1. Resonant states in the shell model Andrey Shirokov (Moscow State Univ.) In collaboration with Alexander Mazur (Pacific National Univ.) Pieter Maris and James Vary (Iowa State Univ.) INT, Seattle, June 8, 2011

2. Conventional: bound state energies are associated with variational minimum in shell model, NCSM, etc., calculations • Is it also true for resonant states? Can we get resonance width from such calculations? Problem

3. Resonant states: should we, probably, study excitation energies instead? Or the energies E = EA− EA − 1for n−(A−1) scattering (or, generally, with A1 + A2 = A)? • Is it important for them to be stable with respect to ħΩ or Nmax variation? Problem

4. Would be nice to have a simple answer from conventional calculations without doing, say, complicated NCSM−RGM calculations. • So, what are the general properties of eigenstates in continuum consistent with resonance at the energy Er and width Γ? • Some observations, examples follow; this is still work in progress. Problem

5. So, what are the general properties of eigenstates in continuum consistent with resonance at the energy Er and width Γ? • Some observations, examples follow; this is still work in progress. • I discuss some general properties for oscillator basis calculations; this is the only relavance to NCSM Problem

6. O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010): • I.M.Lifshitz (1947): “Pushing” point

7. O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010): • I.M.Lifshitz (1947): So, the phase shift at the eigenenergies Ej can be easily calculated! “Pushing” point

8. O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, 064003 (2010): • I.M.Lifshitz (1947): Unfortunately, this does not work: The dimensionality of the matrix is small, the average spacing between the levels is not well-defined. One needs sometimes Dj value below the lowest Ej0 “Pushing” point

9. J-matrix formalism:scattering in the oscillator basis Direct and inverse problem

10. J-matrix

11. J-matrix: universal function f(E)

12. J-matrix: compare with Lifshitz

13. J-matrix: compare with Lifshitz

14. How it works

15. J-matrix inverse N-nucleusscattering analysis suggests values for resonant and non-resonant states that should be compared with that obtained in NCSM Immediate result: inverse scattering

16. Nαinverse scattering and NCSM

17. Nαnon-resonant inverse scattering and NCSM

18. J-matrix: Direct problem

19. J-matrix: Direct problemnα scattering

21. Convergence…

22. J-matrix: Let us try to extract resonance information from Eλbehavior only Resonance

23. Eλshould increase with ħΩ Within narrow resonance Eλisnearly ħΩ-independent The slope of Eλ(ħΩ) depends however on Nmax, l, Eλ value ħΩ dependence

24. ħΩ dependence

25. Breit-Wigner: • Simple approximation: φ=0 Resonance parameters Derivatives calculated through

26. Breit-Wigner: • Simple approximation: φ=0 Do not expect to get a reasonable result for Eror Γ if Γ/2Δ is small! If |Γ/2Δ| is large, we get good results for Er, Γ and φ. Resonance parameters Derivatives calculated through

27. Resonance

28. Resonance

29. Resonance

30. Resonance

31. What can we do if we obtain Eλin a non-resonant region above the resonance? • We can extrapolate energies to larger (finite) Nmax value when Eλis in the resonant region. • Expected dependence is Resonance parameters

32. This works. However this extrapolation seems to be unstable and inconvenient Resonance parameters

33. More convenient is an exponential extrapolation. Resonance parameters

34. We get stable Erand Γ; Γ is too small as compared with experiment. Resonance in nα scattering

35. I discussed general features of continuum states obtained in many-body calculations with oscillator basis. • The best way to compare the calculated results with experiment is to use “experimental” phase shifts and get Eλ consistent with scattering data using simple inverse scattering technique. • Studying ħΩ dependence of Eλ obtained in NCSM, one can get resonance energy and width.However, usually an extrapolation to a reasonable Nmax value is required. Conclusions

36. Thank you