  For charged particle transport the linear Boltzmann transport equation (BTE)
turns out to be a partial hyper-singular integro-differential operator. This is
due to the fact that the related differential cross-sections
$\sigma(x,\omega',\omega,E',E)$ may have hyper-singularities. In these cases
the energy integral appearing in the collision terms must be interpreted as the
Hadamard finite part integral leading to hyper-singular integral operators. The
article considers a refined expression for the exact transport operator and
related variational formulations of the inflow initial boundary value problem
for one particle equation containing hyper-singularities. We find that the
exact BTE contains the first-order partial derivatives with respect to energy
combined by partial Hadamard (first-order) singular integral operators. In
addition, it contains the second-order partial derivatives with respect to
angle and some mixed terms. The analysis will be carried out only for the so
called M{\o}ller-type interaction (scattering) which is a kind of prototype of
hyper-singular interactions. The generalizations to other type of collisions,
such as to Bremsstrahlung, go analogously. We also expose a weak form (the
variational formulation) of the hyper-singular transport problem. Another
variant variational formulation decreases the level of singularities in the
integration (appearing in the due bilinear form) containing only singularities
of order one that is, singularities like ${1\over{E'-E}}dE' dE$. The
variational formulation is an essential step in order to show the existence of
generalized solutions e.g. by Lions-Lax-Milgram Theorem based methods
(proceedings for solution spaces and existence theory are omitted here). The
corresponding approximative transport operator is deduced. It turns out to be a
CSDA-Fokker-Planck type operator.
