Scattering Theory with Special Regularization

Abstract

We present an alternative, but equivalent, approach to the regularization of the reference problem in the J-matrix method of scattering. After identifying the regular solution of the reference wave equation with the "sine-like" solution in the J-matrix approach we proceed by direct integration to find the expansion coefficients in an L hasis set that ensures a tridiagonal representation of the reference Hamiltonian. A äfferential equation in the energy is then deduced for these coefficients. The second independent solution of this equation, called the "cosine-like" solution, is derived by requiring it to pertain to the space. These requirements lead to solutions that are exactly identical to those obtained in the classical J-matrix approach. We find the present approach to be more direct and transparent than the classical differential approach of the J-matrix method. In three dimensional scattering, the energy continuum wavefunction is obtained by utilizing two independent solutions of the reference wave equation. One of them is typically singular (usually, near the origin of configuration space). Both are asymptotically regular and sinusoidal with a phase difference (shift) that contains information about the scattering potential. Therefore, both solutions are essential for cattering calculations. Various regularization techniques were developed to handle the singular solution leading to different well-established scattering methods. To simplify the calculation the regularized solutions are usually constructed in a space that diagonalizes the reference Hamiltonian. In this work, we start by proposing solutions that are already regular. We write them as infinite series of square integrable basis functions that are compatible with the domain of the reference Hamiltonian. However, we relax the diagonal constraint on the representation by requiring that the basis supports an infinite tridiagonal matrix representation of the wave operator. The hope is that by relaxing this constraint on the slation space a larger freedom is achieved in regularization such that a natural choice merges as a result. We find that one of the resulting two independent wavefunctions is, in fact, the regular solution of the reference problem. The other is uniquely regularized in the sense that it solves the reference wave equation only outside a dense region covering the singularity in configuration space. However, asymptotically it is identical to the irregular solution. We show that this tural and special regularization is equivalent to that already used in the J-matrix method of scattering. In the present thesis "Scattering Theory with Special Regularization": We start with the three dimensional problem with spherical symmetry where the ference Hamiltonian is the partial -wave free Hamiltonian 2. Then we propose an L basis compatible with the regular domain of H, and pand the exact regular solution of the reference problem in this basis. We then take advantage of a recently obtained integral formula to calculate the sine-like expansion coefficients. We then obtain a second order differential equation in the energy satisfied by s(E) he second independent solution of this equation is identified with the expansion coefficients of the "regularized" cosine-like wave function. we show that this wavefunction satisfies the regularized reference wave equation and that the original irregular solution and this regularized wave function are symptotically equal, as required on physical ground.