problem 3 April, 2009 Department of Computational Physics, St Petersburg State University Numerical solution of the three body scattering problem

Transcription

1 Numerical Department of Computational Physics, St Petersburg State University 3 April, 2009

2 Outline Introduction The Three- Faddeev Equation Solution expansion Malfliet-Tjon potential Numerical calculations: eigenvalues, eigenfunctions, geometrical connections Conclusions

3 Introduction

4 Introduction Expanding of the tree wave function Basis of hyperspherical adiabatic harmonics The standard projection procedure leads to effective equations for the expansion coefficients How can be solved? Have to find out properties of the basis functions

5 Introduction Geometrical connections A ij = φ i, ρ φ j Develop a formalism to study different models - of a particle off a two bound state - of a particle off a resonant state of a pair of particles Investigate eigenvalues λ k, eigenfunctions φ k, and geometrical connections A ij of the operator h(ρ) = ρ 2 θ + V (ρ cos θ).

6 Faddeev Equation in {ρ, θ} Coordinates

7 Faddeev equation in {ρ, θ} coordinates The Faddeev s-wave equation in polar coordinates ( ρ 2 1 ) 4ρ 2 U(ρ, θ) + ( 1ρ ) 2 2θ + V (ρ cos θ) U(ρ, θ) 3 θ+(θ) EU(ρ, θ) = V (ρ cos θ) dθ U(ρ, θ ) 4 θ (θ) where θ + (θ) = π/2 π/6 θ and θ (θ) = π/3 θ

8 Faddeev equation in {ρ, θ} coordinates The boundary conditions should be imposed for the solution: Regularity of the solution U(0, θ) = U(ρ, 0) = U(ρ, π/2) = 0 Asymptotics as ρ U(ρ, θ) ϕ(x)ρ 1/2 sin(qy)/q + +a(q)ϕ(x)ρ 1/2 exp(iqy) + A(θ, E) exp(i Eρ), where x = ρ cos θ.

9 Faddeev equation in {ρ, θ} coordinates The bound-state wave function of the two target system ϕ(x) obeys the equation { 2 x + V (x)}ϕ(x) = ɛϕ(x) It is assumed that the discrete spectrum of the Hamiltonian 2 x + V (x) consists of one negative eigenvalue ɛ at most

10 Basis functions

11 Expansion basis As basis we choose the eigenfunctions set of the operator h(ρ) ( ρ 2 2 θ + V (ρ cos θ)) φ k (θ ρ) = λ k (ρ)φ k (θ ρ) ρ is an external parameter for h(ρ) EV and EF of h(ρ) inherit parametrical dependence on ρ

12 Spectral properties The main spectral properties of the operator h(ρ) as ρ and of the two Hamiltonian 2 x + V (x) λ = lim ρ λ 0(ρ) = ɛ, φ 0 (θ ρ) = ρ ϕ(ρ cos θ)(1 + O(ρ µ )) Excited states φ k (θ ρ), k 1, when ρ the asymptotic behavior is given by the formulas λ k (ρ) ( ) 2k 2, ρ φ k (θ ρ) 2 π sin(2kθ)

13 Perturbation Theory

14 Perturbation theory - introduction If the parameter ρ is large the support of the potential V (ρ cos θ) is small on the interval [0, π/2] The estimates for EV and EF of the operator h(ρ) can be obtained by the perturbation theory considering the potential V (ρ cos θ) as a small perturbation λ k λ 0 k + λ1 k, φ k (θ ρ) φ 0 k (θ ρ) + φ1 k (θ ρ) The leading terms λ 0 k and φ0 k for k 1 are given by λ 0 k = (2k)2 /ρ 2, φ 0 k (θ ρ) = 2 π sin(2kθ)

15 Correction terms The first order correction terms λ 1 k and φ1 k are defined in terms of the matrix elements of the potential V (ρ cos θ) π/2 V nk (ρ) = φ 0 n V φ 0 k = dθ φ 0 n(θ ρ)v (ρ cos θ)φ 0 k (θ ρ) 0 As the model situation, the function V (x) is taken in the form of Yukawa potential V (x) = C exp( µx) x where C and µ are some parameters.

16 Calculation of corrections As we set ρ, we can approximately calculate integral n+k 16nkC V nk (ρ) ( 1) π [1 exp( µr)(1 + µr)] µ 2 1 ρ 3. The first order correction term λ 1 k is λ 1 k = V kk ρ 3 The second order correction term λ 2 k is λ 2 k = n k V nk 2 λ (0) k λ (0), n λ 2 k ρ 4

17 Calculation of corrections For eigenfunctions the first correction term is given by φ 1 k = n k V nk λ (0) k λ n (0) φ 0 n ρ 1 From equation ( 1ρ 2 2θ + V (ρ cos θ) ) φ k (θ ρ) = λ k (ρ)φ k (θ ρ) one can obtain formula for A ij A ij = φ i V ρ φ j λ i λ j

18 A ij Rough calculation of A ij leads to dependence A ij ρ 2, i, j 0 A ij ρ 5/2, i = 0 or j = 0

19 Solution Expansion

20 Solution expansion The operator h(ρ) is Hermitian on [0, π/2] interval and its eigenfunction set {φ k (θ ρ)} 0 is complete. U(ρ, θ) = φ 0 (θ ρ)f 0 (ρ) + k 1 φ k (θ ρ)f k (ρ) Substituting this expansion in Faddev equation and projecting on basis functions lead to the following system of equations for F k (ρ), k = 0, 1,... ( ρ 2 1 4ρ 2 + λ k(ρ) E i=0 ) F k (ρ) = ( F i (ρ) 2A ki + B ki (ρ)f i (ρ) W ki (ρ)f i (ρ) ρ )

21 Solution expansion The nonadiabatical matrix elements A ki (ρ), B ki (ρ) and potential coupling matrix W ki (ρ) are given by integrals A ki (ρ) = φ k ρ φ i = B ki (ρ) = φ k 2 ρ φ i = W ki (ρ) = 3 4 π/2 0 π/2 0 π/2 0 dθφ k (θ ρ) φ i(θ ρ), ρ dθφ k (θ ρ) 2 φ i (θ ρ) ρ 2, θ+ dθφ (θ) k (θ ρ)v (ρ cos θ) dθ φ i (θ ρ) θ (θ)

22 Solution expansion Presence of first derivative F i (ρ) in the right part of the expression Matrix form of equation ( ρ 2 1 ) 4ρ 2 + Λ(ρ) E F (ρ) = 2AF (ρ)+(b W )F (ρ)

23 Transform equation in form Solution expansion F (ρ) + PF (ρ) + QF (ρ) = 0 ( ) where P = 2A and Q = 1 Λ(ρ) + E + B W are 4ρ 2 two-dimensional infinite matrices Introduce F = UG and elements of G are new unknown quantities

24 Solution expansion UG + ( 2U + PU ) G + ( U + PU + QU ) G = 0 Equation for matrix U U = 1 PU = AU 2 As ρ U(ρ) = I + ρ AUdρ

25 Solution expansion Form without first derivative ( UG P 1 ) 4 P2 + Q UG = 0 Substitute expressions for P and Q G + U 1 ( A (ρ) A 2 (ρ) + 1 4ρ 2 Λ(ρ) + E + B W )UG = 0

26 Solution expansion As ρ asymptotic behavior U(ρ) I ( ρ 2 1 ) 4ρ 2 I + Λ(ρ) E G = = ( A (ρ) A 2 (ρ) + B(ρ) W (ρ) ) G The right hand side terms vanish for large ρ faster than ρ 2. Hence this terms can be neglected. ( ρ 2 1 ) 4ρ 2 + λas k (ρ) E G k (ρ) = 0 here λ as 0 (ρ) = ɛ and λas k (ρ) = (2k)2 /ρ 2, k 1

27 Solution expansion Solutions to these equations with appropriate asymptotics can be expressed in terms of the Bessel (Y, J) functions and the Hankel (H (1) ) functions of the first kind as following πqρ G 0 (ρ) 2 πqρ a 0 (q) G k (ρ) a k (E) Y 0 (qρ) J 0 (qρ) 2q + 2 H(1) 0 (qρ)eiπ/4 π Eρ H (1) 2 2k ( Eρ)e i(π/4+kπ)

28 Malfliet-Tjon Potential

29 Malfliet-Tjon potential A potential, which describes the modeling nucleon-nucleon interaction acting in the triplet state with the spin 3/2 Form of the Yukawa terms superposition V (x) = V 1 exp( µ 1 x) x with parameters listed in the table + V 2 exp( µ 2 x) x Coefficient Value V 1 626, 885MeV V , 72MeV µ 1 1, 55 Fm 1 µ 2 3, 11 Fm 1

30 3 Malfliet-Tjon potential V(x) 2 V(x), Fm^(-2) x, Fm

31 Unit system Proton-neutron interaction ) ( 2 2µ 2 x + V (x) ψ(x) = Eψ(x), Reduced mass [V ] = [E] = MeV µ = m pm n m p + m n m2 2m = m 2, m p m n = m ( 2 x + m 2 V (x) m 2 E ) ψ(x) = 0

32 Unit system 2 m 41, 47 Fm2 MeV Obtain equation ( ) x 2 + Ṽ (x) Ẽ ψ(x) = 0 where [Ṽ ] = [Ẽ] = Fm 2 Set x = ρ cos θ in polar coordinates and ρ is parameter

33 30 ρ = 1 V(rho*cos(Theta)), rho= V(rho*cos(Theta)) ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

34 3 ρ = 10 V(rho*cos(Theta)), rho=10 2 V(rho*cos(Theta)) ,2 1,4 1,6 Theta

35 4 ρ = 100 V(rho*cos(Theta)), rho=100 3 V(rho*cos(Theta)) ,52 1,53 1,54 1,55 1,56 1,57 1,58 Theta

36 Numerical Calculations

37 Finite-difference approach Numerical calculations φ k (θ h) + 2φ k (θ) φ k (θ + h) ρ 2 h 2 + +V (ρ cos θ)φ k (θ) = λ k φ k (θ) where h = θ i+1 θ i, i = 1,..., N and N Boundary conditions φ k (0) = φ k (π/2) = 0

38 Software CLAPACK Intel Math Kernel Library for Unix DSTEVX : computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices.

39 Eigenvalues

40 Asymptotics k = 0 k 1 lim λ 0(ρ) = ɛ ρ λ k (ρ) ( ) 2k 2 ρ

41 0,04 λ 0 (ρ) lambda0 0,02 0,00-0,02-0,04-0,06-0, rho

42 λ 10 (ρ) lambda10 asymptotic rho

43 Eigenvalues 0,30 0,25 0,20 0,15 0,10 0,05 0,00-0,05 Eigenvalues lambda0 lambda1 lambda2 lambda3 lambda4 lambda5-0, rho

44 Deviation from asymptotics ρ 10 deviation is big and varies from 10% up to 90%. For small k the absolute deviation is bigger than for big k. ρ 100 deviation is about 5 7% ρ = 700 deviation is less than 1% ρ 1000 deviation is about 0, 7%

45 Eigenfunctions

46 1,4 1,2 φ 0 (θ), ρ = 1 phi0(theta), rho=1 1,0 0,8 0,6 0,4 0,2 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

47 1,4 1,2 φ 0 (θ), ρ = 3 phi0(theta), rho=3 1,0 0,8 0,6 0,4 0,2 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

48 1,4 1,2 1,0 0,8 0,6 0,4 0,2 φ 0 (θ), ρ = 5 phi0(theta), rho=5 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

49 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 φ 0 (θ), ρ = 10 phi0(theta), rho=10 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

50 3,0 φ 0 (θ), ρ = 30 phi0(theta), rho=30 2,5 2,0 1,5 1,0 0,5 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

51 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

52 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

53 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

54 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

55 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

56 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

57 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

58 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

59 φ 0 (θ), ρ = ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

60 rho=1 rho=3 rho=5 rho=10 rho=30 rho=100 rho=300 rho=500 rho=1000 Step-by-step ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

61 rho=1 rho=3 rho=5 rho=10 rho=30 rho=100 rho=300 rho=500 rho=1000 Step-by-step ,48 1,50 1,52 1,54 1,56 1,58 Theta

62 φ 0 (θ): conclusions As ρ eigenfunction φ 0 transforms into δ-function Computational s: decreasing of spacing, time of computation, size of data, useless zero-area

63 φ k (θ), k 1 1,0 0,5 phi_k(theta), rho=1 k=1 k=2 k=3 k=4 k=5 0,0-0,5-1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

64 1,0 0,5 phi_k(theta), rho=3 φ k (θ), k 1 k=1 k=2 k=3 k=4 k=5 0,0-0,5-1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

65 1,0 0,5 phi_k(theta), rho=5 φ k (θ), k 1 k=1 k=2 k=3 k=4 k=5 0,0-0,5-1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

66 φ k (θ), k 1 1,0 phi_k(theta), rho=10 k=1 k=2 k=3 k=4 k=5 0,5 0,0-0,5-1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

67 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6-0,8-1,0-1,2 phi_k(theta), rho=30 φ k (θ), k 1 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta k=1 k=2 k=3 k=4 k=5

68 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6-0,8-1,0-1,2 phi_k(theta), rho=100 φ k (θ), k 1 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta k=1 k=2 k=3 k=4 k=5

69 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6-0,8-1,0-1,2 phi_k(theta), rho=300 φ k (θ), k 1 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta k=1 k=2 k=3 k=4 k=5

70 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6-0,8-1,0-1,2 phi_k(theta), rho=500 φ k (θ), k 1 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta k=1 k=2 k=3 k=4 k=5

71 1,2 1,0 0,8 0,6 0,4 0,2 0,0-0,2-0,4-0,6-0,8-1,0-1,2 phi_k(theta), rho=1000 φ k (θ), k 1 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta k=1 k=2 k=3 k=4 k=5

72 1,0 0,5 0,0 Step-by-step φ 1 (θ) phi1(theta) rho=1 rho=3 rho=5 rho=10 rho=30 rho=100 rho=300 rho=500 rho=1000-0,5-1,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

73 Orthogonality and deviations Check orthogonality on each step When does eigenfunction reach its asymptotic behavior? max φ k,ρ (θ) sin(2kθ) θ

74 k max deviation 1 0, , , , , , , , , , , , , , ρ = 500

75 0,5 Deviation in case k = 20, ρ = 500 phi_20 - sin(2k*theta), k=20 0,4 0,3 0,2 0,1 0,0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

76 k max deviation 1 0, , , , , , , , , , , , , , ρ = 1000

77 Deviation in case k = 20, ρ = ,25 phi_k - sin(2k*theta), k=20 0,20 0,15 0,10 0,05 0,00 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 Theta

78 Geometrical connections

79 Geometrical connections Off-diagonal matrix of geometrical connections A with elements A ki (ρ) = φ k φ i = π/2 0 dθ φ k (θ ρ) φ i(θ ρ) ρ Determines behavior of right part of equation ( ρ 2 1 ) 4ρ 2 I + Λ(ρ) E G = = ( A (ρ) A 2 (ρ) + B(ρ) W (ρ) ) G

80 Geometrical connections A ki = A ik and A kk = 0 A ki = φ k V ρ φ i λ k λ i

81 1,0 0,8 A ki, k = 0 or i = 0 num. calc. O(rho^(-2,5)) O(rho^(-2)) 0,6 A01 0,4 0,2 0, rho

82 3 A ki, k 0 and i 0 num. calc. O(rho^(-2)) O(rho^(-1,5)) 2 A rho

83 A13 0,0-0,2-0,4-0,6-0,8-1,0-1,2-1,4-1,6-1,8-2,0 A ki, k 0 and i 0 num. calc. O(rho^(-2)) O(rho^(-1,5)) rho

84 The last note about A ki The behavior of geometrical connections A ki is regular (no singularity or special points) in case of Malfliet-Tjon potential.

85 Thank you!