Semiclassical Szegö Limit of Eigenvalue Clusters for the Hydrogen Atom Zeeman Hamiltonian

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 L²(R³) , 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 ℓ₃ 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 hL₃ is the third component of the usual angular momentum operator and is the quantization of ℓ₃. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states Ψ _{α , N}, and their derivatives L₃(h) Ψ _{α , N}, in the large quantum number regime N→ ∞.