The J-Matrix Method potx

[...]... 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... 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... 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. .. 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. .. scattering In the following chapter [1], we will apply the method to electronhydrogen elastic s-wave scattering below the n = 2 threshold, and to inelastic radiallimit scattering calculations above and below the ionization threshold Our basic approach is to treat an uncoupled Hamiltonian H0 exactly in the space spanned by the complete L 2 basis The remaining part of the Hamiltonian (i.e., the E.J Heller... 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... are both the components of the wave function ψn at the boundary of their respective inner spaces Recent work seems to indicate that the R-matrix method works best using eigenfunctions of the scattering H0 as a basis [10] In this basis, H0 is of course diagonal, and may be treated exactly by the addition of the Buttle correction In our basis set, H0 is tridiagonal and is also treated exactly Other types... tridiagonal and is also treated exactly Other types of basis sets can also be used in the R-matrix method In general, however, the ability to account for H0 , exactly is lost The same is true in the present method, in the analogous situation when other basis functions not belonging to the tridiagonal set {φn } are used As in the R-matrix approach, we expect to find no Kohn-type pseudo-resonances [11] appearing... method, H0 is accounted for exactly independent of basis size Thus we start with a large 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... 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 of the method to all partial waves for both Laguerre and Hermite basis sets has been derived and will be the subject of a future publication The case where H0 , contains the term α /r (i.e., the Coulomb case) is also worked out for Laguerre . use the J-matrix method as a universal approach for the description of the process involving the ionization of atoms. They succeed in ad- dressing the difficult problem of correctly describing the. in the scattering of three charged particles using the J-matrix method. Papp exploits the J-matrix structure of the Coulomb Green’s matrix to solve the Faddeev-type integral equations of the. of the reference Hamiltonian H 0 by giving a full solution of the reference problem in the complete basis. Furthermore, in the spirit of the R-matrix method, Heller and Yamani proposed that the