Three-dimensional eigenfunctions of the one-speed transport operator

The Case singular eigenfunction expansion method to solve, by separation of variables, the one-dimensional transport equation in plane geometry has widely been used in different fields of physics theoretical methods applications. Using the rigged Hilbert spaces approach, a modified one-dimensional transport equation has been derived to prove the singular eigenfunctions completeness. In this paper, motivated by the two approaches, an attempt is made to generalize it to a three-dimensional (3D) transport equation. To begin, an equivalent 3D transport equation with self-adjoint modified transport operator with compact inverse has been derived. A general solution to the equation is being expressed by exponential function of the modified transport operator.