[...]... from the fact that V N is defined to be zero in this region of Hilbert space, and because (H0 − E) is tridiagonal in the basis {φn }, ˜ and therefore does not connect the N terms in the expansion of ⌽ or the first N ˜ ˜ terms in the expansion of S and C with φm for m ≥ N + 1 Furthermore, for each m ≥ N + 1 the right-hand side of Eq (21) leads to the three-term recursion relation (8a) and (8c) for the. .. for m > Nβ in direct analogy with the single-channel case, because of the three-term recursion relation satisfied by the coefficients of Sβ and Cβ The remainder of the equations; i.e., Eq (41) for m ≤ Nβ , lead to the same number of conditions as there are unknowns The totality of these equations can be organized in a similar fashion as done in the form (22), with the L 2 matrix elements ˜ of V appearing... part of the problem “diagonalized” and the full analytic structure of the S matrix is built into the problem, raising the hope that quite small basis sets will be sufficient for many problems The analytic nature of the solutions allows variational corrections to be made and provides a solid footing for further theoretical work We now summarize the steps necessary to perform a calculation with the J -matrix... coefficients of the sine-like and cosine-like functions in terms of the basis sets is outlined In Section 2.2, the general method is illustrated in detail for the case of the radial kinetic energy in a Laguerre basis The analogous results for the oscillator basis and for the Coulomb problem are outlined in Sections 2.3 and 2.4, respectively The details of the Coulomb derivation are given in the Appendix... by these N basis functions, we expect the sine-like and the cosine-like solutions, derived in Section 2.1, to be valid Therefore we write our solution as ˜ ˜ ˜ ψ E = ⌽ + S + t C, (17) 8 E.J Heller, H.A Yamani N −1 0 ˜ ˜ ˜ where ⌽ = n=0 an φn , S is the sine-like expansion χ E of Eq (10), and C is the ˜ cosine-like solution of Eq (13) The unknown coefficient t, then, corresponds to the ˜ tangent of the. .. obeys the same differential equation with different boundary conditions is then constructed The fact that both J -matrix solutions obey the same recurrence scheme is essential to the success of the method as an efficient technique for solving scattering problems [1] The program of the chapter is as follows: In Section 2.1, the generalized H0 problem is considered and a general procedure for obtaining the. .. channel α ≤ Nc The right-hand side driving term and the solution α “vector” containing the aβn ’s and Rαα ’s have as many columns as open channels The R matrix can then be obtained by solving the resulting linear equations As before, the calculation may be facilitated by a pre-diagonalization of the inner block using an energy-independent transformation 5 Discussion The comparison between the approach... theoretical interest 2 The H0 Problem The problem examined in this section is the “solution” of the equation, H0 − k 2 2 ⌿ = 0 (9) within the framework of the L 2 function space {⌽n } in such a manner as to obtain both an asymptotically sine-like and asymptotically cosine-like function The two ˜ ˜ J -matrix solutions, S(r ) and C(r ), form the basis for the asymptotic representation of the scattering wavefunction... We will not consider the four cases for m as is done in the single-channel case, but show instead that for m > Nβ the left-hand side of Eq (41) also vanishes ˜ identically First, the potential term vanishes by the definition of V for m > Nβ 1 Second, − 2 Ѩ2 Ѩr 2 − E − E β is tridiagonal in the φ (β) Thus, there will be (β) (β) no overlap between φm and the first Nβ functions φn Then Eq (41) becomes... applied to the inner matrix as in Section 2 If desired, the matrix elements of H0 + V N − E can be evaluated in the Slater set (λr )n e−λr /2 , n = 1, 2, , N, since these are just transformed Laguerres Then a different transformation ⌫ will be necessary to pre-diagonalize the inner matrix In the following chapter we apply the method presented here to s-wave electronhydrogen scattering model The generalization . alt="" The J-Matrix Method Abdulaziz D. Alhaidari · Eric J. Heller · Hashim A. Yamani · Mohamed S. Abdelmonem Editors The J-Matrix Method Developments and Applications Foreword. regularization approach in the J-matrix method and demonstrate the resulting improvements. On the other hand, a method for the accurate evaluation of the S-matrix for