\label{thm:self-adjointness-h-l}
The operator $H_L$ is self-adjoint: $H_L^* = H_L$.

We compute the boundary form. For $f, g \in \mathrm{Dom}(H_L)$, integration by parts on each edge gives
\begin{align*}
\langle H_L f, g \rangle - \langle f, H_L g \rangle
&= -\int_{e_B} f''_B \overline{g_B} \, dx - \int_{e_C} f''_C \overline{g_C} \, dx \\
&\quad + \int_{e_B} f_B \overline{g''_B} \, dx + \int_{e_C} f_C \overline{g''_C} \, dx \\
&= -\left[ f'_B \overline{g_B} - f_B \overline{g'_B} \right]_0^1 - \left[ f'_C \overline{g_C} - f_C \overline{g'_C} \right]_0^1.
\end{align*}
Evaluating at the endpoints $0$ and $1$ (both corresponding to $\omega$), continuity gives $f_B(\omega) = f_C(\omega)$ and $g_B(\omega) = g_C(\omega)$, so
\begin{align*}
\langle H_L f, g \rangle - \langle f, H_L g \rangle
&= -\Big[ f'_B(\omega) \overline{g_B(\omega)} - f_B(\omega) \overline{g'_B(\omega)} \\
&\quad\quad + f'_C(\omega) \overline{g_C(\omega)} - f_C(\omega) \overline{g'_C(\omega)} \Big] \\
&\quad + \text{(terms at $x=0$ cancel by periodicity)} \\
&= -\overline{g_B(\omega)} \Big( f'_B(\omega) + f'_C(\omega) \Big) + f_B(\omega) \overline{\Big( g'_B(\omega) + g'_C(\omega) \Big)}.
\end{align*}
The Kirchhoff condition forces $f'_B(\omega) + f'_C(\omega) = 0$ and $g'_B(\omega) + g'_C(\omega) = 0$, so both terms vanish:
\[
\langle H_L f, g \rangle - \langle f, H_L g \rangle = 0.
\]
Thus $H_L \subset H_L^*$. For the reverse inclusion, standard von Neumann theory~\cite{ReedSimon1980} shows that the Kirchhoff condition is \emph{maximal} among boundary conditions yielding symmetric operators on $L$. Therefore $H_L^* = H_L$.