Blaschke-term interpretation of multiple-π geometric phases

We consider a general slowly (adiabatically) moving system represented by a two-component wave function. Variation in the slowness of motion along the trajectory can result in (odd) multiple-π jumps in the geometric phase, in place of the usual ±π phases. These large multiple phase jumps have recently been obtained by simulation of molecular trajectories. By deriving a perturbational solution, exact in the adiabatic limit, we here quantitatively equate them with the relative decrease of the speed of motion at the instant of the jump. These jumps are further identified with Blaschke terms in the Hilbert transform expression for the time (t) dependent phase, terms that arise in the ground state from wave-function-zeros in the lower half of the complex t-plane.