GENERALIZED DEMKOV MODEL - STRONG-COUPLING APPROXIMATION

A recent discussion and derivation of the strong-coupling form of the transition probability in the generalized Demkov model is shown to be incorrect. By the same token, our previous derivations of the strong-c oupling forms of the transition probabilities for both the generalized Demkov and the three-parameter exponential Nikitin models are shown t o be correct. The recent error is traced to an inconsistent differenti ation of the amplitude (as against the phase) of the semiclassical wav efunction at the classical turning point, at which by construction the non-analytic total wavefunction is discontinuous in its second-order derivative, which follows from the cusp-like discontinuity in the firs t-order derivative of at least the off-diagonal Hamiltonian potential matrix elements. It is concluded that caution must be exercised when i nvoking comparison equations and their asymptotics and that our recent generalizations to complex energies and interactions are also correct . It is also pointed out that the hyperbolic sine component of the tra nsition amplitude in the recent incorrect discussion does not appear i n the analytic Rosen-Zener model, and would otherwise deny the existen ce of Stueckelberg oscillations. It is concluded also that Stueckelber g phase-integral derivations are essential if transition probabilities and total cross sections are not to be under-estimated at large impac t parameters, due to a neglect of the bending of double Stokes' lines. Also consistent with our established phase-integral analysis for the non-crossing collision, we confirm the small second-order correction t o the Stueckelberg frequency of oscillation, which was derived by Vita nov, and which we interpret as the argument of a Stokes' constant; we use the method of steepest descent for this derivation, avoiding the u nnecessary reversion of series by Debye and Watson.