Reflection principles for functions of Neumann and Dirichlet Laplacians on open reflection invariant subsets of R

For an open subset Ω of R, symmetric with respect to a hyperplane and with positive part Ω+, we consider the Neumann/Dirichlet Laplacians −∆N/D,Ω and −∆N/D,Ω+ . Given a Borel function Φ on [0,∞) we apply the spectral functional calculus and consider the pairs of operators Φ(−∆N,Ω) and Φ(−∆N,Ω+), or Φ(−∆D,Ω) and Φ(−∆D,Ω+). We prove relations between the integral kernels for the operators in these pairs, which in the particular cases of Ω+ = Rd−1 × (0,∞) and Φt(u) = exp(−tu), u ≥ 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.