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

On linear hypersingular Boltzmann transport equation and its variational formulation