Relative Periodic Orbits in the Spatial Anisotropic Kepler Problem

The spatial anisotropic Kepler problem comes from quantum mechanics, which models electron motion in semiconductors with donor impurities and depends on the anisotropic parameter $\beta \in (-1,+\mathrm{\infty })$. After reducing the system modulo rotational symmetry, we investigate periodic orbits in this two-degree-of-freedom setting. Combining an index comparison in \cite{HLOQS26}, the volume formula in \cite{CHHL23} and Franks’ theorem, we prove that the system possesses infinitely many periodic orbits on any fixed compact and regular energy surface for $\beta \in (-1,0]$.