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For a two-dimensional quantum mechanical problem, we obtain a generalizedpower-series expansion of the S-matrix that can be done near an arbitrary pointon the Riemann surface of the energy, similarly to the standard effective rangeexpansion. In order to do this, we consider the Jost-function and analyticallyfactorize its momentum dependence that causes the Jost function to be amulti-valued function. The remaining single-valued function of the energy isthen expanded in the power-series near an arbitrary point in the complex energyplane. A systematic and accurate procedure has been developed for calculatingthe expansion coefficients. This makes it possible to obtain a semi-analyticexpression for the Jost-function (and therefore for the S-matrix) near anarbitrary point on the Riemann surface and use it, for example, to locate thespectral points (bound and resonant states) as the S-matrix poles. The methodis applied to a model simlar to those used in the theory of quantum dots.

Analytic structure and power-series expansion of the Jost function for the two-dimensional problem