Abelian integrals, algebraic functions and compact Riemann surfaces in the semiclassical theory of non-adiabatic collisions

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Some beautiful classical results in the mathematics of Abelian integrals, algebraic functions and compact Riemann surfaces, which are relatively little used in physics applications (except in string theory), are shown to be relevant in the semiclassical theory of non-adiabatic collisions. These results are discussed in the context of the adiabatic theorem for the time-dependent Schrödinger equation describing multi-(electronic) level molecular collision systems. It is found that the different potential energy surfaces governing nuclear motion can be regarded as a single algebraic function (t) of the complex time variable t. The topological and analytic properties of the compact Riemann t-surface for this algebraic function then determine the nature of the non-adiabatic quantum transitions. The amplitudes of these transitions are given by a generalization of Dykhne's formula, whose main feature is an Abelian (action) integral of the Abelian differential . These ideas are applied to a rederivation of the Landau - Zener formula, which is shown to result from a model with a genus zero Riemann surface. A genus one model, illustrating more substantially the interplay between topology, analyticity and quantum transitions, is also discussed in detail.