On the two-dimensional Coulomb-like potential with a central point interaction

Details On the two-dimensional Coulomb-like potential with a central point interaction On the two-dimensional Coulomb-like potential with a central point interaction

2010-06-30

arxiv.org/abs/1006.5952v2

J. Phys. A: Math. Theor. 43 (2010) art. no. 474020

Physics

Mathematical Physics

Scientific paper

In the first part of the paper, we introduce the Hamiltonian $-\Delta-Z/\sqrt{x^2+y^2}$, Z>0, as a selfadjoint operator in $L^2(R^2)$. A general central point interaction combined with the two-dimensional Coulomb-like potential is constructed and properties of the resulting one-parameter family of Hamiltonians is studied in detail. The construction is also reformulated in the momentum representation and a relation between the coordinate and the momentum representation is derived. In the second part of the paper we prove that the two-dimensional Coulomb-like Hamiltonian can be derived as a norm resolvent limit of the Hamiltonian of a Hydrogen atom in a planar slab as the width of the slab tends to zero.

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