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The general semiclassical time-dependent two-state problem is considered fora specific field configuration referred to as the generalized Rosen-Zenermodel. This is a rich family of pulse amplitude- and phase-modulation functionsdescribing both non-crossing and term-crossing models with one or two crossingpoints. The model includes the original constant-detuning non-crossingRosen-Zener model as a particular case. We show that the governing system ofequations is reduced to a confluent Heun equation. When inspecting theconditions for returning the system to the initial state at the end of theinteraction with the field, we reformulate the problem as an eigenvalue problemfor the peak Rabi frequency and apply the Rayleigh-Schr\"odinger perturbationtheory. Further, we develop a generalized approach for finding the higher-orderapproximations, which is applicable for the whole variation region of theinvolved input parameters of the system. We examine the general surface in the3D space of input parameters, which defines the position of the n-th orderreturn-resonance, and show that the section of the general surface isaccurately approximated by an ellipse. We find a highly accurate analyticdescription through the zeros of a Kummer confluent hypergeometric function.From the point of view of the generality, the analytical description ofmentioned curve for the whole variation range of all involved parameters is themain result of the present paper.

Complete-return spectrum for a generalized Rosen-Zener two-state term-crossing model