Ellipticity of quantum mechanical Hamiltonians in the edge algebra

Abstract

We have studied the ellipticity of quantum mechanical Hamiltonians, in particular of the helium atom, in order to prove existence of a parametrix and corresponding Green operator. The parametrix is considered in local neighbourhoods of coalescence points of two particles. We introduce appropriate hyperspherical coordinates where the singularities of the Coulomb potential are considered as embedded edge/corner-type singularities. This shows that the Hamiltonian can be written as an edge/corner degenerate differential operator in a pseudo-differential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.