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Introduction to Multichannel Scattering

R. H. H. B. H = R  B. t = 0. t > 0. Introduction to Multichannel Scattering. Martin Čížek Charles University, Prague. Channels - example. Channel hamiltonian and channel interaction:. Asymptotic condition.

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Introduction to Multichannel Scattering

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  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

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