# On beautiful analytic structure of the S-matrix

## TL;DR

This paper thoroughly analyzes the analytic structure of the s-wave S-matrix for exponentially decaying potentials, revealing detailed pole configurations, hidden structures through domain coloring, and clarifying the nature of physical and redundant poles.

## Contribution

It provides a detailed characterization of the S-matrix's analytic structure, including pole positions, residues, and the origin of redundant poles, enhancing understanding of scattering theory.

## Key findings

- All poles and residues of the S-matrix can be precisely determined.
- Redundant poles are linked to analytic continuation peculiarities.
- Redundant poles contribute oscillating terms to the completeness relation.

## Abstract

For an exponentially decaying potential, analytic structure of the $s$-wave S-matrix can be determined up to the slightest detail, including position of all its poles and their residues. Beautiful hidden structures can be revealed by its domain coloring. A fundamental property of the S-matrix is that any bound state corresponds to a pole of the S-matrix on the physical sheet of the complex energy plane. For a repulsive exponentially decaying potential, none of infinite number of poles of the $s$-wave S-matrix on the physical sheet corresponds to any physical state. On the second sheet of the complex energy plane, the S-matrix has infinite number of poles corresponding to virtual states and a finite number of poles corresponding to complementary pairs of resonances and anti-resonances. The origin of redundant poles and zeros is confirmed to be related to peculiarities of analytic continuation of a parameter of two linearly independent analytic functions. The overall contribution of redundant poles to the asymptotic completeness relation, provided that the residue theorem can be applied, is determined to be an oscillating function.