BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908457201827840 |
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| author | Gu, Qingsong Qiu, Hua |
| author_facet | Gu, Qingsong Qiu, Hua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14858 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians Gu, Qingsong Qiu, Hua Functional Analysis 28A80, 31E05 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. |
| title | BGD domains in p.c.f. self-similar sets II: spectral asymptotics for Laplacians |
| topic | Functional Analysis 28A80, 31E05 |
| url | https://arxiv.org/abs/2507.14858 |