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### Session S30 - Mathematical Methods in Quantum Mechanics

Thursday, July 15, 19:30 ~ 19:55 UTC-3

## On limiting eigenvalue distribution theorems for clusters and sub-clusters of the hydrogen atom in a weak constant magnetic field

### Carlos Villegas-Blas

We consider the hydrogen atom under the action of a weak constant magnetic field $H_V(\hbar,B) = - \frac{\hbar^2}{2} \Delta - \frac{1}{|x|} - \frac{\epsilon (\hbar)B}{2}\hbar L_3 + \frac{(\epsilon (\hbar)B)^2}{8}(x_1^2 + x_2^2)$ with $\hbar$ the Planck parameter, and the constant magnetic field ${\bf B}(\hbar)=(0,0,\epsilon (\hbar)B)$ with $B>0$ fixed and $\epsilon (\hbar)=\hbar^q$ used to control the strenght of the magnetic field with respect to $\hbar$.
We study the eigenvalue distribution in suitable defined clusters and sub-clusters of the Hamiltonian $H_V(\hbar,B)$ around the energy $E=-1/2$ in the semiclassical regime $\hbar\rightarrow{0}$. We show that for $q>33/2$ and $q>19$, the clusters and sub-clusters are well defined. For the clusters, we show that the limiting eigenvalue distribution is given by an explicit measure determined by the values of $\frac{B}{2} \ell_3 ({\bf x, p})$ along the classical Kepler orbits on the surface energy $E=-1/2$ where $\ell_3 ({\bf x,p}) = x _1p_2 - x_2 p_1$ is the third component of the classical angular momentum vector. For the sub-clusters case, we show that the limiting eigenvalue distribution involves averages of $\frac{B^2}{8}(x_1^2+x_2^2)$ along the classical Kepler orbits with energy $E=-1/2$ and a previously chosen value of $\ell_3 ({\bf x,p})$.