Analytic Properties of the Jost Functions via the Poincar\'e-Picard Theorem

Abstract:The analytic properties of the Jost functions are fundamental in quantum scattering theory and in the analytic continuation of the scattering matrix into the complex energy plane. In this work, the analyticity of the Jost functions is investigated from the perspective of parameter-dependent ordinary differential equations. Starting from the radial Schrödinger equation for a short-range central potential, a first-order differential system is derived for the coefficient functions associated with the Ricatti--Bessel and Ricatti--Neumann solutions. The multivalued dependence on the energy variable is shown to originate from the square-root relation between energy and momentum. By explicitly factorizing the momentum-dependent branching terms, the scattering problem is transformed into a differential system whose coefficients are single-valued analytic functions of the complex energy. Using the classical theory of analytic dependence of solutions of ordinary differential equations on parameters, it is shown that the transformed Jost functions are single-valued analytic functions of the energy variable for finite radial distance. The geometric interpretation of the factorization procedure is also discussed in terms of the topology of the associated Riemann surface.

21 pages, we corrected the variable transformation scaling factors in Sections 4 and 5 to satisfy the energy-independent boundary condition requirements of the Poincaré-Picard theorem. Structural results and physical conclusions remain unchanged

arXiv:2605.28859 [quant-ph]

(or arXiv:2605.28859v2 [quant-ph] for this version)