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Christian Hainzl

Publications and source records attributed to Christian Hainzl.

77 records · Page 5Linked to original sources
arXiv2002

Increase of the binding energy of an electron by coupling to a photon field

We look at an electron in the field of an arbitrary external potential $V$, such that the Schrödinger operator $p^2 + V$ has at least one eigenvalue, and show that by coupling to a quantized radiation field the binding energy increases, at least for small enough values of the coupling constant $α$. Moreover, we provide concrete numbers for $α$, the ultraviolet cut-off $Λ$, and the radiative correction for which our procedure works.

math-ph↗
arXiv2001

Semiclassics in the lowest Landau band

This paper deals with the comparison between the strong Thomas-Fermi theory and the quantum mechanical ground state energy of a large atom confined to lowest Landau band wave functions. Using the tools of microlocal semiclassical spectral asymptotics we derive precise error estimates. The approach presented in this paper suggests the definition of a modified strong Thomas-Fermi functional, where the main modification consists in replacing the integration over the variables perpendicular to the magnetic field by an expansion in angular momentum eigenfunctions. The resulting DSTF theory is studied in detail in the second part of the paper.

math-ph↗
arXiv2000

Gradient corrections for semiclassical theories of atoms in strong magnetic fields

This paper is divided into two parts. In the first one the von Weizsäcker term is introduced to the Magnetic TF theory and the resulting MTFW functional is mathematically analyzed. In particular, it is shown that the von Weizsäcker term produces the Scott correction up to magnetic fields of order $B \ll Z^2$, in accordance with a result of V. Ivrii on the quantum mechanical ground state energy. The second part is dedicated to gradient corrections for semiclassical theories of atoms restricted to electrons in the lowest Landau band. We consider modifications of the Thomas-Fermi theory for strong magnetic fields (STF), i.e. for $B \ll Z^3$. The main modification consists in replacing the integration over the variables perpendicular to the field by an expansion in angular momentum eigenfunctions in the lowest Landau band. This leads to a functional (DSTF) depending on a sequence of one-dimensional densities. For a one-dimensional Fermi gas the analogue of a Weizsäcker correction has a negative sign and we discuss the corresponding modification of the DSTF functional.

math-ph↗
arXiv2000

A discrete density matrix theory for atoms in strong magnetic fields

This paper concerns the asymptotic ground state properties of heavy atoms in strong, homogeneous magnetic fields. In the limit when the nuclear charge Z tends to infinity with the magnetic field B satisfying B >> Z^{4/3} all the electrons are confined to the lowest Landau band. We consider here an energy functional, whose variable is a sequence of one-dimensional density matrices corresponding to different angular momentum functions in the lowest Landau band. We study this functional in detail and derive various interesting properties, which are compared with the density matrix (DM) theory introduced by Lieb, Solovej and Yngvason. In contrast to the DM theory the variable perpendicular to the field is replaced by the discrete angular momentum quantum numbers. Hence we call the new functional a discrete density matrix (DDM) functional. We relate this DDM theory to the lowest Landau band quantum mechanics and show that it reproduces correctly the ground state energy apart from errors due to the indirect part of the Coulomb interaction energy.

math-ph↗