Semiclassical Szeg\"o 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 $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$,definedon$L^2 (R^3)$,inaconstantmagneticfield${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,\epsilon(h)B)$intheweakfieldlimit$\epsilon(h) \rightarrow 0$as$h\rightarrow{0}$\mathrm{.}WeconsiderthePlanc{k}^{\mathrm{\prime }}sparameter$h$takingvaluesalongthesequence$h=1/(N+1)$,with$N=0,1,2,\ldots$,and$N\rightarrow\infty$\mathrm{.}Weproveasemiclassical$N \rightarrow \infty$ LEDT of the Szeg"o-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an expression involving the regularized classical Kepler orbits with energy $E=-1\mathrm{/}2$ and ({\bf ii}) a weak limit measure that involves the component ${\mathrm{\ell }}_3$ of the angular momentum vector in the direction of the magnetic field. This LEDT extends results of Szeg"o-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 $w(h, B) = \frac{(\epsilon(h)B)^2}{8} (x_1^2 + x_2^2) - \frac{ \epsilon(h)B}{2} hL_3 ,$wheretheoperator$hL_3$ is the third component of the usual angular momentum operator and is the quantization of ${\mathrm{\ell }}_3$. The unbounded Zeeman perturbation is controlled using localization properties of both the hydrogen atom coherent states ${\mathrm{\Psi }}_{\alpha ,N}$, and their derivatives $L_3(h){\mathrm{\Psi }}_{\alpha ,N}$, in the large quantum number regime $N\to \mathrm{\infty }$.