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Ellipticity of quantum mechanical Hamiltonians in the edge algebra

Pages: 420 - 429, Issue Special, September 2011

 Abstract        Full Text (377.2K)              

Heinz-Jürgen Flad - Institut für Mathematik, Technische Universität Berlin, Strabe des 17. Juni 136, D-10623 Berlin, Germany (email)
Gohar Harutyunyan - Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D-26111 Oldenburg, 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 di erential operator in a pseudo-di erential 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 di erential operator in a pseudo-di erential operator algebra. In the edge degenerate case, we prove the ellipticity of the Hamiltonian.

Keywords:  Schrödinger equation, helium atom, edge singularity, ellipticity
Mathematics Subject Classification:  Primary: 35J10, 81V45; Secondary: 35J47, 35A17, 35A20

Received: July 2010;      Revised: February 2011;      Published: October 2011.