### Abstract:

For a two-dimensional quantum-mechanical problem, we obtain a generalized
power series expansion of the S-matrix that can be done near an arbitrary point
on the Riemann surface of the energy, similar to the standard effective-range
expansion. In order to do this, we consider the Jost function and analytically
factorize its momentum dependence that causes the Jost function to be a multivalued
function. The remaining single-valued function of the energy is then
expanded in the power series near an arbitrary point in the complex energy
plane. A systematic and accurate procedure has been developed for calculating
the expansion coefficients. This makes it possible to obtain a semi-analytic
expression for the Jost function (and therefore for the S-matrix) near an arbitrary
point on the Riemann surface and use it, for example, to locate the spectral
points (bound and resonant states) as the S-matrix poles. The method is applied
to a model similar to those used in the theory of quantum dots.