The J-Matrix Method Episode 4 docx

78 A.D. Alhaidari 5040 M = 2 30 2010 0.0 (a) –1.0 –2.0 –3.0 –4.0 –5.0 0 10.08.06.04.02.0 0.0 (b) –1.0 –2.0 –3.0 –4.0 –5.0 0 M = 2 1 5.04.03.02.01.0 0.0 (c) –1.0 –2.0 –3.0 –4.0 –5.0 0 M = 2 2 Fig. 2 Same as Fig. 1, except that the potential is now two-term separable with V 00 = 5, V 11 = 3, and V 01 = V 10 =−1 Scattering Phase Shift for Relativistic Separable Potentials 79 50403020100 −5.0 −4.0 −3.0 −2.0 −1.0 0.0 (a) (b) 10.08.06.04.02.00 −5.0 −4.0 −3.0 −2.0 −1.0 0.0 (c) 5.04.03.02.01.00 –5.0 –4.0 –3.0 –2.0 –1.0 0.0 M = 3 = 0 M = 3 = 1 M = 3 = 2 Fig. 3 Same as Fig. 1, except that the potential is now three-term separable with V = ⎛ ⎝ 5 −10 −13 2 02−1 ⎞ ⎠ 80 A.D. Alhaidari M = 4 50403020100 –5.0 –4.0 –3.0 –2.0 –1.0 0.0 (a) (b) (c) 10.08.06.04.02.00 0.0 5.04.03.02.01.00 −5.0 −4.0 −3.0 −2.0 −1.0 −5.0 −4.0 −3.0 −2.0 −1.0 0.0 = 0 M = 4 = 1 M = 4 = 2 Fig. 4 Same as Fig. 1, except that the potential is now four-term separable with the potential parameters V nm given by (28) and the phase shift is obtained numerically using the relativistic J-matrix method [2–4, 6] Scattering Phase Shift for Relativistic Separable Potentials 81 Acknowledgments I am grateful to Prof. H. A. Yamani for fruitful discussions and suggestions for improvements on the original manuscript. References 1. A. D. Alhaidari, J. Phys. A: Math. Gen. 34, 11273 (2001); 37, 8911 (2004) 2. A. D. Alhaidari, J. Math. Phys. 43, 1129 (2002) 3. P. Horodecki, Phys. Rev. A 62, 052716 (2000) 4. A. D. Alhaidari, H. A. Yamani, and M. S. Abdelmonem, Phys. Rev. A 63, 062708 (2001) 5. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Func- tions of Mathematical Physics, 3rd edition (Springer-Verlag, New York, 1966) pp. 239–249 6. A. D. Alhaidari, H. Bahlouli, A. Al-Hasan, and M. S. Abdelmonem, Phys. Rev. A 75, 062711 (2007) 7. H. A. Yamani and L. Fishman, J. Math. Phys. 16, 410 (1975) Accurate Evaluation of the S-Matrix for Multi-Channel Analytic and Non-Analytic Potentials in Complex L 2 Bases H.A. Yamani and M.S. Abdelmonem Abstract We describe an efficient and accurate scheme to compute the S-matrix elements for a given multi-channel analytic and non-analytic potentials in complex- scaled orthonormal Laguerre or oscillator bases using the J-matrix method. As ex- amples of the utilization of the scheme, we evaluate the cross section of two-channel square wells in an oscillator basis and find the resonance position for the same po- tential using the Laguerre basis. We also find resonance positions of a two-channel analytic potential for several angular momenta  using both bases. Additionally, we evaluate the effect of including the Coulomb term (z/r) when employing the Laguerre basis. The work reported here is a continuation of the authors’ effort [1–7] to utilize the advantages of the J-matrix method of scattering. In particular, we have provided a solution for the Lippman–Schwinger equation in a closed form for a model finite- rank potential. This yields a T-matrix and S-matrix in terms of certain coefficients that are related to the H 0 -problem, with the matrix elements of Green’s matrix being evaluated in the basis set chosen. In this chapter, we answer the question of how to accurately and easily evaluate the S-matrix elements for such a potential from analytic and non-analytic general potentials. This chapter is organized as follows. In Sect. 2, we review the defining state- ments of the various functions that come into play in the calculation of the S-matrix elements. In Sect. 3, we explicitly outline the various methods available for the calculation of these functions. It is important to be certain of the accuracy of the potential element V nm for high values of n and m. We specifically show that for the complex-scaled bases, the Gauss quadrature scheme of evaluating V nm has several desired features. It is easy to implement, makes full use of the nature of the basis, and works for general potential. Furthermore, its accuracy is assured, even for high n and m indices, through the proper choice of a parameter K, which specifies the order of the quadrature scheme utilized. We plan to achieve the goal of accurately H.A. Yamani Ministry of Commerce & Industry, P.O. Box 5729, Riyadh 11127, Saudi Arabia e-mail: haydara@sbm.net.sa A.D. Alhaidari et al. (eds.), The J-Matrix Method, 83–102. 83 C  Springer Science+Business Media B.V. 2008 84 H.A. Yamani, M.S. Abdelmonem calculating the matrix elements V nm in stages. In Sect. 3.1 we propose an approx- imate scheme to determine the matrix elements of a one-channel central analytic potential in the Laguerre basis using a real and complex scale parameter λ.As this scheme does not work for non-analytic potentials, in Sect. 3.2 we develop an approximate scheme that may be applied for both kinds of potentials. In Sect. 3.3 and 3.4 we generalize the scheme in the preceding two sub-sections to find the matrix elements  φ (α) n   V αβ    φ (β) m  of a multi-channel potential V αβ (r), where the scale parameters λ α and λ β associated with the channel basis are allowed to differ, and have the form λ α = | λ α | e −iθ α , and λ β =   λ β   e −iθ β . The results of Sect. 3 that are associated with the use of the oscillator basis are summarized in Sect. 3.5. Finally, in Sect. 4 we discuss the accuracy of the proposed scheme and illustrate its application by considering two examples. In the first we evaluate the cross section of two-channel square wells in the oscillator basis, and find the complex resonance energy of the same potential using the Laguerre basis. In the second example we find the complex resonance energy for a specific two-channel analytical potential (without a Coulomb term) for several angular momenta  using both bases. We then consider the same problem with the Coulomb term included in the calculation using the Laguerre basis. 1ReviewoftheJ-Matrix Method Here we consider the scattering of a structureless, spinless particle by a target with M internal states labeled by the threshold energies E 1 , E 2 , , E M . We assume that the given scattering potential, V αβ , is of a short range. We also assume that the reference Hamiltonian of the particle, H 0 , may include, in addition to the th partial wave kinetic energy operator, the Coulomb term (z/r); that is, H 0 =− 1 2 d 2 dr 2 +  ( +1) 2r 2 + z r . (1) The multi-channel Schr¨odinger equation for the given scattering problem can be written as M  β=1  [H 0 − (E − E α )] δ αβ + V αβ    ψ αβ  = 0. (2) The basis chosen in each channel ␣ is either the Laguerre basis φ (α) n (r) = d (L) n e −λ α r/2 (λ α r) ν/2 L ν n (λ α r) , (3) Accurate Evaluation of the S-Matrix 85 or the oscillator basis φ (α) n (r) = d (o) n e −λ 2 α r 2 /2 (λ α r) ν+1/2 L ν n  λ 2 α r 2  . (4) In the Laguerre basis, ν = 2 + 2 and the normalization factor is d (L) n =  λ α n!  ⌫ (n + ν + 1). In the oscillator basis, ν =  + 1/2 and the normalization factor is d (o) n =  2λ α n!  ⌫ (n + ν + 1). Here, L ν n (ζ ) is the generalized Laguerre polynomial [8], λ is a free complex scale parameter, and  is the partial wave angular momentum quantum number. The J-matrix method of scattering finds the exact S-matrix to the model multi- channel scattering potential, ˜ V . The model potential is obtained from the given po- tential, V , by restricting its infinite matrix representation in a complete L 2 basis to a finite representation. That is, the model potential ˜ V is defined as ˜ V αβ = N α −1  n=0 N β −1  m=0   ¯ φ (α) n  φ (α) n   V αβ   φ (β) m  ¯ φ (β) m   (5) where    ¯ φ (α) n  ∞ n=0 is the orthogonal complement to the basis    φ (α) n  ∞ n=0 in the αth channel in the sense that  ¯ φ (α) n   φ (α) m  =  φ (α) m   ¯ φ (α) n  = δ nm . The exact multi-channel S-matrix can be written as [9] S αβ (E) = T N α δ αβ −      J (α) N α −1,N α J (β) N β −1,N β  R + N α R + N β   T N α −1 − T N α ×  D (+) αβ (E) ⌬ (+) (E)   T N β −1 − T N β (6) Here ⌬ (+) (E) =          1 −Y 1,1  −Y 1,2 −Y 1,M −Y 2,1  1 −Y 2,2  −Y 2,M −Y M,1 −Y M,2  1 −Y M,M          , (7) and D (+) αβ (E) =           0 δ α,1 δ α,2 δ α,M δ β,1 δ β,2 ⌬ (+) (E) δ β,M           , (8) 86 H.A. Yamani, M.S. Abdelmonem where Y α,β =−g (α,β) N α −1,N β −1 J (β) N β −1,N β R + N β (9) and in (8) we show the first row and column, while the remainder is represented by ⌬ (+) (E) of equation (7). Additionally, R ± N (α)andT N−1 (α)aretheJ-matrix ratios associated with H 0 , which are given by R ± n (α) = c (α) n ±is (α) n c (α) n−1 ± is (α) n−1 , and T n (α) = c (α) n −is (α) n c (α) n +is (α) n . (10)  s (α) n  is the set of coefficients of the expansion of the sine-like eigenfunction of H 0 in the basis    φ (α) n  . Similarly,  c (α) n  is the set of coefficients of the expansion of the asymptotically cosine-like eigenfunction of H 0 in the same basis [1]. Furthermore, J (β) N β −1,N β =  φ N β −1    H 0 − E + E β    φ N β  and g (α,β) N α −1,N β −1 represent the (N α –1, N β – 1) element of the (α,β) sub-matrix of g, the inverse of the total N c × N c Hamiltonian matrix. The matrix element g (α,β) N α −1,N β −1 takes the following explicit forms: (i) In the Laguerre basis g (α,β) N α −1,N β −1 (E) =  ˜a N α −1 ˜a N β −1 a N α −1 a N β −1  N c  μ=1 ⌳ (α) N α −1,μ ⌳ (β) N β −1,μ E μ − E , (11) where ˜a n =  λn! ⌫ (n + ν + 1) and a n =  λn! ⌫ (n + ν) . (12) (ii) In the oscillator basis g (α,β) N α −1,N β −1 (E) = N c  μ=1 ⌳ (α) N α −1,μ ⌳ (β) N β −1,μ E μ − E . (13) In both bases, E μ is the eigenvalue of the full Hamiltonian, and  ⌳ (1) 0,μ , ⌳ (1) 1,μ , ,⌳ (1) N 1 −1,μ , ⌳ (2) 0,μ , ,⌳ (2) N 2 −1,μ , ,⌳ (M) 0,μ , ,⌳ (M) N M −1,μ  (14) is the associated eigenvector. Here, N c = M  α=1 N α is the dimension of the Hamilto- nian matrix. This is the same as the total number of L 2 functions used to diagonalize the scattering Hamiltonian. Accurate Evaluation of the S-Matrix 87 Thus, the calculation of the exact S(E) for the model potential requires: (1) The matrix element J N−1,N (E), the J-matrix ratios R ± N (E)andT N−1 (E), and (2) The matrix element g (α,β) N α −1,N β −1 (E). 2 J N–1,N (E), R ± N ( E ) , T N–1 ( E ) and the Matrix Elements of H 0 in Both Bases We first find expressions for the matrix elements of the J matrix, which are sym- metric and tridiagonal in both bases. (1) In the Laguerre basis: J n,n = λz −  E − E α − λ 2  8  (2n + 2 +2) (15) J n,n−1 =  E − E α + λ 2  8   n (n + 2 + 1) (16) (2) In the oscillator basis: J n,n =  2n +  +3  2  λ 2  2  − (E − E α ) (17) J n,n−1 =  λ 2  2   n (n +  +1/2) (18) We then find expressions for the J-matrix ratios R ± N (E)andT N−1 (E)intheLa- guerre or oscillator bases. In order to do that, we first define the following functions u n =−J n,n /J n,n−1 , and v n =−J n,n−1 /J n,n+1 (19) It is subsequently determined that the recursion relation for R ± N (E)is R ± n+1 = u n + v n R ± n for n ≥ 1. (20) The J-matrix ratio, T n , can be written as: T n = T 0 ⎛ ⎝ n  j=1 R − j R + j ⎞ ⎠ . (21) Thus, for the evaluation of the ratios R ± N (E)andT N−1 (E), we only need to calcu- late R ± 1 (E)andT 0 (E), where R ± 1 = (c 1 ±is 1 )  (c 0 ±is 0 )andT 0 = (c 0 −is 0 )  (c 0 +is 0 ). (22) The explicit forms of R ± 1 (E)andT 0 (E) in the Laguerre and oscillator bases are presented as continued fraction expressions elsewhere [9]. 88 H.A. Yamani, M.S. Abdelmonem We may then determine the matrix elements of the reference Hamiltonian in both bases. In addition to the th partial wave kinetic energy operator, the reference Hamiltonian H 0 may include the Coulomb term (z/r), H 0 (r, z) =− 1 2 d 2 dr 2 +  ( +1) 2r 2 + z r . (23) The bases (3) and (4) remain complete in L 2 (0, ∞)forcomplexλ, as long as Re(λ) > 0 (Laguerre) and Re(λ 2 ) > 0 (oscillator). If we write λ = | λ | e −iθ ,the restriction will be −π/2 <θ<π/2 for the Laguerre basis and −π/4 <θ<π/4 for the oscillator basis. In fact, by considering θ as a complex rotation angle, the above bases with λ = | λ | e −iθ become orthonormal complex-scaled bases. We note that the matrix elements of the reference Hamiltonian  φ θ n   H 0   φ θ m  are calculated once, and used with every model potential that is considered. The matrix elements of H 0 have the following representation: (i) For the Laguerre basis: (H 0 ) nm = λ 2 4  c n> c n<  1 + 2n < ν + 1 + 4(z/λ) ν  − 1 2 δ nm  , (24) where c n =  λn!  ⌫ (n + ν + 1), ν = 2 + 2, and n > (n < )isthelarger (smaller) of the two indices n and m, respectively. (ii) For the oscillator basis (z = 0 only): (H 0 ) n,m =  λ 2 /2   (2n + ν +1)δ nm +  n (n +ν)  δ n,m+1 + δ n,m−1   , (25) where ν =  + 1/2. 3 Evaluation of Potential Matrix Elements Using the Gauss Quadrature Scheme 3.1 The Case of a Single-Channel Central Analytic Potential The potential matrix element V nm =  φ n | V | φ m  in the Laguerre basis can be written explicitly as V nm = ∞  0 φ n (r)V (r) φ m (r) dr. (26) Depending on the form of the potential, some skill may be required to find reli- able numerical values for the above matrix elements, especially for large values of [...]... et al [15] N ε (Oscillator) ε (Laguerre) 0 0 0 0 0 Ref [15] 10 20 30 40 50 4. 7681 94 − i0.0007085 4. 7681 94 − i0.0007099 4. 7681 94 − i0.0007098 4. 7681 94 − i0.0007098 4. 7681 94 − i0.0007098 4. 7682 − i0.000710 4. 768303 − i0.0020759 4. 768199 − i0.0007107 4. 768197 − i0.0007101 4. 768197 − i0.0007101 4. 768197 − i0.0007101 1 1 1 1 1 10 20 30 40 50 6.70372 − i 0.125652 6.70372 − i0.125653 6.70372 − i0.125653 6.70372... |λ1 | = |λ2 | = 5.0, rotational angles θ1 = θ2 = 0.5rad The potential matrix elements are calculated using equation (56) ε N z = −1 10 20 30 40 z = +1 2.81003 − i2 .49 8e − 4 2.81150 − i1.832e − 4 2.81150 − i1.814e − 4 2.81150 − i1.814e − 4 6.27899 − i1.8507e − 2 6.278 04 − i1. 843 2e − 2 6.278 04 − i1. 843 3e − 2 6.278 04 − i1. 843 3e − 2 Acknowledgments The support of this work by King Fahd University of Petroleum... problem [13] Furthermore, we find the position of the resonance of the same potential using the complex rotation method We chose the Laguerre basis with Nα = Nβ = N The scaling parameters |λα | and λβ are considered to be the same and equal 6.0 The rotation angles are taken to be θα = θβ = 0.1 rad We also chose the order of approximation K to equal 60 We then evaluate the matrix elements of the potential,... eigenvectors of the infinite submatrix Hi(2) The S-matrix and j scattering phase shifts are defined through the asymptotics of the functions (5), hence they are governed by the structure of the submatrix Hi(2) only The energy j E I of IS and its other features are dictated by the structure of the other submatrix Hi(1) that is generally independent from Hi(2) Therefore we cannot expect that the j j S-matrix... Performing the diagonalization of the finite matrix J(K) of equation ( 34) once, we have the eigenvalues ζk(K ) K −1 k=0 (K vk ) and eigenvectors K −1 k=0 The summand in equation (38) is the value of the potential at the scaled positions ζk(K ) /λ , weighted by the product (K of the nth and mth coefficient of the eigenvector vk ) Since the argument of the potential in the sum (38) is real, this approximation... be found by the formula X n (E) = ℘n N (E) X N +1 (E) ( 14) We are considering the case when the Hamiltonian matrix is block-diagonal We note that the kinetic energy matrix in the oscillator basis is tridiagonal Hence with the interaction (6) we can obtain the Hamiltonian matrix in the oscillator basis of the type (4) that has the structure shown in Fig 1 Solid lines schematically show the infinite tridiagonal... bring us to the 3 H system with three bound states with extremely large (few GeV) binding energies; with larger E I values we obtain the 3 H nucleus with two bound states (the binding energy of the ground state is of the order of few hundreds MeV); the further increase of E I results in the further decrease with E I of the binding energy of the 3 H nucleus which has a single bound state; if the IS energy... to Fnm of equation (47 ) In fact the above result reduces to that of the one-channel case (46 ) As has been noticed in the one-channel case, the similarities and differences between the evaluation of the one-channel potential matrix elements in the Laguerre and oscillator bases are preserved in the multi-channel case Thus, the matrix eleαβ ments of multi-channel potentials, Vnm , in the oscillator basis... example, we find the complex resonance energy for a specific two-channel analytical potential (with no Coulomb term) for several angular momenta using both bases We then consider the same potential when the Coulomb term is included using the Laguerre basis The matrix elements of the reference Hamiltonian are given by equation ( 24) We apply the proposed method to the characterization of the s-wave narrow... characterized completely by the S-matrix [3] Hence the structure of the S-matrix is governed by the infinite-dimensional submatrix H (2) that is independent from the submatrix (1) H (1) The IS energies εμ , on the other hand, are controled by the submatrix H (1) (1) and are independent from the submatrix H (2) Varying matrix elements Hnn of the (1) (1) submatrix H one causes variation of the IS energies εμ . Alhaidari M = 4 5 040 3020100 –5.0 4. 0 –3.0 –2.0 –1.0 0.0 (a) (b) (c) 10.08.06. 04. 02.00 0.0 5. 04. 03.02.01.00 −5.0 4. 0 −3.0 −2.0 −1.0 −5.0 4. 0 −3.0 −2.0 −1.0 0.0 = 0 M = 4 = 1 M = 4 = 2 Fig. 4 Same. using both bases. We then consider the same problem with the Coulomb term included in the calculation using the Laguerre basis. 1ReviewoftheJ-Matrix Method Here we consider the scattering of a. along the ray re −iθ . Due to the restriction 0 ≤ θ<π/2 on the values of θ, the application of the Cauchy integral theo- rem shows that the integral along the stated ray is equivalent to the