Ellipticity of quantum mechanical Hamiltonians in the edge algebra
Heinz-Jürgen Flad - Institut für Mathematik, Technische Universität Berlin, Strabe des 17. Juni 136, D-10623 Berlin, Germany (email) 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 dierential operator in a pseudo-dierential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.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 dierential operator in a pseudo-dierential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.
Keywords: Schrödinger equation, helium atom, edge singularity, ellipticity
Received: July 2010; Revised: February 2011; Published: October 2011. |