BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians

Let $K$ be a p.c.f. self-similar set equipped with a strongly recurrent Dirichlet form. Under a homogeneity assumption, for an open set $Ω\subset K$ whose boundary $\partial Ω$ is a graph-directed self-similar set, we prove that the eigenvalue counting function $ρ^Ω(x)$ of the Laplacian with Dirichlet or Neumann boundary conditions (Neumann only for connected $Ω$) has an explicit second term as $x\to +\infty$, beyond the dominant Weyl term. If $\partialΩ$ has a strong iterated structure, we establish that \begin{equation*} ρ^Ω(x)=ν(Ω)G\Big(\frac{\log x}2\Big)x^{\frac{d_S}2}+κ(\partialΩ)G_1\Big(\frac{\log x}2\Big)x^{\frac d2}+o\big(x^{\frac d2}\big), \end{equation*} where $G$ and $G_1$ are bounded periodic functions, $ν$ and $κ$ are certain reference measures, and $d_S$ and $d$ are dimension-related parameters.