Ellipticity of quantum mechanical Hamiltonians in the edge algebra

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

Ellipticity of quantum mechanical Hamiltonians in the edge algebra

Heinz-Jürgen Flad ^1, and Gohar Harutyunyan ^2,

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

Abstract
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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: edge singularity, helium atom, Schrödinger equation, ellipticity.

Mathematics Subject Classification: Primary: 35J10, 81V45; Secondary: 35J47, 35A17, 35A20.

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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