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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2012.03045 |
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| _version_ | 1866913187871326208 |
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| author | Stempak, Krzysztof |
| author_facet | Stempak, Krzysztof |
| contents | Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $Ω_+=Ω\cap C_+$ be the positive part of $Ω$. We define a family $\{-Δ_η^+\}$ of self-adjoint extensions of the Laplacian $-Δ_{Ω_+}$, labeled by homomorphisms $η\colon W\to \{1,-1\}$. In the construction of these $η$-Laplacians $η$-symmetrization of functions on $Ω$ is involved. The Neumann Laplacian $-Δ_{N,Ω_+}$ is included and corresponds to $η\equiv1$. If $H^{1}(Ω)=H^{1}_0(Ω)$, then the Dirichlet Laplacian $-Δ_{D,Ω_+}$ is either included and corresponds to $η={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_03045 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Finite reflection groups and symmetric extensions of Laplacian Stempak, Krzysztof Functional Analysis Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $Ω_+=Ω\cap C_+$ be the positive part of $Ω$. We define a family $\{-Δ_η^+\}$ of self-adjoint extensions of the Laplacian $-Δ_{Ω_+}$, labeled by homomorphisms $η\colon W\to \{1,-1\}$. In the construction of these $η$-Laplacians $η$-symmetrization of functions on $Ω$ is involved. The Neumann Laplacian $-Δ_{N,Ω_+}$ is included and corresponds to $η\equiv1$. If $H^{1}(Ω)=H^{1}_0(Ω)$, then the Dirichlet Laplacian $-Δ_{D,Ω_+}$ is either included and corresponds to $η={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$. |
| title | Finite reflection groups and symmetric extensions of Laplacian |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2012.03045 |