# Analyticity of resolvents of elliptic operators on quantum graphs with small
edges

D.I. Borisov

###### Abstract

We consider an arbitrary metric graph, to which we glue another graph with
edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small
positive parameter. On such graph, we consider a general self-adjoint second
order differential operator $\mathcal{H}_{\varepsilon}$ with varying
coefficients subject to general vertex conditions; all coefficients in
differential expression and vertex conditions are supposed to be analytic in
$\varepsilon$. We introduce a special operator on a certain graph obtained by
rescaling the aforementioned small edges and assume that it has no embedded
eigenvalues at the threshold of its essential spectrum. Under such assumption,
we show that certain parts of the resolvent of $\mathcal{H}_{\varepsilon}$ are
analytic in $\varepsilon$. This allows us to represent the resolvent of
$\mathcal{H}_{\varepsilon}$ by an uniformly converging Taylor-like series and
its partial sums can be used for approximating the resolvent up to an
arbitrary power of $\varepsilon$. In particular, the zero-order approximation
reproduces recent convergence results by G. Berkolaiko, Yu. Latushkin, S.
Sukhtaiev and by C. Cacciapuoti, but we additionally show that next-to-leading
terms in $\varepsilon$-expansions of the coefficients in the differential
expression and vertex conditions can contribute to the limiting operator
producing the Robin part at the vertices, to which small edges are incident.
We also discuss possible generalizations of our model including both the cases
of a more general geometry of the small parts of the graph and a non-analytic
$\varepsilon$-dependence of the coefficients in the differential expression
and vertex conditions.

Keywords: graph, small edge, resolvent, analyticity, Taylor series,
approximation

Institute of Mathematics, Ufa Federal Research Center, Russian Academy of
Sciences, Ufa, Russia,

&

Bashkir State University, Ufa, Russia,

&

University of Hradec Králové, Hradec Králové, Czech Republic

<EMAIL_ADDRESS>
##  1 Introduction

The theory of quantum graphs or, in other words, the spectral theory of
differential operators on graphs is actively developing nowadays and a rather
new direction is devoted to studying quantum graphs with small edges. One of
early studied problems came from the homogenization theory and it was devoted
to models of woven membrane kind, see [9], [13]. The well-known result states
that an elliptic operator on a graph looking as a net with small edges
converge to a two-dimensional operator on an Euclidean domain approximated in
a natural sense by such graph.

Another situation, which attracted quite a lot of attention, concerned the
graphs with finitely many small edges. As a rather early publication on this
subject we mention paper [8], in which it was shown that a general vertex
condition in a graph can be approximated in the norm resolvent sense via
ornamenting by small edges supporting a magnetic field and with delta-coupling
at the vertices.

The case of a general graph with finitely small edges was addressed in [1]. On
such graph, a Schrödinger operator subject to general vertex conditions was
considered. The main result stated that under a certain non-resonance
condition, see Condition 3.2 in [1], the considered operator converged in the
norm resolvent sense to a Schrödinger operator on the limiting graph without
small edges, which were replaced by certain conditions at the limiting
vertices. The vertex conditions in [1] were described in terms of sympletic
geometry and Lagrangian planes; the limiting vertex conditions were also
formultated in the same terms. A similar study was made in [7], where there
was considered a star graph, in which the vertex was replaced by a small
rescaled finite graph. The operator on this perturbed graph was designed so
that after the rescaling the small graph back to a finite size, the
differential expression and vertex conditions turned out to be independent of
a small parameter characterizing the size of the small graph. The main result
of [7] provided the leading terms in the asymptotics for the perturbed
resolvent and the estimates for the error term. The limiting condition at the
vertex of the limiting star graph contained only the Neumann and Dirichlet
parts.

The above described results motivated further studies of the graphs with small
edges aimed on better understanding and describing the behavior of the
resolvents. A natural question, apart of determining the limiting operator, is
to find several next terms in the asymptotics for the resolvent or even its
complete asymptotic expansion. This question was addressed in few recent
papers [4], [5], [6] for some toy models represented by very simple graphs. A
main feature of the considered models was the possibility of singular
dependence of the coefficients in the differential expression on a small
parameter governing the lengths of small edges. The main results obtained for
these models stated that the resolvents were analytic in certain sense with
respect to the small parameter and in the limit, the condition at the vertex
to which the small edges shrank, could involve a Robin part. This Robin part
was generated by the next-to-leading terms in the expansions of the
coefficients of the differential expression with respect to the small
parameter. For the resolvents, several first terms in their asymptotic
expansions were found. The analytic dependence of the resolvent on the small
parameter was in some sense surprising since small edges is an example of a
singular perturbation and usually, while dealing with singular perturbation,
one can hope only to find an asymptotic expansion for the resolvent but not to
prove that this expansion is a converging Taylor series.

The present work is motivated by papers [1], [7]. We consider a general graph
$\Gamma$, to which a small graph $\gamma_{\varepsilon}$ is glued. This small
graph is obtained by rescaling a given graph $\gamma$ into $\varepsilon^{-1}$
times, where $\varepsilon$ is a small parameter, see Figures 1, 2. On this
perturbed graph with small edges we define a second order scalar differential
operator $\mathcal{H}_{\varepsilon}$ depending on $\varepsilon$. The
differential expression is of a general form involving first order terms and
varying coefficients. The vertex conditions are also of a general matrix form
with arbitrary not necessarily unitary matrix coefficients. All coefficients
in the differential expression and the vertex conditions depend arbitrarily on
$\varepsilon$. For the coefficients in the vertex conditions and in the
differential expression on non-small edges in $\Gamma$, this dependence is
analytic in $\varepsilon$, while the coefficients in the differential
expression on the small edges in $\gamma_{\varepsilon}$ are meromorphic in
$\varepsilon$. On the coefficients in the differential expression and the
vertex conditions of $\mathcal{H}_{\varepsilon}$ we impose conditions ensuring
the self-adjointness of this operator. Then we attach finitely many leads to
the graph $\gamma$ and denote such extended graph by $\gamma_{\infty}$, see
Figure 2. On the graph $\gamma_{\infty}$ we consider a self-adjoint operator
$\mathcal{H}_{\infty}$ with differential expression and vertex conditions
generated in a certain way by those of the operator
$\mathcal{H}_{\varepsilon}$. Our main assumption is that the operator
$\mathcal{H}_{\infty}$ has no embedded eigenvalues at the bottom of its
essential spectrum; this does not exclude the presence of possible virtual
levels at the same bottom. This assumption is of a similar nature as Non-
resonance condition in work [1]. Under this condition, our main result states
that the resolvent of the operator $\mathcal{H}_{\varepsilon}$ is in some
sense analytic in $\varepsilon$. Namely, we consider the restrictions of this
resolvent to the graphs $\Gamma$ and $\gamma_{\varepsilon}$ and the latter
restriction then is sandwiched between the rescaling operators; such parts of
the resolvent turn out to be analytic in $\varepsilon$. As a corollary, this
provides a Taylor-like series for the resolvent. All these results are
established for the resolvent of $\mathcal{H}_{\varepsilon}$ regarded not only
as operator acting in $L_{2}$, but also as acting from $L_{2}$ into the
Sobolev spaces $W_{2}^{2}$ and into the spaces of continuous functions
$C^{2}$.

The mentioned Taylor-like series for the resolvent allows us to find a
limiting operator on the graph $\Gamma$, the resolvent of which approximates
the resolvent of $\mathcal{H}_{\varepsilon}$ in the sense of the above norms.
This is a convergence result as in [1] and [7] but for our general model.
Similar to the toy models in [4], [5], [6], our general model also has a
special feature related with the limiting vertex condition at $M_{0}$. Namely,
we show that the next-to-leading terms in the $\varepsilon$-expansions of the
coefficients in the differential expression and vertex conditions of
$\mathcal{H}_{\varepsilon}$ generate a Robin part in the limiting vertex
conditions at a vertex $M_{0}$, through which the graph $\gamma_{\varepsilon}$
is glued to the graph $\Gamma$. Such case was excluded by the formulation of
the problem in [7]. In [1], the Robin part could appear in the limit, but only
due to the fact that, in our terms, the conditions at the vertices of the
small edges were independent of $\varepsilon$.

Finally, we also conclude that despite the small edges are a singular
perturbation, which can not be treated by means of the regular perturbation
theory [11], the resolvent of the operator $\mathcal{H}_{\varepsilon}$ depends
analytically in $\varepsilon$ and behaves similar to the situation, when the
edges vary but do not vanish. The case with non-vanishing edges was considered
in [3] and the main result stated that the resolvent of the Schrödinger
operator on such graphs depended analytically on the edges lengths and the
matrices involved in the vertex conditions.

Figure 1: Examples of initial graphs $\Gamma$ and $\gamma$. The edges incident
to $M_{0}$ are split into three groups ($n=3$).

In conclusion of this section, let us briefly describe the structure of the
paper. In the next section we formulate the problem, our main assumptions and
the main results. In the third section we discuss how far our results can be
extended to the case, when several small graphs are glued to the graph
$\Gamma$, see Figure 4, including the case of the dependence on several
independent small parameters. And we also discuss the case, when the assumed
analyticity in the small parameter for the coefficients in the differential
expression and vertex conditions is replaced by the infinite differentiability
in $\varepsilon$ or just by the existence of power asymptotic expansions. In
the fourth section we discuss the self-adjointness of the operator we deal
with. In the fifth section we introduce and study some auxiliary operator on
$\gamma$, which is employed then in proving the main results in the sixth
section.

##  2 Problem and main results

###  2.1 Problem

Let $\Gamma$ be a finite metric graph having no isolated vertices. The lengths
of its edges can be both finite or infinite. We fix arbitrarily a vertex in
the graph $\Gamma$ and denote it by $M_{0}$. The degree of this vertex is
finite, that is, there exist finitely many edges $\mathrm{e}_{i}$,
$i=1,\ldots,d_{0}$ incident to $M_{0}$. Each loop in the family
$\\{\mathrm{e}_{i}\\}_{i=1,\ldots,d_{0}}$ is counted twice. By $\gamma$ we
denote one more graph, which is supposed to be finite, metric, having only
edges of finite lengths and no isolated vertices or edges, see Figure 1.

Figure 2: Graph $\Gamma_{\varepsilon}$ with a glued small graph and graph
$\gamma_{\infty}$. The small edges in the graph $\Gamma_{\varepsilon}$ are
indicated by thin lines. The leads in the graph $\gamma_{\infty}$ are of light
gray color.

Letting $\varepsilon$ to be a small positive parameter, we scale each edge of
the graph $\gamma$ by $\varepsilon$ and the resulting graph is denoted by
$\gamma_{\varepsilon}$. More precisely, the vertices of the graph
$\gamma_{\varepsilon}$ are the same as those of $\gamma$, while each edge
$\mathrm{e}\in\gamma$ is replaced by the edge of the length
$\varepsilon|\mathrm{e}|$.

We chose arbitrary vertices $M_{j}$, $j=1,\ldots,n$, in the graph
$\gamma_{\varepsilon}$ and then partition the aforementioned edges
$\mathrm{e}_{i}$, $i=1,\ldots,d_{0}$, in the graph $\Gamma$ into $n$ non-empty
groups $\\{\mathrm{e}_{i}\\}_{i\in J_{j}}$, $j=1,\ldots,n$, where $J_{j}$ are
some non-empty disjoint sets of indices and
$\bigcup\limits_{j=1}^{n}J_{j}=\\{1,\ldots,d_{0}\\}$, $1\leqslant n\leqslant
d_{0}$. Then we replace the vertex $M_{0}$ in the graph $\Gamma$ by its $n$
copies, one copy for each group $\\{\mathrm{e}_{i}\\}_{i\in J_{j}}$. Finally,
we introduce a graph, which will be the main object of our study, as the union
of the graphs $\Gamma$ and $\gamma_{\varepsilon}$, where the vertices $M_{j}$,
$j=1,\ldots,n$, are identified with the aforementioned copies of the vertex
$M_{0}$ in the graph $\Gamma$, see Figure 2. After the described gluing of the
graphs $\Gamma$ and $\gamma_{\varepsilon}$, the edges $e_{i}$,
$i=1,\ldots,d_{0}$, are no longer connected at the vertex $M_{0}$ but instead,
each group $\\{\mathrm{e}_{i}\\}_{i\in J_{j}}$ is incident to the vertex
$M_{j}$ in the graph $\gamma_{\varepsilon}$. Hereafter, we shall often
identify the graphs $\Gamma$ and $\gamma_{\varepsilon}$ with the corresponding
subgraphs in $\Gamma_{\varepsilon}$. In particular, in the sense of this
identification, each function defined on $\Gamma_{\varepsilon}$ is also
supposed to be defined on $\Gamma$ and $\gamma_{\varepsilon}$ and vice versa.
On each edge in $\Gamma$ and $\gamma$ we fix a direction and consequently, a
variable on it. Then the chosen direction is naturally transferred to the
graph $\Gamma_{\varepsilon}$.

The main object of our study is an unbounded operator
$\mathcal{H}_{\varepsilon}$ in $L_{2}(\Gamma_{\varepsilon})$ with a
differential expression $\hat{\mathcal{H}}(\varepsilon)$, the action of which
is defined as

$\hat{\mathcal{H}}(\varepsilon)u:=-\frac{d\
}{dx}p_{\varepsilon}\frac{du}{dx}+\mathrm{i}\left(\frac{d\
}{dx}(q_{\varepsilon}u)+q_{\varepsilon}\frac{du}{dx}\right)+V_{\varepsilon}u,$
(2.1)

where $\mathrm{i}$ is the imaginary unit and the coefficients are

$\begin{gathered}p_{\varepsilon}:=\left\\{\begin{aligned}
&p_{\Gamma}(\,\cdot\,,\varepsilon)&&\text{on}\quad\Gamma,\\\
\mathcal{S}_{\varepsilon}&p_{\gamma}(\,\cdot\,,\varepsilon)&&\text{on}\quad\gamma_{\varepsilon},\end{aligned}\right.\qquad
q_{\varepsilon}:=\left\\{\begin{aligned}
&q_{\Gamma}(\,\cdot\,,\varepsilon)&&\text{on}\quad\Gamma,\\\
\varepsilon^{-1}&\mathcal{S}_{\varepsilon}q_{\gamma}(\,\cdot\,,\varepsilon)\quad&&\text{on}\quad\gamma_{\varepsilon},\end{aligned}\right.\\\
V_{\varepsilon}:=\left\\{\begin{aligned}
&V_{\Gamma}(\,\cdot\,,\varepsilon)&&\text{on}\quad\Gamma,\\\
\varepsilon^{-2}&\mathcal{S}_{\varepsilon}V_{\gamma}(\,\cdot\,,\varepsilon)&&\text{on}\quad\gamma_{\varepsilon}.\end{aligned}\right.\end{gathered}$
(2.2)

Here $p_{\Gamma}=p_{\Gamma}(\,\cdot\,,\varepsilon)\in W_{\infty}^{1}(\Gamma)$,
$q_{\Gamma}=q_{\Gamma}(\,\cdot\,,\varepsilon)\in W_{\infty}^{1}(\Gamma)$,
$V_{\Gamma}=V_{\Gamma}(\,\cdot\,,\varepsilon)\in L_{2}(\Gamma)$ and
$p_{\gamma}=p_{\gamma}(\,\cdot\,,\varepsilon)\in W_{\infty}^{1}(\gamma)$,
$q_{\gamma}=q_{\gamma}(\,\cdot\,,\varepsilon)\in W_{\infty}^{1}(\gamma)$,
$V_{\gamma}=V_{\gamma}(\,\cdot\,,\varepsilon)\in L_{2}(\gamma)$ are some real
functions defined on the graphs $\Gamma$ and $V_{\gamma}$ and these functions
are analytic in $\varepsilon$ in the sense of the norms in the mentioned
spaces. The symbol $\mathcal{S}_{\varepsilon}$ stands for a linear operator
mapping $L_{2}(\gamma)$ onto $L_{2}(\gamma_{\varepsilon})$ by the rule

$(\mathcal{S}_{\varepsilon}u)(x):=u\left(\frac{x}{\varepsilon}\right)\quad\text{as}\quad
x\in\mathrm{e}_{\varepsilon}$

on each edge $\mathrm{e}_{\varepsilon}$ in the graph $\gamma_{\varepsilon}$.
We also assume an uniform ellipticity condition for the expression
$\hat{\mathcal{H}}(\varepsilon)$, namely, the existence of a fixed constant
$c_{\mathcal{H}}>0$ independent of $x\in\Gamma$, $\xi\in\gamma$ such that

$p_{\Gamma}(x,0)\geqslant c_{\mathcal{H}}\quad\text{a.e. on}\quad\Gamma,\qquad
p_{\gamma}(\xi,0)\geqslant c_{\mathcal{H}}\quad\text{a.e. on}\quad\gamma$

Thanks to the assumed analyticity in $\varepsilon$, this condition implies the
same condition for $p_{\Gamma}(x,\varepsilon)$ and $p_{\gamma}(x,\varepsilon)$
for all sufficiently small $\varepsilon$ with the constant $c$ replaced by
$c/2$.

The vertex conditions on the graph $\Gamma_{\varepsilon}$ are imposed as
follows. Let $M$ be an arbitrary vertex in $\Gamma_{\varepsilon}$ of a degree
$d(M)>0$ and $\mathrm{e}_{i}(M)$, $i=1,\ldots,d(M)$, be the edges incident to
this vertex and $u_{i}:=u\big{|}_{\mathrm{e}_{i}(M)}$, $i=1,\ldots,d(M)$, be
the restrictions of a function $u$ to the edges $\mathrm{e}_{i}(M)$. We
introduce two $d(M)$-dimensional vectors

$\mathcal{U}_{M}(u):=\begin{pmatrix}u_{1}(M)\\\ \vdots\\\
u_{d(M)}(M)\end{pmatrix},\qquad\mathcal{U}_{M}^{\prime}(u):=\begin{pmatrix}\frac{du_{1}}{dx_{1}}(M)\\\
\vdots\\\ \frac{du_{d(M)}}{dx_{d(M)}}(M)\end{pmatrix},$ (2.3)

where $x_{i}$ is the variable on the edge $\mathrm{e}_{i}$; this variable is
chosen according the above discussed direction. At each vertex
$M\in\Gamma_{\varepsilon}$ of a degree $d(M)>0$ we impose a general vertex
condition

$\mathrm{A}_{M}(\varepsilon)\mathcal{U}_{M}(u)+\mathrm{B}_{M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)=0,$
(2.4)

where $\mathrm{A}_{M}(\varepsilon)$ and $\mathrm{B}_{M}(\varepsilon)$ are some
matrices of size $d(M)\times d(M)$ analytic in $\varepsilon$.

The matrices $\mathrm{A}_{M}(\varepsilon)$ and $\mathrm{B}_{M}(\varepsilon)$
in conditions (2.5) are defined non-uniquely up to the left multiplication by
an arbitrary non-degenerate $d(M)\times d(M)$ matrix. In view of this fact, we
denote $r(M):=\operatorname{rank}\mathrm{B}_{M}(0)$ and without loss of
generality we additionally assume that for each $M\in\Gamma_{\varepsilon}$,
the first $r(M)$ rows in the matrix $\mathrm{B}_{M}(0)$ are linearly
independent and the other rows are zero, while each row among last $d(M)-r(M)$
ones in the matrix $\mathrm{A}_{M}(0)$ is non-zero.

The domain of the operator $\mathcal{H}_{\varepsilon}$ consists of the
functions in $\dot{W}_{2}^{2}(\gamma_{\varepsilon})$ satisfying the imposed
vertex conditions; hereinafter, for an arbitrary graph we denote
$\dot{W}_{2}^{j}(\,\cdot\,):=\bigoplus\limits_{\mathrm{e}\in\,\cdot\,}W_{2}^{j}(\mathrm{e})$,
$j=1,2$. The action of the operator $\mathcal{H}_{\varepsilon}$ on such
functions is defined by differential expression (2.1).

We want the introduced operator $\mathcal{H}_{\varepsilon}$ to be self-adjoint
and this is ensured by the two following assumptions. The first of them says
that the matrix

$\mathrm{A}_{M}(\varepsilon)\Pi_{M}^{-1}(\varepsilon)\mathrm{B}_{M}^{*}(\varepsilon)+\mathrm{i}\mathrm{B}_{M}(\varepsilon)\Pi_{M}^{-1}(\varepsilon)\Theta_{M}(\varepsilon)\Pi_{M}^{-1}(\varepsilon)\mathrm{B}_{M}^{*}(\varepsilon)$
(2.5)

is self-adjoint, where

$\displaystyle\Pi_{M}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M)p_{\varepsilon}\big{|}_{\mathrm{e}_{i}(M)}(M)\big{\\}}_{i=1,\ldots,d(M)},$
(2.6)
$\displaystyle\Theta_{M}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M)q_{\varepsilon}\big{|}_{\mathrm{e}_{i}(M)}(M)\big{\\}}_{i=1,\ldots,d(M)},$

and $\mathrm{e}_{i}(M)$ are the edges incident to the vertex $M$, while
$\nu_{i}(M):=1$ if the inward direction on the edge $\mathrm{e}_{i}(M)$ at the
vertex $M$ coincides with the above chosen direction on this edge and
$\nu_{i}(M):=-1$ otherwise.

The second assumption is the rank condition
$\operatorname{rank}\big{(}\mathrm{A}_{M}(0)\ \;\mathrm{B}_{M}(0)\big{)}=d(M)$
imposed at each vertex $M\in\Gamma_{\varepsilon}$. In view of the above
conditions for the structure of matrices $\mathrm{A}_{M}$ and
$\mathrm{B}_{M}$, this condition is obviously equivalent to

$\operatorname{rank}\big{(}\mathrm{A}_{M}(\varepsilon)\
\;\mathrm{B}_{M}(\varepsilon)\big{)}=d(M)$ (2.7)

for all sufficiently small $\varepsilon$.

###  2.2 Notation and assumption

To formulate our main results, we need some additional notations. By
$\gamma_{\infty}$ we denote a graph obtained by attaching leads (edges of
infinite lengths) $\mathrm{e}_{i}^{\infty}$, $i\in J_{j}$, $j=1,\ldots,n$, to
each vertex $M_{j}$, $j=1,\ldots,n$, in the graph $\gamma$; the vertex $M_{j}$
serves as the origin for the attached edges $\mathrm{e}_{i}$. The variable on
the graph $\gamma_{\infty}$ is denoted by $\xi$.

The introduced graph appears in a very natural way. Namely, we consider the
subgraph $\gamma_{\varepsilon}$ in $\Gamma_{\varepsilon}$ with the edges
$\mathrm{e}_{i}$ incident to the vertices $M_{j}$. Then we rescale this
subgraph and the edges by means of $\varepsilon$ so that the graph
$\gamma_{\varepsilon}$ turns into the graph $\gamma$. Under such rescaling,
the edges $\mathrm{e}_{i}$ become ones with lengths of order
$O(\varepsilon^{-1})$ and in the limit $\varepsilon\to+0$, we just replace
them by the leads $\mathrm{e}_{i}^{\infty}$.

The described scaling corresponds to the change of the variables
$\xi=x\varepsilon^{-1}$ and we need to see how the part of the operator
$\mathcal{H}_{\varepsilon}$ corresponding to $\gamma_{\varepsilon}$ is
transformed under such change. Namely, we first replace the coefficients of
the differential expression $\hat{\mathcal{H}}_{\varepsilon}$ on the edges
$\mathrm{e}_{i}$ by their values at $M_{0}$. Then we multiply the obtained
differential expression by $\varepsilon^{2}$, pass to the variable $\xi$ and
$\varepsilon=0$. This gives rise to the differential expression

$\displaystyle\hat{\mathcal{H}}_{\infty}u:=-\frac{d\
}{d\xi}p_{\gamma}(\,\cdot\,,0)\frac{du}{d\xi}+\mathrm{i}\left(\frac{d\
}{d\xi}(q_{\gamma}(\,\cdot\,,0)u)+q_{\gamma}(\,\cdot\,,0)\frac{du}{d\xi}\right)+V_{\gamma}(\,\cdot\,,0)u\quad\text{on}\quad\gamma,$
(2.8)
$\displaystyle\hat{\mathcal{H}}_{\infty}u:=-\mathsf{p}_{i}(0)\frac{d^{2}u}{d\xi^{2}}\quad\text{on}\quad\mathrm{e}_{i}^{\infty},\quad
i\in J_{j},\quad
j=1,\ldots,n,\qquad\mathsf{p}_{i}(\varepsilon):=p_{\Gamma}\big{|}_{\mathrm{e}_{i}}(M_{0},\varepsilon).$

Then we do a similar rescaling in the vertex conditions, but we multiply them
by

$\tilde{\mathrm{E}}_{M}(\varepsilon):=\begin{pmatrix}\mathrm{I}_{r(M)}&0\\\
0&\varepsilon^{-1}\mathrm{I}_{d(M)-r(M)}\end{pmatrix}.$

We see that

$\displaystyle\varepsilon\tilde{\mathrm{E}}_{M}(\varepsilon)\mathrm{A}_{M}(\varepsilon)=\mathrm{A}_{M}^{(0)}+O(\varepsilon),\qquad\tilde{\mathrm{E}}_{M}(\varepsilon)\mathrm{B}_{M}(\varepsilon)=\mathrm{B}_{M}^{(0)}+O(\varepsilon),\quad$
(2.9) $\displaystyle\mathrm{A}_{M}^{(0)}:=\begin{pmatrix}0\\\
\mathrm{A}_{M}^{-}(0)\end{pmatrix},\qquad\mathrm{B}_{M}^{(0)}:=\begin{pmatrix}\mathrm{B}_{M}^{+}(0)\\\
\frac{d\mathrm{B}_{M}^{-}}{d\varepsilon}(0)\end{pmatrix}$ (2.10)

and $\mathrm{A}_{M}^{-}(\cdot)$ and $\mathrm{B}_{M}^{-}(\cdot)$ are the
matrices formed by last $d(M)-r(M)$ rows of respectively the matrices
$\mathrm{A}_{M}(\cdot)$ and $\mathrm{B}_{M}(\cdot)$, while a matrix
$\mathrm{B}_{M}^{+}(\cdot)$ is formed by first $r(M)$ rows of the matrix
$\mathrm{B}_{M}$. Passing then to the limit as $\varepsilon\to+0$, we end up
with the following vertex conditions:

$\mathrm{A}_{M}^{(0)}\mathcal{U}_{M}(u)+\mathrm{B}_{M}^{(0)}\mathcal{U}_{M}^{\prime}(u)=0\quad\text{at
each}\quad M\in\gamma_{\infty},$ (2.11)

where the vectors $\mathcal{U}_{M}(u)$ and $\mathcal{U}_{M}^{\prime}(u)$ are
introduced as in (2.3) with the derivatives $\frac{du_{i}}{dx_{i}}$
substituted by $\frac{du_{i}}{d\xi_{i}}$.

The described procedure gives rise to an auxiliary operator
$\mathcal{H}_{\infty}$ on the graph $\gamma_{\infty}$ with differential
expression (2.8) subject to the vertex conditions (2.11). The domain of the
operator $\mathcal{H}_{\infty}$ consists of the functions in
$\dot{W}_{2}^{2}(\gamma)$ satisfying the imposed vertex conditions. It will be
shown later, see Section 4.2, that the operator $\mathcal{H}_{\infty}$ is
self-adjoint.

Since the graph $\gamma$ is finite and all its edges are of finite lengths,
the only edges of infinite length in the graph $\gamma_{\infty}$ are the leads
$\mathrm{e}_{i}^{\infty}$ attached to the vertices $M_{j}$, $j=1,\ldots,n$.
The differential expression of the introduced operator $\mathcal{H}_{\infty}$
on these leads is just the negative Laplacian multiplied by non-zero constants
$\mathsf{p}_{i}$, see (2.8). This is why it is straightforward to confirm that
the essential spectrum of the operator $\mathcal{H}_{\infty}$ is the half-line
$[0,+\infty)$.

The main condition we assume in the work is as follows.

  1. (A).

The operator $\mathcal{H}_{\infty}$ has no embedded eigenvalue at the bottom
of its essential spectrum.

At the bottom of its essential spectrum, the operator $\mathcal{H}_{\infty}$
can have virtual level, namely, there can be a bounded non-trivial solution
$\psi\in\dot{W}_{2}^{2}(\gamma)\oplus\bigoplus\limits_{i\in J_{j},\
j=1,\ldots,n}W_{2,loc}^{2}(\mathrm{e}_{i}^{\infty})$ of the boundary value
problem

$\hat{\mathcal{H}}_{\infty}\psi=0\quad\text{on}\quad\gamma_{\infty},\qquad\mathrm{A}_{M}^{(0)}\mathcal{U}_{M}(\psi)+\mathrm{B}_{M}^{(0)}\mathcal{U}^{\prime}_{M}(\psi)=0\quad\text{at
each}\quad M\in\gamma_{\infty}.$ (2.12)

Again in view of the definition of the differential expression
$\hat{\mathcal{H}}_{\infty}$ on the leads $\mathrm{e}_{i}^{\infty}$, the
aforementioned boundedness condition for $\psi$ is equivalent to the
identities $\psi=const$ on $\mathrm{e}_{i}^{\infty}$, $i\in J_{j}$,
$j=1,\ldots,n$, where $const$ stands for some constants depending in general
on the choice of the edge $\mathrm{e}_{i}^{\infty}$.

Given a function $u$ defined and continuous on the edges
$\mathrm{e}_{i}^{\infty}$ in the vicinity of the points $M_{j}$, we denote

$\mathcal{U}_{\gamma}(u):=\begin{pmatrix}u\big{|}_{\mathrm{e}_{i}^{\infty}}(M_{j})\end{pmatrix}_{i\in
J_{j},\ j=1,\ldots,n}$

where $u\big{|}_{\mathrm{e}_{i}^{\infty}}$ is the restriction of $u$ to the
edge $\mathrm{e}_{i}^{\infty}$. In view of the said above, Condition (A) is
equivalent to the following: each non-trivial solution $\psi$ to problem
(2.12) satisfies

$\mathcal{U}_{\gamma}(\psi)\neq 0.$ (2.13)

Let $\psi^{(j)}$, $j=1,\ldots,k$, be linearly independent bounded non-trivial
solutions to problem (2.12) satisfying condition (2.13). It is clear that
$k\leqslant d_{0}$ since once $k>d_{0}$ it is possible to find a linear
combination of the functions $\psi^{(j)}$, for which condition (2.13) is
violated. If the operator $\mathcal{H}_{\infty}$ has no virtual level at the
bottom of its essential spectrum, we let $k:=0$.

We denote: $\varPsi^{(j)}:=\mathcal{U}_{\gamma}(\psi^{(j)})$, $j=1,\ldots,k$.
We choose the functions $\psi^{(j)}$ so that the associated vectors
$\varPsi^{(j)}$ are orthonormalized in $\mathds{C}^{d_{0}}$. If $k<d_{0}$, we
choose arbitrary vectors $\varPsi^{(j)}\in\mathds{C}^{d_{0}}$,
$j=k+1,\ldots,d_{0}$, so that the vectors
$\varPsi^{(j)}\in\mathds{C}^{d_{0}}$, $j=1,\ldots,d_{0}$, form an
orthonormalized basis in $\mathds{C}^{d_{0}}$. Hence, the matrix
$\Psi:=\begin{pmatrix}\varPsi^{(1)}&\ldots&\varPsi^{(k)}&\varPsi^{(k+1)}&\ldots&\varPsi^{(d_{0})}\end{pmatrix}$
is unitary.

###  2.3 Parts of the resolvent and main result

Our main result concerns certain operators, which are naturally regarded as
some parts of the resolvent $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$. In
this subsection we introduce these operators and their limits as
$\varepsilon\to+0$ and we formulate our main result.

Let $\mathcal{P}_{\Gamma}:L_{2}(\Gamma_{\varepsilon})\to L_{2}(\Gamma)$ and
$\mathcal{P}_{\gamma_{\varepsilon}}:L_{2}(\Gamma_{\varepsilon})\to
L_{2}(\gamma_{\varepsilon})$ be the operators of restriction to the graphs
$\Gamma$ and $\gamma_{\varepsilon}$, that is, on each $f\in
L_{2}(\Gamma_{\varepsilon})$ they act as follows:
$\mathcal{P}_{\Gamma}f:=f\big{|}_{\Gamma}$,
$\mathcal{P}_{\gamma_{\varepsilon}}f:=f\big{|}_{\gamma_{\varepsilon}}$. In the
sense of the decomposition $L_{2}(\Gamma_{\varepsilon})=L_{2}(\Gamma)\oplus
L_{2}(\gamma_{\varepsilon})$ these operators satisfy the identity

$\mathcal{P}_{\Gamma}\oplus\mathcal{P}_{\gamma_{\varepsilon}}=\mathcal{I}_{\Gamma_{\varepsilon}},$
(2.14)

where $\mathcal{I}_{\Gamma_{\varepsilon}}$ is the identity mapping in
$L_{2}(\Gamma_{\varepsilon})$.

For each $\lambda\in\mathds{C}\setminus\mathds{R}$ the resolvent
$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ is well-defined. The operators

$\mathcal{R}_{\Gamma}(\varepsilon,\lambda):=\mathcal{P}_{\Gamma}(\mathcal{H}_{\varepsilon}-\lambda)^{-1}(\mathcal{I}_{\Gamma}\oplus\mathcal{S}_{\varepsilon}),\qquad\mathcal{R}_{\gamma}(\varepsilon,\lambda):=\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}(\mathcal{H}_{\varepsilon}-\lambda)^{-1}(\mathcal{I}_{\Gamma}\oplus\mathcal{S}_{\varepsilon})$

are also well-defined, linear and are bounded as acting from
$L_{2}(\Gamma)\oplus L_{2}(\gamma)$ into $\dot{W}_{2}^{2}(\Gamma)$ and
$\dot{W}_{2}^{2}(\gamma)$; here the direct sum is understood in the sense of
identity (2.14).

Let us clarify the action of the operators $\mathcal{R}_{\Gamma}$ and
$\mathcal{R}_{\gamma}$. Given $f\in L_{2}(\Gamma_{\varepsilon})$ and
$u_{\varepsilon}:=(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f$, we consider the
restrictions of these functions to the subgraphs $\Gamma$ and
$\gamma_{\varepsilon}$. The restrictions to $\gamma_{\varepsilon}$ are
additionally rescaled by means of the operator $\mathcal{S}_{\varepsilon}$,
that is, they are simply regarded as functions of the rescaled variable
$\xi:=x\varepsilon^{-1}$ defined on $\gamma$. These restrictions, with the
rescaling taken into account, are obviously $\mathcal{P}_{\Gamma}f$,
$\mathcal{P}_{\Gamma}u_{\varepsilon}$ and
$\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}f$,
$\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}u_{\varepsilon}$.
Then the operators $\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ map a pair
$(\mathcal{P}_{\Gamma}f,\,\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}f)$
respectively into $\mathcal{P}_{\Gamma}u_{\varepsilon}$ and
$\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}u_{\varepsilon}$.
We also observe an obvious identity implied immediately by the definition of
the operators $\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$:

$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}=\big{(}\mathcal{R}_{\Gamma}(\varepsilon,\lambda)\oplus\mathcal{S}_{\varepsilon}\mathcal{R}_{\gamma}(\varepsilon,\lambda)\big{)}(\mathcal{P}_{\Gamma}\oplus\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}).$
(2.15)

Now we introduce an operator, which will serve as a limiting one for
$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$. Since the operator
$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ represents the action of the
resolvent $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ on the subgraph $\Gamma$,
its limit should be the resolvent $(\mathcal{H}_{0}-\lambda)^{-1}$, where
$\mathcal{H}_{0}$ is an operator in $L_{2}(\Gamma)$ with the differential
expression $\hat{\mathcal{H}}(0)$ subject to the vertex conditions

$\mathrm{A}_{M}^{(0)}\mathcal{U}_{M}(u)+\mathrm{B}_{M}^{(0)}\mathcal{U}_{M}^{\prime}(u)=0\quad\text{at
each vertex}\quad M\in\Gamma.$ (2.16)

For the vertices $M\neq M_{0}$, the matrices in the above vertex condition are
defined as $\mathrm{A}_{M}^{(0)}:=\mathrm{A}_{M}(0)$,
$\mathrm{B}_{M}^{(0)}:=\mathrm{B}_{M}(0)$; the domain of the operator
$\mathcal{H}_{0}$ is formed by the functions from $\dot{W}_{2}^{2}(\Gamma)$
satisfying vertex conditions (2.16).

The structure of the matrices $\mathrm{A}_{M_{0}}^{(0)}$,
$\mathrm{B}_{M_{0}}^{(0)}$ is more complicated and heuristically they can be
found as follows. Given $f\in L_{2}(\Gamma_{\varepsilon})$, let
$u_{\varepsilon}:=(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f$. Since the
coefficients in the differential expression and boundary conditions of
$\mathcal{H}_{\varepsilon}$ depend analytically in $\varepsilon$ and the graph
$\gamma_{\varepsilon}$ is also obtained via rescaling by means of
$\varepsilon$, it is natural to expect that on $\Gamma_{\varepsilon}$, we have
two approximations

$u_{\varepsilon}(x)=u_{0}(x)+\ldots\quad\text{on}\quad\Gamma,\qquad
u_{\varepsilon}(x)=v_{0}(\xi)+\varepsilon
v_{1}(\xi)+\ldots\quad\text{on}\quad\gamma_{\varepsilon},\qquad\xi:=x\varepsilon^{-1},$
(2.17)

where $u_{0}$, $v_{0}$, $v_{1}$ are some functions. Since these are
approximations for the same function $u_{\varepsilon}$, which is continuously
differentiable on the edges $\mathrm{e}_{i}$, $i=1,\ldots,d_{0}$, we should
obviously have

$\mathcal{U}_{M_{0}}(u_{0})=\mathcal{U}_{\gamma}(v_{0}),\qquad\mathcal{U}_{M_{0}}^{\prime}(u_{0})=\begin{pmatrix}v_{1}^{\prime}\big{|}_{\mathrm{e}_{i}^{\infty}}(M_{j})\end{pmatrix}_{i\in
J_{j},\
j=1,\ldots,n},\qquad\begin{pmatrix}v_{0}^{\prime}\big{|}_{\mathrm{e}_{i}^{\infty}}(M_{j})\end{pmatrix}_{i\in
J_{j},\ j=1,\ldots,n}=0.$ (2.18)

We substitute then (2.17) into the equation
$(\mathcal{H}_{\varepsilon}-\lambda)u_{\varepsilon}=f$ considered on
$\gamma_{\varepsilon}$, pass to the variables $\xi$, and equate the
coefficients at the like powers of $\varepsilon$. This gives immediately the
boundary value problem for $v_{0}$

$\hat{\mathcal{H}}_{\infty}v_{0}=0\quad\text{on}\quad\gamma_{\infty},\qquad\mathrm{A}_{M}^{(0)}\mathcal{U}_{M}(v_{0})+\mathrm{B}_{M}^{(0)}\mathcal{U}_{M}^{\prime}(v_{0})=0\quad\text{at}\quad
M\in\gamma_{\infty},$ (2.19)

and for $v_{1}$

$\displaystyle\hat{\mathcal{H}}_{\infty}v_{1}=\frac{d\
}{d\xi}\frac{dp_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\frac{dv_{0}}{d\xi}-\mathrm{i}\left(\frac{d\
}{d\xi}\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)v_{0}+\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\frac{dv_{0}}{d\xi}\right)-\frac{dV_{\gamma}^{(0)}}{d\varepsilon}(\,\cdot\,,0)v_{0}\quad\text{on}\quad\gamma,$
(2.20)
$\displaystyle\mathrm{A}_{M}^{(0)}\mathcal{U}_{M}(v_{1})+\mathrm{B}_{M}^{(0)}\mathcal{U}_{M}^{\prime}(v_{1})=-\mathrm{A}_{M}^{(1)}\mathcal{U}_{M}(v_{0})-\mathrm{B}_{M}^{(1)}\mathcal{U}_{M}^{\prime}(v_{0})\quad\text{at}\quad
M\in\gamma_{\infty},$

where $\mathrm{A}_{M}^{(1)}$ and $\mathrm{B}_{M}^{(1)}$ are the matrices in
the next terms in expansions (2.9); they read as

$\mathrm{A}_{M}^{(1)}:=\begin{pmatrix}\mathrm{A}_{M}^{+}(0)\\\
\frac{d\mathrm{A}_{M}^{-}}{d\varepsilon}(0)\end{pmatrix},\qquad\mathrm{B}_{M}^{(1)}:=\begin{pmatrix}\frac{d\mathrm{B}_{M}^{+}}{d\varepsilon}(0)\\\
\frac{1}{2}\frac{d^{2}\mathrm{B}_{M}^{-}}{d\varepsilon^{2}}(0)\end{pmatrix}$

In view of the first and second conditions in (2.18), the solution of problem
(2.19) reads as

$v_{0}=\sum\limits_{i=1}^{k}c_{i}\psi^{(i)},\quad\begin{pmatrix}c_{1}(f)&\ldots&c_{k}(f)\end{pmatrix}^{t}:=\begin{pmatrix}\varPsi^{(1)}&\ldots&\varPsi^{(k)}\end{pmatrix}\mathcal{U}_{M_{0}}\big{(}(\mathcal{H}_{\Gamma}-\lambda)^{-1}f\big{)}.$
(2.21)

In order to ensure the solvability of problem (2.20), we multiply the equation
for $v_{1}$ by $\psi^{(i)}$, $i=1,\ldots,k$, and integrate twice by parts over
$\gamma$. In view of formula (2.21) and the second identity in (2.18), such
procedure gives certain solvability condition relating the vectors
$\mathcal{U}_{M_{0}}(u_{0})$ and $\mathcal{U}_{M_{0}}^{\prime}(u_{0})$. The
relation obtained in this way is exactly the desired vertex condition for the
operator $\mathcal{H}_{0}$ at the vertex $M_{0}$.

The corresponding matrices $\mathrm{A}_{M_{0}}^{(0)}$ and
$\mathrm{B}_{M_{0}}^{(0)}$ turn out to be of form

$\mathrm{A}_{M_{0}}^{(0)}:=\begin{pmatrix}\mathrm{Q}&0\\\
0&\mathrm{I}_{d_{0}-k}\end{pmatrix}\Psi^{*}+\mathrm{i}\begin{pmatrix}\mathrm{I}_{k}&0\\\
0&0\end{pmatrix}\Psi^{*}\Theta_{\Gamma,M_{0}}(0),\qquad\mathrm{B}_{M_{0}}^{(0)}:=-\begin{pmatrix}\mathrm{I}_{k}&0\\\
0&0\end{pmatrix}\Psi^{*}\Pi_{\Gamma,M_{0}}(0),$ (2.22)

where the symbol $0$ in the first row of the matrix $\mathrm{A}_{M_{0}}$
stands for the zero matrix of the size $k\times(d_{0}-k)$, while in the second
row the same symbol denotes the zero matrix of the size $(d_{0}-k)\times k$.
In the definition of the matrix $\mathrm{B}_{M_{0}}$, the first matrix is of
the size $d_{0}\times d_{0}$ and the symbols $0$ stands for the zero matrices
of respectively the sizes $k\times(d_{0}-k)$, $(d_{0}-k)\times k$, and
$(d_{0}-k)\times(d_{0}-k)$. The symbol $\mathrm{I}_{d}$ denotes the unit
matrix of size $d\times d$. The matrices $\Pi_{\Gamma,M_{0}}$,
$\Theta_{\Gamma,M_{0}}$ are defined as

$\Pi_{\Gamma,M_{0}}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M_{0})p_{\Gamma}\big{|}_{\mathrm{e}_{i}}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d_{0}},\qquad\Theta_{\Gamma,M_{0}}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M_{0})q_{\Gamma}\big{|}_{\mathrm{e}_{i}}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d_{0}},$

where $\mathrm{e}_{i}$ are the edges incident to the vertex $M_{0}$, while
$\nu_{i}(M_{0})$ is defined in the same way as in (2.6).

The matrix $\mathrm{Q}$ is exactly one appearing via integrating by parts,
while obtaining the aforementioned solvability condition of problem (2.20):

$\mathrm{Q}:=\begin{pmatrix}Q^{(11)}&\ldots&Q^{(k1)}\\\ \vdots&&\vdots\\\
Q^{(1k)}&\ldots&Q^{(kk)}\end{pmatrix},$

with the entries

$\displaystyle\begin{aligned}
Q^{(ij)}:=Q_{\gamma}^{(ij)}+\sum\limits_{M\in\gamma_{\infty}}Q_{M}^{(ij)},\end{aligned}$
(2.23) $\displaystyle\begin{aligned}
Q_{\gamma}^{(ij)}:=&\left(\frac{dp_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\frac{d\psi^{(i)}}{d\xi},\frac{d\psi^{(j)}}{d\xi}\right)_{L_{2}(\gamma)}+\left(\frac{d\psi^{(i)}}{d\xi},\mathrm{i}\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\psi^{(j)}\right)_{L_{2}(\gamma)}\\\
&+\left(\mathrm{i}\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\psi^{(i)},\frac{d\psi^{(j)}}{d\xi}\right)_{L_{2}(\gamma)}+\left(\frac{dV_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\psi^{(i)},\psi^{(j)}\right)_{L_{2}(\gamma)},\end{aligned}$
(2.24) $\displaystyle
Q_{M}^{(ij)}:=\big{(}\mathcal{L}_{M}(\psi^{(i)}),\mathcal{U}_{M}(\psi^{(j)})\big{)}_{\mathds{C}^{d(M)}}-\frac{\mathrm{i}}{2}\Big{(}\mathcal{E}_{M}(\psi^{(i)}),\mathcal{V}_{M}(\psi^{(j)})\Big{)}_{\mathds{C}^{d(M)}},$
(2.25)
$\displaystyle\mathcal{L}_{M}(\psi^{(i)}):=\frac{d\Pi_{\gamma,M}}{d\varepsilon}(0)\mathcal{U}_{M}^{\prime}(\psi^{(i)})-\mathrm{i}\frac{d\Theta_{\gamma,M}}{d\varepsilon}(0)\mathcal{U}_{M}(\psi^{(i)})+\big{(}\mathrm{U}_{M}^{(0)}+\mathrm{I}_{d(M)}\big{)}^{-1}\mathrm{P}_{M,\bot}^{(0)}\mathcal{E}_{M}(\psi^{(i)}),$
(2.26)
$\displaystyle\mathcal{E}_{M}(\,\cdot\,):=2\mathrm{i}\big{(}\tilde{\mathrm{A}}_{M}^{(0)}-\mathrm{i}\tilde{\mathrm{B}}_{M}^{(0)}\big{)}^{-1}\Big{(}\mathrm{A}_{M}^{(1)}\mathcal{U}_{M}(\,\cdot\,)+\mathrm{B}_{M}^{(1)}\mathcal{U}_{M}^{\prime}(\,\cdot\,)\Big{)},$
(2.27)
$\displaystyle\mathcal{V}_{M}(\,\cdot\,):=\Pi_{\gamma,M}(0)\mathcal{U}_{M}^{\prime}(\,\cdot\,)-\mathrm{i}\Theta_{\gamma,M}(0)\mathcal{U}_{M}(\,\cdot\,),$
$\displaystyle\begin{aligned}
\tilde{\mathrm{A}}_{M}^{(0)}:=\mathrm{A}_{M}^{(0)}+\mathrm{i}\mathrm{B}_{M}^{(0)}\Pi_{\gamma,M}^{-1}(0)\Theta_{\gamma,M}(0),\qquad\tilde{\mathrm{B}}_{M}^{(0)}:=\mathrm{B}_{M}^{(0)}\Pi_{\gamma,M}^{-1}(0),\end{aligned}$
$\displaystyle\mathrm{U}_{M}^{(0)}:=-\big{(}\tilde{\mathrm{A}}_{M}^{(0)}-\mathrm{i}\tilde{\mathrm{B}}_{M}^{(0)}\big{)}^{-1}\big{(}\tilde{\mathrm{A}}_{M}^{(0)}+\mathrm{i}\tilde{\mathrm{B}}_{M}^{(0)}\big{)},$
(2.28) $\displaystyle\begin{aligned}
&\Pi_{\gamma,M}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M)p_{\gamma}\big{|}_{\mathrm{e}_{i}(M)}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d(M)},\\\
&\Theta_{\gamma,M}(\varepsilon):=\operatorname{diag}\big{\\{}\nu_{i}(M)q_{\gamma}\big{|}_{\mathrm{e}_{i}(M)}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d(M)},\end{aligned}$
(2.29)

where $\mathrm{e}_{i}(M)$ are the edges incident to the vertex $M$, the
numbers $\nu_{i}(M)$ are defined as in (2.6), and while applying formula
(2.29) to $M=M_{j}$, the functions $V_{\gamma}^{(l)}$ are supposed to be
continued on $\mathrm{e}_{i}^{\infty}$, $i\in J_{j}$, $j=1,\ldots,n$, as
follows: $p_{\gamma}(\,\cdot\,,\varepsilon)\equiv\mathsf{p}_{i}(\varepsilon)$,
$q_{\gamma}(\,\cdot\,,\varepsilon)\equiv\varepsilon\mathsf{q}_{i}(\varepsilon)$,
$\mathsf{q}_{i}(\varepsilon):=q_{\Gamma}\big{|}_{\mathrm{e}_{i}}(M_{0},\varepsilon)$.
We shall show that the matrix $\mathrm{U}_{M}^{(0)}$ defined in (2.28) is
unitary, see Lemma 4.1. By $\mathrm{P}_{M}^{(0)}$ we denote the projector in
$\mathds{C}^{d(M)}$ onto the eigenspace of the matrix $\mathrm{U}_{M}^{(0)}$
associated with the eigenvalue $-1$, and we let
$\mathrm{P}_{M,\bot}^{(0)}:=\mathrm{I}_{d(M)}-\mathrm{P}_{M}^{(0)}$.

The second identity in (2.17) means that the function $v_{0}$ should represent
the limit of the operator $\mathcal{R}_{\gamma}$ as $\varepsilon\to+0$.
Namely, in view of formula (2.21), we introduce one more operator
$\mathcal{R}_{\gamma}^{(0)}(\lambda):L_{2}(\Gamma)\to\dot{W}_{2}^{2}(\gamma)$
acting on each $f\in L_{2}(\Gamma)$ by the rule:

$\displaystyle\mathcal{R}_{\gamma}^{(0)}(\lambda)f:=\sum\limits_{i=1}^{k}c_{i}(f)\psi^{(i)},$
$\displaystyle\begin{pmatrix}c_{1}(f)&\ldots&c_{k}(f)\end{pmatrix}^{t}:=\begin{pmatrix}\varPsi^{(1)}&\ldots&\varPsi^{(k)}\end{pmatrix}\mathcal{U}_{M_{0}}\big{(}(\mathcal{H}_{0}-\lambda)^{-1}f\big{)}.$

We shall show later that this operator is indeed the limiting one for
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$.

We introduce the following spaces of continuous functions on graphs:

$\displaystyle\mathfrak{C}(\,\cdot\,):=\mathfrak{C}^{0}(\,\cdot\,):=\bigoplus\limits_{\mathrm{e}\in\,\cdot\,}C(\overline{\mathrm{e}})\cap
L_{\infty}(\mathrm{e}),\qquad\mathfrak{C}^{1}(\,\cdot\,):=\bigoplus\limits_{\mathrm{e}\in\,\cdot\,}C^{1}(\overline{\mathrm{e}})\cap
W_{\infty}^{1}(\mathrm{e}),$
$\displaystyle\mathfrak{C}^{2}(\,\cdot\,):=\bigoplus\limits_{\mathrm{e}\in\,\cdot\,}C^{1}(\overline{\mathrm{e}})\cap
W_{\infty}^{2}(\mathrm{e}),\qquad\|u\|_{\mathfrak{C}^{i}(\overline{\mathrm{e}})}:=\sum\limits_{\mathrm{e}\in\cdot}\|u\|_{W_{\infty}^{i}(\mathrm{e})}.$

If the functions $p_{\Gamma}$, $q_{\Gamma}$, $V_{\Gamma}$ and $p_{\gamma}$,
$q_{\gamma}$, $V_{\gamma}$ have an additional smoothness
$p_{\Gamma},q_{\Gamma}\in\mathfrak{C}^{1}(\Gamma)$,
$V_{\Gamma}\in\mathfrak{C}^{0}(\Gamma)$,
$p_{\gamma},q_{\gamma}\in\mathfrak{C}^{1}(\gamma)$,
$V_{\gamma}\in\mathfrak{C}^{0}(\gamma)$ and are analytic in $\varepsilon$ in
the norms of these spaces, then we let
$\mathfrak{C}^{2}(\,\cdot\,):=\bigoplus\limits_{\mathrm{e}\in\,\cdot\,}C^{2}(\overline{\mathrm{e}})\cap
W_{\infty}^{2}(\mathrm{e})$.

Now we are in position to formulate our main result.

######  Theorem 2.1.

Assume that the matrices $\mathrm{A}_{M}(\varepsilon)$,
$\mathrm{B}_{M}(\varepsilon)$ satisfy the above formulated conditions. Then
the operators $\mathcal{H}_{\varepsilon}$ and $\mathcal{H}_{0}$ are self-
adjoint. Suppose that Condition (A) holds. The operators
$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ are linear and bounded as acting
from $L_{2}(\Gamma)\oplus L_{2}(\gamma)$ into $\mathfrak{C}^{2}(\Gamma)$ and
$\mathfrak{C}^{2}(\gamma)$. For each $\lambda\in\mathds{C}\setminus\mathds{R}$
there exists $\varepsilon_{0}(\lambda)>0$ such that as
$\varepsilon<\varepsilon_{0}(\lambda)$, the operators
$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ are analytic in $\varepsilon$ as
operators from $L_{2}(\Gamma)\oplus L_{2}(\gamma)$ into
$\dot{W}_{2}^{2}(\Gamma)$ and $\dot{W}_{2}^{2}(\gamma)$ and into
$\mathfrak{C}^{2}(\Gamma)$ and $\mathfrak{C}^{2}(\gamma)$. In both cases, the
leading terms of the Taylor series for these operators read as

$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)=(\mathcal{H}_{0}-\lambda)^{-1}\mathcal{P}_{\Gamma}+O(\varepsilon),\qquad\mathcal{R}_{\gamma}(\varepsilon,\lambda)=\mathcal{R}_{\gamma}^{(0)}(\lambda)\mathcal{P}_{\Gamma}+O(\varepsilon),$
(2.30)

where the estimates for the error terms depend on $\lambda$.

##  3 Discussion and generalizations

###  3.1 Discussion of main results

In this subsection we discuss the main features of our problem and the main
result. First of all we stress that our operator is very general. Namely, its
differential expression involves not only the potential, but also the first
order terms and a varying coefficient at a higher derivatives, see (2.1). On
the small edges, the coefficients can depend singularly on $\varepsilon$
because of the presence of the negative powers of $\varepsilon$ in (2.2). The
vertex conditions are also of general form, see (2.4). The self-adjointness of
the matrix in (2.5) and rank condition (2.7) are very natural and in fact,
they are a criterion ensuring the self-adjointness of the considered operator.

Our main result, Theorem 2.1, states that under the assumptions made for the
coefficients in the differential expression and the vertex conditions and
under Condition (A), the resolvent of the operator $\mathcal{H}_{\varepsilon}$
is analytic in $\varepsilon$ in certain sense. Namely, the operators
$\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ are analytic in $\varepsilon$.
Since the resolvent $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ can be
recovered from the operators $\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$ and
$\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ via formula (2.15), Theorem 2.1 in
fact states the analyticity of the parts of the resolvent
$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ corresponding to subgraphs $\Gamma$
and $\gamma_{\varepsilon}$. As we have already mentioned, the presence of
small edges is a singular perturbation, which can not be treated by the
methods from the regular perturbation theory [11]. Usually, for singularly
perturbed problems one can expect only the convergence result and a chance to
construct some asymptotic series. For instance, this is a standard situation
for elliptic operators on Euclidean domains or manifolds with singular
localized perturbation [10], [12]. A classical example is an elliptic problem
in a domain with a small hole, see [10, Ch. III], [12, Ch. II, Sec. 2.2].
It is possible to construct complete asymptotic expansions for the
corresponding resolvent provided it acts on a sufficiently smooth function,
but already simple explicitly solvable examples of such problems show that
these asymptotic series are not analytic in the small parameter. From this
point of view, our result on analyticity is of a special nature: despite of
the presence of a singular perturbation, the parts of the resolvent
$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ are analytic in $\varepsilon$ in
the operator norms and these norms are strongest possible.

Once we know that the operators $\mathcal{R}_{\Gamma}(\varepsilon,\lambda)$
and $\mathcal{R}_{\gamma}(\varepsilon,\lambda)$ are analytic in $\varepsilon$,
they are represented by their converging Taylor series. Then we can recover
the resolvent $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ via formula (2.15)
and obtain a converging Taylor-like series for the resolvent. Namely, given
the Taylor series

$\displaystyle\mathcal{R}_{\Gamma}(\varepsilon,\lambda)(f_{\Gamma},f_{\gamma})=\sum\limits_{l=0}^{\infty}\varepsilon^{l}u_{l}^{\Gamma},$
$\displaystyle u_{0}^{\Gamma}:=(\mathcal{H}_{0}-\lambda)^{-1}f_{\Gamma},$
(3.1)
$\displaystyle\mathcal{R}_{\gamma}(\varepsilon,\lambda)(f_{\Gamma},f_{\gamma})=\sum\limits_{l=0}^{\infty}\varepsilon^{l}u_{l}^{\gamma},\quad$
$\displaystyle
u_{0}^{\gamma}:=\sum\limits_{i=1}^{k}c_{i}(f_{\Gamma})\psi^{(i)},$

converging respectively in
$\dot{W}_{2}^{2}(\Gamma)\cap\mathfrak{C}^{2}(\Gamma)$ and
$\dot{W}_{2}^{2}(\gamma)\cap\mathfrak{C}^{2}(\gamma)$, we immediately obtain

$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f=\sum\limits_{l=0}^{\infty}\varepsilon^{l}u_{l}^{\Gamma}\oplus\mathcal{S}_{\varepsilon}^{-1}u_{l}^{\gamma},\qquad
f_{\Gamma}:=\mathcal{P}_{\Gamma}f,\qquad
f_{\gamma}:=\mathcal{S}_{\varepsilon}\mathcal{P}_{\gamma_{\varepsilon}}f,$
(3.2)

where the series converges in
$\dot{W}_{2}^{2}(\Gamma_{\varepsilon})\cap\mathfrak{C}^{2}(\Gamma_{\varepsilon})$.
The remainder in this series can be estimated by means of standard estimates
for the remainders of the Taylor series in (3.1); one just should take into
consideration the following obvious relations:

$\big{\|}f_{\Gamma}\oplus f_{\gamma}\big{\|}_{L_{2}(\Gamma)\oplus
L_{2}(\gamma)}=\left(\|f\|_{L_{2}(\Gamma)}^{2}+\varepsilon^{-1}\|f\|_{L_{2}(\gamma_{\varepsilon})}^{2}\right)^{\frac{1}{2}}\leqslant\varepsilon^{-\frac{1}{2}}\|f\|_{L_{2}(\Gamma_{\varepsilon})}.$

In particular, in this way we obtain:

$\Big{\|}(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f-(\mathcal{H}_{0}-\lambda)^{-1}f\Big{\|}_{\dot{W}_{2}^{2}(\Gamma)\cap\mathfrak{C}^{2}(\Gamma)}\leqslant
C\varepsilon^{\frac{1}{2}}\|f\|_{L_{2}(\Gamma_{\varepsilon})};$ (3.3)

hereinafter the symbol $C$ denotes various inessential constants independent
of $f$, $f_{\Gamma}$, $f_{\gamma}$ and $\varepsilon$. By straightforward
calculations it is easy to confirm that

$\|\mathcal{S}_{\varepsilon}^{-1}u_{0}^{\gamma}\|_{L_{2}(\gamma_{\varepsilon})}^{2}=\varepsilon\|u_{0}^{\gamma}\|_{L_{2}(\gamma)}^{2}\leqslant
C\varepsilon\|f_{\Gamma}\|_{L_{2}(\Gamma)}^{2}\leqslant
C\varepsilon\|f\|_{L_{2}(\Gamma_{\varepsilon})}^{2},$

and then it follows from (3.2) that

$\big{\|}(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f\big{\|}_{L_{2}(\gamma_{\varepsilon})}\leqslant
C\varepsilon^{\frac{1}{2}}\|f\|_{L_{2}(\Gamma_{\varepsilon})}.$

The latter estimate and (3.3) are exactly a convergence result similar to ones
established in papers [1] and [7]. However, in these papers the difference of
the resolvents $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f$ and
$(\mathcal{H}_{0}-\lambda)^{-1}f$ was estimated in $L_{2}$-norm only, while
our inequality (3.3) provides the bounds in stronger
$\dot{W}_{2}^{2}(\Gamma)$-norm and $\mathfrak{C}^{2}(\Gamma)$-norm. Moreover,
we have one more estimate

$\bigg{\|}(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f-\mathcal{S}_{\varepsilon}u_{0}^{\gamma}\bigg{\|}_{\mathfrak{C}(\gamma_{\varepsilon})}\leqslant
C\varepsilon^{\frac{1}{2}}\|f\|_{L_{2}(\Gamma_{\varepsilon})}.$

This estimate shows how to approximate the resolvent
$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f$ on $\gamma_{\varepsilon}$ to get a
small error term. In particular, in view of formula for $u_{0}^{\gamma}$ in
(3.1), we see that the resolvent $(\mathcal{H}_{\varepsilon}-\lambda)^{-1}f$
is approximated by
$\sum\limits_{i=1}^{k}c_{i}(f_{\Gamma})\mathcal{S}_{\varepsilon}\psi^{(i)}$ in
$\mathfrak{C}(\gamma_{\varepsilon})$-norm with an error of order
$\varepsilon^{\frac{1}{2}}$.

In comparison with [1] and [7], our convergence result has one more important
feature. This feature is related with the limiting condition at the vertex
$M_{0}$ for the operator $\mathcal{H}_{0}$, namely, with the definition of the
matrices $\mathrm{A}_{M_{0}}^{(0)}$ and $\mathrm{B}_{M_{0}}^{(0)}$. As
formulae (2.22) show, if the operator $\mathcal{H}_{\infty}$ has a virtual
level at the bottom of its essential spectrum, the matrix
$\mathrm{A}_{M_{0}}^{(0)}$ involves the matrix $\mathrm{Q}$. The presence of
the latter produces a Robin part in the vertex condition at the vertex
$M_{0}$. According to formulae (2.23), (2.24), (2.25), (2.26), the matrix
$\mathrm{Q}$ is determined by the functions
$\frac{dV_{\gamma}^{(i)}}{d\varepsilon}(\,\cdot\,,0)$ and by the matrices
$\mathrm{A}_{M}^{(1)}$, $\mathrm{B}_{M}^{(1)}$, $M\in\gamma$.

We also observe that the functions
$\frac{dV_{\gamma}^{(i)}}{d\varepsilon}(\,\cdot\,,0)$ and the matrices
$\mathrm{A}_{M}^{(1)}$, $\mathrm{B}_{M}^{(1)}$ are in fact next-to-leading
terms in the above differential expression and vertex conditions. Hence, we
conclude that the Robin part in the limiting condition at the vertex $M_{0}$
is generated solely by the above next-to-leading terms. Once such terms
vanish, only the Dirichlet and Neumann parts in the vertex condition at
$M_{0}$ can be present. In particular, this explains why only the Dirichlet
and Neumann parts appeared in the limit in [7]; the next-to-leading terms were
apriori assumed to be zero. The results of [1] allowed a Robin part in the
limiting vertex condition, but in terms of our notations this was because the
matrices $\mathrm{A}_{M}$ and $\mathrm{B}_{M}$ in (2.9) were assumed to be
independent of $\varepsilon$ and this is why the matrices
$\mathrm{A}_{M}^{(1)}$ were in general non-zero. However, the issue when and
why the Robin part appears in the limiting condition at $M_{0}$ was not
addressed in [1] and moreover, since the main result in [1] was formulated in
terms of Lagrangian planes, the presence and the origination of the Robin part
in the limiting vertex condition was not so explicit as in our case. We also
stress that our result shows how the Robin condition can be generated in the
general case by the coefficients both in the differential expression and
vertex conditions of $\mathcal{H}_{\varepsilon}$.

In order to demonstrate the difference between our convergence result and
those in [1], [7], we consider a simple model of a star graph with two fixed
edges and one small edge of the length $\varepsilon$, which was studied in
[6], see Figure 3. On the fixed edges we consider a Schrödinger operator
$-\frac{d^{2}}{dx^{2}}+V_{\pm}$ with some continuous potentials, while on the
small edge the operator reads as
$-\frac{d^{2}}{dx^{2}}+\varepsilon^{-1}V_{0}(\,\cdot\,\varepsilon)$, where
$V_{0}\in C[0,1]$ is some given function. At the vertex $M_{0}$ the Kirchhoff
condition is imposed, while the vertex $M_{\varepsilon}$ is subject to the
Dirichlet or Neumann condition. The limiting operator then is just the same
operator on finite edges, while at the vertex $M_{0}$ we have the Dirichlet
condition if the same is imposed at $M_{\varepsilon}$. But if the vertex
$M_{\varepsilon}$ is subject to the Neumann condition, then in the limit at
$M_{0}$ we have a delta-interaction, that is, the Kirchhoff condition with a
coupling:

$u\big{|}_{\mathrm{e}_{-}}(M_{0})=u\big{|}_{\mathrm{e}_{-}}(M_{0})=:u(M_{0}),\qquad
u\big{|}_{\mathrm{e}_{-}}(M_{0})+u\big{|}_{\mathrm{e}_{-}}(M_{0})=cu(M_{0}),\qquad
c:=\int\limits_{0}^{1}V_{0}(t)\,dt.$

As we see, the coupling constant $c$ is determined solely by the potential
$V_{0}$.

It is also interesting to compare our Condition (A) with similar conditions in
[1] and [7]. All these conditions are close but there are some differences. In
our terms, roughly speaking, Non-resonance condition 3.2 in [1] says that
there are no embedded eigenvalues at the bottom of the essential spectrum for
all $\varepsilon>0$. Namely, we should repeat the rescaling we employed while
obtaining the operator $\mathcal{H}_{\infty}$ in (2.8), (2.11) but without
letting $\varepsilon=0$. This gives an operator depending on $\varepsilon$ and
coinciding with $\mathcal{H}_{\infty}$ as $\varepsilon=0$. Such operator can
not have an embedded eigenvalue at the bottom of its essential spectrum for
$\varepsilon>0$. This does not imply that the same holds for $\varepsilon=0$
and this is the main difference between our Condition (A) and Non-resonance
condition in [1]. Once we make the aforementioned rescaling of small edges for
the model studied in [7], we arrive at an operator on the graph
$\gamma_{\infty}$, which is independent of $\varepsilon$ and coincides with
our operator $\mathcal{H}_{\infty}$. In this specific situation, in [7], there
were considered two cases: generic and non-generic ones. The generic case
corresponds to the absence of embedded eigenvalues. In the non-generic case,
there can be both virtual level and embedded eigenvalues at the bottom of the
essential spectrum. But since the model in [7] was quite specific, it was
possible to study the the convergence of the resolvent in the non-generic case
as well.

In the present work we do not study the case when Condition (A) is violated
and the operator has an embedded eigenvalue at the bottom of the essential
spectrum. The reason is that although our technique is sufficient to treat
such case, it involves more technical details. The nature of the expected
results also suggests that this case deserves an independent study, which we
postpone for the next work.

###  3.2 Generalizations of model

Here we discuss possible ways of extending our results to more general models.

The first possibility concerns a case, when instead of one graph
$\gamma_{\varepsilon}$, several similar small rescaled graphs
$\gamma_{\varepsilon}^{(j)}$ are glued at different vertices of the graph
$\Gamma$, see Figure 4. Each of the small graphs $\gamma_{\varepsilon}^{(j)}$
can be introduced via rescaling by means of an associated small parameter
$\varepsilon^{(i)}$; then coefficients $p_{\varepsilon}^{(j)}$,
$q_{\varepsilon}^{(j)}$, $V_{\varepsilon}^{(j)}$ of the differential
expression on each such graph $\gamma_{\varepsilon}^{(j)}$ are still defined
by formula (2.2), but now on each graph $\gamma_{\varepsilon}^{(j)}$ the
rescaling is made by means of a corresponding operator
$\mathcal{S}_{\varepsilon}^{(j)}$. We can suppose first that all parameters
$\varepsilon^{(j)}$ are analytic functions in a single small parameter
$\varepsilon$. In this situation, all our results remain valid with obvious
minor changes; in particular, the form of the limiting vertex condition at a
vertex $M_{0}^{(j)}$, to which the graph $\gamma_{\varepsilon}^{(j)}$ is
attached, is determined only by the graph $\gamma_{\varepsilon}^{(j)}$ and
there is no influence by the neighbouring ones.

Figure 3: Star graph with a small edge and the limiting graph without the
small edge.

A more general situation is when in the same setting we assume that the
parameters $\varepsilon^{(j)}$ are independent and all coefficients in the
differential expression and vertex conditions of the operator
$\mathcal{H}_{\varepsilon}$ depend analytically in all $\varepsilon^{(j)}$. In
such case, we first of all should assume additionally that for each attached
graph $\gamma_{\varepsilon}^{(j)}$ and each vertex
$M\in\gamma_{\infty}^{(j)}$, identities similar to (2.9) hold:

$\displaystyle\varepsilon^{(j)}\tilde{\mathrm{E}}_{M}(\varepsilon^{(j)})\mathrm{A}_{M}(\varepsilon)=\big{(}\varepsilon^{(j)}\big{)}^{\mu}\big{(}\mathrm{A}_{M}^{(0)}+\varepsilon^{(j)}\mathrm{A}_{M}^{(1)}\big{)}+O(|\varepsilon|^{2+\mu}),$
$\displaystyle\tilde{\mathrm{E}}_{M}(\varepsilon^{(j)})\mathrm{B}_{M}(\varepsilon)=\big{(}\varepsilon^{(j)}\big{)}^{\mu}\big{(}\mathrm{B}_{M}^{(0)}+\varepsilon^{(j)}\mathrm{B}_{M}^{(1)}\big{)}+O(|\varepsilon|^{2+\mu}),$

where $\mu=0$ if $\mathrm{B}_{M}(0)\neq 0$ and $\mu=1$ if
$\mathrm{B}_{M}(0)=0$ and
$\varepsilon=(\varepsilon^{(1)},\varepsilon^{(2)},\ldots)$. For the
coefficients of the differential expression we should assume that

$p_{\gamma^{(j)}}(\,\cdot\,,\varepsilon)=p_{\gamma^{(j)}}(\,\cdot\,,0)+\varepsilon^{(j)}\frac{dp_{\gamma^{(j)}}}{d\varepsilon^{(j)}}(\,\cdot\,,0)+O(|\varepsilon|^{2}),$
(3.4)

and the same should hold for the other coefficients. Then the corresponding
matrices $Q^{(j)}$ are to be calculated for each graph $\gamma^{(j)}$ by
formulae (2.23), (2.24), (2.25), (2.26). If for some $j$ the matrix $Q^{(j)}$
turns out to have a zero eigenvalue, then assumptions (2.9), (3.4) should be
replaced by stricter ones:

$\displaystyle\varepsilon^{(j)}\tilde{\mathrm{E}}_{M}(\varepsilon^{(j)})\mathrm{A}_{M}(\varepsilon)=\big{(}\varepsilon^{(j)}\big{)}^{\mu}\big{(}\mathrm{A}_{M}^{(0)}+\varepsilon^{(j)}\mathrm{A}_{M}^{(1)}\big{)}+O(|\varepsilon|^{2+\mu}),$
$\displaystyle\tilde{\mathrm{E}}_{M}(\varepsilon^{(j)})\mathrm{B}_{M}(\varepsilon)=\big{(}\varepsilon^{(j)}\big{)}^{\mu}\big{(}\mathrm{B}_{M}^{(0)}+\varepsilon^{(j)}\mathrm{B}_{M}^{(1)}\big{)}+O(|\varepsilon|^{2+\mu}),$
$\displaystyle
p_{\gamma^{(j)}}(\,\cdot\,,\varepsilon)=p_{\gamma^{(j)}}(\,\cdot\,,0)+\varepsilon^{(j)}\frac{dp_{\gamma^{(j)}}}{d\varepsilon^{(j)}}(\,\cdot\,,0)+\frac{\big{(}\varepsilon^{(j)}\big{)}^{2}}{2}\frac{d^{2}p_{\gamma^{(j)}}}{d(\varepsilon^{(j)})^{2}}(\,\cdot\,,0)+O(|\varepsilon|^{3}).$

The above conditions mean that the coefficients in the differential expression
and vertex conditions can depend on all small parameters, but the leading
terms in their Taylor should be as specified above. Under such assumptions,
all our results remain true and now the analyticity holds with respect to all
parameters $\varepsilon^{(i)}$. Once these assumptions fail, in the general
situation, the result on analyticity is no longer valid. The reason for this
situation is in fact a well-known phenomenon in the multi-parametric
perturbation theory, which says that the eigenvalues of a matrix analytic with
respect to more than one small parameter are not necessary analytic, see [11,
Ch. II, Sect. 5.7]. The same also concerns the case, when several graphs
$\gamma^{(j)}$ are rescaled by means of different independent small parameters
and are glued to the same vertex in the graph $\Gamma$. In such general
situation one can expect only a convergence result as one proved in [1]. Of
course, this does not exclude the situations, when for some particular graphs
and specific choices of small edges, differential expression and vertex
conditions, the resolvent has the analyticity property, but this can happen
only due to some specific features of the considered models.

Figure 4: Graph $\Gamma$ with several glued small graphs

One more important way of extending our model is to assume that all
coefficients in the differential expression and the vertex conditions are not
analytic in $\varepsilon$ but either infinite differentiable or just possess
power asymptotic expansions in $\varepsilon$. In these cases all our results,
including ones just discussed above for several glued small graphs, remain
true with appropriate modifications. Namely, if the mentioned coefficients are
infinite differentiable in $\varepsilon$, then the statement on the
analyticity property is to be replaced by the infinite differentiability in
$\varepsilon$ for the operators $\mathcal{R}_{\Gamma}$ and
$\mathcal{R}_{\gamma}$. If the coefficients in the differential expression and
the vertex conditions just possess power asymptotic expansions in
$\varepsilon$, then the operators $\mathcal{R}_{\Gamma}$ and
$\mathcal{R}_{\gamma}$ also have power asymptotic expansions in $\varepsilon$.
The way of proving such results is very simple and is based on the analyticity
case treated above. Namely, we represent the matrices in the vertex conditions
as

$\mathrm{A}_{M}(\varepsilon)=\mathrm{A}_{M}^{(0)}+\varepsilon\mathrm{A}_{M}^{(1)}+\varepsilon^{2}\mathrm{A}_{M}^{(2)}+\varepsilon^{3}\breve{\mathrm{A}}_{M}(\varepsilon),\qquad\mathrm{B}_{M}(\varepsilon)=\mathrm{B}_{M}^{(0)}+\varepsilon\mathrm{B}_{M}^{(1)}+\varepsilon^{2}\mathrm{B}_{M}^{(2)}+\varepsilon^{3}\breve{\mathrm{B}}_{M}(\varepsilon),$
(3.5)

where the matrices $\breve{\mathrm{A}}_{M}(\varepsilon)$ and
$\breve{\mathrm{B}}_{M}(\varepsilon)$ are either infinitely differentiable in
$\varepsilon$ or have power in $\varepsilon$ asymptotic expansions. For the
coefficients of the differentiable expression, a similar representation is to
be employed. The terms $\breve{\mathrm{A}}_{M}$ and $\breve{\mathrm{B}}_{M}$
are to be treated as frozen coefficients independent of $\varepsilon$ and the
matrices in (3.5) should be treated as second order polynomials in
$\varepsilon$. Then it is possible to track the dependence on
$\breve{\mathrm{A}}_{M}$ and $\breve{\mathrm{B}}_{M}(\varepsilon)$ and on
similar terms for the coefficients in the differential expression in the proof
of Theorem 2.1. As a result, then one can see that the operators
$\mathcal{R}_{\Gamma}$ and $\mathcal{R}_{\gamma}$ as well as the resolvent
$(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ are represented by convergent
series (3.1) and (3.2), but now with coefficients depending on $\varepsilon$.
These coefficients are either infinitely differentiable in $\varepsilon$ or
possess power in $\varepsilon$ asymptotic expansions. In the first case we
conclude immediately that all the operators in question are infinitely
differentiable in $\varepsilon$, while in the other case we can easily find
the asymptotic expansions for these operators from their series.

##  4 Self-adjointness

Our main result includes a statement on the self-adjointness of both perturbed
and limiting operators and in the proofs, we shall also use the same property
for certain auxiliary operators. In this section we first establish a
criterion of the self-adjointness for a general operator. Then we shall employ
it to prove the same property for the operators $\mathcal{H}_{0}$ and
$\mathcal{H}_{\gamma}$.

###  4.1 Criterion of self-adjointness

Let $\Xi$ be a finite metric graph having no isolated vertices; the lengths of
its edges can be both finite or infinite. On each edge we fix arbitrarily an
orientation and as above, by $d(M)$ we denote the degree of a vertex
$M\in\Xi$.

We consider an unbounded operator $\mathcal{H}$ in $L_{2}(\Xi)$ with the
differential expression

$\hat{\mathcal{H}}u:=-\frac{d\ }{dx}p\frac{du}{dx}+\mathrm{i}\left(\frac{d\
}{dx}(qu)+q\frac{du}{dx}\right)+Vu$

subject to the vertex conditions

$\mathrm{A}_{M}\mathcal{U}_{M}(u)+\mathrm{B}_{M}\mathcal{U}_{M}^{\prime}(u)=0\quad\text{at
each vertex $M$}.$ (4.1)

Here $p,\,q\in W_{\infty}^{1}(\Xi)$, $V\in L_{2}(\Xi)$ are real-valued
functions, while $\mathrm{A}_{M}$ and $\mathrm{B}_{M}$ are some matrices of
size $d(M)\times d(M)$. The ellipticity condition is assumed: $p\geqslant c>0$
uniformly on $\Xi$. The domain of the operator $\mathcal{H}$ consists of the
functions in $\dot{W}_{2}^{2}(\Xi)$ obeying vertex condition (4.1).

By $\Pi_{M}$, $\Theta_{M}$ we denote the following diagonal matrices with real
entries:

$\Pi_{M}:=\operatorname{diag}\big{\\{}\nu_{i}(M)p\big{|}_{\mathrm{e}_{i}(M)}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d(M)},\qquad\Theta_{M}:=\operatorname{diag}\big{\\{}\nu_{i}(M)q\big{|}_{\mathrm{e}_{i}(M)}(M,\varepsilon)\big{\\}}_{i=1,\ldots,d(M)},$

where $\mathrm{e}_{i}(M)$ are the edges incident to the vertex $M$ and the
numbers $\nu_{i}(M)$ are defined in the same way as in (2.6).

The main statement of this subsection is the following lemma.

######  Lemma 4.1.

The operator $\mathcal{H}$ is self-adjoint if and only if for each vertex
$M\in\Xi$ the matrices $\mathrm{A}_{M}$ and $\mathrm{B}_{M}$ satisfy the rank
condition

$\operatorname{rank}(\mathrm{A}_{M}\ \mathrm{B}_{M})=d(M)$ (4.2)

and the matrix

$\mathrm{A}_{M}\Pi_{M}^{-1}\mathrm{B}_{M}^{*}+\mathrm{i}\mathrm{B}_{M}\Pi_{M}^{-1}\Theta_{M}\Pi_{M}^{-1}\mathrm{B}_{M}^{*}$
(4.3)

is self-adjoint. Under these restrictions, vertex condition (4.1) can be
equivalently rewritten as

$\mathrm{i}(\mathrm{U}_{M}-\mathrm{I}_{d(M)})\mathcal{U}_{M}(u)+(\mathrm{U}_{M}+\mathrm{I}_{d(M)})\big{(}\Pi_{M}\mathcal{U}_{M}^{\prime}(u)-\mathrm{i}\Theta_{M}\mathcal{U}_{M}(u)\big{)}=0,$
(4.4)

where the matrix

$\mathrm{U}_{M}:=-\big{(}\mathrm{A}_{M}+\mathrm{i}\mathrm{B}_{M}\Pi_{M}^{-1}(\Theta_{M}-\mathrm{I}_{d(M)})\big{)}^{-1}\big{(}\mathrm{A}_{M}+\mathrm{i}\mathrm{B}_{M}\Pi_{M}^{-1}(\Theta_{M}+\mathrm{I}_{d(M)})\big{)}$

is well-defined and is unitary.

###### Proof.

In a particular case $p\equiv 1$, $q\equiv 0$, $V\equiv 0$, the lemma states a
well-known result, see Theorem 1.1.4 in [2, Ch. 1, Sect. 1.4.]. In fact, the
proof of this theorem can be adapted also for our more general case up to some
minor changes. Below we describe how to do this.

First of all we observe that vertex condition (4.1) can be equivalently
rewritten as

$\displaystyle\tilde{\mathrm{A}}_{M}\mathcal{U}_{M}(u)+\tilde{\mathrm{B}}_{M}\big{(}\Pi_{M}\mathcal{U}_{M}^{\prime}(u)-\mathrm{i}\Theta_{M}\mathcal{U}_{M}(u)\big{)}=0,$
(4.5)
$\displaystyle\tilde{\mathrm{A}}_{M}:=\mathrm{A}_{M}+\mathrm{i}\mathrm{B}_{M}\Pi_{M}^{-1}\Theta_{M},\qquad\tilde{\mathrm{B}}_{M}:=\mathrm{B}_{M}\Pi_{M}^{-1}.$
(4.6)

Let $u$ be an arbitrary function from the domain of the operator $\mathcal{H}$
and $v$ be an arbitrary function from
$\bigoplus\limits_{\mathrm{e}\in\Xi}C^{\infty}(\mathrm{e})$ vanishing outside
a small neighbourhood of an arbitrarily fixed vertex $M\in\Xi$. Integrating
twice by parts, we get:

$\displaystyle(\mathcal{H}u,v)_{L_{2}(\Xi)}=$
$\displaystyle\big{(}\Pi_{M}\mathcal{U}_{M}^{\prime}(u),\mathcal{U}_{M}(v)\big{)}_{\mathds{C}^{d(M)}}-\big{(}\mathcal{U}_{M}(u),\Pi_{M}\mathcal{U}_{M}^{\prime}(v)\big{)}_{\mathds{C}^{d(M)}}+(u,\hat{\mathcal{H}}v)_{L_{2}(\Xi)}$
$\displaystyle=$
$\displaystyle(\mathsf{F}_{M}^{\prime},\mathsf{G}_{M})_{\mathds{C}^{d(M)}}-(\mathsf{F}_{M},\mathsf{G}_{M}^{\prime})_{\mathds{C}^{d(M)}}+(u,\hat{\mathcal{H}}v)_{L_{2}(\Xi)},$

where

$\displaystyle\mathsf{F}_{M}^{\prime}:=\Pi_{M}\mathcal{U}_{M}^{\prime}(u)-\mathrm{i}\Theta_{M}\mathcal{U}_{M}(u),\qquad\mathsf{G}_{M}:=\mathcal{U}_{M}(v),$
$\displaystyle\mathsf{F}_{M}:=\mathcal{U}_{M}(u),\qquad\mathsf{G}_{M}^{\prime}:=\Pi_{M}\mathcal{U}_{M}^{\prime}(v)-\mathrm{i}\Theta_{M}\mathcal{U}_{M}(v).$

Hence, the function $v$ is in the domain of the adjoint operator
$\mathcal{H}^{*}$ if

$(\mathsf{F}_{M}^{\prime},\mathsf{G}_{M})_{\mathds{C}^{d(M)}}-(\mathsf{F}_{M},\mathsf{G}_{M}^{\prime})_{\mathds{C}^{d(M)}}=0.$

The latter identity is exactly equation (1.4.16) in the proof of Theorem 1.1.4
in [2, Ch. 1, Sect. 1.4] but with different $\mathsf{F}_{M}$,
$\mathsf{F}_{M}^{\prime}$, $\mathsf{G}_{M}$, $\mathsf{G}_{M}^{\prime}$.
However, the specific form of these quantities were not important in that
proof in [2]. This is why the arguing from [2] can be reproduced literally
with above introduced $\mathsf{F}_{M}$, $\mathsf{F}_{M}^{\prime}$,
$\mathsf{G}_{M}$, $\mathsf{G}_{M}^{\prime}$; as the matrices $A$ and $B$ used
in that proof, our matrices $\tilde{\mathrm{A}}_{M}$ and
$\tilde{\mathrm{B}}_{M}$ serve. Rank condition (4.2) is equivalent to similar
condition (1.4.10) in [2, Ch. 1, Sect. 1.4] since thanks to the non-degeneracy
of the matrix $\Pi_{M}^{-1}$ we have

$\displaystyle\operatorname{rank}(\tilde{\mathrm{A}}_{M}\
\;\tilde{\mathrm{B}}_{M})=$
$\displaystyle\operatorname{rank}\big{(}\mathrm{A}_{M}+\mathrm{i}\mathrm{B}_{M}\Pi_{M}^{-1}\Theta_{M}\
\;\mathrm{B}_{M}\Pi_{M}^{-1}\big{)}$ $\displaystyle=$
$\displaystyle\operatorname{rank}\big{(}\mathrm{A}_{M}\
\;\mathrm{B}_{M}\Pi_{M}^{-1}\big{)}=\operatorname{rank}\big{(}\mathrm{A}_{M}\
\;\mathrm{B}_{M}\big{)}=d(M).$

Condition (1.4.11) for the matrices $A$ and $B$ from Theorem 1.1.4 in [2, Ch.
1, Sect. 1.4.] is exactly our condition (4.3), while equivalent formulation
(4.4) of the vertex condition are implied immediately by similar formulations
(1.4.13) in [2]. The proof is complete. ∎

###  4.2 Self-adjointness of operators $\mathcal{H}_{0}$ and
$\mathcal{H}_{\infty}$

In this subsection we prove that under the assumptions of Theorem 2.1 the
operators $\mathcal{H}_{0}$ and $\mathcal{H}_{\infty}$ are self-adjoint. In
order to prove this statement for the operator $\mathcal{H}_{0}$, we shall
need the self-adjointness of the matrix $\mathrm{Q}$ involved in the matrix
$\mathrm{A}_{M_{0}}^{(0)}$, see (2.22). This fact is provided by the following
lemma, which will be independently proved later in a separate subsection, see
Subsection 4.3.

######  Lemma 4.2.

The matrix $\mathrm{Q}$ is self-adjoint.

Let us show that the operators $\mathcal{H}_{0}$ and $\mathcal{H}_{\infty}$
are self-adjoint. The matrix in (2.5) is self-adjoint as $\varepsilon=0$ for
each vertex $M\in\Gamma$, $M\neq M_{0}$. Hence, the matrices
$\mathrm{A}_{M}^{(0)}$, $\mathrm{B}_{M}^{(0)}$ satisfy condition (4.2), (4.3)
for each vertex $M\in\Gamma$, $M\neq M_{0}$. Thanks to the self-adjointness of
the matrix $\mathrm{Q}$ stated in Lemma 4.2, the self-adjointness of the
matrix in (4.3) corresponding to the matrices $\mathrm{A}_{M_{0}}^{(0)}$ and
$\mathrm{B}_{M_{0}}^{(0)}$ is confirmed by simple straightforward
calculations. In view of the non-degeneracy of the matrix
$\Pi_{\Gamma,M_{0}}(0)$, condition (4.2) corresponding to the same matrices is
checked as follows:

$\displaystyle\operatorname{rank}\big{(}\mathrm{A}_{M_{0}}^{(0)}\
\mathrm{B}_{M_{0}}^{(0)}\big{)}=$
$\displaystyle\operatorname{rank}\Big{(}\mathrm{A}_{M_{0}}^{(0)}+\mathrm{i}\mathrm{B}_{M_{0}}^{(0)}\Pi_{\Gamma,M_{0}}^{-1}(0)\Theta_{\Gamma,M_{0}}(0)\Psi\quad-\mathrm{B}_{M_{0}}^{(0)}\Pi_{\Gamma,M_{0}}^{-1}(0)\Psi\Big{)}$
$\displaystyle=$
$\displaystyle\operatorname{rank}\left(\begin{pmatrix}\mathrm{Q}&0\\\
0&\mathrm{I}_{d_{0}-k}\end{pmatrix}\quad\begin{pmatrix}\mathrm{I}_{k}&0\\\
0&0\end{pmatrix}\right)=d_{0}.$

Hence, we can apply Lemma 4.1 and we see that the operator $\mathcal{H}_{0}$
is self-adjoint.

We proceed to proving the self-adjointness of the operator
$\mathcal{H}_{\infty}$. In view of the assumption on the matrices
$\mathrm{A}_{M}(0)$ and $\mathrm{B}_{M}(0)$ made in Subsection 2.1, see the
explanation after (2.4), we see easily that rank condition (2.7) implies the
same for the matrices $\mathrm{A}_{M}^{(0)}$ and $\mathrm{B}_{M}^{(0)}$
defined in (2.10). Then we multiply the matrix in (2.5) by $\varepsilon$ and
then by $\tilde{\mathrm{E}}_{M}(\varepsilon)$ from left and right keeping its
self-adjointness and then we pass to the limit as $\varepsilon\to+0$. By
identities (2.9) we then arrive to the self-adjointness of the matrix

$\mathrm{A}_{M}^{(0)}\Pi_{\gamma,M}^{-1}(0)\big{(}\mathrm{B}_{M}^{(0)}\big{)}^{*}+\mathrm{i}\mathrm{B}_{M}^{(0)}\Pi_{\gamma,M}^{-1}(0)\Theta_{\gamma,M}(0)\Pi_{\gamma,M}^{-1}(0)\big{(}\mathrm{B}_{M}^{(0)}\big{)}^{*}.$

Applying now Lemma 4.1, we conclude that the operator $\mathcal{H}_{\infty}$
is self-adjoint.

###  4.3 Self-adjointness of $\mathrm{Q}$

In this section we prove Lemma 4.2. The proof is based on obtaining certain
representations for the operators $\mathcal{L}_{M}$ and $\mathcal{E}_{M}$ used
in (2.25), (2.26), (2.27), which will imply then the desired self-adjointness.
These representations are based on an appropriate transformation of vertex
conditions (2.4) at $M\in\gamma_{\varepsilon}$ and we begin our proof with
describing them.

Given a vertex $M\in\gamma_{\varepsilon}$, including vertices $M=M_{j}$,
$j=1,\ldots,n$, we rescale the variables as $\xi=x\varepsilon^{-1}$ passing in
fact to the graph $\gamma_{\infty}$. Then vertex condition (2.4) becomes

$\varepsilon\mathrm{A}_{M}(\varepsilon)\mathcal{U}_{M}(u)+\mathrm{B}_{M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)=0,$
(4.7)

where the vectors $\mathcal{U}_{M}(u)$ and $\mathcal{U}_{M}^{\prime}(u)$ are
introduced as above but are treated in terms of the variable $\xi$. According
Lemma 4.1, at each vertex $M\in\gamma_{\infty}$ the above vertex condition can
be equivalently rewritten as

$\mathrm{i}(\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)})\mathcal{U}_{M}(u)+(\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)})\big{(}\Pi_{\gamma,M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)-\mathrm{i}\Theta_{\gamma,M}(\varepsilon)\mathcal{U}_{M}(u)\big{)}=0,$
(4.8)

where, we recall, the matrices $\Pi_{\gamma,M}$ and $\Theta_{\gamma,M}$ were
defined in (2.29), and

$\displaystyle\mathrm{U}_{M}(\varepsilon):=-$
$\displaystyle\Big{(}\varepsilon\mathrm{A}_{M}(\varepsilon)+\mathrm{i}\mathrm{B}_{M}(\varepsilon)\Pi_{\gamma,M}^{-1}(\varepsilon)\big{(}\Theta_{\gamma,M}M(\varepsilon)-\mathrm{I}_{d(M)}\big{)}\Big{)}^{-1}$
(4.9)
$\displaystyle\cdot\Big{(}\varepsilon\mathrm{A}_{M}(\varepsilon)+\mathrm{i}\mathrm{B}_{M}(\varepsilon)\Pi_{\gamma,M}^{-1}(\varepsilon)\big{(}\Theta_{\gamma,M}(\varepsilon)+\mathrm{I}_{d(M)}\big{)}\Big{)}$

is an unitary matrix.

######  Lemma 4.3.

For all $M\in\Gamma$, $M\neq M_{0}$ and all $M\in\gamma_{\infty}$, the matrix
$\mathrm{U}_{M}(\varepsilon)$ is analytic in $\varepsilon$.

###### Proof.

In view of rank condition (2.7) and formulae (2.9) and owing to the
analyticity of $\mathrm{A}_{M}(\varepsilon)$ and $\mathrm{B}_{M}(\varepsilon)$
in $\varepsilon$, the matrix $\mathrm{U}_{M}(\varepsilon)$ is meromorphic in
$\varepsilon$. At the same time, by the unitarity of
$\mathrm{U}_{M}(\varepsilon)$ we have
$\|\mathrm{U}_{M}(\varepsilon)\mathrm{x}\|_{\mathds{C}^{d(M)}}=\|\mathrm{x}\|_{\mathds{C}^{d(M)}}$
for all $\mathrm{x}\in\mathds{C}^{d(M)}$ and sufficiently small $\varepsilon$.
Hence, the matrix $\mathrm{U}_{M}(\varepsilon)$ is bounded uniformly in
$\varepsilon$ and can not have poles. Therefore, it is analytic in
$\varepsilon$. The proof is complete. ∎

Thanks to the unitarity of the matrix $\mathrm{U}_{M}(\varepsilon)$, by the
results in [11, Ch. II, Sec. 4.6], the eigenvalues and the associated
orthonormalized in $\mathds{C}^{d(M)}$ vectors of the matrix
$\mathrm{U}_{M}(\varepsilon)$ are analytic in $\varepsilon$. By
$\mathrm{P}_{M}(\varepsilon)$ we denote the total projector in
$\mathds{C}^{d(M)}$ onto the eigenspace associated with the eigenvalues of the
matrix $\mathrm{U}_{M}(\varepsilon)$ converging to $-1$ as $\varepsilon\to+0$.
We also let
$\mathrm{P}_{M,\bot}(\varepsilon):=\mathrm{I}_{d(M)}-\mathrm{P}_{M}(\varepsilon)$.
By the aforementioned analyticity in $\varepsilon$ of the eigenvalues and the
eigenvectors of $\mathrm{U}_{M}(\varepsilon)$, the projectors
$\mathrm{P}_{M}(\varepsilon)$ and $\mathrm{P}_{M,\bot}(\varepsilon)$ are also
analytic in $\varepsilon$.

We apply the projectors $\mathrm{P}_{M}(\varepsilon)$ and
$\mathrm{P}_{M,\bot}(\varepsilon)$ to vertex condition (4.8) and we rewrite it
equivalently as

$\displaystyle\mathrm{P}_{M}(\varepsilon)\mathcal{U}_{M}(u)+\mathrm{K}_{M}(\varepsilon)\big{(}\Pi_{\gamma,M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)-\mathrm{i}\Theta_{\gamma,M}(\varepsilon)\mathcal{U}_{M}(u)\big{)}=0,$
(4.10)
$\displaystyle\mathrm{P}_{M,\bot}(\varepsilon)\Pi_{\gamma,M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)+\big{(}\mathrm{K}_{M,\bot}(\varepsilon)-\mathrm{i}\mathrm{P}_{M,\bot}(\varepsilon)\Theta_{\gamma,M}(\varepsilon)\big{)}\mathcal{U}_{M}(u)=0,$

where

$\displaystyle\mathrm{K}_{M}(\varepsilon):=-\mathrm{i}\big{(}\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)}\big{)}^{-1}\mathrm{P}_{M}(\varepsilon)\big{(}\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)}\big{)}$
(4.11)
$\displaystyle\mathrm{K}_{M,\bot}(\varepsilon):=\mathrm{i}\big{(}\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)}\big{)}^{-1}\mathrm{P}_{M,\bot}(\varepsilon)\big{(}\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)}\big{)}.$

It is obvious that both matrices $\mathrm{K}_{M}(\varepsilon)$ and
$\mathrm{K}_{M,\bot}(\varepsilon)$ are well-defined and analytic in
$\varepsilon$. Employing the unitarity of the matrix
$\mathrm{U}_{M}(\varepsilon)$, it is straightforward to confirm that both
matrices $\mathrm{K}_{M}(\varepsilon)$ and $\mathrm{K}_{M,\bot}(\varepsilon)$
are self-adjoint. We also observe that

$\mathrm{U}_{M}(0)=\mathrm{U}_{M}^{(0)},\qquad\mathrm{P}_{M}(0)=\mathrm{P}_{M}^{(0)},\qquad\mathrm{K}_{M}(0)=0.$

Since the operator $\mathrm{P}_{M}(\varepsilon)$ is analytic in $\varepsilon$,
by the results in [11, Ch. 2, Sect. 4.2], there exists a transforming function
for this projector, namely, an invertible operator
$\mathrm{S}_{M}(\varepsilon)$ in $\mathds{C}^{d(M)}$, analytic in small
$\varepsilon$ together with its inverse operator such that

$\mathrm{S}_{M}(0)=\mathrm{I}_{d(M)},\qquad\mathrm{S}_{M}^{-1}(\varepsilon)\mathrm{P}_{M}^{(0)}\mathrm{S}_{M}(\varepsilon)=\mathrm{P}_{M}(\varepsilon),\qquad\mathrm{S}_{M}^{-1}(\varepsilon)\mathrm{P}_{M,\bot}^{(0)}\mathrm{S}_{M}(\varepsilon)=\mathrm{P}_{M,\bot}(\varepsilon)$
(4.12)

The inverse operator $\mathrm{S}_{M}^{-1}(\varepsilon)$ is given explicitly by
formula (4.18) in [11, Ch. 2, Sect. 4.2] and this implies

$\mathrm{S}_{M}(\varepsilon)=\big{(}\mathrm{P}_{M}(\varepsilon)\mathrm{P}_{M}^{(0)}+\mathrm{P}_{M,\bot}(\varepsilon)\mathrm{P}_{M,\bot}^{(0)}\big{)}^{-1}\Big{(}\mathrm{I}_{d(M)}-\big{(}\mathrm{P}_{M}(\varepsilon)-\mathrm{P}_{M}^{(0)}\big{)}^{2}\Big{)}^{\frac{1}{2}}.$
(4.13)

In view of formulae (4.12), (4.13), vertex condition (4.10) can be
equivalently rewritten as

$\displaystyle\mathrm{P}_{M}^{(0)}\mathrm{S}_{M}(\varepsilon)\mathcal{U}_{M}(u)+\mathrm{S}_{M}(\varepsilon)\mathrm{K}_{M}(\varepsilon)\big{(}\Pi_{\gamma,M}(\varepsilon)\mathrm{U}_{M}^{\prime}(\varepsilon)-\mathrm{i}\Theta_{\gamma,M}(\varepsilon)\mathcal{U}_{M}(u)\big{)}=0,$
$\displaystyle\mathrm{P}_{M,\bot}^{(0)}\mathrm{S}_{M}(\varepsilon)\Pi_{\gamma,M}M(\varepsilon)\mathcal{U}_{M}^{\prime}(u)+\mathrm{S}_{M}(\varepsilon)\big{(}\mathrm{K}_{M,\bot}(\varepsilon)-\mathrm{i}\mathrm{P}_{M,\bot}(\varepsilon)\Theta_{\gamma,M}(\varepsilon)\big{)}\mathcal{U}_{M}(u)=0.$

Since the first terms in the above identities belong respectively to the
spaces $\mathrm{P}_{M}^{(0)}\mathds{C}^{d(M)}$ and
$\mathrm{P}_{M,\bot}^{(0)}\mathds{C}^{d(M)}$, the same is true for the other
terms in these identities. Therefore, we can apply the projector
$\mathrm{P}_{M,\bot}^{(0)}$ to these identities and we equivalently rewrite
them as

$\mathrm{P}_{M}^{(0)}\tilde{\mathrm{S}}_{M}(\varepsilon)\mathcal{U}_{M}(u)+\tilde{\mathrm{K}}_{M}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)=0,\qquad\mathrm{P}_{M,\bot}^{(0)}\tilde{\mathrm{S}}_{M,\bot}(\varepsilon)\mathcal{U}_{M}^{\prime}(u)+\tilde{\mathrm{K}}_{M,\bot}(\varepsilon)\mathcal{U}_{M}(u)=0,$
(4.14)

where

$\displaystyle\tilde{\mathrm{S}}_{M}(\varepsilon):=\mathrm{S}_{M}(\varepsilon)\big{(}\mathrm{I}_{d(M)}-\mathrm{i}\mathrm{K}_{M}(\varepsilon)\Theta_{\gamma,M}(\varepsilon)\big{)},\qquad\tilde{\mathrm{K}}_{M}(\varepsilon):=\mathrm{P}_{M}^{(0)}\mathrm{S}_{M}(\varepsilon)\mathrm{K}_{M}(\varepsilon)\Pi_{\gamma,M}(\varepsilon),$
(4.15)
$\displaystyle\tilde{\mathrm{S}}_{M,\bot}(\varepsilon):=\mathrm{S}_{M}(\varepsilon)\Pi_{\gamma,M}(\varepsilon),\qquad\tilde{\mathrm{K}}_{M,\bot}(\varepsilon):=\mathrm{P}_{M,\bot}^{(0)}\mathrm{S}_{M}(\varepsilon)\big{(}\mathrm{K}_{M,\bot}(\varepsilon)-\mathrm{i}\mathrm{P}_{M,\bot}(\varepsilon)\Theta_{\gamma,M}(\varepsilon)\big{)}.$

The matrices $\mathrm{S}_{M}(\varepsilon)$,
$\tilde{\mathrm{S}}_{M}(\varepsilon)$,
$\tilde{\mathrm{S}}_{M,\bot}(\varepsilon)$,
$\tilde{\mathrm{K}}_{M}(\varepsilon)$,
$\tilde{\mathrm{K}}_{M,\bot}(\varepsilon)$, $\Pi_{\gamma,M}(\varepsilon)$,
$\Theta_{\gamma,M}(\varepsilon)$ are analytic in $\varepsilon$ and by
$\mathrm{S}_{M,1}$, $\tilde{\mathrm{S}}_{M,1}$,
$\tilde{\mathrm{S}}_{M,\bot,1}$, $\tilde{\mathrm{K}}_{M,1}$,
$\tilde{\mathrm{K}}_{M,\bot,1}$, $\Pi_{\gamma,M,1}(\varepsilon)$,
$\Theta_{\gamma,M,1}(\varepsilon)$ we denote their derivatives in
$\varepsilon$ taken at $\varepsilon=0$. For instance,

$\tilde{S}_{M,1}:=\frac{d\tilde{\mathrm{S}}_{M}}{d\varepsilon}(0),\qquad\tilde{S}_{M,\bot,1}:=\frac{d\tilde{\mathrm{S}}_{M,\bot}}{d\varepsilon}(0).$

In order to prove Lemma 4.2, we shall need several properties of the
introduced matrices, which are summarized in the following auxiliary
statement.

######  Lemma 4.4.

For each vertex $M\in\gamma_{\infty}$, the identities hold:

$\displaystyle\begin{aligned}
&\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1}=\mathrm{P}_{M,1}\mathrm{P}_{M}^{(0)},\qquad\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1}\mathrm{P}_{M,\bot}^{(0)}=0,\\\
&\mathrm{P}_{M}^{(0)}\mathrm{P}_{M,1}=\mathrm{P}_{M,1}\mathrm{P}_{M,\bot}^{(0)},\qquad\mathrm{P}_{M}^{(0)}\mathrm{P}_{M,1}\mathrm{P}_{M}^{(0)}=0,\qquad\mathrm{P}_{M,1}^{*}=\mathrm{P}_{M,1},\end{aligned}$
(4.16) $\displaystyle\begin{aligned}
\mathrm{S}_{M,1}=\mathrm{P}_{M,1}\mathrm{P}_{M,\bot}^{(0)}-\mathrm{P}_{M,1}\mathrm{P}_{M}^{(0)}=\mathrm{P}_{M,1}\mathrm{P}_{M,\bot}^{(0)}-\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1},\end{aligned}$
(4.17)
$\displaystyle\mathrm{P}_{M}^{(0)}\mathrm{K}_{M,1}\mathrm{P}_{M,1}^{(0)}=0.$
(4.18)

###### Proof.

Formulae (4.16) are obtained immediately via calculating the coefficients at
$\varepsilon^{1}$ in the obvious identities:

$\big{(}\mathrm{P}_{M,\bot}(\varepsilon)\big{)}^{2}=\mathrm{P}_{M,\bot}(\varepsilon),\qquad\big{(}\mathrm{P}_{M}(\varepsilon)\big{)}^{2}=\mathrm{P}_{M}(\varepsilon),\qquad\big{(}\mathrm{P}_{M,\bot}(\varepsilon)\big{)}^{*}=\mathrm{P}_{M,\bot}(\varepsilon).$

Formula (4.17) can be confirmed by expanding the right hand side in (4.13)
into the power series in $\varepsilon$, calculating the coefficient at
$\varepsilon^{1}$, and employing then (4.16). It follows from the formula for
$\mathrm{K}_{M}(\varepsilon)$ in (4.11) that

$\displaystyle\mathrm{i}(\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)})\mathrm{P}_{M}(\varepsilon)\mathrm{K}_{M}(\varepsilon)\mathrm{P}_{M,\bot}(\varepsilon)=\mathrm{P}_{M}(\varepsilon)(\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)})\mathrm{P}_{M,\bot}(\varepsilon)=0.$

Expanding this identity into the power series in $\varepsilon$ and calculating
the coefficient at $\varepsilon^{1}$, we arrive at (4.18). The proof is
complete. ∎

Now we are in position to prove Lemma 4.2.

###### Proof of Lemma 4.2.

According definition (4.9) of $\mathrm{U}_{M}(\varepsilon)$, formulae (4.5),
(4.6) applied with

$\tilde{\mathrm{A}}_{M}=\tilde{\mathrm{A}}_{M}(\varepsilon):=\varepsilon\mathrm{A}_{M}(\varepsilon)+\mathrm{i}\mathrm{B}_{M}(\varepsilon)\Pi_{\gamma,M}^{-1}(\varepsilon)\Theta_{\gamma,M}(\varepsilon),\qquad\tilde{\mathrm{B}}_{M}=\tilde{\mathrm{B}}_{M}(\varepsilon):=\mathrm{B}_{M}(\varepsilon)\Pi_{\gamma,M}^{-1}(\varepsilon),$

and the easily checked identities

$\displaystyle-2\mathrm{i}\big{(}\tilde{\mathrm{A}}_{M}(\varepsilon)-\mathrm{i}\tilde{\mathrm{A}}_{M}(\varepsilon)\big{)}^{-1}\tilde{\mathrm{A}}_{M}(\varepsilon)=\mathrm{i}\big{(}\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)}\big{)},$
$\displaystyle-2\mathrm{i}\big{(}\tilde{\mathrm{A}}_{M}(\varepsilon)-\mathrm{i}\tilde{\mathrm{A}}_{M}(\varepsilon)\big{)}^{-1}\tilde{\mathrm{B}}_{M}(\varepsilon)=\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)},$

for each $\psi^{(i)}$ and each vertex $M\in\gamma_{\infty}$ we have:

$\displaystyle\varepsilon\mathrm{A}_{M}(\varepsilon)$
$\displaystyle\mathcal{U}_{M}(\psi^{(i)})+\mathrm{B}_{M}(\varepsilon)\mathcal{U}_{M}^{\prime}(\psi^{(i)})=\frac{\mathrm{i}}{2}\big{(}\tilde{\mathrm{A}}_{M}(\varepsilon)-\mathrm{i}\tilde{\mathrm{B}}_{M}(\varepsilon)\big{)}\bigg{(}\mathrm{i}(\mathrm{U}_{M}(\varepsilon)-\mathrm{I}_{d(M)})\mathrm{P}_{M}(\varepsilon)$
(4.19)
$\displaystyle\cdot\Big{(}\big{(}\mathrm{I}_{d(M)}-\mathrm{i}\mathrm{K}_{M}(\varepsilon)\Theta_{\gamma,M}(\varepsilon)\big{)}\mathcal{U}_{M}(\psi^{(i)})+\mathrm{K}_{M}(\varepsilon)\Pi_{\gamma,M}(\varepsilon)\mathcal{U}_{M}^{\prime}(\psi^{(i)})\Big{)}$
$\displaystyle+\big{(}\mathrm{U}_{M}(\varepsilon)+\mathrm{I}_{d(M)}\big{)}\mathrm{P}_{M,\bot}(\varepsilon)\big{(}\Pi_{\gamma,M}(\varepsilon)\mathcal{U}_{M}^{\prime}(\psi^{(i)})+\big{(}\mathrm{K}_{M,\bot}(\varepsilon)-\mathrm{i}\Theta_{\gamma,M}(\varepsilon)\big{)}\mathcal{U}_{M}(\psi^{(i)})\big{)}\bigg{)}.$

The latter long term in the right hand side in the above identity becomes the
assumed vertex condition for $\psi^{(i)}$ as $\varepsilon=0$ and hence, it
vanishes as $\varepsilon=0$. We expand then identity (4.19) into the Taylor
series in $\varepsilon$ and calculate the coefficient at $\varepsilon^{1}$ if
$\mathrm{B}_{M}(0)\neq 0$ and at $\varepsilon^{2}$ if $\mathrm{B}_{M}(0)=0$.
Employing then the vertex condition for $\psi^{(i)}$, we obtain:

$\displaystyle\mathcal{E}_{M}(\psi^{(i)})=$
$\displaystyle\mathrm{i}(\mathrm{U}_{M}^{(0)}-\mathrm{I}_{d(M)})\mathrm{P}_{M}^{(0)}\Big{(}\mathrm{P}_{M}^{(0)}\mathrm{K}_{M,1}\mathcal{V}_{M}(\psi^{(i)})+\mathrm{P}_{M,1}\mathcal{U}_{M}(\psi^{(i)})\Big{)}$
$\displaystyle+(\mathrm{U}_{M}^{(0)}+\mathrm{I}_{d(M)})\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1}\Big{(}\mathcal{V}_{M}(\psi^{(i)})+\mathrm{K}_{M,\bot}(0)\mathcal{U}_{M}(\psi^{(i)})$
$\displaystyle+\Pi_{\gamma,M,1}\mathcal{U}_{M}^{\prime}(\psi^{(i)})+\big{(}\mathrm{K}_{M,\bot,1}-\mathrm{i}\Theta_{\gamma,M,1}\big{)}\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)})\Big{)}.$

Applying the matrices $\mathrm{P}_{M}^{(0)}$ and
$(\mathrm{U}_{M}^{(0)}+\mathrm{I}_{d(M)})^{-1}\mathrm{P}_{M,\bot}^{(0)}$ to
the above identity and using formulae (4.16) and
$\mathrm{P}_{M}^{(0)}\mathrm{K}_{M,\bot}(0)=0$,
$\mathrm{P}_{M}^{(0)}\mathcal{U}_{M}(\psi^{(i)})=0$, we arrive at

$\displaystyle\begin{aligned}
\mathrm{P}_{M}^{(0)}\mathcal{E}_{M}(\psi^{(i)})=-2\mathrm{i}\mathrm{P}_{M}^{(0)}\mathrm{K}_{M,1}\mathcal{V}_{M}(\psi^{(i)})-2\mathrm{i}\mathrm{P}_{M}^{(0)}\mathrm{P}_{M,1}\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)}),\end{aligned}$
(4.20) $\displaystyle\begin{aligned}
(\mathrm{U}_{M}^{(0)}+\mathrm{I}_{d(M)})^{-1}\mathrm{P}_{M,\bot}^{(0)}\mathcal{E}_{M}&(\psi^{(i)})=\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1}\Big{(}\Pi_{\gamma,M}\mathcal{U}_{M}^{\prime}(\psi^{(i)})-\mathrm{i}\Theta_{\gamma,M}(0)\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)})\Big{)}\\\
&-\mathrm{P}_{M,\bot}^{(0)}\Big{(}\Pi_{\gamma,M,1}\mathcal{U}_{M}^{\prime}(\psi^{(i)})+\big{(}\mathrm{K}_{M,\bot,1}-\mathrm{i}\Theta_{\gamma,M,1}\big{)}\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)})\Big{)}.\end{aligned}$
(4.21)

Owing to identities (4.21) and the vertex conditions for $\psi^{(i)}$, we can
rewrite formula (2.26) for $\mathcal{L}_{M}(\psi^{(i)})$ as

$\displaystyle\mathcal{L}_{M}(\psi^{(i)})=\mathrm{P}_{M,\bot}^{(0)}\mathrm{P}_{M,1}\mathcal{V}_{M}(\psi^{(i)})-\mathrm{P}_{M,\bot}^{(0)}\mathrm{K}_{M,\bot,1}\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)}).$

Substituting this identity and (4.18), (4.20) into formula (2.25) for
$Q_{M}^{(ij)}$, we obtain immediately that

$\displaystyle Q_{M}^{(ij)}=$
$\displaystyle\big{(}\mathrm{P}_{M,1}\mathcal{V}_{M}(\psi^{(i)}),\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(j)})\big{)}_{\mathds{C}^{d(M)}}-\big{(}\mathrm{K}_{M,\bot,1}\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(i)}),\mathrm{P}_{M,\bot}^{(0)}\mathcal{U}_{M}(\psi^{(j)})\big{)}_{\mathds{C}^{d(M)}}$
(4.22)
$\displaystyle+\big{(}\mathrm{K}_{M,1}\mathrm{P}_{M}^{(0)}\mathcal{V}_{M}(\psi^{(i)}),\mathrm{P}_{M}^{(0)}\mathcal{U}_{M}(\psi^{(j)})\big{)}_{\mathds{C}^{d(M)}}+\big{(}\mathrm{P}_{M,1}\mathrm{P}_{M}^{(0)}\mathcal{U}_{M}(\psi^{(i)}),\mathrm{P}_{M}^{(0)}\mathcal{V}_{M}(\psi^{(j)})\big{)}_{\mathds{C}^{d(M)}}.$

Since the matrices $\mathrm{K}_{M,\bot,1}$ and $\mathrm{K}_{M,1}$ are self-
adjoint, we conclude that $Q_{M}^{(ji)}=\overline{Q_{M}^{(ij)}}$. In view of
formulae (2.23), (2.24), this completes the proof. ∎

##  5 Auxiliary operator on graph $\gamma$

In this section we consider an auxiliary operator on certain extension of the
graph $\gamma$ and we study its properties, which will be employed then in the
proof of our main results. The mentioned extension is another graph denoted by
$\gamma_{ex}$ and obtained by attaching additional unit edges
$\mathrm{e}_{i}^{ex}$, $i\in J_{j}$, $j=1,\ldots,n$, to the each vertex
$M_{j}$, $j=1,\ldots,n$, in the graph $\gamma$. The boundary vertices, being
the end-points of the edges $\mathrm{e}_{i}^{ex}$ and not coinciding with
$M_{j}$, are denoted by $M_{i}^{ex}$, $i\in J_{j}$, $j=1,\ldots,n$.

The graph $\gamma_{ex}$ is the very similar to $\gamma_{\infty}$, the only
difference is that instead of leads $\mathrm{e}_{i}^{\infty}$, here we have
the edges $\mathrm{e}_{i}^{ex}$ of finite lengths. This is why, apart of
$M_{i}^{ex}$, $i=1,\ldots,n$, the vertices of $\gamma_{ex}$ coincides with the
vertices in the graph $\gamma_{\infty}$. Bearing this fact in mind, in this
section, while writing $M\in\gamma_{\infty}$, we shall mean that we consider a
vertex in the graph $\gamma_{ex}$ not coinciding with $M_{i}^{ex}$,
$i=1,\ldots,n$.

The aforementioned auxiliary operator on the graph $\gamma_{ex}$ is denoted by
$\mathcal{H}_{ex}(\varepsilon)$. Its differential expression reads as

$\displaystyle\hat{\mathcal{H}}_{ex}(\varepsilon)u:=-\frac{d\
}{d\xi}p_{\gamma}(\cdot,\varepsilon)\frac{du}{d\xi}+\mathrm{i}\left(\frac{d\
}{d\xi}(q_{\gamma}(\,\cdot\,,\varepsilon)u)+q_{\gamma}(\,\cdot\,,\varepsilon)\frac{du}{d\xi}\right)+V_{\gamma}(\,\cdot\,,\varepsilon)u\quad\text{on}\quad\gamma,$
$\displaystyle\hat{\mathcal{H}}_{ex}(\varepsilon)u:=-\mathsf{p}_{i}(\varepsilon)\frac{d^{2}u}{d\xi_{i}^{2}}+2\mathrm{i}\varepsilon\mathsf{q}_{i}(\varepsilon)\frac{du}{d\xi_{i}}\quad\text{on}\quad\mathrm{e}_{i}^{ex},\quad
i\in J_{j},\quad j=1,\ldots,n,$

where $\xi_{i}$ is the variable on the edge $\mathrm{e}_{i}^{ex}$ measured
from the vertex $M_{j}$, while $\mathrm{e}_{i}$ are the edges in the graph
$\Gamma$ incident to the edge $M_{0}$.

At the vertices $M\in\gamma_{\infty}$ we impose vertex conditions (4.7). The
vertices $M_{i}^{ex}$ are subject to the Robin condition

$\Pi_{\Gamma,M_{0}}(\varepsilon)\mathcal{U}^{\prime}_{ex}(u)-\mathrm{i}\varepsilon\Theta_{\Gamma,M_{0}}(\varepsilon)\mathcal{U}_{ex}(u)=0,$
(5.1)

where we denote

$\displaystyle\Pi_{\Gamma,M_{0}}(\varepsilon):=\operatorname{diag}\big{\\{}\mathsf{p}_{i}(\varepsilon)\big{\\}}_{i=1,\ldots,d_{0}},\qquad\Theta_{\Gamma,M_{0}}(\varepsilon):=\operatorname{diag}\big{\\{}\mathsf{q}_{i}(\varepsilon)\big{\\}}_{i=1,\ldots,d_{0}},$
$\displaystyle\mathcal{U}_{ex}^{\prime}(u):=\left(\begin{aligned}
\frac{du\big{|}_{\mathrm{e}_{1}^{ex}}}{\displaystyle d\xi_{1}}&(M_{1}^{ex})\\\
&\vdots\\\
\frac{du\big{|}_{\mathrm{e}_{d_{0}}^{ex}}}{d\xi_{d_{0}}}&(M_{d_{0}}^{ex})\end{aligned}\right),\qquad\hphantom{\mathcal{U}_{ex}(u)}\mathcal{U}_{ex}(u):=\left(\begin{aligned}
u&(M_{1}^{ex})\\\ &\vdots\\\ u&(M_{d_{0}}^{ex})\end{aligned}\right)$

for all functions $u\in\dot{W}_{2}^{2}(\gamma_{ex})$. Condition (5.1) is
obviously a particular case of condition (4.7).

It follows from rank condition (2.7), formulae (2.10) and the analyticity in
$\varepsilon$ of the matrices $\mathrm{A}_{M}(\varepsilon)$ and
$\mathrm{B}_{M}(\varepsilon)$ that
$\operatorname{rank}\big{(}\mathrm{A}_{M}(\varepsilon)\
\mathrm{B}_{M}(\varepsilon)\big{)}=d(M)$ for each vertex $M\in\gamma_{ex}$ and
sufficiently small $\varepsilon$. The self-adjointness of the matrix in (2.5)
implies the same property for the similar matrix associated with each vertex
$M\in\gamma_{ex}$. Hence, the operator $\mathcal{H}_{ex}(\varepsilon)$
satisfies all assumptions of Lemma 4.1 and therefore, it is self-adjoint. Then
its resolvent $(\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)^{-1}$ is
well-defined for $\lambda\in\mathds{C}\setminus\mathds{R}$.

Our main aim in this section is to prove that this resolvent is meromorphic in
$\varepsilon$ and to study the structure of its pole at $\varepsilon=0$. In
order to formulate a needed result, we first introduce some auxiliary
notations. The eigenvectors $\mathrm{y}_{i}$, $i=1,\ldots,k$, of the self-
adjoint matrix $\mathrm{Q}$ form an orthonormalized basis in $\mathds{C}^{k}$
and a matrix

$\mathrm{Y}:=\begin{pmatrix}y_{11}&\ldots&y_{k1}\\\ \vdots&&\vdots\\\
y_{1k}&\ldots&y_{kk}\end{pmatrix},\qquad\mathrm{y}_{i}:=\begin{pmatrix}\mathrm{y}_{i1}\\\
\vdots\\\ \mathrm{y}_{ik}\end{pmatrix},\quad i=1,\ldots,k,$ (5.2)

reduces $\mathrm{Q}$ to a diagonal form:

$\mathrm{Y}^{*}\mathrm{Q}\mathrm{Y}=\operatorname{diag}\\{\lambda_{1}(\mathrm{Q}),\ldots,\lambda_{k}(\mathrm{Q})\\},\qquad\mathrm{Y}^{*}=\mathrm{Y}^{-1},$
(5.3)

where $\lambda_{i}(\mathrm{Q})$ are the eigenvalues of the matrix $\mathrm{Q}$
with associated eigenvectors $\mathrm{y}_{i}$. If zero is an eigenvalue of the
matrix $\mathrm{Q}$, we order the eigenvalues $\lambda_{i}(\mathrm{Q})$ so
that $\lambda_{i}(\mathrm{Q})=0$ for $i=1,\ldots,k_{0}$. If zero is not an
eigenvalue of $\mathrm{Q}$, we let $k_{0}:=0$.

We define

$\tilde{\psi}^{(j)}:=\sum\limits_{i=1}^{k}y_{ji}\psi^{(i)},\qquad\psi^{(i)}=\sum\limits_{j=1}^{k}y_{ji}\tilde{\psi}^{(j)}.$
(5.4)

The introduced functions $\tilde{\psi}^{(j)}$ are non-trivial solutions of
problem (2.12) obeying condition (2.13) and

$\big{(}\mathcal{U}_{\gamma}(\tilde{\psi}^{(i)}),\mathcal{U}_{\gamma}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d_{0}}}=\delta_{ij},$
(5.5)

where $\delta_{ij}$ is the Kronecker delta.

The main statement of this section is as follows.

######  Lemma 5.1.

For each $\lambda\in\mathds{C}\setminus\mathds{R}$, the resolvent
$(\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)^{-1}:L_{2}(\gamma_{ex})\to\dot{W}_{2}^{2}(\gamma_{ex})$
is meromorphic for $\varepsilon$ small enough. This operator can be
represented as

$\big{(}\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda\big{)}^{-1}=\varepsilon^{-2}\mathcal{A}_{-2}(\lambda)+\varepsilon^{-1}\mathcal{A}_{-1}+\mathcal{A}_{0}(\varepsilon,\lambda),$
(5.6)

where
$\mathcal{A}_{0}(\varepsilon,\lambda):L_{2}(\gamma_{ex})\to\dot{W}_{2}^{2}(\gamma_{ex})$
is a bounded operator analytic in $\varepsilon$, while operators
$\mathcal{A}_{-2}$, $\mathcal{A}_{-1}$ are given by the formulae

$\mathcal{A}_{-2}(\lambda)=\sum\limits_{i=1}^{k_{0}}\mathcal{C}^{(-2)}_{i}(\,\cdot\,)\tilde{\psi}^{(i)},\qquad\mathcal{A}_{-1}(\lambda)=\sum\limits_{i=1}^{k_{0}}\mathcal{C}^{(-2)}_{i}(\,\cdot\,)\hat{\psi}^{(i)}+\sum\limits_{i=1}^{k}\mathcal{C}^{(-1)}_{i}(\,\cdot\,)\tilde{\psi}^{(i)},$
(5.7)

where
$\mathcal{C}^{(-2)}_{i},\,\mathcal{C}^{(-1)}_{i}:L_{2}(\gamma_{ex})\to\mathds{C}$
are some bounded linear functionals depending on $\lambda$ and
$\hat{\psi}^{(i)}\in\dot{W}_{2}^{2}(\gamma)$ are some functions. In
particular,

$\displaystyle\begin{pmatrix}\mathcal{C}^{(-2)}_{1}(f)\\\ \vdots\\\
\mathcal{C}^{(-2)}_{k_{0}}(f)\end{pmatrix}:=(\mathrm{X}_{0}-\lambda\mathrm{G}_{0})^{-1}\begin{pmatrix}\big{(}f,\tilde{\psi}^{(1)}\big{)}_{L_{2}(\gamma_{ex})}\\\
\vdots\\\
\big{(}f,\tilde{\psi}^{(k_{0})}\big{)}_{L_{2}(\gamma_{ex})}\end{pmatrix},$
(5.8) $\displaystyle\begin{pmatrix}\mathcal{C}^{(-1)}_{k_{0}+1}(f)\\\
\vdots\\\
\mathcal{C}^{(-1)}_{k}(f)\end{pmatrix}:=\mathrm{Q}_{1}^{-1}\begin{pmatrix}\big{(}f,\tilde{\psi}^{(k_{0}+1)}\big{)}_{L_{2}(\gamma_{ex})}\\\
\vdots\\\
\big{(}f,\tilde{\psi}^{(k)}\big{)}_{L_{2}(\gamma_{ex})}\end{pmatrix}+\mathrm{X}_{1}(\lambda)\begin{pmatrix}\big{(}f,\tilde{\psi}^{(1)}\big{)}_{L_{2}(\gamma_{ex})}\\\
\vdots\\\
\big{(}f,\tilde{\psi}^{(k_{0})}\big{)}_{L_{2}(\gamma_{ex})}\end{pmatrix}.$

Here

$\mathrm{Q}_{1}:=\operatorname{diag}\big{\\{}\lambda_{k_{0}+1}(\mathrm{Q}),\ldots,\lambda_{k}(\mathrm{Q})\big{\\}},\quad\mathrm{G}_{0}:=\begin{pmatrix}(\tilde{\psi}^{(1)},\tilde{\psi}^{(1)})_{L_{2}(\gamma_{ex})}&\ldots&(\tilde{\psi}^{(k_{0})},\tilde{\psi}^{(1)})_{L_{2}(\gamma_{ex})}\\\
\vdots&&\vdots\\\
(\tilde{\psi}^{(1)},\tilde{\psi}^{(k_{0})})_{L_{2}(\gamma_{ex})}&\ldots&(\tilde{\psi}^{(k_{0})},\tilde{\psi}^{(k_{0})})_{L_{2}(\gamma_{ex})}\end{pmatrix},$

$\mathrm{X}_{0}$ is some fixed self-adjoint matrix of size $k_{0}\times k_{0}$
and $\mathrm{X}_{1}(\lambda)$ is some matrix of size $(k-k_{0})\times k_{0}$.

######  Remark 5.1.

It is also possible to find explicitly the functionals
$\mathcal{C}^{(-1)}_{i}$ for $i=1,\ldots,k_{0}$, the functions
$\hat{\psi}^{(i)}$ and the matrix $\mathrm{X}_{1}(\lambda)$ in the above
lemma, but we shall not need them in what follows. This is why we do not
provide such formulae in the formulation of the lemma.

The proof of Lemma 5.1 consists of several main steps, which we present below
in separate subsections.

###  5.1 Meromoprhic dependence on $\varepsilon$

In this section we begin proving Lemma 5.1 and we shall show that the
resolvent $(\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)^{-1}$ is
meromoprhic in $\varepsilon$ and has a pole at $\varepsilon=0$ of at most
second order.

Since the graph $\gamma_{ex}$ is finite, the resolvent of the operator
$\mathcal{H}_{ex}(\varepsilon)$ is compact and hence, this operator has a
discrete spectrum. For small $\varepsilon$ the number $\varepsilon^{2}\lambda$
is also small and this is why we should first check whether the operator
$\mathcal{H}_{ex}(\varepsilon)$ has small eigenvalues and if so, how they
behave. In order to understand this, we consider the operator
$\mathcal{H}_{ex}(0)$. We restrict the functions $\tilde{\psi}^{(j)}$ to the
graph $\gamma_{ex}$ and we see immediately that these restrictions, still
denoted by $\tilde{\psi}^{(j)}$, are the eigenfunctions of the operator
$\mathcal{H}_{ex}(0)$ associated with a zero eigenvalue. And vice versa, if
$\psi$ is an eigenfunction of $\mathcal{H}_{ex}(0)$ associated with the zero
eigenvalue, then in view of the definition of the differential expression
$\mathcal{H}_{ex}(0)$ on the edges $\mathrm{e}_{i}^{ex}$ and the Neumann
condition at their end-points $M_{i}^{ex}$, the function $\psi$ is necessary
constant on each of these edges. Then we replace the edges
$\mathrm{e}_{i}^{ex}$ by infinite edges $\mathrm{e}_{i}^{\infty}$ passing in
this way to the graph $\gamma_{\infty}$ and we continue then the function
$\psi$ by the aforementioned constants on entire edges
$\mathrm{e}_{i}^{\infty}$. The resulting function solves problem (2.12).
Therefore, zero is an eigenvalue of the operator $\mathcal{H}_{ex}(0)$ if and
only if the operator $\mathcal{H}_{\infty}$ has a virtual level at the bottom
of its essential spectrum. The eigenfunctions of $\mathcal{H}_{ex}(0)$
associated with the zero eigenvalues are the aforementioned restrictions
$\tilde{\psi}^{(j)}$ to $\gamma_{ex}$.

The resolvent $(\mathcal{H}_{ex}(\varepsilon)-z)^{-1}$ depends analytically in
small $\varepsilon$ for $z\in\mathds{C}$ separated from the spectrum of the
operator $\mathcal{H}_{ex}(0)$. In a particular case when the operator
$\mathcal{H}_{ex}(\varepsilon)$ is just a Schrödinger operator with a fixed
potential independent of $\varepsilon$, this is a corollary of a more general
result presented in Theorem 3.5 in [3]. For the case of the differential
expression with general varying and $\varepsilon$-dependent coefficients the
technique used in [3] still works as it is said in [3] after Equation (2).
Hence, due to standard results in the analytic perturbation theory, the
operator $\mathcal{H}_{ex}(\varepsilon)$ has exactly $k$ eigenvalues, counting
their multiplicity, which converge to zero as $\varepsilon\to+0$. We denote
these eigenvalues by $\Lambda_{j}^{(\varepsilon)}$, $j=1,\ldots,k$, while the
associated eigenfunctions orthonormalized in $L_{2}(\gamma_{ex})$ are denoted
by $\varphi_{j}^{(\varepsilon)}$. According general Theorems 3.8, 3.9 in [3],
the eigenvalues $\Lambda_{j}(\varepsilon)$ are analytic in $\varepsilon$ and
the eigenfunctions $\varphi_{j}$ can be also chosen being analytic in
$\varepsilon$.

Standard representation for the resolvent of a self-adjoint operator, see [11,
Ch. V, Sect. 3.5], then gives:

$\big{(}\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda\big{)}^{-1}f=\sum\limits_{j=1}^{k}\frac{\big{(}f,\varphi_{j}^{(\varepsilon)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(\varepsilon)}-\varepsilon^{2}\lambda}\varphi_{j}^{(\varepsilon)}+\mathcal{A}_{1}(\varepsilon,\lambda),$
(5.9)

and $\mathcal{A}_{1}$ is the reduced resolvent, which can be represented as

$\mathcal{A}_{1}(\varepsilon,\lambda)=\frac{1}{2\pi\mathrm{i}}\int\limits_{|z|=\delta}(z-\varepsilon^{2}\lambda)^{-1}\big{(}\mathcal{H}_{ex}(\varepsilon)-z\big{)}^{-1}\,dz,$

where $\delta$ is a sufficiently small fixed positive constant such that the
ball $\\{z\in\mathds{C}:\,|z|\leqslant\delta\\}$ contains only zero eigenvalue
of the operator $\mathcal{H}_{ex}(0)$ and is separated from the other
eigenvalues. By the aforementioned analyticity of the resolvent
$(\mathcal{H}_{ex}(\varepsilon)-z)^{-1}$ in $\varepsilon$ and the known
analyticity in $z$, the above formula for the operator $\mathcal{A}_{1}$
implies that it is analytic in $\varepsilon$ for each fixed
$\lambda\in\mathds{C}\setminus\mathds{R}$.

Since the eigenvalues $\Lambda_{j}^{(\varepsilon)}$ and the eigenfunctions
$\varphi_{j}^{(\varepsilon)}$ are also analytic in $\varepsilon$, they are
represented by their Taylor series as follows:

$\Lambda_{j}^{(\varepsilon)}=\varepsilon\Lambda_{j}^{(1)}+\varepsilon^{2}\Lambda_{j}^{(2)}(\varepsilon)+\varepsilon^{3}\Lambda_{j}^{(3)}(\varepsilon)+\varepsilon^{4}\tilde{\Lambda}_{j}^{(\varepsilon)},\qquad\varphi_{j}^{(\varepsilon)}=\varphi_{j}^{(0)}+\varepsilon\varphi_{j}^{(1)}+\varepsilon^{2}\varphi_{j}^{(1)}+\varepsilon^{3}\tilde{\varphi}_{j}^{(\varepsilon)},$
(5.10)

where $\Lambda_{j}^{(i)}$, $i=1,2$, $\varphi_{j}^{(i)}$, $i=0,1,2$, are the
leading terms in the Taylor series and $\tilde{\Lambda}_{j}^{(\varepsilon)}$,
$\tilde{\varphi}_{j}^{(\varepsilon)}$ are remainders analytic in
$\varepsilon$. The functions $\varphi_{j}^{(0)}$ are eigenfunctions of the
operators $\mathcal{H}_{ex}(0)$ associated with the zero eigenvalue and hence,
they are some linear combinations of the functions $\psi^{(j)}$.

We substitute representations (5.10) into the terms in sum (5.9) and expand
the result into the Laurent series in $\varepsilon$. Such expansion is
possible owing to the inequality $\operatorname{Im}\lambda\neq 0$ and it reads
as:

$\displaystyle\begin{aligned}
\frac{\big{(}f,\varphi_{j}^{(\varepsilon)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(\varepsilon)}-\varepsilon^{2}\lambda}\varphi_{j}^{(\varepsilon)}=\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\varepsilon\Lambda_{j}^{(1)}}\varphi_{j}^{(0)}+\mathcal{A}_{2,j}(\varepsilon,\lambda)f\quad\text{if}\quad\Lambda_{j}^{(1)}\neq
0,\end{aligned}$ (5.11) $\displaystyle\begin{aligned}
\frac{\big{(}f,\varphi_{j}^{(\varepsilon)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(\varepsilon)}-\varepsilon^{2}\lambda}\varphi_{j}^{(\varepsilon)}=&\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\varepsilon^{2}(\Lambda_{j}^{(2)}-\lambda)}\varphi_{j}^{(0)}-\frac{\Lambda_{j}^{(3)}\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\varepsilon(\Lambda_{j}^{(2)}-\lambda)^{2}}\varphi_{j}^{(0)}\\\
&+\frac{1}{\varepsilon(\Lambda_{j}^{(2)}-\lambda)}\bigg{(}\big{(}f,\varphi_{j}^{(1)}\big{)}_{L_{2}(\gamma_{ex})}\varphi_{j}^{(0)}+\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}\varphi_{j}^{(1)}\bigg{)}\\\
&+\mathcal{A}_{2,j}(\varepsilon,\lambda)f\quad\text{if}\quad\Lambda_{j}^{(1)}=0,\end{aligned}$
(5.12)

where
$\mathcal{A}_{2,j}(\varepsilon,\lambda):L_{2}(\gamma_{ex})\to\dot{W}_{2}^{2}(\gamma)$
are some linear bounded operators analytic in $\varepsilon$ for each fixed
$\lambda$. We observe that the denominators in the quotients in (5.12) are
non-zero since $\Lambda_{j}^{(1)}$, $\Lambda_{j}^{(2)}$ are real, while the
imaginary part of $\lambda$ is non-zero.

We substitute representations (5.11), (5.12) into (5.9) and we conclude that
the resolvent $(\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)^{-1}$ is
meromoprhic in $\varepsilon$ and has a pole at $\varepsilon=0$ of at most
second order. In order to determine the coefficients at $\varepsilon^{-2}$ and
$\varepsilon^{-1}$ in the corresponding Laurent series for
$(\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)^{-1}$, we need to find
$\Lambda_{j}^{(1)}$, $\Lambda_{j}^{(2)}$, $\varphi_{j}^{(1)}$ and this will be
done in the next subsection.

###  5.2 Expansions for eigenvalues and eigenfunctions

In this subsection we find the leading coefficients in expansions (5.10). In
order to do this, we substitute these expansions into the eigenvalue equation
$\mathcal{H}_{ex}(\varepsilon)\varphi_{j}^{(\varepsilon)}=\Lambda_{j}^{(\varepsilon)}\varphi_{j}^{(\varepsilon)}$,
expand the coefficients into the powers series in $\varepsilon$ and equate the
coefficients at the like powers. This gives certain boundary value problems
for $\varphi_{j}^{(\varepsilon)}$ and $\Lambda_{j}^{(\varepsilon)}$. Before
doing this, as in Section 4.3, we rewrite vertex condition (4.7) at each
vertex $M\in\gamma_{\infty}$ to (4.14), (4.15) and we employ the projectors
$\mathrm{P}_{M}(\varepsilon)$ and $\mathrm{P}_{M,\bot}(\varepsilon)$ and the
transforming function $\mathrm{S}_{M}(\varepsilon)$ obeying (4.12), (4.13).
Then, after the above described procedure of substituting the expansions into
the eigenvalue equation, we arrive at the equation

$\mathcal{H}_{ex}(0)\varphi_{j}^{(0)}=0$ (5.13)

and at the following boundary value problem for $\varphi_{j}^{(1)}$:

$\displaystyle\hat{\mathcal{H}}_{ex}(0)\varphi_{j}^{(1)}=g_{j}^{(1)}+\Lambda_{j}^{(1)}\varphi_{j}^{(1)}\quad\text{on}\quad\gamma_{ex},\qquad\Pi_{\Gamma,M_{0}}(0)\mathcal{U}_{ex}^{\prime}(\varphi_{j}^{(1)})=\mathrm{g}_{ex,j}^{(1)},$
(5.14)
$\displaystyle\mathrm{P}_{M}^{(0)}\mathcal{U}_{M}(\varphi_{j}^{(1)})=\mathrm{g}_{M,j}^{(1)},\qquad\mathrm{P}_{M,\bot}^{(0)}\mathcal{V}_{M}(\varphi_{j}^{(1)})+\mathrm{K}_{M,\bot}(0)\mathcal{U}_{M}(\varphi_{j}^{(1)})=\mathrm{g}_{M,\bot,j}^{(1)}\quad\text{at}\quad
M\in\gamma_{\infty},$

where

$\displaystyle
g_{j}^{(1)}:=-\frac{d\hat{\mathcal{H}}_{ex}(\varepsilon)\varphi_{j}^{(i)}}{d\varepsilon}\bigg{|}_{\varepsilon=0},\qquad\mathrm{g}_{ex,j}^{(1)}:=\Pi_{\Gamma,M_{0},1}^{(1)}\,\mathcal{U}_{ex}(\varphi_{j}^{(0)}),$
(5.15)
$\displaystyle\mathrm{g}_{M,\bot,j}^{(1)}:=-\mathrm{P}_{M,\bot}^{(0)}\sum\limits_{M\in\gamma_{\infty}}\big{(}\tilde{\mathrm{S}}_{M,\bot,1}\,\mathcal{U}_{M}^{\prime}(\varphi_{j}^{(0)})+\tilde{\mathrm{K}}_{M,\bot,1}\,\mathcal{U}_{M}(\varphi_{j}^{(0)})\big{)},$
$\displaystyle\mathrm{g}_{M,j}^{(1)}:=-\mathrm{P}_{M}^{(0)}\sum\limits_{M\in\gamma_{\infty}}\big{(}\tilde{\mathrm{S}}_{M,1}\,\mathcal{U}_{M}(\varphi_{j}^{(0)})+\tilde{\mathrm{K}}_{M,1}\mathcal{U}_{M}^{\prime}(\varphi_{j}^{(0)})\big{)}.$

Equation (5.13) is solved immediately:

$\varphi_{j}^{(0)}=\sum\limits_{i=1}^{k}\alpha_{ji}\tilde{\psi}^{(i)},$ (5.16)

where $\alpha_{ji}$ are some constants such that the functions
$\varphi_{j}^{(0)}$ are orthonormalized in $L_{2}(\gamma_{ex})$. This
condition means that the matrices

$\alpha:=\begin{pmatrix}\alpha_{11}&\ldots&\alpha_{k1}\\\ \vdots&&\vdots\\\
\alpha_{1k}&\ldots&\alpha_{kk}\end{pmatrix},\qquad\mathrm{G}:=\begin{pmatrix}(\tilde{\psi}^{(1)},\tilde{\psi}^{(1)})_{L_{2}(\gamma_{ex})}&\ldots&(\tilde{\psi}^{(k)},\tilde{\psi}^{(1)})_{L_{2}(\gamma_{ex})}\\\
\vdots&&\vdots\\\
(\tilde{\psi}^{(k)},\tilde{\psi}^{(1)})_{L_{2}(\gamma_{ex})}&\ldots&(\tilde{\psi}^{(k)},\tilde{\psi}^{(k)})_{L_{2}(\gamma_{ex})}\end{pmatrix}$

should satisfy the equation
$\mathrm{G}\alpha\alpha^{*}=\alpha^{*}\mathrm{G}\alpha=\mathrm{I}_{k}$.

In order to analyze problem (5.14), we shall employ the following auxiliary
lemma.

######  Lemma 5.2.

Assume that Condition (A) holds. Given an arbitrary family of vectors
$\mathrm{g}_{M}\in\mathrm{P}_{M}^{(0)}\mathds{C}^{d(M)}$,
$\mathrm{g}_{M,\bot}\in\mathrm{P}_{M,\bot}\mathds{C}^{d(M)}$ for each vertex
$M\in\gamma_{\infty}$, an arbitrary vector
$\mathrm{g}_{ex}\in\mathds{C}^{d_{0}}$, and an arbitrary function $g\in
L_{2}(\gamma_{ex})$, the boundary value problem

$\displaystyle\hat{\mathcal{H}}_{ex}(0)u=g\quad\text{on}\quad\gamma_{ex},\qquad\Pi_{\Gamma,M_{0}}(0)\mathcal{U}_{ex}^{\prime}(u)=\mathrm{g}_{ex},$
(5.17)
$\displaystyle\mathrm{P}_{M}^{(0)}\mathcal{U}_{M}(u)=\mathrm{g}_{M},\qquad\mathrm{P}_{M,\bot}^{(0)}\mathcal{V}_{M}(u)+\mathrm{K}_{M,\bot}(0)\mathcal{U}_{M}(u)=\mathrm{g}_{M,\bot}\quad\text{at}\quad
M\in\gamma_{\infty},$

is solvable in $\dot{W}_{2}^{2}(\gamma_{ex})$ if and only if

$\displaystyle(g,\tilde{\psi}^{(j)})_{L_{2}(\gamma_{ex})}=$
$\displaystyle-\big{(}\mathrm{g}_{ex},\mathcal{U}_{\gamma}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d_{0}}}+\sum\limits_{M\in\gamma_{\infty}}\big{(}\mathrm{g}_{M,\bot},\mathcal{U}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}-\sum\limits_{M\in\gamma_{\infty}}\big{(}\mathrm{g}_{M},\mathcal{V}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}$
(5.18)

for each $j=1,\ldots,k$.

###### Proof.

Throughout the proof by $C$ we denote various inessential constants
independent of $g$, $\mathrm{g}_{ex}$, $\mathrm{g}_{M}$,
$\mathrm{g}_{M,\bot}$. Let $u_{0}$ be a function on $\gamma_{ex}$ satisfying
the boundary conditions in (5.17). Such function can be easily constructed on
each edge as a linear function with appropriate coefficients multiplied by a
cut-off function. We seek a solution to problem (5.17) as $u=u_{0}+\tilde{u}$
and for a new unknown function $\tilde{u}$ we obtain the equation

$\mathcal{H}_{ex}(0)\tilde{u}=\tilde{g},\qquad\tilde{g}:=g-\hat{\mathcal{H}}_{ex}(0)u_{0}$
(5.19)

Since zero is an eigenvalue of the operator $\mathcal{H}_{ex}(0)$, equation
(5.19) is solvable if and only if its right hand side orthogonal to all
eigenfunctions associated with the zero eigenvalue:

$(\tilde{g},\psi^{(j)})_{L_{2}(\gamma_{ex})}=0,\qquad j=1,\ldots,k.$ (5.20)

Integrating by parts and employing the definition of the functions
$\psi^{(j)}$ and the vertex conditions for the function $u_{0}$, it is
straightforward to confirm that

$\displaystyle\big{(}\hat{\mathcal{H}}_{ex}(0)u_{0},\tilde{\psi}^{(j)}\big{)}_{L_{2}(\gamma_{ex})}=$
$\displaystyle-\sum\limits_{i=1}^{d_{0}}\mathsf{p}_{i}(0)\frac{du_{0}\big{|}_{\mathrm{e}_{i}}}{d\xi_{i}}(M_{i}^{ex})\overline{\tilde{\psi}^{(j)}}\big{|}_{\mathrm{e}_{i}}(M_{i}^{ex})$
$\displaystyle+\sum\limits_{M\in\gamma_{\infty}}\big{(}\mathcal{V}_{M}(u_{M}),\mathcal{U}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}-\sum\limits_{M\in\gamma_{\infty}}\big{(}\mathcal{U}_{M}(u_{M}),\mathcal{V}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}$
$\displaystyle=$
$\displaystyle-\big{(}\mathrm{g}_{ex},\mathcal{U}_{\gamma}(\psi^{(j)})\big{)}_{\mathds{C}^{d_{0}}}+\sum\limits_{M\in\gamma}\big{(}\mathrm{g}_{M,\bot},\mathcal{U}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}$
$\displaystyle-\sum\limits_{M\in\gamma_{\infty}}\big{(}\mathrm{g}_{M},\mathcal{V}_{M}(\tilde{\psi}^{(j)})\big{)}_{\mathds{C}^{d(M)}}.$

Substituting these identities into (5.20), we conclude that equation (5.19),
and hence, problem (5.17) is solvable if and only if conditions (5.18) are
satisfied. The proof is complete. ∎

In order to check the solvability of problem (5.14), we should substitute
formulae (5.16) into (5.15) and calculate then the terms in solvability
condition (5.18). To avoid unnecessary technicalities, we first do such
calculations for a particular case, when instead of $\varphi_{j}^{(0)}$, we
substitute $\tilde{\psi}^{(j)}$ into (5.15). Once we obtain solvability
conditions (5.18) for such particular case, the needed condition is just a
their linear combinations with coefficients $\alpha_{ij}$.

For $j=1,\ldots,k$ we denote

$\displaystyle{g}_{j}=-\frac{d\hat{\mathcal{H}}_{ex}(\varepsilon)\tilde{\psi}^{(j)}}{d\varepsilon}\bigg{|}_{\varepsilon=0},\qquad$
$\displaystyle{\mathrm{g}}_{M,j}=-\mathrm{P}_{M}^{(0)}\big{(}\tilde{\mathrm{S}}_{M,1}\,\mathcal{U}_{M}(\tilde{\psi}^{(j)})+\tilde{\mathrm{K}}_{M,1}\,\mathcal{U}_{M}^{\prime}(\tilde{\psi}^{(j)})\big{)},$
(5.21)
$\displaystyle{\mathrm{g}}_{ex,j}=\mathrm{i}\Theta_{\Gamma,M_{0}}(0)\mathcal{U}_{ex}(\tilde{\psi}^{(j)}),\qquad$
$\displaystyle{\mathrm{g}}_{M,\bot,j}=-\mathrm{P}_{M,\bot}^{(0)}\big{(}\tilde{\mathrm{S}}_{M,\bot,1}\,\mathcal{U}_{M}^{\prime}(\tilde{\psi}^{(j)})+\tilde{\mathrm{K}}_{M,\bot,1}\,\mathcal{U}_{M}(\tilde{\psi}^{(j)})\big{)}.$

We observe that

$\displaystyle
g_{j}^{(1)}=\sum\limits_{i=1}^{k}\alpha_{ji}g_{i},\qquad\hphantom{\mathrm{g}_{ex}}{\mathrm{g}}_{ex,j}^{(1)}=\sum\limits_{i=1}^{k}\alpha_{ji}{\mathrm{g}}_{ex,i},$
(5.22)
$\displaystyle{\mathrm{g}}_{M,j}^{(1)}=\sum\limits_{i=1}^{k}\alpha_{ji}{\mathrm{g}}_{M,i},\qquad{\mathrm{g}}_{M,\bot,j}^{(1)}=\sum\limits_{i=1}^{k}\alpha_{ji}{\mathrm{g}}_{M,\bot,i}.$

Integrating by parts and taking into consideration the vertex conditions for
$\psi^{(i)}$, we obtain:

$\displaystyle-({g}_{j},\tilde{\psi}^{(l)})_{L_{2}({\gamma_{ex}})}=$
$\displaystyle\left(\frac{dp_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\frac{d\tilde{\psi}^{(j)}}{d\xi},\frac{d\tilde{\psi}^{(l)}}{d\xi}\right)_{L_{2}(\gamma)}+\left(\frac{d\tilde{\psi}^{(l)}}{d\xi},\mathrm{i}\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\tilde{\psi}^{(l)}\right)_{L_{2}(\gamma)}$
(5.23)
$\displaystyle+\left(\mathrm{i}\frac{dq_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\tilde{\psi}^{(j)},\frac{d\tilde{\psi}^{(l)}}{d\xi}\right)_{L_{2}(\gamma)}+\left(\frac{dV_{\gamma}}{d\varepsilon}(\,\cdot\,,0)\tilde{\psi}^{(j)},\tilde{\psi}^{(l)}\right)_{L_{2}(\gamma)}$
$\displaystyle+\sum\limits_{M\in\gamma_{\infty}}\Big{(}\Pi_{\gamma,M,1}\,\mathcal{U}_{M}^{\prime}(\tilde{\psi}^{(j)})-\mathrm{i}\Theta_{\gamma,M,1}\,\mathcal{U}_{M}(\tilde{\psi}^{(j)}),\mathcal{U}_{M}(\tilde{\psi}^{(l)})\Big{)}_{\mathds{C}^{d(M)}}$
$\displaystyle-\mathrm{i}\big{(}\Theta_{\Gamma,M_{0}}\,\mathcal{U}_{\gamma}(\tilde{\psi}^{(j)}),\mathcal{U}_{\gamma}(\tilde{\psi}^{(l)})\big{)}_{\mathds{C}^{d_{0}}}.$

The definition of the matrices $\tilde{\mathrm{S}}_{M,\bot}(\varepsilon)$,
$\tilde{\mathrm{K}}_{M,\bot}(\varepsilon)$ in (4.15), the definition of
$\mathrm{K}_{M,\bot}(0)$, the first identity in (4.12) and identities (4.16),
(4.17) imply that

$\displaystyle\tilde{\mathrm{S}}_{M,1}=\mathrm{S}_{M,1}-\mathrm{i}\mathrm{K}_{M,1}\Theta_{\gamma,M,1},$
$\displaystyle\tilde{\mathrm{K}}_{M,1}=\mathrm{P}_{M}^{(0)}\mathrm{K}_{M,1}\Pi_{\gamma,M}(0),$
$\displaystyle\tilde{\mathrm{S}}_{M,\bot,1}=\Pi_{\gamma,M,1}+\mathrm{S}_{M,1}\Pi_{\gamma,M}(0),\quad$
$\displaystyle\tilde{\mathrm{K}}_{M,\bot,1}=\mathrm{P}_{M,\bot}^{(0)}\big{(}\mathrm{K}_{M,\bot,1}-\mathrm{i}\Theta_{\gamma,M,1}+\mathrm{i}\mathrm{P}_{M,1}\Theta_{\gamma,M}(0)\big{)}.$

We substitute the latter formulae and (5.23), (5.21) into the right hand side
in (5.24) and use identities (4.22), (5.4), (5.5). Comparing then the result
with (2.23), (2.24), (2.25) and employing Lemma 4.4, we see that

$\displaystyle\lambda_{j}(\mathrm{Q})\delta_{jl}=$
$\displaystyle-({g}_{j},\tilde{\psi}^{(l)})_{L_{2}(\gamma_{ex})}-\big{(}{\mathrm{g}}_{ex,j},\mathcal{U}_{\gamma}(\tilde{\psi}^{(l)})\big{)}_{\mathds{C}^{d_{0}}}$
(5.24)
$\displaystyle+\sum\limits_{M\in\gamma_{\infty}}\big{(}{\mathrm{g}}_{M,\bot,j},\mathcal{U}_{M}(\tilde{\psi}^{(l)})\big{)}_{\mathds{C}^{d(M)}}-\sum\limits_{M\in\gamma_{\infty}}\big{(}{\mathrm{g}}_{M,j},\mathcal{V}_{M}(\tilde{\psi}^{(l)})\big{)}_{\mathds{C}^{d(M)}}.$

It follows from these identities and (5.22) that solvability conditions (5.18)
for problem (5.14) can be written as

$\alpha_{jl}\lambda_{l}(\mathrm{Q})=\Lambda_{j}^{(1)}\sum\limits_{i=1}^{k}(\psi^{(i)},\psi^{(l)})_{L_{2}(\gamma_{ex})}\alpha_{ji},\qquad
j,l=1,\ldots,k.$

The above identities mean that $\Lambda_{j}^{(1)}$ and the vectors
$\alpha_{j}:=\big{(}\alpha_{j1}\ \ldots\ \alpha_{jk}\big{)}^{t}$ are the
eigenvalues and the associated eigenvectors of the following equivalent
eigenvalue equations:

$\displaystyle\operatorname{diag}\big{\\{}\lambda_{1}(\mathrm{Q}),\ldots,\lambda_{k}(\mathrm{Q})\big{\\}}\alpha_{j}=\Lambda_{j}^{(1)}\mathrm{G}\alpha_{j},$
(5.25)
$\displaystyle\big{(}\mathrm{G}^{-\frac{1}{2}}\operatorname{diag}\big{\\{}\lambda_{1}(\mathrm{Q}),\ldots,\lambda_{k}(\mathrm{Q})\big{\\}}\mathrm{G}^{-\frac{1}{2}}-\Lambda_{j}^{(1)}\big{)}\mathrm{G}^{\frac{1}{2}}\alpha_{j}=0.$

Here we have also employed the fact that the Gram matrix $\mathrm{G}$ is self-
adjoint and positive definite that ensures the existence of
$\mathrm{G}^{\pm\frac{1}{2}}$. The matrix in the second equation in (5.25) is
self-adjoint and hence, its eigenvalues $\Lambda_{j}^{(1)}$ are real and the
associated eigenvectors can be chosen orthonormalized in $\mathds{C}^{k}$.
These eigenvectors produce the eigenvectors of the first equation in (5.25)
via multiplication by $\mathrm{G}^{-\frac{1}{2}}$. We also observe that
$\Lambda_{j}^{(1)}=\lambda_{j}(\mathrm{Q})=0$, $j=1,\ldots,k_{0}$. In view of
the aforementioned condition $\alpha^{*}\mathrm{G}\alpha=\mathrm{I}_{k}$ and
the first equation in (5.25) we then get easily that

$\displaystyle\alpha_{ji}=0,\hphantom{(\mathrm{G})_{ji}}\qquad
j=1,\ldots,k_{0},\qquad i=k_{0}+1,\ldots,k,$ (5.26)
$\displaystyle(\mathrm{G}\alpha)_{ji}=0,\qquad j=k_{0}+1,\ldots,k,\qquad
i=1,\ldots,k_{0},$

where $(\mathrm{G}\alpha)_{ji}$ is the entry of the matrix $\mathrm{G}\alpha$
in the $i$th row and $j$th column. We also observe that the first equation in
(5.25) can be rewritten for the matrix $\alpha$ as

$\operatorname{diag}\big{\\{}\lambda_{1}(\mathrm{Q}),\ldots,\lambda_{k}(\mathrm{Q})\big{\\}}\alpha=\mathrm{G}\alpha\operatorname{diag}\big{\\{}\Lambda_{1}^{(1)},\ldots,\Lambda_{k}^{(1)}\big{\\}}.$
(5.27)

###  5.3 Structure of pole

In accordance with representation (5.9), the singularity of the operator
$\big{(}\mathcal{H}_{ex}(\varepsilon)-\varepsilon^{2}\lambda\big{)}^{-1}$ at
$\varepsilon=0$ is determined by the sums over $j$ of the corresponding
singular terms in (5.11), (5.12). Let us find these sums.

The condition $\Lambda_{j}^{(1)}\neq 0$ in (5.11) is obviously satisfied only
for $j\geqslant k_{0}+1$. By formula (5.16) we get

$\sum\limits_{j=k_{0}+1}^{k}\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(1)}}\varphi_{j}^{(0)}=\sum\limits_{j=k_{0}+1}^{k}\sum\limits_{i,l=1}^{k}\frac{\alpha_{ji}\overline{\alpha_{jl}}}{\Lambda_{j}^{(1)}}(f,\tilde{\psi}^{(l)})_{L_{2}(\gamma_{ex})}\tilde{\psi}^{(i)}.$
(5.28)

The numbers
$\sum\limits_{j=k_{0}+1}^{k}\frac{\alpha_{ji}\overline{\alpha_{jl}}}{\Lambda_{j}^{(1)}}$
are the entries of the following matrix product

$\begin{pmatrix}\alpha_{k_{0}+11}&\ldots&\alpha_{k1}\\\ \vdots&&\vdots\\\
\alpha_{k_{0}+1k}&\ldots&\alpha_{kk}\end{pmatrix}\operatorname{diag}\big{\\{}(\Lambda_{k_{0}+1}^{(1)})^{-1},\ldots,(\Lambda_{k}^{(1)})^{-1}\big{\\}}\begin{pmatrix}\overline{\alpha_{k_{0}+11}}&\ldots&\overline{\alpha_{k_{0}+1k}}\\\
\vdots&&\vdots\\\
\overline{\alpha_{k1}}&\ldots&\overline{\alpha_{kk}}\end{pmatrix}.$ (5.29)

Employing equation (5.27) and the aforementioned orthonormality condition
$\mathrm{G}\alpha\alpha^{*}=\mathrm{I}_{k}$, it is straightforward to confirm
that matrix product (5.29) is equal to
$\operatorname{diag}\big{\\{}\lambda_{k_{0}+1}^{-1}(\mathrm{Q}),\ldots,\lambda_{k}^{-1}(\mathrm{Q})\big{\\}}$.
Hence, identity (5.28) can be continued as

$\sum\limits_{j=k_{0}+1}^{k}\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(1)}}\varphi_{j}^{(0)}=\sum\limits_{j=k_{0}+1}^{k}\frac{(f,\tilde{\psi}^{(j)})_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(1)}}\tilde{\psi}^{(j)}.$
(5.30)

A similar sum comes from (5.12):

$\sum\limits_{j=1}^{k_{0}}\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(2)}-\lambda}\varphi_{j}^{(0)}=\sum\limits_{j=1}^{k_{0}}\sum\limits_{i,l=1}^{k_{0}}\frac{\alpha_{ji}\overline{\alpha_{jl}}}{\Lambda_{j}^{(2)}-\lambda}(f,\tilde{\psi}^{(l)})_{L_{2}(\gamma_{ex})}\tilde{\psi}^{(i)},$
(5.31)

where we have employed the first identities from (5.26). Here the numbers
$\sum\limits_{j=1}^{k_{0}}\frac{\alpha_{ji}\overline{\alpha_{jl}}}{\Lambda_{j}^{(2)}-\lambda}$
can be regarded as the entries of the following matrix product

$\beta\operatorname{diag}\big{\\{}(\Lambda_{1}^{(2)}-\lambda)^{-1},\ldots,(\Lambda_{k_{0}}^{(2)}-\lambda)^{-1}\big{\\}}\beta^{*},\quad\text{where}\quad\beta:=\begin{pmatrix}\alpha_{11}&\ldots&\alpha_{k_{0}1}\\\
\vdots&&\vdots\\\ \alpha_{1k_{0}}&\ldots&\alpha_{k_{0}k_{0}}\end{pmatrix}.$

We note that the numbers $\Lambda_{j}^{(2)}$ are real, while the imaginary
part of $\lambda$ is non-zero and this is why the diagonal matrix in the above
product is well-defined. In view of the orthonormalization condition
$\alpha^{*}\mathrm{G}\alpha=\mathrm{I}_{k}$ and the first identities in (5.26)
we have $\beta^{*}\mathrm{G}_{0}\beta=\mathrm{I}_{k_{0}}$,
$\mathrm{G}_{0}=(\beta^{-1})^{*}\beta^{-1}$ and hence,

$\displaystyle\beta\operatorname{diag}$
$\displaystyle\big{\\{}(\Lambda_{1}^{(2)}-\lambda)^{-1},\ldots,(\Lambda_{k_{0}}^{(2)}-\lambda)^{-1}\big{\\}}\beta^{*}=\Big{(}(\beta^{-1})^{*}\operatorname{diag}\big{\\{}\Lambda_{1}^{(2)}-\lambda,\ldots,\Lambda_{k_{0}}^{(2)}-\lambda\big{\\}}\beta^{-1}\Big{)}^{-1}$
$\displaystyle=$
$\displaystyle\Big{(}(\beta^{-1})^{*}\operatorname{diag}\big{\\{}\Lambda_{1}^{(2)},\ldots,\Lambda_{k_{0}}^{(2)}\big{\\}}\beta^{-1}-\lambda\mathrm{G}_{0}\Big{)}^{-1}$

and the matrix
$\mathrm{X}_{0}:=(\beta^{-1})^{*}\operatorname{diag}\big{\\{}\Lambda_{1}^{(2)},\ldots,\Lambda_{k_{0}}^{(2)}\big{\\}}\beta^{-1}$
is obviously self-adjoint. Therefore, it follows from (5.31) that

$\sum\limits_{j=1}^{k_{0}}\frac{\big{(}f,\varphi_{j}^{(0)}\big{)}_{L_{2}(\gamma_{ex})}}{\Lambda_{j}^{(2)}-\lambda}\varphi_{j}^{(0)}=\sum\limits_{i=1}^{k_{0}}\mathcal{C}^{(-2)}_{i}(f)\tilde{\psi}^{(i)},$

where the functionals $\mathcal{C}^{(-2)}_{i}$ are defined in (5.8).
Substituting the obtained identity, (5.30) into (5.11), (5.12) and emplyoing
then representation (5.9), we arrive at representation (5.6), (5.7), (5.8).
This completes the proof of Lemma 5.1.

##  6 Analyticity of resolvent

In this section we prove Theorem 2.1. Our main idea is to reduce the equation
for the resolvent, that is,
$(\mathcal{H}_{\varepsilon}-\lambda)u_{\varepsilon}=f$, to a linear system of
equations and to analyze then the dependence of its solution on $\varepsilon$.
Since our graph $\Gamma_{\varepsilon}$ consists of two subgraphs $\Gamma$ and
$\gamma_{\varepsilon}$ containing respectively finite and small edges, it is
natural to consider the function $u_{\varepsilon}$ as a solution to
appropriate boundary value problems on these two graphs and then to match
these solutions on the edges $\mathrm{e}_{i}$, $i\in J_{j}$, $j=1,\ldots,n$,
incident to the vertices $M_{j}$.

Let $f_{\Gamma}\in L_{2}(\Gamma)$, $f_{\gamma}\in L_{2}(\gamma)$ be two
arbitrary functions. In terms of these functions we introduce one more
function $f\in L_{2}(\Gamma_{\varepsilon})$ as $f:=f_{\Gamma}$ on $\Gamma$ and
$f:=\mathcal{S}_{\varepsilon}f_{\gamma}$ on $\gamma_{\varepsilon}$. We let
$u_{\varepsilon}:=(\mathcal{H}_{\varepsilon}-\lambda)^{-1}$ and we see
immediately that the restriction of the function $u_{\varepsilon}$ to the
graph $\Gamma$, that is, the function
$W_{\varepsilon}:=\mathcal{P}_{\Gamma}u_{\varepsilon}$, solves the boundary
value problem for the equation

$(\hat{\mathcal{H}}(\varepsilon)-\lambda)W_{\varepsilon}=f_{\Gamma}\quad\text{on}\quad\Gamma$
(6.1)

subject to homogeneous vertex conditions (2.4) at all vertices $M\in\Gamma$
except for $M_{0}$, while at the vertex $M_{0}$ an inhomogeneous vertex
condition

$\mathcal{U}_{M_{0}}(W_{\varepsilon})=\mathrm{a}(\varepsilon),\qquad\mathrm{a}(\varepsilon):=\begin{pmatrix}a_{1}(\varepsilon)\\\
\vdots\\\ a_{d_{0}}(\varepsilon)\end{pmatrix}$ (6.2)

holds with some constants $a_{i}=a_{i}(\varepsilon)$, $i=1,\ldots,d_{0}$.
These constants are, of course, given by the left hand side in (6.2), but we
treat them as unknown. If we find them, then we can recover the function
$u_{\varepsilon}$ on the subgraph $\Gamma$ as a solution to the above boundary
value problem.

Similarly, we consider the restriction
$\mathcal{P}_{\gamma_{\varepsilon}}u_{\varepsilon}$ of the function
$u_{\varepsilon}$ to the graph $\gamma_{\varepsilon}$ and we rescale it by
means of $\mathcal{S}_{\varepsilon}$ letting
$w_{\varepsilon}:=\mathcal{S}_{\varepsilon}^{-1}\mathcal{P}_{\gamma_{\varepsilon}}u_{\varepsilon}$.
The function $w_{\varepsilon}$ solves the equation
$(\hat{\mathcal{H}}_{ex}(\varepsilon)-\varepsilon^{2}\lambda)w_{\varepsilon}=\varepsilon^{2}f_{\gamma}$
on $\gamma$ and satisfies vertex conditions (4.7). Our next aim is to continue
the function $w_{\varepsilon}$ on the edges $\mathrm{e}_{i}^{ex}$ so that the
continuation solves certain boundary value problem again written in terms of
$\hat{\mathcal{H}}_{ex}(\varepsilon)$ and vertex condition (5.1). In view of
the latter vertex condition, we denote

$a^{\prime}_{i}(\varepsilon):=\mathsf{p}_{i}(\varepsilon)\frac{dw_{\varepsilon}\big{|}_{\mathrm{e}_{i}}}{d\xi_{i}}(M_{j})-\mathrm{i}\varepsilon\mathsf{q}_{i}(\varepsilon)w_{\varepsilon}\big{|}_{\mathrm{e}_{i}}(M_{j}),\qquad\mathrm{a}^{\prime}(\varepsilon):=\begin{pmatrix}a_{1}^{\prime}(\varepsilon)\\\
\vdots\\\ a_{d_{0}}^{\prime}(\varepsilon)\end{pmatrix},$

where $w_{\varepsilon}\big{|}_{\mathrm{e}_{i}}$ is the restriction of
$w_{\varepsilon}$ to an edge $\mathrm{e}_{i}$, $i\in J_{j}$, incident to a
vertex $M_{j}$, $j=1,\ldots,n$. We also observe that
$w_{\varepsilon}\big{|}_{\mathrm{e}_{i}}(M_{j})=a_{i}(\varepsilon)$, where
$a_{i}$ are exactly the constants introduced in (6.2).

We continue the function $w_{\varepsilon}$ on the edges $\mathrm{e}_{i}^{ex}$
as follows: