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

* 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?

* Resonant states: should we, probably, study excitation energies instead? Or the energies E = E A − E A − 1 for n−(A−1) scattering (or, generally, with A 1 + A 2 = A)? * Is it important for them to be stable with respect to ħΩ or N max variation?

* 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 E r and width Γ? * Some observations, examples follow; this is still work in progress.

* So, what are the general properties of eigenstates in continuum consistent with resonance at the energy E r 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

* O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, (2010): * I.M.Lifshitz (1947):

* O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, (2010): * I.M.Lifshitz (1947): So, the phase shift at the eigenenergies E j can be easily calculated!

* O.Rubtsova, V.Kukulin, V.Pomerantsev, JETP Lett. 90, 402 (2009); Phys. Rev. C 81, (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 D j value below the lowest E j 0

Direct and inverse problem

* J-matrix inverse N-nucleus scattering analysis suggests values for resonant and non-resonant states that should be compared with that obtained in NCSM

* J-matrix: Let us try to extract resonance information from E λ behavior only

E λ should increase with ħΩ Within narrow resonance E λ is nearly ħΩ-independent The slope of E λ (ħΩ) depends however on N max, l, E λ value

* Breit-Wigner: * Simple approximation: φ=0 Derivatives calculated through

* Breit-Wigner: * Simple approximation: φ=0 Derivatives calculated through Do not expect to get a reasonable result for E r or Γ if Γ/2Δ is small! If |Γ/2Δ| is large, we get good results for E r, Γ and φ.

* What can we do if we obtain E λ in a non- resonant region above the resonance? * We can extrapolate energies to larger (finite) N max value when E λ is in the resonant region. * Expected dependence is

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

More convenient is an exponential extrapolation.

* We get stable E r and Γ; Γ is too small as compared with experiment.

* 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 E λ consistent with scattering data using simple inverse scattering technique. * Studying ħΩ dependence of E λ obtained in NCSM, one can get resonance energy and width. However, usually an extrapolation to a reasonable N max value is required.