Friday, May 11, 2012

1205.2260 (Matěj Tušek)

Atoms confined by very thin layers    [PDF]

Matěj Tušek
We consider the Hamiltonian of an atom with $N$ electrons and a fixed nucleus in a very thin plane-parallel layer. Projecting this Hamiltonian on the lowest transverse mode we obtain the so-called effective Hamiltonian that acts on $L^{2}(\R^{2N})$ and whose potential part depends on the width, $a$, of the layer. We prove that this effective Hamiltonian tends, in the norm resolvent sense, to the Hamiltonian of a two-dimensional atom (with the three-dimensional Coulomb potential) as $a\to 0+$. Finally we demonstrate how to localize the bottom of spectrum of the initial full Hamiltonian with the knowledge of the spectrum of the latter one. The analyticity and the monotonicity of eigenvalues in $a$ is also discussed.
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