M^\ast$, the set of embedded eigenvalues of the operator ${{-\Delta}}_{\widetilde M}$ in $\left( t_-(k),t_+(k) \right)$ is finite.

On a diagonalization of the operator $L$ for a regular tree graph {#sec3}
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In this section, we summarize some results and tools we need and which are developed in [@af; @al]. We shall essentially follow [@af Section 3] and we refer to the cited papers for more details.

Define the operator $\Pi$ on $\ell^2(\mathcal{V})$ by $$\label{eq3,1}
(\Pi \phi)(v) := \sum_{w \, : \, w \rightarrow v} \phi(w),$$ where for two vertices $v$ and $w$, $v \rightarrow w$ means that they are connected by an edge with $\vert w \vert = \vert v \vert + 1$. Using the inner product defined on the Hilbert space $\ell^2(\mathcal{V})$ by , it can be easily checked that the adjoint operator $\Pi^\ast$ is given by $$\label{eq3,2}
(\Pi^\ast \phi)(v) := \sum_{w \, : \, v \rightarrow w} \phi(w).$$ If we let $L_S$ be the spherical Laplacian defined on $\ell^2(\mathcal{V})$ by $$\label{eq3,3}
(L_S \phi)(v) := \sum_{\substack{w \, : \, w \sim v \\ \vert w \vert = \vert v \vert}} \phi(w),$$ then the operator $L$ given by can be written as $$\label{eq3,4}
L = \Pi + \Pi^\ast + L_S.$$ 