Finite reflection groups and symmetric extensions of Laplacian
Volume 261 / 2021
Abstract
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 $\Omega $ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\Omega _+=\Omega \cap C_+$ be the positive part of $\Omega $. We define a family $\{-\Delta _{\eta }^+\}$ of self-adjoint extensions of the Laplacian $-\Delta _{\Omega_+ }$, labeled by homomorphisms $\eta \colon W\to \{1,-1\}$. In the construction of these $\eta $-Laplacians, $\eta $-symmetrization of functions on $\Omega $ is involved. The Neumann Laplacian $-\Delta _{N,\Omega _+}$ is included and corresponds to $\eta \equiv 1$. If $H^{1}(\Omega )=H^{1}_0(\Omega )$, then the Dirichlet Laplacian $-\Delta _{D,\Omega _+}$ is also included and corresponds to $\eta ={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\Psi (-\Delta _{N,\Omega })$ and $\Psi (-\Delta _{\eta }^+)$, or $\Psi (-\Delta _{D,\Omega })$ and $\Psi (-\Delta _{D,\Omega _+})$, where $\Psi $ 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 $W$.
