Resonances in the complex k plane  Summary and Q&A
TL;DR
Resonances and poles in the scattering amplitude can be understood by analyzing the complex plane, with poles representing points of infinite scattering amplitudes and complex wave numbers having implications for bound states and antibound states.
Questions & Answers
Q: How can resonances and poles be analyzed in the context of scattering amplitudes?
Resonances and poles can be analyzed by solving the equation tan(delta) = i, which guarantees a pole in the scattering amplitude. This allows us to identify points in the complex plane where the scattering amplitude becomes infinite.
Q: What is the significance of complex wave numbers in the imaginary axis?
Complex wave numbers in the imaginary axis, represented as k = i * kappa, can indicate bound states. The energy corresponding to these wave numbers is negative, suggesting the presence of confined states.
Q: What are antibound states and how do they relate to the complex plane?
Antibound states are solutions of the scattering amplitude that match a pure increasing exponential. In the complex plane, these states are represented by poles in the negative k region. Some nuclear states are associated with antibound states.
Q: How does the complex plane provide a valuable framework for understanding scattering solutions?
The complex plane accommodates scattering solutions, resonances, bound states, and antibound states. By analyzing the complex plane, we gain a more comprehensive understanding of the behavior of scattering amplitudes and the properties of these states.
Summary & Key Takeaways

Resonances and poles in the scattering amplitude can be analyzed in the complex plane, with poles representing points where the scattering amplitude becomes infinite.

Complex wave numbers in the imaginary axis can represent bound states, while poles in the complex plane can also represent antibound states.

The complex plane provides a rich framework for understanding scattering solutions, resonances, bound states, and antibound states.