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

TL;DR

This paper develops a systematic power-series expansion method for the Jost function and S-matrix in two-dimensional quantum problems, enabling semi-analytic analysis near any complex energy point, including spectral states.

Contribution

It introduces a generalized power-series expansion of the S-matrix near arbitrary points on the Riemann surface for 2D quantum systems, with a systematic calculation of expansion coefficients.

Findings

Accurate semi-analytic expressions for the Jost function and S-matrix near arbitrary energy points.

Method successfully applied to a quantum dot-like model.

Facilitates locating spectral points such as bound and resonant states.

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, similarly 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 multi-valued 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 simlar to those used in the theory of quantum dots.