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

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 ${H_V(h,B) = (1/2)( - \imath h {{\nabla }} - {{\mathbf {A}}}(h))^2 - |\mathbf{x}|^{-1}}$, defined on $L^2 (\mathbb {R}^3)$, in a constant magnetic field ${\mathbf {B}}(h) = {{\nabla }} \times {{\mathbf {A}}}(h)=(0,0,\epsilon (h)B)$ in the weak field limit $\epsilon (h) \rightarrow 0$ as $h\rightarrow {0}$. We consider the Planck’s parameter h taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow \infty$. We prove a semiclassical $N \rightarrow \infty$ 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 $\ell _3$ 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 $w(h, B) = \frac{(\epsilon (h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon (h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third component of the usual angular momentum operator and is the quantization of $\ell _3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states ${\Psi _{{\varvec{\alpha }},N}}$, and their derivatives ${L_3(h)\Psi _{{\varvec{\alpha }},N}}$, in the large quantum number regime $N\rightarrow \infty$.