Few-body problems via Faddeev-Yakubovsky equations formalism

Slides:



Advertisements
Similar presentations
A brief introduction D S Judson. Kinetic Energy Interactions between of nucleons i th and j th nucleons The wavefunction of a nucleus composed of A nucleons.
Advertisements

1 Eta production Resonances, meson couplings Humberto Garcilazo, IPN Mexico Dan-Olof Riska, Helsinki … exotic hadronic matter?
Spectroscopy at the Particle Threshold H. Lenske 1.
LLNL-PRES This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.
LLNL-PRES-XXXXXX This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.
Introduction to Molecular Orbitals
Owned and operated as a joint venture by a consortium of Canadian universities via a contribution through the National Research Council Canada Propriété.
Rimantas LAZAUSKAS, 07/07/2009 Elastic p- 3 H scattering below the n- 3 He threshold.
Coupled-Channel analyses of three-body and four-body breakup reactions Takuma Matsumoto (RIKEN Nishina Center) T. Egami 1, K. Ogata 1, Y. Iseri 2, M. Yahiro.
Table of contents 1. Motivation 2. Formalism (3-body equation) 3. Results (KNN resonance state) 4. Summary Table of contents 1. Motivation 2. Formalism.
The R-matrix method and 12 C(  ) 16 O Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium 1.Introduction 2.The R-matrix formulation:
Completeness of the Coulomb eigenfunctions Myles Akin Cyclotron Institute, Texas A&M University, College Station, Texas University of Georgia, Athens,
Andrey Shirokov (Moscow State Univ.) In collaboration with Alexander Mazur (Pacific National Univ.) Pieter Maris and James Vary (Iowa State Univ.) INT,
Systémes à petit nombre de corps (Few-body systems) IPHC Strasbourg, France Rimantas Lazauskas, IPHC Strasbourg, France.
Solution of Faddeev Integral Equations in Configuration Space Walter Gloeckle, Ruhr Universitat Bochum George Rawitscher, University of Connecticut Fb-18,
Nicolas Michel CEA / IRFU / SPhN Shell Model approach for two-proton radioactivity Nicolas Michel (CEA / IRFU / SPhN) Marek Ploszajczak (GANIL) Jimmy Rotureau.
Quantum calculation of vortices in the inner crust of neutron stars R.A. Broglia, E. Vigezzi Milano University and INFN F. Barranco University of Seville.
Chiral condensate in nuclear matter beyond linear density using chiral Ward identity S.Goda (Kyoto Univ.) D.Jido ( YITP ) 12th International Workshop on.
Sedimentation of a polydisperse non- Brownian suspension Krzysztof Sadlej IFT UW IPPT PAN, May 16 th 2007.
Three-Body Scattering Without Partial Waves Hang Liu Charlotte Elster Walter Glöckle.
Application of correlated basis to a description of continuum states 19 th International IUPAP Conference on Few- Body Problems in Physics University of.
Eiji Nakano, Dept. of Physics, National Taiwan University Outline: 1)Experimental and theoretical background 2)Epsilon expansion method at finite scattering.
Low-energy scattering and resonances within the nuclear shell model
L.D. Blokhintsev a, A.N. Safronov a, and A.A. Safronov b a Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow, Russia b Moscow State.
THEORETICAL PREDICTIONS OF THERMONUCLEAR RATES P. Descouvemont 1.Reactions in astrophysics 2.Overview of different models 3.The R-matrix method 4.Application.
Study on the Mass Difference btw  and  using a Rel. 2B Model Jin-Hee Yoon ( ) Dept. of Physics, Inha University collaboration with C.Y.Wong.
Nicolas Michel CEA / IRFU / SPhN / ESNT April 26-29, 2011 Isospin mixing and the continuum coupling in weakly bound nuclei Nicolas Michel (University of.
Faddeev Calculation for Neutron-Rich Nuclei Eizo Uzu (Tokyo Univ. of Science) Collaborators Masahiro Yamaguchi (RCNP) Hiroyuki Kamada (Kyusyu Inst. Tech.)
Adiabatic hyperspherical study of triatomic helium systems
Theoretical Study of the 4 He(γ,n) 3 He and 4 He(γ,p) 3 H reactions The 22 nd European Conference on Few-Body Problems in Physics Krakow Nir.
多体共鳴状態の境界条件によって解析した3α共鳴状態の構造
Paul Vaandrager Supervisor: Prof S. Rakitianski 10 July 2012 A Study of Resonant- and Bound-State Dependence on the Variables of a Step-Potential for a.
Three-body calculation of the 1s level shift in kaonic deuterium
Unbound states in the shell model and the tetraneutron resonance
One Dimensional Quantum Mechanics: The Free Particle
Hadron excitations as resonant particles in hadron reactions
Two-body force in three-body system: a case of (d,p) reactions
H. Kamada (Kyushu Institute of Technology, Japan) H. Witala , J
The Strong Force: NN Interaction
Int’l Workshop on SDE TT
Active lines of development in microscopic studies of
Theory of Scattering Lecture 2.
Overview of Molecular Dynamics Simulation Theory
Open quantum systems.
6/25/2018 Nematic Order on the Surface of three-dimensional Topological Insulator[1] Hennadii Yerzhakov1 Rex Lundgren2, Joseph Maciejko1,3 1 University.
Nuclear Structure Tools for Continuum Spectroscopy
Rimantas LAZAUSKAS, IPHC Strasbourg, France
Structure and dynamics from the time-dependent Hartree-Fock model
Three-body resonance in the KNN-πYN coupled system
Workshop on Precision Physics and Fundamental Constants
The continuum time-dependent Hartree-Fock method for Giant Resonances
Ch. Elster, L. Hlophe TORUS collaboration Supported by: U.S. DOE
Elements of Quantum Mechanics
Assistant professor at Isfahan University of Technology, Isfahan, Iran
The Weak Structure of the Nucleon from Muon Capture on 3He
Quantum Two.
Daejeon16 for light nuclei
Three-nucleon reactions with chiral forces
PHY 752 Solid State Physics Superconductivity (Chap. 18 in GGGPP)
R.R. Silva, M.E. Bracco, S.H. Lee, M. Nielsen
Nuclear Forces - Lecture 3 -
有限密度・ 温度におけるハドロンの性質の変化
少数体ケイオン核の課題 August 7, 2008 Y. Akaishi T. Yamazaki, M. Obu, M. Wada.
Meson Production reaction on the N* resonance region
Di-nucleon correlations and soft dipole excitations in exotic nuclei
Kazuo MUTO Tokyo Institute of Technology
Ab-initio nuclear structure calculations with MBPT and BHF
R. Lazauskas Application of the complex-scaling
Quantum One.
Presentation transcript:

Few-body problems via Faddeev-Yakubovsky equations formalism 04/08/2019 R. Lazauskas

Contents Solution of the 3-5 body Faddeev-Yakubovsky equations for nuclear systems Some applications n-4He scattering Resonances in 5H 04/08/2019 04/08/2019 2 2 R. Lazauskas R. Lazauskas 2

Introduction R. Lazauskas R. Lazauskas Collisions How to take care of the boundary condition? Conceptual difficulties to uncouple different particle channel, to constrain assymptotes of the solutions in all directions and thus get unique (physical) solution to the Schrödinger eq. It is ok, as long as there is single particle channel (elastic plus target/projectile excitations) Mathematically Ill-conditioned problem when several particle channels are open Faddeev-Yakubovsky equations efficiently separates asymptotes of the binary channels In configuration space wave functions extend to infinity! Increasingly complex asymptotic behaviour for A>2 systems!! … L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39, 1459 (1960). [Sov. Phys. JETP 12, 1014(1961)]. O. A. Yakubovsky, Sov. J. Nucl. Phys. 5, 937 (1967). 04/08/2019 04/08/2019 3 3 R. Lazauskas R. Lazauskas 3

Properties of the rigorous scattering eq. Should separate all possible scattering channels to incorporate proper asymptotes! Number of binary channels increases ~2N Should be systematically reducible to smaller subsystems, in order to built proper asymptotic solutions and to be consistent to its subsystems (tree-like structure) a c 𝑟 𝑎𝑏,𝑐 𝑟 𝑎𝑏 b 04/08/2019 04/08/2019 4 4 R. Lazauskas R. Lazauskas 4

Faddeev-Yakubovsky eq 3-body (Faddeev eq.) 4-body (Faddeev-Yakubovsky eq.) 2 1 j i j i 3 k k l 𝜙 12 = 𝐺 0 𝑉 12 Ψ l 3 2 1 K-type (12 components) H-type (6 components) 𝜙 23 = 𝐺 0 𝑉 23 Ψ 𝜙 𝑗𝑘 = 𝐺 0 𝑉 𝑗𝑘 Ψ 1 3 𝐾 𝑖𝑗,𝑘 𝑙 = 𝐺 𝑖𝑗 𝑉 𝑖𝑗 𝜙 𝑗𝑘 + 𝜙 𝑖𝑘 ; 𝐻 𝑖𝑗 𝑘𝑙 = 𝐺 𝑖𝑗 𝑉 𝑖𝑗 𝜙 𝑘𝑙 2 𝜙 31 = 𝐺 0 𝑉 31 Ψ Ψ= 𝑖<𝑗 𝜙 𝑖𝑗 = 𝑖<𝑗 (𝐾 𝑖𝑗,𝑘 𝑙 + 𝐾 𝑖𝑗,𝑙 𝑘 +𝐻 𝑖𝑗 𝑘𝑙 ) Ψ= 𝜙 12 + 𝜙 23 + 𝜙 31 Equations for short-ranged pairwise interactions 04/08/2019 04/08/2019 5 5 R. Lazauskas R. Lazauskas 5

5-body Faddeev-Yakubovski eq 2 1 2 1 2 4 3 5 1 2 1 4 3 5 5 04/08/2019 04/08/2019 6 6 R. Lazauskas R. Lazauskas 6

Faddeev-Yakubovsky eq 2 1 2 1 2 4 3 5 1 2 1 4 3 5 5 Merits: Handling of symmetries Boundary conditions for binary channels Built-in reduction to subsystems Price Overcomplexity J.W. Waterhouse : « Pandora » Problem Number eq. (identical particles) Number eq. (different particles) A=2 1 A=3 3 A=4 2 18 A=5 5 180 A=6 15 2700 A=N nint( 2 𝑁−1 ! (𝜋/2) 𝑁 ) 𝑁! 𝑁−1 ! 2 𝑁−1 04/08/2019 04/08/2019 7 7 R. Lazauskas R. Lazauskas 7

5-body Faddeev-Yakubovski eq 2 1 2 1 2 4 3 5 1 2 1 4 3 5 5 NUMERICAL SOLUTION *R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003). PW decomposition of the components K,H,T,S,F Radial parts expanded using Lagrange-mesh method D. Baye, Physics Reports 565 (2015) 1 Resulting linear algebra problem solved using iterative methods Observables extracted using integral relations 04/08/2019 04/08/2019 8 8 R. Lazauskas R. Lazauskas 8

Numerical costs R. Lazauskas R. Lazauskas NUMERICAL SOLUTION 2 1 2 1 2 4 3 5 1 2 1 4 3 5 5 Problem Number eq. (ident particles) Number eq. (diff. particles) PW basis. Radial disc. 2N 1 2 ~N 3N 3 ~100 ~N2 4N 18 ~104 ~N3 5N 5 180 ~106 ~N4 NUMERICAL SOLUTION *R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003). PW decomposition of the components K,H,T,S,F Radial parts expanded using Lagrange-mesh method D. Baye, Physics Reports 565 (2015) 1 Resulting linear algebra problem solved using iterative methods Observables extracted using integral relations 04/08/2019 04/08/2019 9 9 R. Lazauskas R. Lazauskas 9

Short overview of nuclear problems by FY eq’s 3N-problem (Faddeev eq.) 1st solution: A. Laverne and C. Gignoux: Nucl. Phys. A 203 (1973) 597 G. Gignoux, A. Laverne, and S. P. Merkuriev Phys. Rev. Lett. 33 (1974) 1350 Review: W. Glockle et al., Physics Reports 274 ( 1996) 107-285 4N-problem (Faddeev eq.) 1st solution: S. P. Merkuriev, S.L. Yakovlev, C. Gignoux, Nucl. Phys. A 431 (1984) 125. Benchmarks: A. Nogga, et al., Phys. Rev. C 65 (2002) 054003 (bound state) R. Lazauskas et al.,Phys. Rev. C 71 (2005) 034004 (n3H scattering) M. Viviani et al., Phys. Rev. C 84 (2011) 054010 (p3He scattering) M. Viviani et al., arXiv:1610.09140 (p3H,n3He scattering) 𝑝 + 3 𝐻↔𝑛 + 3 𝐻𝑒 04/08/2019 04/08/2019 10 10 R. Lazauskas R. Lazauskas 10

5N problem: n-4He scattering 04/08/2019 04/08/2019 11 11 R. Lazauskas R. Lazauskas 11

n-4He scattering R. Lazauskas R. Lazauskas NCSMC: P. Navratil et al., Physica Scripta 91 (2016) 053002 04/08/2019 04/08/2019 12 12 R. Lazauskas R. Lazauskas 12

n-4He scattering R. Lazauskas R. Lazauskas 04/08/2019 04/08/2019 13 13 NCSM: P. Navratil et al., Physica Scripta 91 (2016) 053002 GFMC: K.M. Nollett et. al., Phys.Rev.Lett.99:022502,2007 NCSM-HORSE: A.M. Shirokov et. al., Phys. Rev. C 94, 064320 (2016) RGM: J. Kircher, PhD work, GWU (2011) 04/08/2019 04/08/2019 13 13 R. Lazauskas R. Lazauskas 13

n-4He scattering R. Lazauskas R. Lazauskas Idaho N3LO pot. 04/08/2019 K.M. Nollett et. al., Phys.Rev.Lett.99:022502,2007 P. Navratil et al., Physica Scripta 91 (2016) 053002 Idaho N3LO pot. 04/08/2019 04/08/2019 14 14 R. Lazauskas R. Lazauskas 14

Case of little interest: S-wave 04/08/2019 04/08/2019 15 15 R. Lazauskas R. Lazauskas 15

Case of little interest: S-wave Experimental n-4He scattering length … nothing should be as easy to measure… NIST (Neutron News 3, 1992) Coh a (fm) Inc b (fm) 1H -3.7406(11) -3.79406(11) 25.274(9) 2H 6.671(4) 4.04(3) 3H 4.792(27) -1.04(17) 3He 5.74(7)-1.483(2)i -2.5(6)+2.568(3)i 4He 3.26(3) 3.07(2) TUNL TUNL: D.R. Tilley et al., Nucl. Phys. A708 (2002) 3 NIST: https://www.ncnr.nist.gov Experimental data: D.C.Rorer et al., Nucl. Phys. A 133 (1969) 410 S.F.Mughabghab, Atlas of Neutron Resonances (2006) R.Genin et al., Journal de Physique 24 (1963) 21 04/08/2019 04/08/2019 16 16 R. Lazauskas R. Lazauskas 16

Case of little interest: S-wave TUNL: D.R. Tilley et al., Nucl. Phys. A708 (2002) 3 NIST: https://www.ncnr.nist.gov S. Ali PSA: S. Ali et al., Rev. Mod. Phys. 57 (1985) 923 Bang-Gignoux pot: J. Bang, C. Gignoux, Nucl. Phys. A 313 (1979) 119 NCSMC: P. Navratil et al., Physica Scripta 91 (2016) 053002 GFMC: K.M. Nollett, PRL99, 022502 (2007) 04/08/2019 04/08/2019 17 17 R. Lazauskas R. Lazauskas 17

5H resonances? R. Lazauskas R. Lazauskas 04/08/2019 04/08/2019 18 18

Approximate dynamics of 3H, no full 5-body dynamics 5H resonances? Approximate dynamics of 3H, no full 5-body dynamics How to handle resonances? Direct methods with Kapur-Peierls bc .. complicated 3-body boundary condition Complex scaling expensive, difficulties with broad resonances 04/08/2019 04/08/2019 19 19 R. Lazauskas R. Lazauskas 19

Extrapolation is a flourishing business 04/08/2019 04/08/2019 20 20 R. Lazauskas R. Lazauskas 20

Algebraic branching point 𝜆 =𝜆 0 Extrapolation is a flourishing business In QM scattering problem the « Energy manifold » E is not an axis, but a two-dimensional Riemann sheet! When moving from bound state region (axis!) to continuum one should take into account the proper analytical behavior: branching points + non-trivial dependence in 𝒌(𝝀𝑽) Trajectory of the S-matrix pole with a coupling constant 𝜆 with 𝑘 𝜆 0 =0 Algebraic branching point 𝜆 =𝜆 0 Bound state 𝜆 >𝜆 0 Anti-resonance 𝜆 <𝜆 0 Resonance 𝜆 <𝜆 0 Analytic behavior in the vicinity of the branching point 𝑘(𝜆→ 𝜆 0 )~𝑖 𝜆− 𝜆 0 04/08/2019 04/08/2019 21 21 R. Lazauskas R. Lazauskas 21

Extrapolation is a flourishing business In QM scattering problem the « Energy manifold » E is not an axis, but a two-dimensional Riemann sheet! When moving from bound state region (axis!) to continuum one should take into account the proper analytical behavior: branching points + non-trivial dependence in 𝒌(𝝀𝑽) If it is not done properly… there are some risks 04/08/2019 04/08/2019 22 22 R. Lazauskas R. Lazauskas 22

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 =−236.5 𝑒 −0.6 𝑟 2 𝜆𝑉(𝑟)=𝜆 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 23 23 R. Lazauskas R. Lazauskas 23

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 =−236.5 𝑒 −0.6 𝑟 2 𝜆𝑉(𝑟)=𝜆 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 24 24 R. Lazauskas R. Lazauskas 24

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 =−236.5 𝑒 −0.6 𝑟 2 𝜆𝑉(𝑟)=𝜆 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 25 25 R. Lazauskas R. Lazauskas 25

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 =−236.5 𝑒 −0.6 𝑟 2 𝜆𝑉(𝑟)=𝜆 𝑒 − 𝑟 2 /2 04/08/2019 04/08/2019 26 26 R. Lazauskas R. Lazauskas 26

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 = 1.8 𝑟 1438.72𝑒 −3.11𝑟 −626.885𝑒 −1.55𝑟 𝜆𝑉(𝑟)=𝜆 𝑟 2 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 27 27 R. Lazauskas R. Lazauskas 27

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 = 1.8 𝑟 1438.72𝑒 −3.11𝑟 −626.885𝑒 −1.55𝑟 𝜆𝑉(𝑟)=𝜆 𝑟 2 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 28 28 R. Lazauskas R. Lazauskas 28

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 = 1.8 𝑟 1438.72𝑒 −3.11𝑟 −626.885𝑒 −1.55𝑟 𝜆𝑉(𝑟)=𝜆 𝑟 2 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 29 29 R. Lazauskas R. Lazauskas 29

ACCC method R. Lazauskas R. Lazauskas ACCC method (V.I. Kukulin et al., « Theory of resonances », Kluwer AP 1989) Add artificial binding potential to Hamiltonian 𝑉 𝑟 Calculate several binding energies of the system Ei(i) Determine accurately 0 such that E(0)=0 ! Extrapolate Eres=E(=0) using Ei(i) and 0 values, knowing that 𝐸 𝑟𝑒𝑠 (→  0 ) ~𝑖 −  0 Padé extrapolation is used: 𝐸 𝑟𝑒𝑠 (→  0 ) =𝑖𝑃 𝑛,𝑚 −  0 𝑃 𝑛,𝑚 (𝑞)= 𝑎 1 𝑞+ 𝑎 1 𝑞 2 +..+ 𝑎 𝑛 𝑞 𝑛 1+ 𝑏 1 𝑞+..+ 𝑏 𝑚 𝑞 𝑚 2b-example 𝑉 𝑙=1 = 1.8 𝑟 1438.72𝑒 −3.11𝑟 −626.885𝑒 −1.55𝑟 𝜆𝑉(𝑟)=𝜆 𝑟 2 𝑒 − 𝑟 2 /3.5 04/08/2019 04/08/2019 30 30 R. Lazauskas R. Lazauskas 30

Back to 5H(J=1/2+) R. Lazauskas R. Lazauskas INOY Potential N3LO Potential E(5H)-E(3H)=1.65(5)-i1.26(6) E(5H)-E(3H)=1.8(1)-i1.15(15) 04/08/2019 04/08/2019 31 31 R. Lazauskas R. Lazauskas 31