Resonance states of relativistic two-particle systems Kapshai V.N., Grishechkin Yu.A. F. Scorina Gomel State University

Plan of the talk resonance states in nonrelativistic QM equations of quasipotential type integral equations in the RCR and scattering amplitudes integral equations for resonance states in the RCR and in the MR solving method complex scaling in the RCR and in the MR results of solving conclusion 2

Resonance states of two-particle systems in quantum theory Nonrelativistic theory Scattering states Bound states Resonant states -complex value 0 Re q Im q Bound states Resonances 3

- complex value 4 Relativistic theory Bound states Bound states Scattering states Scattering states Resonant states Resonant states

Relativistic two-particle equations of quasipotential type 5 Quasipotential equations In the momentum representation (MR) In the relativistic configurational representation (RCR) integral form difference form integral form

Integral equations in the MR 6 Three dimensional equations for scattering states j=1 – the Logunov-Tavkhelidze equation j=2 – the Kadyshevsky equation j=3 – the modified Logunov-Tavkhelidze equation j=4 – the modified Kadyshevsky equation Green functions (GF) 2E q – two-particle system energy m – mass of each particle – non-relativistic limit

Integral equations in the RCR 7 r – coordinate in the RCR Transformation of wave function Three-dimensional equations for scattering states in the RCR V(r) – spherically symmetric potential in the RCR RCR is as expansion over functions

8 Partial equations in the RCR for the s-states Equations for the scattering s-states χ q >0 – rapidity Green functions in the RCR — the Logunov-Tavkhelidze — the Kadyshevsky — the modified Logunov-Tavkhelidze — the modified Kadyshevsky – non-relativistic limit

Green functions in the RCR 9 Green functions 1) 2) Symmetry properties Asymptotic behavior — the Logunov-Tavkhelidze — the Kadyshevsky — the modified Logunov-Tavkhelidze — the modified Kadyshevsky

Integral equations in the RCR and scattering amplitude 10 At r → ∞ using GF asymptotic one obtain Scattering amplitude Scattering cross section N on-relativistic limit

Partial equations for s-states in the MR 11 Equations for scattering s-states Green functions — the Logunov-Tavkhelidze — the Kadyshevsky — the modified Logunov-Tavkhelidze — the modified Kadyshevsky

12 Equations for bound and resonance s-states in the RCR Equations for bound states Equations for resonance states Rapidity – is imaginary value V(r) o r Bound states Resonance states Rapidity – is complex value - n on-relativistic limit

13 Partial equations for s-states in the MR Equations for bound states Equations for resonance states - n on-relativistic limit

Solving method 14 Quadrature formula

Model potential 15 В РКП В ИП

Nonrelativistic case 16 Solving of the Schrödinger equation in the coordinate representation at m=1, V 2 =15, α = 1

17 Relativistic case Cross section for j=1 at m=1, V 2 =15, α = 1 Zeros of determinant

Complex scaling method 18 0 Re r Im r r → r e i θ θ Scaling in the RCR

19 Scaling in the MR (non-relativistic case) p → pe -iθ θ Im p Re p 0

20 χ q → e -iθ θ Im χ q Re χ q 0 Scaling in the MR (relativistic case)

Results 21 Zeros of determinant after complex scaling on 30 º Solving in the RCRSolving in the MR

Resonance rapidity in the case of the Kadyshevsky equation 22 θ=0 Решение в РКПРешение в ИП θ=30 º Resonances not determine

23 Resonance rapidy in the case of modified Logunov-Tavkhelidze equation Solving in the RCRSolving in the MR θ=0 θ=30 º Resonances not determine

Conclusion Complex scaling method can be used for resonance states finding on the basis relativistic two-particle equations solving both in the RCR and in the MR. 24

Thank you for your attention! 25