  We study the nodal properties of many-body eigenstates of stationary
Schr\"odinger equation that affect the accuracy of real-space quantum Monte
Carlo calculations. In particular, we introduce weighted nodal domain averages
that provide a new probe of nodal surfaces beyond the usual expectations.
Particular choices for the weight function reveal, for example, that the
difference between two arbitrary fermionic eigenvalues is given by the nodal
hypersurface integrals normalized by overlaps with the bosonic ground state of
the given Hamiltonian. Noninteracting and fully interacting Be atom with
corresponding almost exact and approximate wave functions are used to
illustrate several aspects of these concepts. Variational formulations that
employ different weights are proposed for prospective improvement of nodes in
variational and fixed-node diffusion Monte Carlo calculations.
