Asymptotics of Spectral Clusters for a Perturbation of the Hydrogen Atom

Abstract

We consider perturbations of the Schrödinger operator of the hydrogen atom, of size $O(\hbar^{1+\delta})$ with $\delta > 0$ . We show that, if $\hbar$ is restricted to take values along a certain sequence converging to zero, the multiple eigenvalue −1/2 of the unperturbed Hamiltonian breaks into a cluster of eigenvalues disjoint from the rest of the spectrum. Then we obtain a Szegö limit theorem for the spectral shifts in this cluster, as $\hbar\to 0$ . Specifically, we prove: The weak limit of the normalized spectral measure of the spectral shifts is the push-forward of Liouville measure of the unperturbed energy surface H =  −1/2 by the averaged symbol of the perturbation.