The Zeeman effect for the 'anisotropic hydrogen atoms' in the complex WKB approximation: II. Quantization of two-dimensional Lagrangian tori (with focal points) for the Pauli operator with spin-orbit interaction

Abstract

For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.