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

Abstract

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 HamiltonianWe 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), 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 Ψα,N , and their derivatives L3(h)Ψα,N , in the large quantum number regime N → ∞.