Influence of Auxiliary Equation on Wave Functions for Time-Dependent Pauli Equation in Presence of Aharonov-Bohm Effect

Invariant operator method for discrete or continuous spectrum eigenvalue andunitary transformation approach are employed to study the two-dimensionaltime-dependent Pauli equation in presence of the Aharonov-Bohm effect (AB)and external scalar potential. For the spin particles the problem with themagnetic field is that it introduces a singularity into wave equation at theorigin. A physical motivation is to replace the zero radius flux tube by oneof radius R, with the additional condition that the magnetic field beconfined to the surface of the tube, and then taking the limit R→0 at the end of the computations. We point that the invariant operator must contain the step function θ ( r-R ). Consequently, the problem becomes more complicated. In order to avoid this difficulty, we replace the radius R by ρ( t)R, where ρ (t) is a positive time-dependent function. Then at the end of calculations we take the limit R→0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter $C$ appearing in the nonlinear auxiliaryequation satisfied by ρ( t), i.e., C>0, C=0, and C<0. Following the C's values the spectrum of quantum states is discrete (C>0 ) or continuous (C≤0).