Abstract
We demonstrate that it is possible to construct elementary solutions (eigenfunctions) of the linear transport equation for certain types of continuously varying spatial media. In general, both discrete and continuum modes result, which appear to be complete on the half-range. A detailed analysis is given for an ’’exponential’’ medium, including numerical results and a half-range completeness proof. A ’’linear’’ medium is also considered. A general method is presented for constructing, jointly, the spatial variation of a medium and the corresponding functional forms of the eigenfunctions. Our results represent a partial generalization of the singular eigenfunction technique to media with continuous spatial variation.
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