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Introduction to Multichannel Scattering

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R

H

H

B

H=RB

t = 0

t > 0

Introduction to Multichannel Scattering

Martin Čížek

Charles University, Prague

Channel hamiltonian and channel interaction:

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:

The theory is said to be asymptoticly complete if

ΣαΩα+(H) = ΣαΩα-(H) =

R (orthogonal complement to bound states B)

R

Sα

B

H

H

Scattering operator

R

Sα

B

H

H

Has=ΣαSα

Intertwining relations:

Corollary

i.e. we can define “On-Shell T-matrix”

Two body two body, for example: e + AB → A + B-

Three body break up, for example: e + AB → A + B + e

It is possible to show (from definition of Ω±):

From S in terms of Ω±:

Lippmann-Schwinger

Lippmann-Schwinger for T:

- Coupled channels radial equations
- Analytic properties
- Rieman sheet

- Resonances

- Threshold singularities