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

For an open subset Omega of 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, infinity) 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 the particular cases of Omega(+) = Rd-1 x (0, infinity) and Phi(t)(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.