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For an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to ahyperplane and with positive part $\Omega_+$, we consider the Neumann/DirichletLaplacians $-\Delta_{N/D,\Omega}$ and $-\Delta_{N/D,\Omega_+}$. Given a Borelfunction $\Phi$ on $[0,\infty)$ we apply the spectral functional calculus andconsider the pairs of operators $\Phi(-\Delta_{N,\Omega})$ and$\Phi(-\Delta_{N,\Omega_+})$, or $\Phi(-\Delta_{D,\Omega})$ and$\Phi(-\Delta_{D,\Omega_+})$. We prove relations between the integral kernelsfor the operators in these pairs, which in particular cases of$\Omega_+=\mathbb{R}^{d-1}\times(0,\infty)$ and $\Phi_{t}(u)=\exp(-tu)$, $u\geq 0$, $t>0$, were known as reflection principles for the Neumann/Dirichletheat kernels. These relations are then generalized to the context of symmetrywith respect to a finite number of mutually orthogonal hyperplanes.

Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of $\mathbb{R}^d$