# On the Heisenberg condition in the presence of redundant poles of the S-matrix

## TL;DR

This paper analytically investigates the impact of redundant poles in the S-matrix on the Heisenberg condition and asymptotic completeness, demonstrating their contributions and origins in analytic continuation.

## Contribution

It provides explicit formulas for Jost functions and residues for redundant poles, showing the Heisenberg condition remains valid despite S-matrix singularities.

## Key findings

- Redundant poles do not violate the Heisenberg condition.
- Explicit expressions for residues of redundant poles are derived.
- Redundant poles' contribution to asymptotic completeness is analytically determined.

## Abstract

For the same potential as originally studied by Ma [Phys. Rev. {\bf 71}, 195 (1947)] we obtain analytic expressions for the Jost functions and the residui of the S-matrix of both (i) redundant poles and (ii) the poles corresponding to true bound states. This enables us to demonstrate that the Heisenberg condition is valid in spite of the presence of redundant poles and singular behaviour of the S-matrix for $k\to \infty$. In addition, we analytically determine the overall contribution of redundant poles to the asymptotic completeness relation, provided that the residuum theorem can be applied. The origin of redundant poles and zeros is shown to be related to peculiarities of analytic continuation of a parameter of two linearly independent analytic functions.