Introduction to multichannel scattering
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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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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

Channels - example

Channel hamiltonian and channel interaction:


Asymptotic condition

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:


Introduction to multichannel scattering

The theory is said to be asymptoticly complete if

ΣαΩα+(H) = ΣαΩα-(H) =

R (orthogonal complement to bound states B)

R

B

H

H


Introduction to multichannel scattering

Scattering operator

R

B

H

H

Has=ΣαSα


Energy conservation

Energy conservation

Intertwining relations:

Corollary

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


The cross section

The Cross Section

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

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


Symmetries rotational invariance

Symmetries: Rotational Invariance


Time reversal invariance

Time-Reversal Invariance


Time independent picture

Time-Independent Picture

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

From S in terms of Ω±:

Lippmann-Schwinger


T operator

T-operator

Lippmann-Schwinger for T:


Additional topics

Additional topics

  • Coupled channels radial equations

  • Analytic properties

    - Rieman sheet

    - Resonances

  • Threshold singularities


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