  We investigate quasi-free Hadamard states defined via characteristic initial
data on null cones centred at the axis of symmetry in spherically symmetric
space-times. We characterize the necessary singular behaviour of null boundary
two-point functions such that one can define non-linear observables at this
null boundary and give formulas for the calculation of these observables. These
results extend earlier characterizations of null boundary states defining
Hadamard states in the bulk of the null cone. As an application of our derived
formulas, we consider their implications for the semi-classical Einstein
equations and calculate the vacuum polarization associated with Hawking
radiation near a collapsing body.
