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

2. Time evolution: Formal solution: Formal Scattering Theory - Intro t = 0 t > 0

3. Formal Theory: Asymptotes Hamiltonian: t = 0 t < 0 Definition: in-asymptote

4. Formal Theory: Asymptotes t = 0 t > 0 Definition: out-asymptote

5. Asymptotic condition The theory is said to fulfill the asymptotic condition if for every exists for which: + the same for t → +∞ Møller wave operators:

6. Asymptotic completeness The theory is said to be asymptoticly complete if Ω + ( H ) = Ω - ( H ) = R (orthogonal complement to B ) B R H H H=RBH=RB

7. Scattering operator B R H H In asymptotesOut asymptotes Unitarity: S + S = S S + =1

8. Energy conservation Intertwining relations: Corollary i.e. we can define “On-Shell T-matrix” S 1 + remainder

9. The Cross Section

10. It can be shown, that the procedure does not depend on the shape of ψ in =φ(p), provided φ(p) is sharply peaked around p 0 Key assumption is

11. Rotation described by Rotational invariance: [H,R(α)] = [H 0, R(α)] = 0 Consequence: [R(α),Ω ± ] = [R(α),S] = 0 Symmetries: Rotational Invariance S-matrix is diagonal in basis formed by common eigenvectors of operators H 0, J 2, J 3

12. Symmetries: Rotational Invariance

13. More on symmetries Parity invariance: Time reversal:

14. Resolvent – Green’s operator Significance: 1. Green’s function for time-independent Schrodinger equation 2. Fourier transform of evolution operator Resolvent (Lippmann-Schwinger) equation: G=G 0 +G 0 VG

15. Green and Møller It is possible to show: i. e.

16. Green’s and T-operator Lippmann-Schwinger for T: Born series

17. Scattering and T-operators Recall:

18. Stationary scattering states Recall: We apply: Spatial representation: Lippmann-Schwinger eqution

19. Partial wave expansions Free wave (just question to expand |p  in terms of |E,l,m  ) Stationary scattering state: Note:

20. Partial waves – integral eqation

21. Normalized / Regular solution Jost function

22. Analytic properties φ l,p (r) is analytic function of p, λ Jost function is analytic in λ Jost function is analytic in p in upper half-plain Zeros of Jost function = poles of s-matrix: Re p Im p bound states virtual state resonances physical sheet

23. Rieman surface for the energy Re E E=p 2 /2m

24. S, T, K - matrix Recall: (1-S) and 2πiT was the same “on shell” more generally T(z)=V+VG(z)V Def: M=i(1-S)(1+S) -1 i. e. S=(1+iM)(1-iM) -1 K-matrix (reaction or Heitler’s matrix): note: it is possible to define it in analogy for T with standing-wave G

25. Literature J. R. Taylor: Scattering Theory, R. G. Newton: Scattering Theory of Waves and Particles, P. G. Burke: Potential Scattering in Atomic Physics