## Abstract

We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian H_{V}(h, B) = (1 / 2) (- ıh∇ - A(h)) ^{2}- | x| ^{- 1}, defined on L^{2}(R^{3}) , in a constant magnetic field B(h) = ∇ × A(h) = (0 , 0 , ϵ(h) B) in the weak field limit ϵ(h) → 0 as h→ 0. We consider the Planck’s parameter h taking values along the sequence h= 1 / (N+ 1) , with N= 0 , 1 , 2 , … , and N→ ∞. We prove a semiclassical N→ ∞ LEDT of the Szegö-type for the scaled eigenvalue shifts and obtain both (i) an expression involving the regularized classical Kepler orbits with energy E= - 1 / 2 and (ii) a weak limit measure that involves the component ℓ_{3} of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szegö-type for eigenvalue clusters for bounded perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this work is that the perturbation involves the unbounded, first-order, partial differential operator w(h,B)=(ϵ(h)B)28(x12+x22)-ϵ(h)B2hL3, where the operator hL_{3} is the third component of the usual angular momentum operator and is the quantization of ℓ_{3}. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states Ψ _{α} _{,} _{N}, and their derivatives L_{3}(h) Ψ _{α} _{,} _{N}, in the large quantum number regime N→ ∞.

Original language | English |
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Pages (from-to) | 3933-3973 |

Number of pages | 41 |

Journal | Annales Henri Poincare |

Volume | 18 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2017 |

### Bibliographical note

Funding Information:PDH was partially supported by NSF Grants 0803379 and 1103104 during the time this work was done. CV-B was partially supported by the projects PAPIIT-UNAM IN106812, PAPIIT-UNAM IN104015 and thanks the members of the Department of Mathematics of the University of Kentucky for their hospitality during a visit. MA-C was supported by a fellowship of DGAPA-UNAM, by Project PAPIIT-UNAM IN106812, and by CONACYT under the Grants 219631, CB-2013-01 and 258302, CB-2015-01. The authors want to

Publisher Copyright:

© 2017, Springer International Publishing AG.

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics