# American Institute of Mathematical Sciences

2011, 2011(Special): 420-429. doi: 10.3934/proc.2011.2011.420

## Ellipticity of quantum mechanical Hamiltonians in the edge algebra

 1 Institut für Mathematik, Technische Universität Berlin, Strabe des 17. Juni 136, D-10623 Berlin, Germany 2 Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D-26111 Oldenburg, Germany

Received  July 2010 Revised  February 2011 Published  October 2011

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.
Citation: Heinz-Jürgen Flad, Gohar Harutyunyan. Ellipticity of quantum mechanical Hamiltonians in the edge algebra. Conference Publications, 2011, 2011 (Special) : 420-429. doi: 10.3934/proc.2011.2011.420
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