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

Abstract

We have studied possible applications of a particular pseudo-differential algebra in singular analysis for the construction of fundamental solutions and Green's functions of a certain class of elliptic partial differential operators. The pseudo-differential algebra considered in the present work, comprises degenerate partial differential operators on stretched cones which can be locally described as Fuchs type differential operators in appropriate polar coordinates. We present a general approach for the explicit construction of their parametrices, which is based on the concept of an asymptotic parametrix, introduced in \cite{FHS16}. For some selected partial differential operators, we demonstrate the feasibility of our approach by an explicit calculation of fundamental solutions and Green's functions from the corresponding parametrices. In our approach, the Green's functions are given in separable form, which generalizes the Laplace expansion of the Green's function of the Laplace operator in three dimensions. As a concrete application in quantum scattering theory, we construct a fundamental solution of a single-particle Hamilton operator with singular Coulomb potential.