\label{prop:discrete-spectrum-h-l}
The spectrum of $H_L$ consists of isolated eigenvalues
\[
\operatorname{Spec}(H_L) = \{ \Lambda_0, \Lambda_1, \Lambda_2, \ldots \}
\]
with $0 \leq \Lambda_0 < \Lambda_1 < \Lambda_2 < \cdots$ and $\Lambda_n \to +\infty$ as $n \to \infty$. Each eigenvalue has finite multiplicity.

The metric graph $L$ is compact (total length $2$), so the embedding $H^2(L) \hookrightarrow L^2(L)$ is compact by the Rellich--Kondrachov theorem. The resolvent $(H_L - \lambda)^{-1}$ is compact for $\lambda \notin \operatorname{Spec}(H_L)$, hence $H_L$ has discrete spectrum~\cite{ReedSimon1980}.

Eigenvalues are real by self-adjointness (Theorem~\ref{thm:self-adjointness-h-l}). The lowest eigenvalue $\Lambda_0 = 0$ corresponds to constant functions (which satisfy Kirchhoff automatically). Positivity of higher eigenvalues follows from the variational characterization
\[
\Lambda_n = \inf_{\substack{f \in \mathrm{Dom}(H_L) \\ \langle f, \psi_k \rangle = 0, \, k < n}} \frac{\langle H_L f, f \rangle}{\langle f, f \rangle} = \inf \frac{\int_L |f'|^2}{\int_L |f|^2},
\]
where the infimum is over functions orthogonal to $\psi_0, \ldots, \psi_{n-1}$ (the lower eigenfunctions). Since $|f'|^2 \geq 0$, all $\Lambda_n \geq 0$.