Explicit Green operators for quantum mechanical Hamiltonians.~II.~ Edge type singularities of the helium atom

TL;DR

This paper develops explicit Green operators for quantum Hamiltonians with edge singularities, specifically applied to the helium atom, enabling detailed analysis of eigenfunction behavior near coalescence points.

Contribution

It extends asymptotic parametrix construction methods from conical to edge-type singularities for quantum Hamiltonians, providing explicit formulas for Green operators.

Findings

Calculated symbols of asymptotic parametrix up to second order.

Derived explicit formulas for Green operators near edges.

Applied methods to helium atom and related two-electron systems.

Abstract

We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second order we have calculated the symbols of an asymptotic parametrix of the nonrelativisic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behaviour of the eigenfunctons near an edge.