Defect in the Joint Spectrum of Hydrogen due to Monodromy

Abstract: In addition to the well known case of spherical coordinates the hydrogen atom separates in three further coordinate systems. Separating in a particular coordinate system defines a system of three commuting operators. We show that the joint spectrum of the Hamilton operator, and the $z$-components of the angular momentum and quantum Laplace-Runge-Lenz vectors obtained from separation in prolate spheroidal coordinates has quantum monodromy for energies sufficiently close to the ionization threshold. This means that one cannot globally assign quantum numbers to the joint spectrum. Whereas the principal quantum number $n$ and the magnetic quantum number $m$ correspond to the Bohr-Sommerfeld quantization of globally defined classical actions a third quantum number cannot be globally defined because the third action is globally multi valued.