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We have studied possible applications of a particular pseudo-differentialalgebra in singular analysis for the construction of fundamental solutions andGreen's functions of a certain class of elliptic partial differentialoperators. The pseudo-differential algebra considered in the present work,comprises degenerate partial differential operators on stretched cones whichcan be locally described as Fuchs type differential operators in appropriatepolar coordinates. We present a general approach for the explicit constructionof their parametrices, which is based on the concept of an asymptoticparametrix, introduced in \cite{FHS16}. For some selected partial differentialoperators, we demonstrate the feasibility of our approach by an explicitcalculation of fundamental solutions and Green's functions from thecorresponding parametrices. In our approach, the Green's functions are given inseparable form, which generalizes the Laplace expansion of the Green's functionof the Laplace operator in three dimensions. As a concrete application inquantum scattering theory, we construct a fundamental solution of asingle-particle Hamilton operator with singular Coulomb potential.

Fundamental Solutions and Green's Functions for certain Elliptic Differential Operators from a Pseudo-Differential Algebra