k$-backward path between them is also true.

Proof.

We prove the lemma for $v,u\in{S_{n}}$ that share a $k$-forward path. The proof for the case that they share a $k$-backward path is symmetric.

By Lemma 3.5 we have

[TABLE]

In addition, by the same lemma, for every $w_{1},w_{2}\in A_{v,u}$,

[TABLE]

Denote \widetilde{\phi}=E_{n+k}\cdots E_{n}(\text{\phi$)$\in\ell}^{2}(S_{n+k}). By Lemma 3.6, for every $w\in A_{v,u}$,

[TABLE]

Now, by Corollary 3.4, the factor $k_{w}(x)$ in the sum is constant, so we may write

[TABLE]

where $c_{1}$ is $k_{w}(x)$ for some $x\in B_{v,u}$. Now, again by Corollary 3.4, the constant $c_{1}$ is the same for every $w_{1},w_{2}\in A_{v,u}$, and that means that $\widetilde{\phi}$ is constant on $A_{v,u}$. By definition, we have

[TABLE]

We apply Corollary 3.4 twice again, and get that $k^{v}(x)$ is constant on $A_{v,u}$, and that constant is the same for every $w\in B_{v,u}$, so we have

[TABLE]

which proves the first part of the lemma. The second part follows by observing that $c_{1},c_{2}$ are natural constants greater than [math], and the same procedure can be applied to every vertex that shares a $k$-forward path with $v$. ∎

Lemma 3.10**.**

Let $G$ be a family preserving graph, and let $r\in\mathbb{N}$. Then for every $k\in{\left\{-n(r),-n(r)+1,\ldots,0,1,\ldots\right\}}$ and for every $v\in S_{n(r)+k}$, either $\underset{x\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(x)=0$ or $\phi_{0}^{r}|_{G_{v}\cap S_{n(r)}}\equiv c$ for some constant $c\in\mathbb{C}$.

Proof.

We prove the lemma for $k\geq 0$, the proof for $k<0$ is symmetric. By Theorem 2.7, $\phi_{0}^{r}$ is an eigenvector of the operator $\Lambda_{n,+k}$. Assume $\Lambda_{n,+k}\left(\phi_{0}^{r}\right)=\lambda_{n,+k}\phi_{0}^{r}$ for some $\lambda_{n,+k}\in\mathbb{C}$. Let $x,y\in G_{v}\cap S_{n(r)}$. The set $G_{v}\cap S_{n(r)}$ is exactly the set of vertices in $S_{n(r)}$ which share a $k$-forward path with $x$ (or with $y$), which means (by Lemma 3.9) that $\Lambda_{n,+k}(\phi_{0}^{r})$ is constant on that set. Combining that with the fact that $\phi_{0}^{r}$ is an eigenvector of $\Lambda_{n,+k}$, we get that for every $x\in G_{v}\cap S_{n(r)}$,

[TABLE]

Where $c$ is a natural number greater than [math]. If $\lambda_{n,+k}=0$, we get $\underset{y\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(y)=0$. Otherwise, dividing $(3.4)$ by $\lambda_{n,+k}$ proves the result. ∎

Lemma 3.11**.**

Let $G$ be a family preserving graph, and let $r,k\in\mathbb{N}$. Assume that $\phi_{k}^{r}\neq 0$. Then for every $v\in S_{n(r)+k}$, $\phi_{0}^{r}$ is constant on $G_{v}\cap S_{n(r)}$.

Proof.

Let $v\in S_{n(r)+k}$ be such that $\phi_{k}^{r}(v)\neq 0$. By Remark 2.10, this means that $\Delta_{d}^{k}(\phi_{0}^{r})(v)\neq 0$, and by Remark 3.8 and $(3.1)$, this means that $\underset{x\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}\neq 0$. By the previous lemma, we get that $\phi_{0}^{r}$ is constant on $G_{v}\cap S_{n(r)}$. By Theorem 2.7 and the second part of Lemma 3.9, $\phi_{0}^{r}$ is an eigenvector of $\Lambda_{n,+k}$ with an eigenvalue $\lambda_{n,+k}\neq 0$. Let $w\in S_{n(r)+k}$ be such that $G_{w}\cap S_{n(r)}\neq G_{v}\cap S_{n(r)}$. Assume that $\phi_{0}^{r}$ is not constant on $G_{w}\cap S_{n(r)}$, and let $x\in G_{w}\cap S_{n(r)}$ be such that $\phi_{0}^{r}(x)\neq 0$. By Lemma 3.10, $\underset{y\in G_{w}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(y)=0$. By the second part of Lemma 3.9, $\Lambda_{n,+k}(\phi_{0}^{r})(x)=c\underset{y\in G_{w}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(y)=0$. On the other hand, $\Lambda_{n,+k}(\phi_{0}^{r})(x)=\lambda_{n,+k}\phi_{0}^{r}(x)\neq 0$, which is a contradiction. Thus we have that $\phi_{0}^{r}$ is constant on $G_{w}\cap S_{n(r)}$, as required. ∎

Proposition 3.12**.**

Let $G$ be a family preserving graph and $r,k\in\mathbb{N}$ such that $\phi_{k}^{r}\neq 0$. then $F_{r}=F_{r,k}$.

Proof.

It is sufficient to show that there exists $c\in\mathbb{C}$ such that for every $v\in S_{n(r)+k}$, $\phi_{0}^{r}|_{G_{v}\cap S_{n(r)}}\equiv c\phi_{k}^{r}(v)$, and by Remark 2.10 we may examine $\Delta_{d}^{k}(\phi_{0}^{r})$. By Lemma 3.11, for every $v\in S_{n(r)+k}$, $\phi_{0}^{r}$ is constant on $G_{v}\cap S_{n(r)}$. Furthermore, by $(3.2)$ and remark 3.8, $\Delta_{d}^{k}(\phi_{0}^{r})(v)=c\underset{x\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(v)$, where $c$ is the number of descending paths from some $x\in G_{v}\cap S_{n(r)}$ to $v$. By spherical symmetry of $G$, for every $w\in S_{n(r)+k}$, the number of descending paths from some $x\in G_{w}\cap S_{n(r)}$ is exactly $c$, and we get the result. ∎

3.2 Some Results Regarding the Projections $(P_{n,r})_{n,r\in\mathbb{N}}$

With the lemmas presented we are almost ready to prove Theorem 2.13. As a final preliminary we show that for every $r\in\mathbb{N}$, the transformation $P_{r}$ defined by $(2.3)$ is indeed an orthogonal projection.

Lemma 3.13**.**

Let $\Gamma$ be a family preserving metric graph and let $r,k\in\mathbb{N}$. Then for every $0\leq t<h(\Gamma)$, either $||h_{t}^{r,k}||_{\mathbb{C}^{g_{\Gamma}(t)}}=1$ or $h_{t}^{r,k}=0$.

Proof.

Let $0\leq t<h(\Gamma)$. Assume that $h_{t}^{r,k}\neq 0$. Then there exists some $x\in\Gamma$ with $|x|=t$ and $\underset{v\in G_{x}\cap S_{n(r)+k}}{\sum}\phi_{0}^{r}(v)\neq 0$. This means, by Lemma 3.11, that $\phi_{0}^{r}$ is constant on $G_{y}\cap S_{n(r)}$ for every $y\in\Gamma$ with $|y|=t$. This implies that $\phi_{k}^{r}$ is constant on $G_{y}\cap{S_{n(r)+k}}$. Now, we have

[TABLE]

[TABLE]

where $v_{x}$ is some representative of $G_{x}\cap S_{n(r)+k}$. Now, note that

[TABLE]

and if we divide $S_{n(r)+k}$ to the classes of $G_{x}\cap S_{n(r)+k}$ for $x$ with $|x|=t$, say $S_{n(r)+k}=\underset{i=1}{\overset{k}{\bigsqcup}}G_{x_{i}}\cap S_{n(r)+k}$, then

[TABLE]

Now, each summand in the right hand side of $(3.4)$ appears $g_{n(r)+k}(t)$ times in $(3.3)$, and we have

[TABLE]

as required. ∎

Lemma 3.14**.**

For every $r,k\in{\mathbb{N}}$, the image of $L^{2}(\Gamma)$ under $P_{r,k}$ (as defined by $(2.5)$) is contained in $L^{2}(\Gamma)$. Furthermore, $P_{r,k}$ is an orthogonal projection.

Proof.

The first part of the lemma follows from the fact that for every $0\leq t<h(\Gamma)$ we have $\left\|P_{r,k}(f)_{t}\right\|_{\mathbb{C}^{g_{\Gamma}(t)}}\leq\left\|f_{t}\right\|_{\mathbb{C}^{g_{\Gamma}(t)}}$. Additionally, for every $x\in\Gamma$ we have

[TABLE]

thus $P_{r,k}$ is indeed a projection. Finally, we need to show that $P_{r,k}$ is self adjoint. Let $f\in L^{2}(\Gamma)$. We need to show that $\left\langle P_{\phi}(f),f-P_{\phi}(f)\right\rangle_{L^{2}(\Gamma)}=0$. Indeed

[TABLE]

where the next to last equality follows since $||h_{t}^{r,k}||_{\mathbb{C}^{g_{\Gamma}(t)}}=1$. ∎

3.3 Proof of Theorem 2.13

Proof.

Let $\Gamma$ be a locally balanced family preserving metric graph.

$(i)$ We first prove orthogonality. Note that for every $t\geq 0$, $\left(P_{r}(f)\right)_{t}$ is the orthogonal projection of $f_{t}$ to the span of $h_{t}^{r}$. In addition, for every $r_{1}\neq r_{2}$, the orthogonality of $\phi_{0}^{r_{1}}$ and $\phi_{0}^{r_{2}}$ implies that $h_{t}^{r_{1}}$ and $h_{t}^{r_{2}}$ are also orthogonal in $\mathbb{C}^{g_{\Gamma}(t)}$. So given $f\in F_{r_{1}}$, $f=P_{r_{1}}(f)$, $g\in F_{r_{2}}$, $g=P_{r_{2}}(g)$, we have that $\underset{|x|=t}{\sum}P_{r_{1}}(f)(x)\overline{P_{r_{2}}(g)(x)}=0$ which implies that $\langle f,g\rangle_{L^{2}}=0$, as required.

Now, let $f\in L^{2}(\Gamma)$ such that $supp(f)\subseteq e=(u,v)$, $gen(e)=n$. For every $e^{\prime}\in E(\Gamma)$ such that $gen(e^{\prime})=gen(e)$, define $f_{e^{\prime}}$ on $e^{\prime}$ to be the translation of $f$ to the edge $e^{\prime}$. That is, for $x$ on $e^{\prime}$, $f_{e^{\prime}}(x)=f(y)$ where $y\in e$ such that $|y|=|x|$.

Define $H_{f}$ to be the subspace of $L^{2}(\Gamma)$ spanned by the set $\{f_{e^{\prime}}\,:\,gen(e^{\prime})=gen(e)\}$. Since $\Gamma$ is locally balanced, either

[TABLE]

or

[TABLE]

Assume first that $\#\{e\in E\,:\,gen(e)=n\}=\#\{v\in V\,:\,gen(v)=n\}$ and let $\{\phi_{1},\ldots,\phi_{|S_{n}|}\}\subseteq\{\phi_{j}^{r}\,:\,j,r\in\mathbb{N}\}$ be a basis of $\ell^{2}(S_{n})$. For every $1\leq i\leq|S_{n}|$, define $g^{i}\in H_{f}$ in the following way:

[TABLE]

where $e_{x}$ is the edge containing $x$, and $v_{x}=i(e_{x})$ is the initial vertex of $e_{x}$. In other words,

[TABLE]

so clearly, $g^{i}\in H_{f}$. Now, we have (note that if $gen(e^{\prime})=n$ then $|G_{x}\cap S_{n}|=1$)

[TABLE]

where the third equality follows from the fact that $|x|=|y|\Rightarrow f_{e_{x}}(x)=f_{e_{y}}(y)$, and the fourth from the fact that $||\phi_{i}||_{\ell^{2}}=1$. This implies that $g^{i}\in\textrm{Im}\left(P_{\phi^{i}}\right)$ and so that $g^{i}\in{\oplus}F_{r}$. Moreover, by the independence of the $\phi^{i}$, it is clear that the functions $\{g^{1},\ldots,g^{|S_{n}|}\}$ are linearly independent. Thus, by dimension considerations it follows that $H_{f}$ is spanned by $\{g^{1},\ldots,g^{|S_{n}|}\}$. This implies that $H_{f}\subseteq\oplus F_{r}$, and in particular $f\in\underset{r\in\mathbb{N}}{\oplus}F_{r}$. Since linear combinations of such $f$’s are dense in $L^{2}(\Gamma)$ we obtain $L^{2}(\Gamma)=\underset{r\in\mathbb{N}}{\oplus}F_{r}$ as required.

The proof in the case that $\#\{e\in E\,:\,gen(e)=n\}=\#\{v\in V\,:\,gen(v)=n+1\}$ is similar.

$(ii)$ First, since differentiation is a local action and $h^{r}$ is constant on edges, it is easy to see that $\Delta P_{r}=P_{r}\Delta$ on $D(\Delta)$. Moreover, for the same reason differentiability properties of $\varphi$ on the edges are unchanged by $P_{r}$. Therefore to show that $P_{r}(D(\Delta))\subseteq D(\Delta)$ we only need to check the gluing conditions at the vertices. Thus let $\varphi\in{D(\Delta)}$, $f=P_{r}(\varphi)$, and let $v\in S_{n(r)+k}$, where $k\in\{-n(r)+1,-n(r),\ldots,0,1,\ldots\}$. We divide into cases.

Assume first that $\underset{u\in{G_{w}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(u)=0$ for every $w\in{S_{n(r)+k}}$. In that case $k\neq 0$. We treat the case that $k\geq 1$. The proof for $k\leq-1$ is symmetric. Note that for $x\in G_{v}$ such that $|x|>|v|$, ${G_{v}\cap{S_{n(r)}}}\subseteq G_{x}\cap{S_{n(r)}}$. Thus, by Lemma 3.10 and the definition of $P_{r}$, $f$ vanishes on the set $\left\{x\in{\Gamma}\ |\ |x|\geq|v|\right\}$.

In order to check the gluing conditions, we first check continuity at $v$. We need to verify that $\underset{x\nearrow v}{lim}f(x)=0$. For convenience, we denote $g_{n(r)}(|x|)$ by $g_{n(r)}$, and $|G_{x}\cap S_{n(r)}|$ by $w_{n(r)}$ since they are constant along segments between vertices.

[TABLE]

where the next to last equality follows from continuity of $\varphi$, and the rest follow from changing the order of summation. As for the derivative matching condition, let $e\in{E(\Gamma)}$ such that $t(e)=v$. Note that $h^{r}$ is constant on $e$, so for $x(e)\in{e}$ we denote $h^{r}(x(e))=c_{e}$. Now, for every $x\in{e}$ $(f|_{e})(x)=c_{e}\left<\varphi_{|x|},h_{|x|}^{r}\right>$. Thus, we may write

[TABLE]

Now note that

[TABLE]

as it is some multiple of $\underset{u\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(u)$. This is a consequence of the fact that $G_{v}\cap{S_{n}(r)}=\underset{e:t(e)=v}{\bigcup}G_{x(e)}\cap{S_{n}(r)}$ and for every $e_{1},e_{2}\in{E(\Gamma)})$ such that $t(e_{1})=t(e_{2})=v$,

[TABLE]

which in turn follows from Corollary 3.4. Thus, by our assumption, the above sum is [math] as required.

We now treat the case in which there exists some $w\in{S_{n(r)+k}}$ for which $\underset{u\in{G_{w}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(u)\neq 0$. This is impossible for $k<0$ since $H_{r}$ is orthogonal to $\ell^{2}(S_{n})$ for every $n<n(r)$, and the above sum is a multiple of $\left<\Delta_{d}^{|k|}\phi_{0}^{r},\delta_{w}\right>$. Thus $k\geq 0$. The fact that the above sum is not [math] implies that $\phi_{k}^{r}\neq 0$. Thus, by Lemma 3.12, $F_{r}=F_{r,k}$. By the uniqueness of the orthogonal projection and by Lemma 3.14, we may write $f=P_{r,k}(\varphi)$. For sufficiently small $\epsilon>0$, for every $x\in\Gamma$ with $|x|=|v|\pm\epsilon$ it holds that $|G_{x}\cap{S_{n(r)+k}}|=1$, thus

[TABLE]

where $v_{x}\in S_{n(r)+k}$ and $x$ lies on the edge $e$ for which either $i(e)=v_{x}$ or $t(e)=v_{x}$. Thus, for $v\in S_{n(r)+k}$ and $x\in\Gamma$ which lies on an edge $(u,v)\in E(\Gamma)$, we have $v_{x}=v$ and so

[TABLE]

Now, when $x$ tends to $v$, $\varphi(y)$ tends to $\varphi(w)$ and so

[TABLE]

It follows that

[TABLE]

which means that $f$ is continuous.

We now prove that the derivative matching condition holds. In order to do so, we divide the edges of which $v$ is a part into two sets. The set of edges which terminate in $v$ will be denoted as $E_{t}$, and the set of edges emanating from $v$ will be denoted as $E_{s}$. We also denote $g_{n(r)+k}(|x|)\coloneqq b_{n(r)+k}^{in}$ for $|u|<|x|<|v|$ with $u\in{S_{n(r)+k-1})}$ (note that $b_{n(r)+k}^{in}$ is the number of edges terminating in $v$), and $g_{n(r)+k}(|x|)\coloneqq b_{n(r)+k}^{out}$ for $|u|>|x|>|v|$ with $u\in{S_{n(r)+k+1}}$ ($b_{n(r)+k}^{out}$ is the number of edges emanating from $v$). Finally, for every edge $e\in{E(\Gamma)}$ denote by $x(e)$ the selection of some point $x\in{e}$. Now, we calculate the sum $\underset{u\sim v}{\sum}(P_{r,k}(\varphi)|_{(u,v)})(x(u,v))$:

[TABLE]

Note that if we pick $|x(e_{1})|=|x(e_{2})|=t_{1}$ for every $e_{1},e_{2}\in E_{t}$ and $|x(e_{3})|=|x(e_{4})|=t_{2}$ for every $e_{3},e_{4}\in E_{s}$, then we may write the above sum as

[TABLE]

Since $\varphi\in D(\Delta)$, by taking the limit $|t_{1}|$ and $|t_{2}|$ to $|v|$ and differentiating, we see that the derivative matching condition holds. ∎

The proof of The proof of the second part of Theorem 2.13, along with a description of the spaces $\left(F_{r}\right)_{r\in{\mathbb{N}}}$, is given in Section 4. second part of Theorem 2.13 is given in Section 4.

4 Decomposing $\Delta$

Let $\Gamma$ be a family preserving metric graph. In this section, we describe an algorithm with which one can produce the one dimensional components of the Laplacian on $\Gamma$. We will also demonstrate each step on an example, denoted by $\widetilde{\Gamma}$ and presented in Figure 4.1, which will accompany us throughout the section. The analysis of two more examples (trees and antitrees) is given in Section 5.

4.1 Step 0 - Generating the equivalent locally balanced graph

In the case that $\Gamma$ is not locally balanced, one should turn it into a locally balanced graph by applying the following procedure: For every $n\in{\mathbb{N}}$, check whether $\#\left\{e\in{E(\Gamma)}\,|\,gen(e)=n\right\}=|S_{n-1}|$ or $\#\left\{e\in{E(\Gamma)}\,|\,gen(e)=n\right\}=|S_{n}|$. If not, add dummy vertices on all of the edges of generation $n$. The vertices are all located at an equal distance from $S_{n-1}$, and kirchhoff boundary conditions are attached to them. This process is illustrated in Figure 4.2. In the appendix we show that this procedure can be carried out to any family preserving metric graph, and that the resulting operator is unitarily equivalent to the original one.

4.2 Step 1 - Obtaining the discrete basis

In this step we consider the discrete graph $G_{\Gamma}$. The purpose of this step is to extract the elements $\left(\phi_{0}^{r}\right)_{r\in{\mathbb{N}}}$ from the basis given by the process described in the proof of Theorem 2.7 ([7]). Note that due to Proposition 3.12 the other elements of the basis are irrelevant to the construction of $\left(F_{r}\right)_{r\in\mathbb{N}}$. We now provide a short description of this process. For simplicity, for a compactly supported $f\in{\ell^{2}(G_{\Gamma})}$ we denote $span\{f,\Delta_{d}\}\coloneqq\overline{span\left\{f,\Delta_{d}f,\Delta_{d}^{2}f,\ldots\right\}}$, where $\Delta_{d}$ is the discrete Laplacian acting on $\ell^{2}(G_{\Gamma})$. Note that for every $n\in{\mathbb{N}}$, $\Delta_{d}^{n}(f)$ is also compactly supported and thus is in the domain of $\Delta_{d}$

• •

Define $\phi_{0}^{0}\coloneqq{\delta_{o}}$, where $o$ is the root of $G_{\Gamma}$. Also, define $W_{0}\coloneqq sp\left\{\delta_{o},\Delta_{d}\right\}$.

• •

Now, let $n_{1}\in{\mathbb{N}}$ be minimal such that $\ell^{2}(S_{n_{1}})\nsubseteq W_{0}$. Let $\left\{\phi_{0}^{1},\ldots,\phi_{0}^{r_{1}}\right\}$ be an orthonormal basis of $\left(\ell^{2}(S_{n_{1}})\cap{W_{0}}\right)^{\perp}$ such that for every $1\leq i\leq r_{1}$ and every $j\geq 0$, $\phi_{0}^{i}$ commutes with $\Lambda_{n,+j}$ (the existence of such a basis when the graph is family preserving is proven in [7]). Define $W_{1}\coloneqq W_{0}\oplus\left(\underset{k=1,\ldots,r_{1}}{\oplus}sp\left\{\phi_{0}^{r_{k}},\Delta_{d}\right\}\right)$.

• •

Proceed inductively.

With the labeling presented in Figure 4.2, the first elements of the basis in our example are:

[TABLE]

Note that if the graph is continued by straight lines, these are the only elements in the basis.

Having obtained the elements $\left(\phi_{0}^{r}\right)_{r\in{\mathbb{N}}}$, the next step is to determine the spaces generated by them, as described in Section 2.2.

4.3 Step 2 - Determining the spaces $\left(F_{r}\right)_{r\in{\mathbb{N}}}$

Let $r\in\mathbb{N}$. Recall that for $x\in{\Gamma}$,

[TABLE]

where $S_{n(r)}$ is the sphere on which $\phi_{0}^{r}$ is supported. Let

[TABLE]

and

[TABLE]

Now let $f\in F_{r}$ (so that $f=P_{r}(f)$), and let $\varphi_{f}(x)=\left\langle f_{|x|},h_{|x|}^{r}\right\rangle_{\mathbb{C}^{g_{\Gamma}(|x|)}}$. Then for every $f\in F_{r}$,

[TABLE]

Clearly, $supp(f)\subseteq\left\{x\in\Gamma\,:\,|x|\in[a_{r},b_{r}]\right\}$555If $b_{r}=\infty$, one should replace $[a_{r},b_{r}]$ in $[a_{r},b_{r})$ etc. and $|x|=|y|\Rightarrow\varphi_{f}(x)=\varphi_{f}(y)$.

Recall that $t_{k}$ is the distance in $\Gamma$ of a vertex $\in S_{k}$ from the root $o$. Let $k\in\mathbb{N}$ be such that $a_{r}=t_{k}$. If $b_{r}=\infty$, let $l=\infty$, and otherwise let $l$ be the maximal $j\in\mathbb{N}$ such that $t_{j}\leq b_{r}$. Denote $A_{r}=\{k,k+1,\ldots,(l-1)\}$. By the fact that $supp(\phi_{k}^{r})\subseteq{S_{n(r)+k}}$, one can conclude that for every $n<n(r)$, and every $v\in{S_{n}}$, $\underset{u\in{G_{v}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(v)=0$. Taking this and the definition of $P_{r}$ into consideration, we conclude that $A_{r}=\{n(r)-1,n(r),\ldots,l-1\}$.

Claim 4.1**.**

For $n(r)\leq{k}\in{A_{r}}$, and for $v\in{S_{k}}$, $\phi_{0}^{r}$ is constant on $G_{v}\cap{S_{n(r)}}$. For $v\in{S_{n(r)-1}}$, $\underset{u\in{G_{v}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(u)=0$.

Proof.

The second part of the claim follows from the fact that $H_{r}$ is orthogonal to $\ell^{2}(S_{n(r)-1})$. As for the first part, it is enough to show that there exists $v\in{S_{k}}$ such that $\underset{u\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(u)\neq 0$. Indeed, if this is true, then $\phi_{0}^{r}$ is an eigenvector of $\Lambda_{n(r),+(n(r)-k)}$ with an eigenvalue $\lambda\neq 0$. Now, if there exists some $w$ such that $\phi_{0}^{r}$ is not constant on $G_{w}\cap{S_{n(r)}}$, then by Lemma 3.10, $\underset{u\in{G_{w}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(u)=0$. This means that $\phi_{0}^{r}$ is an eigenvector of $\Lambda_{n(r),+(n(r)-k)}$ with [math] as an eigenvalue, which is a contradiction. Now assume that for every $v\in{S_{k}}$, $\underset{w\in{G_{v}\cap{S_{n(r)}}}}{\sum}\phi_{0}^{r}(w)=0$. Combining Lemma 3.10 and the fact that for $x,y\in\Gamma$ such that $|x|>|y|>t_{n(r)}$ it holds that $G_{y}\cap{S_{n(r)}}\subseteq{G_{x}\cap{S_{n(r)}}}$, we conclude that for every $x\in{\Gamma}$ with $|x|>t_{k}$, $h^{r}(x)=0$. This contradicts the fact that $k\in{A_{r}}$. ∎

By (4.1), the fact that $\varphi_{f}$ is symmetric and the fact that $h^{r}$ is constant along edges, we have that for every $f\in F_{r}$ and $e_{1},e_{2}\in E(\Gamma)$ such that $gen(e_{1})=gen(e_{2})$, $f|_{e_{1}}$ and $f|_{e_{2}}$ are multiples of each other. In other words, the space $F_{r}$ is determined by $a_{r},b_{r}$ and scalar coefficients attached to the graph’s edges. The scalar attached to an edge $e$ is the value $h^{r}$ takes on some $x\in e$, and by Claim 4.1, if we choose some $v\in G_{x}\cap S_{n(r)}$, then $h^{r}(y)=\sqrt{\frac{|G_{x}\cap S_{n(r)}|}{g_{n(r)}(|x|)}}\phi_{0}^{r}(v)$ for every $y\in e$. This description of $F_{r}$ is illustrated on $\widetilde{\Gamma}$ (for $r=1,2$) in Figure 4.3.

4.4 Step 3 - Obtaining the one dimensional components

The last step is to describe the one-dimensional operator to which $\Delta|_{F_{r}}$ is unitarily equivalent. The following computation shows that the scalars attached to the edges in Step 2 disappear under inner product. Let $\psi,\eta\in F_{r}$. Due to the fact that $\varphi_{f}$ is symmetric for every $f\in F_{r}$, for $t\geq 0$ we may define

[TABLE]

for some $x\in\Gamma$ with $|x|=t$. We have

[TABLE]

This implies that $U_{r}:F_{r}\rightarrow L^{2}([a_{r},b_{r}])$ is a unitary operator. In addition, by the same reasoning as in the proof of Theorem 2.13, $(ii)$, we have $U_{r}\Delta=\Delta_{r}U_{r}$, where $\Delta_{r}$ is the unbounded operator whose domain is $U_{r}(F_{r}\cap D(\Delta))$, and on that domain, $\Delta_{r}(f)=-f^{\prime\prime}$. We conclude that $\Delta|_{F_{r}}$ is unitarily equivalent to $\Delta_{r}$, which means that $\Delta\sim\underset{r\in\mathbb{N}}{\oplus}\Delta|_{F_{r}}\sim\underset{r\in\mathbb{N}}{\oplus}\Delta_{r}$. The rest of this subsection deals with analyzing the domains of the operators $\left(\Delta_{r}\right)_{r\in{\mathbb{N}}}$.

Divide the segment $(a_{r},b_{r})$ into the disjoint collection of segments

[TABLE]

Note that for fixed $n$ and every $x\in\Gamma$, $|G_{x}\cap S_{n}|$ depends only on $|x|$. For $|x|=t$ ,let $w_{n}(t)=|G_{x}\cap S_{n}|$. Now, by (4.2), and using (2.4), we may write explicitly

[TABLE]

Note that for every $j\in A_{r}$, the functions $g_{n(r)}$ and $w_{n(r)}$ are constant on $I_{j}$ as they change their values only at the $t_{j}$’s. Thus, $U_{r}(f)|_{I_{j}}=\varphi_{f}|_{I_{j}}$ is a linear combination of functions in $H^{2}(I_{j})$, so we have $U_{r}(f)|_{I_{j}}\in H^{2}(I_{j})$. In addition, we have

[TABLE]

Functions in $D(\Delta)$ satisfy certain matching conditions on vertices, which transform under $U_{r}$ into matching conditions on the connection points of the segments $\left(I_{j}\right)_{j\in A_{r}}$. In order to express these matching conditions, for every $j\in A_{r}$ we compute

[TABLE]

for $f\in F_{r}$. Let $\epsilon>0$.

[TABLE]

The functions $g_{n},\,w_{n}$ are constant on $\left(t_{j},t_{j+1}\right)$, so we may write $g_{n}(t_{j}+\epsilon)=g_{n}^{j}$ and $w_{n}(t_{j}+\epsilon)=w_{n}^{j}$ for sufficiently small $\epsilon$. Now, taking $\epsilon\rightarrow 0$, we have

[TABLE]

Now, $\star$ is constant on every edge, so for every edge $e$ we pick $x(e)\in e$ and we have

[TABLE]

Now note that for every $v\in S_{j}$ we have that $\underset{e:i(e)=v}{\sum}\left(\underset{u\in G_{x(e)}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(u)\right)$ is some multiple of $\underset{u\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(u)$. Thus, for $j=n(r)-1$, by the second part of Claim 4.1 we have that $\varphi_{f}(t_{j}+)=0$. For $j\geq n(r)$, again by Claim 4.1 we have that $\phi_{0}^{r}$ is constant on $G_{v}\cap S_{n(r)}$, and for every $e$ such that $i(e)=v$ and $x\in e$, $G_{x}\cap S_{n(r)}=G_{v}\cap S_{n(r)}$. Thus, $\phi_{0}^{r}|_{G_{x}\cap S_{n(r)}}\equiv\phi_{0}^{r}(u_{v})$ for some $u_{v}\in G_{v}\cap S_{n(r)}$. Now, if we denote (as before) by $b_{j}^{out}$ the number of edges emanating from vertices in $S_{j}$ (which is the same for every vertex due to the symmetry of the graph), we have

[TABLE]

By a similar computation, it can be shown that $\varphi_{f}(b_{r})=0$ (in the case that $b_{r}<\infty$), and (with the proper notations) that for $j\geq n(r)$

[TABLE]

Thus, for $j\geq n(r)$,

[TABLE]

where

[TABLE]

By a similar computation, using the matching condition of the derivative, we also have

[TABLE]

where

[TABLE]

Remark 4.2*.*

In the case that $r=0$, $n(r)=0$ as the function $\phi_{0}^{r}$ is chosen to be $\delta_{o}$. Thus $w_{n(r)}^{j}=1$ for every $j\in{\mathbb{N}}$. In addition, it can be seen that $b_{j}^{in}=\frac{g_{0}^{j-1}}{|S_{j}|}$ and $b_{j}^{out}=\frac{g_{0}^{j}}{|S_{j}|}$. From here, it can be seen that

[TABLE]

and that $c_{j}^{0}=\frac{1}{d_{j}^{0}}$.

To conclude, $D(\Delta_{r})$ consists of all of the functions in $L^{2}(a_{r},h(\Gamma))$ which satisfy the following conditions:

$f(a_{r})=0$. if $b_{r}<\infty$, then also $f(b_{r})=0$. 2. 2.

$\forall j\in A_{r}$ $f|_{I_{j}}\in H^{2}(I_{j})$. 3. 3.

$\underset{j\in A_{r}}{\sum}\int_{I_{j}}|f^{{}^{\prime\prime}}(t)|dt<\infty$. 4. 4.

$f(t_{j}+)=d_{j}^{r}f(t_{j}-)$ for every $n(r)\leq{j}\in A_{r}$. 5. 5.

$f^{\prime}(t_{j}+)=c_{j}^{r}f^{\prime}(t_{j}-)$ for every $n(r)\leq j\in A_{r}$.

In the case that $A_{r}$ is finite, functions in $D(\Delta_{r})$ also satisfy the condition $f(b_{r})=0$.

For the example $\widetilde{\Gamma}$, the one dimensional version of the spaces $F_{1},\,F_{2}$ from Figure 4.3 are shown in Figure 4.4. Note that functions in $D(\Delta_{1})$ are defined on a compact segment (i.e. $b_{1}<\infty$) whereas functions in $D(\Delta_{2})$ may not be compactly supported (for example, this is the case when the graph is continued by straight lines).

To conclude, the operator $\Delta_{r}$ is a Sturm-Liouville operator on a $L^{2}(I)$ where $I\subseteq[0,\infty)$ with boundary conditions along points which correspond with $\left(t_{l}\right)_{l\in{A_{r}}}$. Since locally this operator acts as a second derivative, we henceforth refer to such operators as ‘weighted Laplacians’.

4.5 Further Discussion

In the algorithm presented in the proof of [13, Theorem 2.6], the spaces $\ell_{2}\left(S_{n}\right)$ are spanned iteratively where at each step, one spans the part of $\ell_{2}\left(S_{n}\right)$ which is not already spanned using a basis of mutual eigenfunctions of the collection $\Lambda_{n,+j}$. The functions obtained are the $\phi_{0}^{r}$, which we then use to construct $h^{r}$ above. As the characterization through mutual eigenfunctions of the family $\Lambda_{n,+j}$ is non-local, two natural questions come to mind regarding the description of the functions $\phi_{0}^{r}$ and $h^{r}$.

The first question regards the choice of $\phi_{0}^{r}$ and asks whether there is some generic choice that can be made at each step and would work for any family preserving graph. In Section 5.1, we show that the decomposition described in [28] actually corresponds to a concrete choice of such a basis at each step, in the case where $\Gamma$ is a radial (spherically symmetric) tree. The choice there, however, is possible only for a tree, as it corresponds to cutting the tree into subtrees by deleting an edge between two generations. Another possible choice for $\phi_{0}^{r}$ comes to mind: perhaps it is possible to use roots of unity of a degree corresponding to the size of $S_{n(r)}$ in a universal way which would work for any family preserving graph. Figure 4.5 gives an example of two trees with $|S_{2}|=6$, where it is clear that the choice of $\phi_{0}$ supported on $S_{2}$ depends on the structure of $S_{1}$. By extension, it is clear that any choice of the functions $\phi_{0}^{r}$ on trees depends on the structure of the entire tree up to the relevant sphere. For general family preserving graphs, the choice would depend on the entire structure of the graph also beyond the relevant sphere, showing that a local universal choice is not possible.

Another question regards finding a more immediate connection between the edge weights, $h^{r}$, and the graph structure. In particular, since $h^{r}$ are constant on edges it is natural to consider them on the line graph and ask whether they can be obtained as eigenfunctions of some natural discrete operator on that graph, e.g. the adjacency matrix or the discrete Laplacian. While we cannot rule out the possibility of the existence of some operator with this property, Figure 4.6 describes an example where even $h^{0}$ (the weight associated with the space of spherically symmetric functions on the graph) is not an eigenfunction of the adjacency matrix, $A$ or the Laplacian $\Delta_{d}$ on the line graph.

5 Examples

In this section we first describe how the results of [28, 32] are a special case of our analysis here. Then, in order to demonstrate the utility of our method, we apply it to obtain a result on metric antitrees.

5.1 Radial Trees

A metric tree $\Gamma$ is called *radial (or regular) *if for every $x,y\in S_{n}$, $deg(x)=deg(y)$, and for any two edges of the same generation $e_{1},\,e_{2}$, $l(e_{1})=l(e_{2})$.

Claim 5.1**.**

Every regular tree is family preserving.

Proof.

Let $\Gamma$ be a regular tree, and let $v,u\in S_{n}$ be backward neighbors. $\Gamma$ is a tree, so the sets $A_{v}=\left\{w\in V(\Gamma)\,:\,v\preceq w\right\}$, $A_{u}=\left\{w\in V(\Gamma)\,:\,u\preceq w\right\}$ are disjoint. Furthermore, regularity of $\Gamma$ implies that the graphs induced by those sets are isomorphic as rooted graphs, with $u,v$ as roots. Let $\sigma:\,A_{v}\rightarrow A_{u}$ be an isomorphism between the induced graphs. Define

[TABLE]

It is easy to verify that $\tau$ is a rooted graph automorphism which satisfies $\tau|_{S_{n-j}}=Id$ for every $j\geq 1$. $\Gamma$ is a tree, so there are no forward neighbors, and that finishes the proof. ∎

A decomposition for the Laplacian on regular trees was presented in [28, 32]. We will present the decomposition here, and show that it is a special case of the decomposition presented in the proof of Theorem 2.13.

For every $v\in{V(\Gamma)}$, denote by $b(v)$ the number of edges emanating from $v$ (i.e., the edges going ‘away’ from the root), and order the edges $\left(e_{v}^{1},\ldots,e_{v}^{b(v)}\right)$. For every $u\in{V(\Gamma)}$, let $A_{u}=\{w\in V(\Gamma)\,:\,u\preceq w\}$. For every $1\leq{j}\leq{b(v)}$, let $G_{e_{v}^{j}}$ be the subtree of $\Gamma$ generated by $A_{t\left(e_{v}^{j}\right)}\cup\left\{v\right\}$ (see Figure 5.1 for an illustration of these definitions).

Let $\omega$ be the root of unity of order $b(v)$ (i.e. $\omega=e^{\frac{2\pi i}{b(v)}}$). Let $s\in\mathbb{N},\,1\leq s\leq b(v)-1$. Let $M_{v}^{\langle s\rangle}$ be the space of all functions $f\in L^{2}(\Gamma)$ such that $supp(f)\subseteq{\underset{k=1}{\overset{b(v)}{\bigcup}}G_{e_{v}^{k}}}$ and there exists a measurable function $g:\,(t_{gen(v)},\infty)\rightarrow\mathbb{C}$ for which $f(x)=g(|x|)\omega^{js}\iff x\in G_{e_{v}^{j}},\text{ }1\leq{j}\leq{b(v)}$. Also, denote by $M_{\Gamma}$ the space of all functions in $L^{2}(\Gamma)$ that are radially symmetric.

The Naimark-Solomyak decomposition is described in the following theorem from [32].

Theorem 5.2**.**

([32, Theorem 3.2])* Let $\Gamma$ be a regular (radially symmetric and spherically homogeneous) metric tree such that $b(o)=1$. Then the subspaces $M_{v}^{\langle s\rangle},\,v\in{V(\Gamma)},\,b(v)>1,\,1\leq{s}\leq{b(v)-1}$ are mutually orthogonal and orthogonal to $M_{\Gamma}$. Moreover,*

[TABLE]

and this decomposition reduces the Dirichlet-Kirchhoff Laplacian on $\Gamma$.

It is apparent from their description that for every $v\in{V(\Gamma)\setminus{o}}$ and every $1\leq s\leq b(v)$, the space $M_{v}^{\langle s\rangle}$ can be described by attaching scalar coefficients to the tree’s edges (see Figure 5.2) in the same manner shown in Section 4 (Figure 4.3).

In order to show that the above decomposition is a special case of the decomposition we describe here, we first associate with each space $M_{v}^{\langle s\rangle}$ a function defined on the associated discrete graph and supported on a sphere.

Lemma 5.3**.**

For every $v\in V(\Gamma)$, and every $s=1,\ldots,b(v)-1$, the space $M_{v}^{\langle s\rangle}$ is equal to the spaces $F_{\phi_{v}^{s}}$, where $\phi_{v}^{s}$ is defined in the following way:

[TABLE]

where $\omega=e^{\frac{2\pi i}{b(v)}}$.

Proof.

We first show that $M_{v}^{\langle s\rangle}\subseteq{F_{\phi_{v}^{s}}}$. Let $f\in M_{v}^{\langle s\rangle}$ and let $g:(t_{gen(v)},\infty)\rightarrow{\mathbb{C}}$ such that $f(x)=g(|x|)\omega^{js}\iff{x\in{G_{e_{j}^{v}}}}$. Also denote $k=gen(v)+1$. It is sufficient to show that $P_{\phi_{v}^{s}}(f)=f$. Let $x\in{\Gamma}$. Assume first that $x\in{G_{e_{v}^{j}}}$ for some $1\leq{j}\leq{b(v)}$ and that $x\neq{v}$. The fact that $\Gamma$ is a tree implies that $G_{x}\cap{S_{gen(v)+1}}=\{u\}$ where $x\in{G_{(v,u)}}$. Thus, $h^{\phi_{v}^{s}}(x)=\frac{\omega^{js}}{\sqrt{g_{k}(|x|)b(v)}}$. A direct computation shows that $\left\langle{f_{|x|}},{h^{\phi_{v}^{s}}_{|x|}}\right\rangle=g(|x|)\sqrt{g_{k}(|x|)b(v)}$ which means that $P_{\phi_{v}^{s}}(f)(x)=g(|x|)\omega^{js}=f(x)$. for $x\notin{G_{e_{v}^{j}}}$ for every $1\leq{j}\leq{b(v)}$, $h^{\phi_{v}^{s}}(x)=0$ which means that $P_{\phi_{v}^{s}}(f)(x)=0$.

The inclusion ${F_{\phi_{v}^{s}}}\subseteq M_{v}^{\langle s\rangle}$ easily follows by defining $g(|x|)=\frac{1}{\sqrt{g_{k}(|x|)b(v)}}\left\langle{f_{|x|}},{h^{\phi_{v}^{s}}_{|x|}}\right\rangle$ for $f\in{F_{\phi_{v}^{s}}}$. ∎

We conclude our demonstration by showing that the functions $\left\{\phi_{v}^{s}\right\}$ satisfy the properties of $\left\{\phi_{0}^{r}\right\}$ in Theorem 2.7. Thus the decomposition described in Theorem 5.2 is indeed a particular case of the decomposition described in Theorem 2.13.

For every $v\in{V(\Gamma)}$ such that $b(v)>1$ and $1\leq{s}\leq{b(v)-1}$, denote $H_{v}^{s}\coloneqq{\overline{sp\left\{\phi_{v}^{s},\Delta_{d}(\phi_{v}^{s}),\ldots\right\}}}$, $H_{0}\coloneqq{\overline{sp\left\{\delta_{0},\Delta_{d}(\delta_{0}),\ldots\right\}}}$. Also, denote $V^{\prime}=\left\{v\in{V(\Gamma)\setminus\{o\}}:b(v)>1\right\}$.

Theorem 5.4**.**

Let $\Gamma$ be a radially symmetric tree. Then $\ell^{2}(V(\Gamma))=H_{0}\oplus\left(\underset{v\in{V^{\prime}}}{\oplus}H_{v}^{s}\right)$. Furthermore, for every $v\in{V^{\prime}}$ and $1\leq{s}\leq{b(v)-1}$,

$(i)$* There exists $n(v,s)\in{\mathbb{N}}$ such that $supp\left(\phi_{v}^{s}\right)\subseteq{S_{n(v,s)}}$.*

$(ii)$* The set $\left\{\phi_{0}^{v,s},\phi_{1}^{v,s},\ldots\right\}$ obtained from $\left\{\phi_{v}^{s},\Delta_{d}(\phi_{v}^{s}),\ldots\right\}$ by applying the Gram-Schmidt process has the property that $supp\left(\phi_{k}^{v,s}\right)\subseteq{S_{n(v,s)+k}}$.*

$(iii)$* For every $j\in{\mathbb{N}}$, $\phi_{0}^{v,s}$ is an eigenvector of $\Lambda_{n(v,s),\pm{j}}$.*

Proof.

Part $(i)$ follows directly from the definition of $\phi_{v}^{s}$. Parts $(ii)$ and $(iii)$ also follow from the definition of $\phi_{v}^{s}$, using in addition the symmetry of the graph. The mutual orthogonality of the family of spaces $H_{0}$, $H_{v}^{s}$ $\left(v\in V^{\prime}\right)$ is an elementary but somewhat tedious computation and the fact that $\ell^{2}(V(\Gamma))=H_{0}\oplus\left(\underset{v\in{V^{\prime}}}{\oplus}H_{v}^{s}\right)$ is a simple local dimension counting. ∎

Remark 5.5*.*

The procedure of obtaining the functions $\phi_{0}^{r}$ of Theorem 2.7 is a constructive procedure which proceeds by constructing cyclic subspaces recursively. One starts with the cyclic space spanned by the delta function at the root and at each step one chooses a function that is orthogonal to all the previously constructed cyclic subspaces and supported on a sphere not yet ‘covered’ by the procedure. Thus, there often is some freedom in the choice of the functions $\phi_{0}^{r}$. Remarkably, though, due to the symmetry of family preserving graphs, as long as these functions are chosen as eigenfunctions of the appropriate operators (the $\Lambda_{n,\pm{j}}$) the associated Jacobi matrices do not depend on the choice of these functions.

Similarly, in the associated metric case covered by Theorem 2.13, the one-dimensional operators obtained in the decomposition are independent of the choice of the $\phi_{0}^{r}$ described in Theorem 2.7. This can be seen from the fact that these one-dimensional operators as described in Section 4 above, do not depend on the choice of $\phi_{0}^{r}$.

Thus, while the discussion above shows that the method of decomposition of [28, 32] can be realized by a particular choice of $\phi_{0}^{r}$ we actually obtain the somewhat more interesting result that the decomposition itself is independent of the particular choice of these functions (among the possibilities allowed by Theorem 2.7).

5.2 Antitrees

An antitree is a rooted graph, $G$, with all possible edges between any two neighboring spheres (and no other edges). Formally, a rooted graph is an antitree if $\forall n\in\mathbb{N},\ \forall u\in S_{n},\ \forall v\in S_{n+1},\,(u,v)\in E$. In a sense, an antitree is a bipartite antithesis to a tree, as it has all available cycles between vertices of different generations. Works exploiting this structure in the context of spectral properties of the Laplace operator include [20, 23, 34]. The Anderson model on antitrees has been studied in [30, 31].

It is shown in [13] that antitrees are family preserving. A metric graph, $\Gamma$, is called an antitree if $G_{\Gamma}$ is an antitree. In this subsection we would like to demonstrate the applicability of our method by applying it to spherically homogeneous metric antitrees. Thus, let $\Gamma$ be a spherically homogeneous metric antitree.

Antitrees are (generally) not locally balanced, as for every $n\in\mathbb{N}$ we have $|S_{n}|\cdot|S_{n+1}|$ edges of generation $n$ (which, if $|S_{n}|,|S_{n+1}|\geq 2$, is greater than both $|S_{n}|$ and $|S_{n+1}|$). Thus, given an antitree $\Gamma$, we consider its unitarily equivalent graph $\widetilde{\Gamma}$ which is obtained by adding vertices where necessary. For simplicity of notation, we assume here that for every $n\geq 1$, $|S_{n}|\geq 2$ (note that $|S_{0}|=1$ since $S_{0}=\{\delta_{o}\}$–the root). The analysis of the general case is not fundamentally different, but more cumbersome to describe.

Write $V(\widetilde{\Gamma})=V(\Gamma)\cup\widetilde{V}$, and denote by $(\rho_{n})_{n\in\mathbb{N}}$ the set of indices for which $S_{\rho_{n}}\subseteq V(\Gamma)$ (note that $\rho_{1}=0$, $\rho_{2}=1$, $\rho_{3}=3$ etc.). Also, denote by $(\tilde{\rho}_{n})_{n\in\mathbb{N}}$ the set of indices for which $S_{\tilde{\rho}_{n}}\subseteq\widetilde{V}$ (here, $\tilde{\rho}_{1}=2$, $\tilde{\rho}_{2}=4$ etc.). Finally, let $\widetilde{G}$ be the discrete structure of $\widetilde{\Gamma}$ and let $(H_{r})_{r=0}^{\infty}$ be the decomposition of $\ell^{2}(\widetilde{G})$ described in Theorem 2.7, such that $H_{0}=\overline{span\left\{\delta_{o},\Delta_{d}(\delta_{o}),\Delta^{2}_{d}(\delta_{o}),\ldots\right\}}$ and for every $r\in\mathbb{N}$, $H_{r}=\overline{span\left\{\phi_{0}^{r},\Delta_{d}(\phi_{0}^{r}),\Delta^{2}_{d}(\phi_{0}^{r}),\ldots\right\}}$ for $\phi_{0}^{r}\in\ell^{2}(S_{n(r)})$ (i.e. $n(r)$ denotes the sphere on which $\phi_{0}^{r}$ is supported). Recall that for every $r\in\mathbb{N}$, $(\phi_{k}^{r})_{k=0}^{\infty}$ is the set obtained by applying the Gram-Schmidt process on $\left(\Delta^{k}(\phi_{0}^{r})\right)_{k=0}^{\infty}$. We want to describe the decomposition of $\Delta$ on $\widetilde{\Gamma}$. In order to do this, we first need to focus on the corresponding discrete decomposition.

Claim 5.6**.**

For every $n\geq 3$ and for every $r\in\mathbb{N}$, $\rho_{n}\neq n(r)$.

Proof.

By Theorem 2.7, for every $\phi\in H_{r}$ we have that $supp(\phi)\cap S_{n(r)-1}=\emptyset$. Assume that there exists $n\geq 3$ and $r\in\mathbb{N}$ such that $n(r)=\rho_{n}$. Note that, as the sphere $S_{\rho_{n}-1}$ consists of added vertices, every $v\in S_{\rho_{n}-1}$ is connected to exactly one $w\in S_{\rho_{n}}$. Thus, for $w\in S_{\rho_{n}}$ such that $\phi_{0}^{r}(w)\neq 0$ and a neighbor of $w$, $v\in S_{\rho_{n}-1}$, $\Delta_{d}(\phi_{0}^{r})(v)=-\phi_{0}^{r}(w)$ which is a contradiction. ∎

Claim 5.7**.**

For every $3\leq n\in\mathbb{N}$ and for every $r\in\mathbb{N}$ such that $0\leq n(r)\leq\rho_{n}-2$, $H_{r}\cap\ell^{2}(S_{\rho_{n}})=\emptyset$.

Proof.

For every $0\leq k\leq\rho_{n}-2$ and $v\in S_{\rho_{n}}$, $G_{v}\cap S_{k}=S_{k}$. In addition, $\phi_{0}^{r}$ is orthogonal to the symmetric functions. Combining this with the fact that $\Delta_{d}^{\rho_{n}-n(r)}(\phi_{0}^{r})(v)=c\underset{w\in G_{v}\cap S_{n(r)}}{\sum}\phi_{0}^{r}(w)$ for some $c\in\mathbb{N}$, we get the result. ∎

Corollary 5.8**.**

For every $3\leq n\in\mathbb{N}$, $\ell^{2}(S_{\rho_{n}})=span\left\{\phi_{\rho_{n}}^{0},\phi_{1}^{r_{1}},\phi_{1}^{r_{2}},\ldots,\phi_{1}^{r_{|S_{\rho_{n}|-1}}}\right\}$ for $r_{1},\ldots,r_{|S_{\rho_{n}}|-1}$ such that $n(r_{i})=\rho_{n}-1$ for every $1\leq i\leq|S_{\rho_{n}}|-1$. In addition, for every such $i$, $\Delta_{d}\left(\phi_{1}^{r_{i}}\right)|_{S_{\rho_{n}+1}}\neq{0}$.

Proof.

The first part follows from the previous claims and from the fact that $\ell^{2}(S_{\rho_{n}})\subset\underset{r\in\mathbb{N}}{\oplus}H_{r}$. The second part follows from the fact that for $0\neq\phi\in\ell^{2}(S_{\rho_{n}})$, $\Delta_{d}(\phi)|_{S_{\rho_{n}+1}}\neq 0$, as again, every vertex in $S_{\rho_{n}+1}$ is connected to exactly one vertex in $S_{\rho_{n}}$. ∎

Corollary 5.3 implies that for every $n\geq 3$, a subspace of $\ell^{2}\left(S_{\rho_{n}+1}\right)$, of dimension $|S_{\rho_{n}}|-1$, is spanned by $\left\{\phi_{1}^{r_{1}},\ldots,\phi_{1}^{r_{|S_{\rho_{n}}|-1}}\right\}$. For every $r\in\mathbb{N}$ such that $n(r)<\rho_{n}-1$, $H_{r}\cap\ell^{2}(S_{\rho_{n}+1})=\emptyset$. This follows, as before, from the fact that for every $v\in S_{\rho_{n}+1}$, $G_{v}\cap S_{n(r)}=S_{n(r)}$. Another one-dimensional space is spanned by $\phi_{{}_{\rho_{n}+1}}^{0}$. Recalling that $|S_{\rho_{n}+1}|=|S_{\rho_{n}}|\cdot|S_{\rho_{n+1}}|$, this means that $\left|\left\{r\in\mathbb{N}\,|\,n(r)=\rho_{n}+1\right\}\right|=|S_{\rho_{n+1}}|\cdot|S_{\rho_{n}}|-|S_{\rho_{n}}|=|S_{\rho_{n}}|\cdot\left(|S_{\rho_{n+1}}|-1\right)$ (as these span the rest of $\ell^{2}(S_{\rho_{n}+1})$). Noting that for every $n\geq 2$, $\rho_{n}+1=\tilde{\rho}_{n-1}$, we conclude the following.

Proposition 5.9**.**

Denote $k_{n}=|S_{\rho_{n}}|\cdot\left(|S_{\rho_{n+1}}|-1\right)$, and for $n\geq 2$, $B_{n}=\left\{r\in\mathbb{N}\,|\,n(r)=\tilde{\rho}_{n-1}\right\}$. Then for every $n\geq 2$, $\left|B_{n}\right|=k_{n}$. Furthermore, define $K_{n}\coloneqq\left\{r\in B_{n}\,|\,dim(H_{r})=3\right\}$.Then $|K_{n}|=|S_{\rho_{n+1}}|-1$, and for every $r\in B_{n}\setminus K_{n}$, $dim(H_{r})=1$.

Denote by $Q$ the set \left\{r\in\mathbb{N}\,|\,\exists n\in\mathbb{N}\text{ s.t. n(r)=\tilde{\rho}_{n}}\right\}\cup\{0\}, and define $W=\underset{r\in Q}{\oplus}H_{r}$. From all of the above, we conclude the following:

• •

For $n\geq 3$, $\ell^{2}(S_{n})\subseteq W$.

• •

For $n=2$, $dim\left(\ell^{2}(S_{n})\cap W\right)=|S_{1}|\cdot|S_{3}|-|S_{1}|+1$.

• •

For $n=0,1$, the intersection of $W$ with $\ell^{2}(S_{n})$ consists of the symmetric functions on $S_{n}$ (for $S_{0}$ this is actually the whole space, as $|S_{0}|=1$).

• •

Finally, for $r\notin Q$ $H_{r}$ is orthogonal to $W$. Thus, the intersection of $H_{r}$ with $\ell^{2}(S_{n})$ is trivial for every $n\geq 3$ and for $n=0$. This implies that for every $r\notin Q$, $n(r)=1$ and there will be exactly $|S_{1}|-1$ such $r$’s.

By considering the spaces $\left(F_{r}\right)_{r=0}^{\infty}$ given by Theorem 2.13 we can now state the analog of [13, Theorem 1.3]

Theorem 5.10**.**

Let $\Gamma$ be an infinite spherically homogeneous metric antitree. Then the spectrum of $\Delta$ on $\Gamma$ is given by the spectrum of a weighted Laplacian on the whole line and the closure of the union of spectra of countably many weighted Laplacians on compact segments.

Proof.

Recall that for $n\in\mathbb{N}$, $t_{n}$ denotes the metric distance of $v$ from $o$ for some $v\in S_{n}$. In addition, recall that $a_{r}$ and $b_{r}$ denote the part of the real line on which functions in $F_{r}$ are supported.

Now, note that for every $x\in\widetilde{\Gamma}$, $G_{x}\cap S_{0}=\left\{o\right\}$. This means that $h^{\delta_{o}}(x)\neq 0$. Thus, $b_{0}=\infty$ and $\Delta$ restricted to $F_{0}$ is unitarily equivalent to a weighted Laplacian on the positive real line, with matching conditions as described in the previous section on the points $\left(t_{n}\right)_{n\in\mathbb{N}}$.

For every $r_{0}\in\left\{r\in\mathbb{N}\,|\,n(r)=1\right\}$, $F_{r_{0}}$ is orthogonal to the symmetric functions. Thus, as every $v\in S_{3}$ is connected to every vertex in $S_{1}$ (through a vertex in $S_{2}$), $\Delta$ restricted to $F_{r_{0}}$ is unitarily equivalent to a weighted Laplacian on the compact segment $\left[0,t_{3}\right]$.

Finally, let $n\in\mathbb{N}$ and let $r_{0}\in\left\{r\in\mathbb{N}\,|\,n(r)=\tilde{\rho}_{n}\right\}$ and consider two cases.

• •

$dim(H_{r_{0}})=1$: In this case, $b_{r}=t_{\tilde{\rho}_{n}+1}$, and we get a copy of a weighted Laplacian on the compact segment $\left[t_{\tilde{\rho}_{n}-1},t_{\tilde{\rho}_{n}+1}\right]$.

• •

$dim(H_{r_{0}})=3$: In this case, $b_{r}=t_{\tilde{\rho}_{n}+3}$, and we get a copy of a weighted Laplacian on the compact segment $\left[t_{\tilde{\rho}_{n}-1},t_{\tilde{\rho}_{n}+3}\right]$.

∎

Remark 5.11*.*

In an antitree, there is an explicit formula for $g_{0}^{j}$ which is given by $g_{0}^{j}=|S_{j}|\cdot|S_{j+1}|$. Thus, with Remark 4.2 in mind, $d_{j}^{0}=\sqrt{\frac{|S_{j+1}|}{|S_{j-1}|}}$.

As a demonstration of the usefulness of an explicit decomposition to one dimensional operators, we mention Remling’s Theorem [29], whose main message is the fact that absolutely continuous spectrum for one dimensional operators is extremely restrictive. In the context of antitrees, following the discussion above and applying [12, Theorem 6] we immediately obtain that

Theorem 5.12**.**

Let $\Gamma$ be a spherically homogeneous metric antitree and assume that

[TABLE]

and

[TABLE]

Then if $\limsup_{n\rightarrow\infty}(t_{n+1}-t_{n})=\infty$ then the absolutely continuous spectrum of $\Delta$ on $\Gamma$ is empty.

Remark 5.13*.*

Theorem 5.12 coincides with (part of) Theorem 8.1 in the recent preprint [25], which has an extensive analysis of spherically homogeneous metric antitrees, also using a decomposition inspired by [13]. Using a modification of the methods of [12], it is likely one could replace (5.1) with $\underset{n\rightarrow\infty}{\liminf}\ \left|\frac{|S_{n+1}|}{|S_{n-1}|}-1\right|>0$ and still obtain absence of absolutely continuous spectrum.

6 Appendix -Proof of Proposition 2.12

Proof of Proposition 2.12.

We say that $n\in\mathbb{N}$ is “bad” if the number of edges of generation $n$ is greater than the number of vertices of generation $n$ and of the number of vertices of generation $n+1$. For such $n$, and $e=(i(e),t(e))$ such that $gen(e)=n$, we add a vertex $w$, the edges $e_{1}=(i(e),w)$ and $e_{2}=(w,t(e))$ such that $l(e_{i})=\frac{l(e)}{2}$, and remove $e$ from $E(\Gamma)$. In simple words, we divide the edge into two edges with equal length. Note that in the resulting graph, the numbers $n,\,n+1$ are not bad, because for every new edge $e$, there is a unique vertex $w_{e}$ which was added with $e$. In addition, none of the other generations became bad, because we did not remove any vertices from $V(\Gamma)$. Thus, if we follow this procedure for every bad $n$, the resulting graph will be locally balanced. Denote that graph by $\overline{\Gamma}$, and let $\overline{\Delta}=\Delta_{\overline{\Gamma}}$. Note that as measure spaces, there is no difference between $\Gamma$ and $\overline{\Gamma}$. The Kirchhoff boundary conditions assure us that the transformation $T:\,D(\Delta)\rightarrow D(\overline{\Delta})$ defined by $T(\varphi)=\varphi$ is unitary, and maps $D(\Delta)$ bijectively onto $D(\overline{\Delta}).$

It is left to prove that $\overline{\Gamma}$ is family preserving. Denote by $\overline{S_{n}}$ the n’th sphere in $\overline{\Gamma}$. Let $v,u\in\overline{S_{n}}$ such that $v$ and $u$ are forward neighbors. Consider the following cases:

$(i)$ $\overline{S_{n}}$ is not a part of $\Gamma$, meaning we added that sphere during the above process. In this case, $v$ and $u$ were added in the middle of the edges $(w_{1},z)$, $(w_{2},z)$ which means that $w_{1}$ and $w_{2}$ are forward neighbors in $\Gamma$. That implies that there exists a rooted graph automorphism $\tau:\,V(\Gamma)\rightarrow V(\Gamma)$ such that $\tau(w_{1})=\tau(w_{2})$, and $\forall k\geq 0\,\tau|_{S_{n+k}}=Id$. We define $\overline{\tau}:\,V(\overline{\Gamma})\rightarrow V(\overline{\Gamma})$ by $\overline{\tau}|_{V(\Gamma)}=\tau$, and if $x$ was added in the middle of the edge $(v_{1},v_{2})$, $\overline{\tau}(x)$ will be the vertex that was added in the middle of $(\tau(v_{1}),\tau(v_{2}))$ (such vertex exists because of spherical symmetry). Now, we have that $\overline{\tau}(v)=u$, and $\forall k\geq 0\,\overline{\tau}|_{\overline{S_{n+k}}}=Id$ , as required.

$(ii)$ $\overline{S_{n}}$ is a part of $\Gamma$. In that case, $\overline{S_{n+1}}$ is also a part of $\Gamma$, because an added vertex cannot connect vertices from the preceding sphere, so we can define $\overline{\tau}$ exactly as in case $(i)$, and get the desired automorphism.

The proof for backward neighbors is exactly the same. ∎