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

## TL;DR

This paper establishes reflection principles relating the integral kernels of spectral functions of Neumann and Dirichlet Laplacians on symmetric open sets, generalizing known heat kernel reflection principles to multiple hyperplanes.

## Contribution

It introduces new relations between spectral operators of Laplacians on symmetric domains, extending classical heat kernel reflection principles to broader spectral functions and multiple symmetries.

## Key findings

- Derived kernel relations for spectral functions of Laplacians on symmetric domains.
- Generalized reflection principles to multiple hyperplanes.
- Extended classical heat kernel reflection principles to spectral calculus.

## Abstract

For an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to a hyperplane and with positive part $\Omega_+$, we consider the Neumann/Dirichlet Laplacians $-\Delta_{N/D,\Omega}$ and $-\Delta_{N/D,\Omega_+}$. Given a Borel function $\Phi$ on $[0,\infty)$ we apply the spectral functional calculus and consider 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 kernels for 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/Dirichlet heat kernels. These relations are then generalized to the context of symmetry with respect to a finite number of mutually orthogonal hyperplanes.