Introduction to Multichannel Scattering

1. R H H B H=RB t = 0 t > 0 Introduction to Multichannel Scattering Martin Čížek Charles University, Prague

2. Channels - example Channel hamiltonian and channel interaction:

3. Asymptotic condition Definition: Channel space = subspace containing all possible states for given channel; example (χ is any L2 function, φ fixed state) Asymptotic condition: For any |ψin in any channel subspace α, there is vector |ψin in H: Møller’s operators:

4. The theory is said to be asymptoticly complete if ΣαΩα+(H) = ΣαΩα-(H) = R (orthogonal complement to bound states B) R Sα B H H

5. Scattering operator R Sα B H H Has=ΣαSα

6. Energy conservation Intertwining relations: Corollary i.e. we can define “On-Shell T-matrix”

7. The Cross Section Two body two body, for example: e + AB → A + B- Three body break up, for example: e + AB → A + B + e

8. Symmetries: Rotational Invariance

9. Time-Reversal Invariance

10. Time-Independent Picture It is possible to show (from definition of Ω±): From S in terms of Ω±: Lippmann-Schwinger

11. T-operator Lippmann-Schwinger for T:

12. Additional topics • Coupled channels radial equations • Analytic properties - Rieman sheet - Resonances • Threshold singularities