Rydberg atoms in weak magnetic fields

The quadratic Zeeman effect of Rydberg atoms is treated with the help of first-order perturbation theory and a special WKB approximation. The Bohr-Sommerfeld quantization leads to complete elliptic integrals. As a result we found that the spectrum is doubly degenerate if $\frac{n^2}{m^2>5\mathrm{}}$, this part exhibiting a linear Stark effect in an additional weak electric field; the rest of the spectrum ($\frac{n^2}{m^2<5\mathrm{}}$) is nondegenerate, exhibiting a quadratic Stark effect in an additional weak electric field.