# Dynamics of quantum systems

**Authors:** I. Rotter  
**arXiv:** [quant-ph/0105068](https://arxiv.org/abs/quant-ph/0105068)  
**Date:** 2001-05-15  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/quant-ph--0105068v1/

## Key points

- Relationship between eigenvalues and poles of the S matrix in quantum systems
- Applies to many-particle quantum systems with two-body forces or quantum billiards
- Existence of branch points causes avoided crossings and double poles of the S matrix
- Branch points significantly impact dynamics of open and closed quantum systems
- Detailed analysis on a two-level system reveals influence of branch points
- Study contributes to understanding fundamental principles in quantum mechanics
- Potential implications for condensed matter physics, atomic physics, and nuclear physics

## Summary

The study "Dynamics of quantum systems" by I. Rotter explores the relationship between eigenvalues and poles of the S matrix in both isolated and overlapping resonance states. This relationship applies to many-particle quantum systems with two-body forces or quantum billiards without such forces. The existence of branch points in the complex plane is identified as causing avoided crossings between discrete and resonance states, resulting in double poles of the S matrix. These branch points significantly impact the dynamics of open and closed quantum systems. To gain further insight, a detailed analysis is conducted on a two-level system using analytical and numerical methods. This reveals how branch points influence the behavior and evolution of quantum systems. Overall, this study sheds light on the intricate interplay between eigenvalues, poles of the S matrix, and branch points in determining dynamics in various quantum systems. The findings contribute to our understanding of fundamental principles in quantum mechanics and have potential implications for fields such as condensed matter physics, atomic physics, and nuclear physics.